Portfolio Management Flashcards
9.1 Economics and Investment Markets
[Present Value Model]
1- Overview of the Present Value Model:
– Assets derive their value from expected future benefits, primarily in the form of cash flows.
– The model assumes that investors prefer receiving a certain amount today over waiting for the same amount in the future, emphasizing the time value of money.
– Future cash flows are discounted to their present value based on specific risk and return factors.
2- Formula:
– General expression for the value of Asset i at time t:
— “P_i_t = T∑_s=1 [E_t(CF_i_t+s) ÷ (1 + l_t,s + θ_t,s + ρ_i_t,s)^s]”
3- Explanation of Variables:
– “P_i_t”: Present value of the asset’s cash flows at time t.
– “E_t(CF_i_t+s)”: Expected cash flow from Asset i at time t + s.
– “l_t,s”: Yield to maturity on a real default-free investment today (e.g., inflation-linked bonds).
– “θ_t,s”: Expected inflation rate between time t and t + s, accounting for changes in real purchasing power.
– “ρ_i_t,s”: Risk premium added to discount rates to account for risks like default, liquidity, and market volatility.
– “s”: Number of periods into the future.
Key Takeaways
– The present value model incorporates time value, inflation expectations, and risk factors to estimate the value of future cash flows.
– Accurate valuation depends on appropriate estimates for expected cash flows, discount rates, and risk premiums.
[Expectations and Asset Values]
1- Overview of Expectations and Asset Values:
– Asset values are determined by expected future cash flows, not historical cash flows.
– Valuation is based on current information available at the time of valuation, denoted as time t.
2- Key Points:
– The current valuation reflects all known or anticipated information about the asset.
– Valuations are dynamic and may change as new information becomes available and expectations about future cash flows are updated.
Key Takeaways
– Accurate asset valuation requires a focus on expected future cash flows and constant updating of inputs to reflect the most current information.
Purchasing an investment today means a lower consumption today. This is the trade-off between saving and consumption.
[Real Default-Free Interest Rates]
1- Overview of Real Default-Free Interest Rates:
– The real default-free interest rate (lt,s) represents the opportunity cost of consuming today versus in the future, assuming no inflation risk.
– It reflects the inter-temporal rate of substitution, which is the ratio of marginal utility from future consumption to current consumption.
2- Key Drivers of lt,s:
– Marginal utility of consumption decreases with higher income, prompting greater investment today.
– During good economic times, individuals tend to save more as the inter-temporal rate of substitution is high.
3- Formula for Real Default-Free Interest Rate:
– Formula: “lt,1 = (1 - Pt,1) / Pt,1”
— Where:
—- lt,1: Real default-free interest rate for one period.
—- Pt,1: Price of a zero-coupon risk-free bond.
– Formula for Bond Price: “Pt,1 = E(mt,1)”
— Where:
—- mt,1: Inter-temporal rate of substitution for future consumption.
4- Practical Example:
– If an investor’s inter-temporal rate of substitution is 0.9515, they are willing to give up $95.15 today to receive $100 in the future.
– For a zero-coupon bond with a face value of $100 trading at $94.83, the investor will buy the bond, causing their substitution value to decrease over time.
Key Takeaways
– The real default-free interest rate is determined by the tradeoff between current and future consumption.
– Bond prices and the inter-temporal rate of substitution are inversely related to the risk-free rate.
[Uncertainty and Risk Premiums]
1- Overview of Uncertainty and Risk Premiums:
– Uncertainty about future payoffs reduces an investor’s marginal utility because risky cash flows are less valuable to risk-averse individuals.
– Investors demand compensation for taking on risk, meaning riskier future cash flows must have higher expected returns.
2- Wealth Effect on Risk Aversion:
– Wealthier investors exhibit lower levels of risk aversion as they are more willing to purchase risky assets.
– As wealth increases, diminishing marginal utility from additional risky assets drives the market toward equilibrium where all investors share the same willingness to hold risky assets.
– The more you have of something, the less marginal increase it brings
3- Risk Premiums on Risky Assets:
– Formula for a risky asset’s price:
— Formula: “Pt,s = [Et(P_t+1,s-1) ÷ (1 + lt,1)] + covt(P_t+1,s-1, mt,1)”
— Where:
—- Pt,s: Current price of the risky asset.
—- Et(P_t+1,s-1): Expected future price of the asset discounted at the risk-free rate.
—- lt,1: Real risk-free interest rate.
—- covt(P_t+1,s-1, mt,1): Covariance term representing the risk premium.
4- Interpretation of the Formula:
– The first term calculates the risk-neutral present value, reflecting the discounted expected future price of the asset.
– The second term (covariance) adjusts for risk. Negative covariance reduces an asset’s price for risk-averse investors, as they demand compensation for holding risky assets.
5- Economic Conditions and Risk Premiums:
– During bad economic times, risky assets offer low returns while investors’ marginal utility of consumption is high, leading to reduced demand for such assets.
– Conversely, assets with positive covariance during downturns act as hedges, gaining value due to higher demand and requiring lower returns.
Key Takeaways
– Risk premiums compensate investors for holding assets with uncertain payoffs.
– Wealth levels, economic conditions, and covariance with consumption drive variations in asset prices and expected returns.
[Risk Premiums on Risky Assets]
1- Covariance Term and Risk-Averse Investors:
– For most risky assets, the covariance term is negative, reducing the asset’s price because the expected future value is high when the marginal utility of consumption is low.
– Risk-averse investors prefer to consume more today if they expect strong future economic performance.
2- Economic Conditions and Risky Assets:
– Risky assets typically offer low expected returns during bad economic times, which coincides with high marginal utility of future consumption for investors.
– In such times, consumers delay unnecessary spending due to reduced income and higher economic uncertainty.
3- Covariance and Asset Pricing:
– A negative covariance term lowers an asset’s price, reflecting the additional risk premium demanded by investors.
– Risk-free assets have a covariance term of zero because their future value is known with certainty.
– Assets with positive covariance during downturns are valued higher as they hedge against bad economic conditions, requiring lower returns.
Key Takeaways
– The covariance term in asset pricing captures the relationship between future returns and economic conditions.
– Risk-averse investors demand higher returns for assets with negative covariance during uncertain times.
– Assets acting as hedges against economic downturns are more valuable and require lower risk premiums.
[Default-Free Interest Rates and Economic Growth]
1- Impact of Real GDP Growth on Interest Rates:
– Forecasted increases in real GDP growth lead to expectations of more goods and services being available in the future.
– This reduces the willingness of investors to substitute consumption across time, resulting in less saving and more borrowing.
– Lower demand for bonds decreases bond prices, raising the real default-free interest rates.
2- Correlation Between GDP Growth and Interest Rates:
– Interest rates are positively linked to expected GDP growth rates and the volatility of GDP growth.
– Economies with higher trend growth or greater volatility in GDP growth typically exhibit higher real default-free interest rates.
3- Influence of Economic Development:
– Developing countries tend to operate below their steady-state growth, leading to a higher marginal product of capital.
– As a result, developing economies generally experience higher real default-free interest rates.
Key Takeaways
– Real GDP growth directly affects investor saving and borrowing behavior, influencing real interest rates.
– Higher growth expectations and GDP volatility are associated with increased real default-free interest rates.
– Developing economies face higher rates due to their elevated marginal product of capital.
The marginal increase in output from adding an additional unit of capital (holding all else constant) is typically highest in countries with developing economies. As a country’s economy becomes more developed and moves into its steady state, its marginal product of capital and real interest rates are expected to decline.
[Real Default-Free Interest Rates and the Business Cycle]
1- Relationship Between Real Interest Rates and GDP Growth:
– Real interest rates and GDP growth are analyzed using inflation-linked government bonds, such as TIPS in the United States.
– Countries with high GDP growth or high GDP growth volatility are expected to have higher real yields.
2- Limitations of Historical Data:
– Historical evidence offers partial support for this relationship, but real yield data relies on expectations of future growth and volatility rather than actual past values.
– GDP-based variables reflect realized growth and volatility, making past data a potentially unreliable predictor of future relationships.
Key Takeaways
– Real yields are influenced by growth expectations and volatility but are not always consistent with historical GDP-based data.
– Inflation-linked bonds provide useful insights into the connection between real rates and economic cycles.
Real interest rates are typically higher in developing economies compared to developed economies. Additionally, all else equal, real interest rates are higher in countries with more volatile GDP growth rates.
Volatility requires a larger risk premium for investors, or equivalently, a large discount rate to determine the present value of the cash flows that depend on uncertain growth projections.
[Determination of the Real Default-Free Interest Rate]
1- Key Determinants:
– The real default-free interest rate is determined by the aggregate opportunity cost of all investors.
– It equates the amount of money provided by aggregate savers to the amount of money demanded by aggregate borrowers.
2- Balancing Supply and Demand:
– The interest rate functions as a balancing mechanism for the supply and demand of funds.
– This rate fulfills the condition where savings meet borrowing requirements, ensuring equilibrium in the financial system.
Key Takeaways
– The real default-free interest rate represents the equilibrium point in the allocation of capital.
– It is a foundational concept in understanding the interplay between investment and consumption choices.
[Intertemporal Consumption and Economic State]
1- Good Economic State:
– High current income leads to high levels of consumption.
– Marginal utility of current consumption is low as people have already purchased most of what they want.
2- Bad Economic State:
– Low current income results in lower consumption levels.
– Marginal utility of consumption increases since future uncertainty makes current consumption more valuable.
3- Decision-Making Based on Expectations:
– Individuals base decisions on current information and expectations of the future.
– They decide how much to save or consume today, even though future income is uncertain.
Key Takeaways
– Assets that provide higher payoffs during bad economic conditions are valued for their ability to hedge against unfavorable scenarios.
– Marginal utility dynamics play a critical role in consumption and investment decisions across economic cycles.
[Bond Decisions and the Intertemporal Rate of Substitution]
1- Buying the Bond:
– Occurs when the bond price is lower than the investor’s expectation of the intertemporal rate of substitution.
– This implies the present value of the bond’s future value is higher, making it an attractive purchase.
2- Selling the Bond:
– Happens when the bond price is higher than the investor’s expectation of the intertemporal rate of substitution.
– In this case, current consumption is deemed more valuable, motivating the sale of the bond.
Key Takeaways
– Bond investment decisions are guided by the comparison between current bond prices and expected future values.
– The intertemporal rate of substitution reflects the trade-off between present and future consumption.
yield = price / rate
[Default-Free Interest Rates and Economic Growth]
1- Relationship Between Real GDP Growth and Interest Rates:
– An increase in real GDP growth leads to an increase in real default-free interest rates.
– Higher GDP growth reduces the desire to save, increasing demand for funds and raising interest rates.
2- Drivers of Interest Rate Changes:
– Interest rates are positively related to both the expected growth rate of GDP and the expected volatility of GDP growth.
– Greater uncertainty about future GDP growth results in higher default-free interest rates.
3- Observations from Historical Data:
– High GDP growth is often accompanied by increased volatility, as seen in past periods like 1996-2007.
– Countries with higher GDP growth and volatility tend to have higher real yields, reflecting the higher opportunity cost of capital.
Key Takeaways
– Real default-free interest rates are determined by expected economic growth and uncertainty.
– Developing economies, with higher growth volatility, tend to exhibit higher real yields.
[Pricing Default-Free Nominal Coupon-Paying Bonds]
1- Overview of Nominal Bonds:
– Default-free bonds with fixed nominal payouts are subject to inflation risk.
– While the payoff is certain in nominal terms, the real value at maturity remains uncertain due to inflation variability.
2- Formula for Pricing:
– Pricing formula for a default-free nominal coupon-paying bond:
— Formula: “P_i_t = T∑_s=1 (CF_i_t+s ÷ (1 + lt,s + θt,s + πt,s)^s)”
— Where:
—- lt,s: Yield to maturity on a real default-free investment today.
—- θt,s: Expected inflation rate between time t and t+s.
—- πt,s: Compensation for uncertainty in inflation.
3- Components of Pricing:
– Yield to maturity accounts for the real-time opportunity cost of investing.
– Expected inflation adjusts for changes in purchasing power over time.
– Inflation uncertainty adds a premium for potential deviations in inflation expectations.
Key Takeaways
– Nominal default-free bonds ensure nominal payouts but expose investors to inflation-related risks.
– Pricing incorporates real yield, expected inflation, and compensation for inflation uncertainty to account for these risks.
[Short-Term Nominal Interest Rates and the Business Cycle]
1- Overview of T-Bills:
– Treasury bills (T-bills) are nominal zero-coupon government bonds with short maturities.
– Their yields are closely tied to the central bank’s policy rate.
– Inflation uncertainty is minimal due to the short investment horizon.
2- Simplified Pricing Formula:
– The pricing formula for a T-bill simplifies due to the absence of inflation uncertainty:
— Formula: “P_i_t = CFi_t+s ÷ (1 + lt,s + θt,s)^s”
— Where:
—- lt,s: Yield to maturity on a real default-free investment today.
—- θt,s: Expected inflation rate between time t and t+s.
—- CFi_t+s: Cash flow at time t+s.
3- Characteristics of T-Bills:
– T-bills are often used as benchmarks for risk-free rates in short-term investment horizons.
– Minimal inflation risk simplifies the valuation process.
Key Takeaways
– T-bills provide a risk-free investment option for short durations with low inflation uncertainty.
– Their yields reflect central bank policies and are crucial in assessing short-term economic conditions.
[Treasury Bill Rates and the Business Cycle]
1- Overview of T-Bill Rates and Business Cycle:
– Nominal rates equal the real interest rate plus inflation expectations.
– Short-term nominal interest rates are positively correlated with short-term real interest rates and inflation expectations.
– Central banks adjust interest rates based on the economy’s position in the business cycle, reducing rates during low activity and increasing them when inflation risk is high.
2- The Taylor Rule:
– Used by central banks to determine policy rates.
— Formula: “pr_t = lt + ιt + 0.5(ιt - ιt_target) + 0.5(Yt - Yt_target)”
— Where:
—- pr_t: Policy rate.
—- lt: Real short-term interest rate.
—- ιt: Inflation rate.
—- ιt_target: Target inflation rate.
—- Yt: Actual real GDP.
—- Yt_target: Potential real GDP.
– Simplified Formula: “pr_t = lt + 1.5ιt - 0.5ιt_target + 0.5(Yt - Yt_target)”
3- Drivers of Policy Rates (Based on the Taylor Rule):
– Positively related to:
— 1- Real short-term interest rates and inflation rates.
— 2- Excess of inflation over target inflation.
— 3- Excess of actual GDP over potential GDP (output gap).
4- Implications of Misaligned Policy Rates:
– Positive output gap indicates overcapacity; negative gap suggests underperformance.
– Misalignment in policy rates (e.g., rates set too low for too long) can exacerbate the business cycle or lead to financial instability (e.g., credit bubbles).
Key Takeaways
– T-bill rates reflect the balance between inflation expectations, real interest rates, and the economy’s output gap.
– Properly aligned policy rates help stabilize the business cycle and control inflation.
[Break-even Inflation Rates and Default-Free Yield Curve]
1- Break-even Inflation Rates:
– Represent the yield difference between a default-free nominal bond and a default-free real bond.
– Reflect expected inflation and a premium for uncertainty but are not a precise estimate of future inflation.
– T-bills, being short-term and default-risk free, have a low correlation with adverse consumption outcomes, resulting in a low-risk premium.
2- Default-Free Yield Curve:
– Represents yields on bonds of varying maturities, incorporating real and inflationary components.
– Influenced by policy rates, which significantly affect short-term yields.
– The slope of the curve reflects:
— 1- Real interest rates.
— 2- Expected inflation.
— 3- Risk premium for inflation uncertainty.
3- Yield Curve Dynamics and Economic Implications:
– Steep yield curves suggest high inflation expectations; inverted curves often precede recessions.
– Late business cycle stages typically exhibit high inflation and high short-term interest rates.
– A positively sloped curve often indicates a willingness to pay for short-dated bonds due to risk premiums associated with bad times.
– A downward-sloping curve may suggest expected declines in interest rates, while an upward-sloping curve may indicate rising rates or elevated risk premiums.
Key Takeaways
– Break-even inflation rates gauge market expectations for inflation but include uncertainty premiums.
– The yield curve’s shape and slope provide insights into inflation expectations, real interest rates, and economic conditions.
– Government bond risk premiums are tied to their consumption-hedging benefits and bond maturity.
Interpretation of an Upward-Sloping Yield Curve
1- Key Concept: Risk Premiums
– Risk premiums are added to longer-term yields to compensate for uncertainty about future interest rates and inflation.
– These premiums increase with maturity because uncertainty grows over longer periods.
2- Understanding the Yield Curve
– A downward-sloping yield curve clearly indicates expectations of falling interest rates because lower yields at longer maturities reflect both expectations and lower risk premiums.
– An upward-sloping yield curve, however, is not definitive proof that investors expect rates to rise. It reflects a combination of:
— Expectations about future interest rates.
— Maturity-related risk premiums.
3- Example
– Imagine a scenario where the 1-year interest rate is 5% and the 2-year interest rate is 5.5%.
— The upward-sloping yield curve suggests that investors might expect higher interest rates in the future.
— However, it could also mean investors are expecting the 1-year rate to stay at 5% (or even fall) but are charging a 0.5% risk premium for the 2-year rate due to uncertainty about future rates.
— This implies that even if expectations are for flat or declining rates, the yield curve can remain upward-sloping due to the premium required for holding longer-term securities.
Key Takeaways
– An upward-sloping yield curve cannot definitively indicate rising interest rate expectations.
– The curve represents both expectations and the maturity-related risk premium.
– In the example, investors may expect rates to remain at 5% or fall, but the 0.5% risk premium for uncertainty keeps the curve upward-sloping.
[Adjustment of Bond Pricing for Default Risk]
1- Overview of the Concept
– Bonds that were once considered “default-free,” such as U.S. Treasuries or government bonds from Greece and Spain, are now recognized to carry some level of default risk. To reflect this, bond pricing models include adjustments for credit risk.
2- Formula
– Adjusted bond pricing formula:
P_t_i = T∑_s=1 [E_t(CF_t+s_i) ÷ (1 + l_t_s + θ_t_s + π_t_s + γ_t_s_i)^s]
3- Explanation of Variables
– P_t_i: Present value of bond i at time t.
– CF_t+s_i: Expected cash flow from bond i at time t + s.
– l_t_s: Yield to maturity on a real, default-free investment today.
– θ_t_s: Expected inflation rate between time t and t + s.
– π_t_s: Compensation for uncertainty in inflation.
– γ_t_s_i: Credit premium reflecting the bond i’s default risk.
– s: Time period (1, 2, …, T) representing cash flow intervals.
4- Adjusted Formulas for Specific Situations
– If there is no default risk, γ_t_s_i = 0. The formula then simplifies to:
P_t_i = T∑_s=1 [E_t(CF_t+s_i) ÷ (1 + l_t_s + θ_t_s + π_t_s)^s].
– Example: A bond with negligible credit risk, such as some U.S. Treasury bonds, has almost no γ_t_s_i, focusing only on inflation and yield adjustments.
Key Takeaways
– Incorporating credit risk ensures a more accurate bond valuation.
– This formula accounts for inflation expectations, uncertainty, and credit risk premiums, making it applicable to bonds with varying degrees of default risk.
[Calculating the Breakeven Rate of Inflation]
1- Definition of the Breakeven Rate of Inflation:
– The breakeven rate of inflation is the difference in yields between a nominal zero-coupon default-free bond and an equivalent real bond. It reflects the compensation investors demand for both expected inflation and the uncertainty surrounding future inflation.
2- Components of the Breakeven Rate:
– Expected Inflation Rate: 2.0%.
– Inflation Uncertainty Premium: 0.75%.
– The nominal bond’s yield includes both the expected inflation rate and the inflation uncertainty premium, whereas the real bond’s yield excludes them.
3- Calculation:
– Formula:
“Breakeven Rate of Inflation = Expected Inflation Rate + Inflation Uncertainty Premium”
– Substitution:
“Breakeven Rate of Inflation = 2.0% + 0.75% = 2.75%”
4- Conclusion:
– The breakeven rate of inflation is 2.75%.
[Credit Spreads: Scenarios of Narrowing and Widening]
1- Overview of the Concept
– Credit spreads represent the difference in yields between corporate bonds and government bonds or between bonds of different credit ratings. These spreads reflect market perceptions of credit risk and economic conditions.
2- Narrowing Credit Spreads
– Description: When credit spreads narrow, the gap between yields on corporate bonds (both high-rated and low-rated) and government bonds reduces. Similarly, the spread between higher-rated and lower-rated corporate bonds contracts.
– Implications:
— Higher-rated corporate bonds outperform government bonds, as the reduced perception of risk favors higher corporate returns.
— Lower-rated corporate bonds outperform higher-rated corporate bonds, reflecting greater investor confidence in lower-rated debt instruments.
3- Widening Credit Spreads
– Description: When credit spreads widen, the yield gap between corporate bonds and government bonds increases. Additionally, the spread between higher-rated and lower-rated corporate bonds expands.
– Implications:
— Higher-rated corporate bonds underperform relative to government bonds, as increased risk aversion drives investors toward safer government securities.
— Lower-rated corporate bonds underperform higher-rated corporate bonds, as riskier assets are penalized during times of uncertainty.
4- Situations Leading to Widening or Narrowing
– Narrowing Spreads:
— Typically occurs during periods of economic growth or recovery when market confidence improves. Investors are more willing to take on credit risk, reducing yield gaps.
— Examples include post-recession recoveries or optimistic market conditions driven by stable or improving corporate fundamentals.
– Widening Spreads:
— Occurs during economic downturns, financial crises, or periods of heightened uncertainty when risk aversion increases. Investors demand higher yields for riskier assets, leading to wider spreads.
— Examples include market disruptions, global financial crises, or heightened geopolitical risks.
Key Takeaways
– Narrowing credit spreads indicate growing market confidence and favor corporate bonds, especially lower-rated ones.
– Widening credit spreads signal risk aversion, favoring safer assets like government bonds and penalizing riskier corporate bonds.
[Equity Pricing Formula Adjusted for Risk]
1- Overview of the Concept
– Equity cash flows are uncertain in both timing and amount. Dividends can fluctuate or cease entirely if the company faces financial difficulties. The pricing formula for equities incorporates these risks and additional equity premiums.
2- Formula
– Generic pricing formula for equities:
P_t_i = ∞∑_s=1 [E_t(CF_t+s_i) ÷ (1 + l_t_s + θ_t_s + π_t_s + γ_t_s_i + κ_t_s_i)^s]
3- Explanation of Variables
– P_t_i: Present value of equity i at time t.
– CF_t_plus_s_i: Expected cash flow from equity i at time t plus s.
– l_t_s: Yield to maturity on a real, default-free investment today.
– θ_t_s: Expected inflation rate between time t and t plus s.
– π_t_s: Compensation for uncertainty in inflation.
– γ_t_s_i: Credit premium for default risk.
– κ_t_s_i: Equity premium relative to credit-risky bonds.
4- Key Adjustments for Equity Cash Flows
– Equity cash flows do not have a fixed maturity, leading the summation to extend to infinity.
– An equity premium (κ_t_s_i) is added to account for the greater risk of equity relative to bonds, reflecting the subordinated claim of equity holders compared to debt holders.
5- Substitution for Risk Premiums
– When equity risk premiums are expressed relative to default-free government bonds, the formula simplifies by replacing terms:
λ_t_s_i = γ_t_s_i + κ_t_s_i.
Key Takeaways
– Equity pricing models integrate multiple risk components, including inflation, credit risk, and equity-specific premiums.
– The infinite summation reflects the perpetual nature of equity cash flows, distinguishing it from fixed-income pricing models.
[Understanding the P/E Ratio]
1- Overview of the P/E Ratio
– The price-to-earnings (P/E) ratio is calculated as:
P/E_ratio = Current_share_price ÷ Earnings_per_share.
– It measures how much investors are willing to pay for each unit of earnings generated by the firm.
2- Implications of High and Low P/E Ratios
– A low P/E ratio suggests that the market lacks confidence in the company’s future earnings potential.
– A high P/E ratio implies that the market expects significant earnings growth in the future.
3- Factors Influencing the P/E Ratio
– Determining whether a P/E ratio is too high or too low is complex, as it depends on various factors, such as:
— High expected future real earnings growth.
— Falling real interest rates.
— Falling inflation expectations.
— Reduced uncertainty about future inflation.
— A decrease in the equity risk premium.
– It is also important to note that a high P/E ratio might reflect irrational exuberance among equity investors rather than fundamentals.
Key Takeaways
– The P/E ratio is a critical valuation tool, but interpreting it requires careful consideration of market expectations and economic factors.
– Both overly low and high P/E ratios can indicate deeper market sentiments or irrational behavior.
[Calculating the Credit Premium for a Corporate Bond]
1- General Equation for the Value of Risky Cash Flows:
– Formula:
“Pt,i = N∑_s=1 Et[CFt+s,i] ÷ [(1 + lt,s + θt,s + πt,s + γt,s,i)^s]”
Where:
– Pt,i: Price of the risky bond.
– CFt+s,i: Cash flows at time t+s.
– lt,s: Real risk-free yield for the period.
– θt,s: Expected inflation rate.
– πt,s: Premium for inflation uncertainty.
– γt,s,i: Credit risk premium.
For a one-year zero-coupon bond, the formula simplifies to:
“Pt,i = CFt+1 ÷ (1 + lt,1 + θt,1 + πt,1 + γt,1,i)”
2- Rearranging to Solve for γt,1,i:
– Formula:
“γt,1,i = (CFt+1 ÷ Pt,i) - (1 + lt,1 + θt,1 + πt,1)”
– Variables:
— CFt+1 = 100: Maturity value of the bond.
— Pt,i = 93.56: Price of the bond.
— lt,1 = 0.015: Real risk-free rate.
— θt,1 = 0.020: Expected inflation rate.
— πt,1 = 0.0075: Premium for inflation uncertainty.
– Substitution:
“γt,1,i = (100 ÷ 93.56) - (1 + 0.015 + 0.020 + 0.0075)”
“γt,1,i = 1.0689 - 1.0425 = 0.0264 or 2.63%”
3- Conclusion:
– The credit risk premium for the bond is 2.63%.
[Valuation Multiples]
1- Overview of Valuation Multiples
– Valuation multiples are used to compare equity securities. Common examples include:
— Price-to-earnings ratio (P/E ratio).
— Price-to-book ratio (P/B ratio).
2- Price-to-Book Ratio (P/B Ratio)
– Formula:
P/B_ratio = Current_share_price ÷ Book_value_of_net_assets.
– Explanation:
— The P/B ratio indicates what fraction of the share value is covered by the company’s net assets.
— It helps assess the valuation of a company’s equity relative to its book value.
— However, P/B ratios can be misleading because certain assets may not appear on the balance sheet or may not be recorded at their current market value.
3- Real Cyclically Adjusted P/E (CAPE)
– CAPE Formula:
CAPE = P ÷ E.
– Explanation:
— P represents the real (inflation-adjusted) price of the equity market.
— E is the 10-year moving average of the market’s inflation-adjusted earnings.
— CAPE smooths short-term volatility by using long-term averages, offering a more stable measure of market valuation over economic cycles.
Key Takeaways
– Valuation multiples like P/B and CAPE provide insights into equity valuations, but they require careful interpretation to account for limitations such as balance sheet accuracy and short-term market fluctuations.
– CAPE is particularly useful for long-term investment analysis as it adjusts for inflation and cyclical earnings variability.
[Commercial Real Estate Pricing Formula]
1- Overview of Commercial Real Estate Investments
– Cash flows from commercial real estate are primarily generated by rents paid by tenants. Rental agreements are periodically adjusted to reflect current market rates, and rents may also be indexed to inflation.
– Rental income shares similarities with bond coupon payments, as the credit quality of commercial real estate is tied to the creditworthiness of the tenants.
– The future value of the property is variable and depends on its location and the economic environment, which introduces equity-like features into commercial real estate investments.
– Commercial real estate is highly illiquid compared to bonds and equities, often requiring months or years for a sale.
2- Pricing Formula for Commercial Real Estate
– Formula:
P_t_i = N∑_s=1 [E_t(CF_t+s_i) ÷ (1 + l_t_s + θ_t_s + π_t_s + γ_t_s_i + κ_t_s_i + ϕ_t_s_i)^s]
3- Explanation of Variables
– P_t_i: Present value of the commercial real estate investment i at time t.
– CF_t+s_i: Expected cash flow from property i at time t plus s.
– l_t_s: Yield to maturity on a real, default-free investment today.
– θ_t_s: Expected inflation rate between time t and t plus s.
– π_t_s: Compensation for uncertainty in inflation.
– γ_t_s_i: Credit premium related to the tenants’ default risk.
– κ_t_s_i: Premium for uncertain property value at the end of the investment period.
– ϕ_t_s_i: Liquidity risk premium, reflecting the illiquidity of commercial real estate.
4- Key Observations
– The formula incorporates factors such as tenant default risk, property value uncertainty, and liquidity risk.
– These features make commercial real estate a hybrid investment, combining elements of fixed-income securities and equity investments.
Key Takeaways
– Commercial real estate pricing accounts for various risks, including inflation, credit, liquidity, and property value uncertainty.
– These factors differentiate it from simpler asset classes like bonds and equities.
[Discount Factors and Tenant Situations in Commercial Real Estate]
1- Overview of Discount Factors
– The applicable discount factors in commercial real estate vary depending on the type of tenant and the rental agreement. These factors influence the pricing model by accounting for inflation, credit risk, property value uncertainty, and liquidity risk.
2- Specific Tenant Situations and Discount Factors
– Developed economy government tenant with inflation-indexed rental income:
— Applicable discount factors: l_t_s, κ_t_s, ϕ_t_s.
— Similar to a default-free inflation-indexed bond with an uncertain future value and illiquidity.
– Developed economy government tenant with fixed nominal rental income:
— Applicable discount factors: l_t_s, θ_t_s, π_t_s, κ_t_s, ϕ_t_s.
— Similar to a default-free nominal bond with an uncertain future value and illiquidity.
– Corporate tenant with fixed nominal rental income:
— Applicable discount factors: l_t_s, θ_t_s, π_t_s, γ_t_s_i, κ_t_s_i, ϕ_t_s_i.
— Similar to a credit-risky nominal bond with an unknown and illiquid future value.
3- Explanation of Discount Factors
– l_t_s: Yield to maturity on a real default-free investment today.
– θ_t_s: Expected inflation rate between time t and t plus s.
– π_t_s: Compensation for uncertainty in future inflation.
– γ_t_s_i: Credit premium related to tenant default risk.
– κ_t_s_i: Premium for uncertain property value at the end of the investment period.
– ϕ_t_s_i: Liquidity risk premium reflecting the illiquidity of commercial real estate.
Key Takeaways
– The nature of the tenant (government vs. corporate) and the rental terms (inflation-indexed vs. fixed nominal) significantly affect the applicable discount factors.
– Government tenants generally require fewer adjustments for credit risk, while corporate tenants introduce more complexity due to higher risk.
[Present Value Discounting Components for Different Financial Asset Classes]
1- General Asset Valuation
– Components in Denominator:
— lt,s: Real default-free rate.
— θt,s: Expected inflation.
— ρit,s: Risk premium for the asset class.
– Explanation:
Discounting incorporates a baseline real rate, expected inflation, and an asset-specific risk premium. This general form applies flexibly across financial assets, adjusting for characteristics like market, liquidity, or default risk.
2- Default-Free Nominal Coupon Bond
– Components in Denominator:
— lt,s: Real default-free rate.
— θt,s: Expected inflation.
— πt,s: Compensation for uncertainty in inflation.
– Explanation:
Valuation captures inflation expectations and an additional premium for inflation volatility. While credit risk is excluded, the presence of πt,s reflects uncertainty in real value from nominal payments.
3- T-Bill (Single-Payment, Short-Term)
– Components in Denominator:
— lt,s: Real default-free rate.
— θt,s: Expected inflation.
– Explanation:
Given its short maturity and default-free nature, the T-bill’s discount rate omits premiums for inflation uncertainty or credit risk. Only real rate and expected inflation are relevant.
4- Bonds with Default Risk
– Components in Denominator:
— lt,s: Real default-free rate.
— θt,s: Expected inflation.
— πt,s: Compensation for inflation uncertainty.
— γit,s: Credit risk premium.
– Explanation:
Compared to default-free bonds, risky bonds include a credit risk premium to account for potential default. The discounting rate reflects both macroeconomic variables and issuer-specific default risk.
5- Equities
– Components in Denominator:
— lt,s: Real default-free rate.
— θt,s: Expected inflation.
— πt,s: Compensation for inflation uncertainty.
— γit,s: Credit premium.
— κit,s: Equity risk premium over risky bonds.
– Explanation:
Equity valuation requires an additional premium for equity-specific risks like earnings volatility and price uncertainty. The equity premium κit,s reflects compensation beyond credit risk and inflation uncertainty.
6- Commercial Real Estate
– Components in Denominator:
— lt,s: Real default-free rate.
— θt,s: Expected inflation.
— πt,s: Compensation for inflation uncertainty.
— γit,s: Credit premium (tenant default).
— κit,s: Uncertainty in property value.
— φit,s: Liquidity risk premium.
– Explanation:
This asset class incorporates layered risks: macroeconomic, tenant default, uncertain end-period asset value, and illiquidity in resale markets. Each component adds a premium to reflect these exposures in the discount rate.
[Quiz - Component Contributing Most to the Hurdle Rate]
1- Overview of the Valuation Context
– A 7.05% hurdle rate is used to estimate the value of a building.
– This hurdle rate reflects multiple risk and uncertainty premiums including:
— Inflation uncertainty.
— Credit risk.
— Sale price uncertainty.
— Liquidity premium.
2- Nominal Risk-Free Rate Calculation
– Given: Price of nominal risk-free zero-coupon bond = CHF 95.74.
– Formula: Nominal risk-free rate = (100 ÷ 95.74) - 1.
– Result: 4.45%
3- Premium for Inflation Uncertainty
– Formula: Nominal risk-free rate - Real risk-free rate - Expected inflation.
– Calculation: 4.45% - 1.42% - 2.17% = 0.86%
4- Credit Risk Premium
– Formula: Yield on corporate bond - Nominal risk-free rate.
– Calculation: 5.33% - 4.45% = 0.88%
5- Sale Price Uncertainty Premium
– Given:
— Hurdle rate = 7.05%.
— Corporate bond yield = 5.33%.
— Liquidity premium = 0.90%.
– Formula: Residual = Hurdle rate - Corporate bond yield - Liquidity premium.
– Calculation: 7.05% - 5.33% - 0.90% = 0.82%
[Key Differences Between Real Interest Rates and Government Bond Returns]
1- Real Interest Rates:
– Definition: Inflation-adjusted rates representing the true cost of borrowing or return on lending, often derived from inflation-linked instruments like TIPS.
– Drivers:
— 1- Expected GDP growth: Higher growth increases capital demand, raising real rates.
— 2- Growth volatility: Greater uncertainty increases real rates as compensation for risk.
– Relationship with GDP Growth:
— Real interest rates are positively correlated with GDP growth and its volatility, as they reflect productivity and economic risk.
— Dependent on future expectations, not past realized values, with partial historical support.
2- Returns on Government Bonds:
– Definition: Comprise interest payments (coupons) and changes in bond prices, sensitive to interest rates, inflation, and economic cycles.
– Drivers:
— 1- Demand for safety: Bonds are sought during downturns, increasing prices and lowering yields.
— 2- Risk premiums: Long-term bonds may carry higher risk premiums, reducing their hedging effectiveness.
— 3- Economic conditions: Rising interest rates during expansions can depress bond prices.
– Relationship with GDP Growth:
— Returns are negatively correlated with GDP growth, as bonds act as a hedge against economic downturns.
3- Why the Difference Exists:
– Real interest rates reflect long-term economic productivity and risk, tied to the cost of capital for productive investments.
– Bond returns are influenced by short-term market dynamics, including risk aversion and the economic cycle, as investors seek safety during downturns.
4- Example Illustration:
– Real Interest Rates: A country with 4% GDP growth expectation sees rising real rates as borrowers compete for capital to fund productive investments.
– Bond Returns: During a recession, investors flee to bonds for safety, driving prices up and delivering positive returns, even if real rates remain unchanged.
Key Takeaways
– Real interest rates are long-term and growth-driven, reflecting fundamental economic productivity.
– Bond returns are short-term, driven by investor behavior, risk aversion, and cyclical market dynamics.
Understanding Correlation of Bond Returns with GDP Growth
1- Negative Correlation of Bonds with GDP Growth
– Government bonds are often used as a hedge against economic downturns. During periods of low GDP growth or recessions, bond prices typically rise due to increased demand for safer assets.
– Conversely, during high GDP growth, bonds may perform poorly as investors shift to higher-risk assets like equities.
2- Short-Dated vs. Long-Dated Bonds
– Short-dated bonds (e.g., 5-year maturity) tend to have a more negative correlation with GDP growth compared to long-dated bonds (e.g., 20-year maturity).
– Longer-dated bonds carry higher risk premiums because they are considered less effective at hedging against bad economic times. This premium reflects the uncertainty and higher duration risk of these bonds.
3- Key Insights from Yield Curve
– An upward-sloping yield curve, such as in the example with a 3.5% yield on 5-year bonds and 4.2% on 20-year bonds, implies that investors demand higher compensation (risk premiums) for holding longer-term bonds.
– Short-dated bonds, being better hedges against economic downturns, exhibit stronger negative correlation with GDP growth.
Example:
If Country A’s economy experiences a sharp slowdown:
— Short-dated bonds may see a significant price increase as investors seek safety, reflecting a strong negative correlation with GDP growth.
— Long-dated bonds may still rise in price but to a lesser extent, as their longer maturities and risk premiums make them less responsive to short-term economic changes.
Key Takeaways
– Bonds generally exhibit negative correlation with GDP growth, serving as a hedge during downturns.
– Short-dated bonds are more negatively correlated with GDP growth due to their better hedging properties.
– Longer-dated bonds have higher risk premiums, making them less effective in this regard.
9.2 Analysis of Active Portfolio Management
– Describe how value added by active management is measured.
– Calculate and interpret the information ratio (ex post and ex ante) and contrast it to the Sharpe ratio.
– Describe and interpret the fundamental law of active portfolio management, including its component terms—transfer coefficient, information coefficient, breadth, and active risk (aggressiveness).
– Explain how the information ratio may be useful in investment manager selection and choosing the level of active portfolio risk.
– Compare active management strategies, including market timing and security selection, and evaluate strategy changes in terms of the fundamental law of active management.
– Describe the practical strengths and limitations of the fundamental law of active management.
[Choice of Benchmark for Active Management]
1- Purpose of a Benchmark:
– Active managers compare their performance to a benchmark to demonstrate their ability to outperform a passive strategy.
2- Key Qualities of a Relevant Benchmark:
– 1- Representativeness: The benchmark should reflect the assets that the active manager is likely to select.
– 2- Replicability: The benchmark’s positions should be replicable at a low cost.
– 3- Transparency: The benchmark weights must be predefined, and returns can be calculated ex post.
3- Commonly Used Benchmarks:
– Security market indexes, often market-value weighted, are commonly used as benchmarks.
– Float-adjusted capitalization-weighted indexes exclude private holdings when adjusting weights.
4- Formula for Benchmark and Portfolio Returns:
– Benchmark return: “RB = T∑_i=1 (wB,i * Ri)”
— RB: Benchmark return.
— wB,i: Weight of security i in the benchmark.
— Ri: Return of security i.
– Portfolio return: “RP = T∑_i=1 (wP,i * Ri)”
— RP: Portfolio return.
— wP,i: Weight of security i in the portfolio.
Key Takeaways
– The choice of a benchmark is critical to effectively evaluate active management performance.
– Benchmarks should be transparent, cost-effective, and representatively weighted for meaningful comparisons.
[Measuring Value Added]
1- Definition of Active Return:
– Active return (RA) is the excess of the portfolio return over the benchmark return.
– Formula: “RA = RP - RB”
— RP: Portfolio return.
— RB: Benchmark return.
2- Alpha:
– Alpha (αP) is a beta-adjusted measure of active return.
– Formula: “αP = RP - βP * RB”
— βP: Portfolio beta relative to the benchmark.
3- Explanation of Active Return:
– Active return can be positive or negative, depending on portfolio performance compared to the benchmark.
– It arises from differences in portfolio weights and benchmark weights, referred to as active weights (Δwi).
4- Calculation of Active Return:
– Formula: “RA = T∑_i=1 (Δwi * Ri)”
— Δwi = wp,i - wB,i: Difference between portfolio weight and benchmark weight for asset i.
— Ri: Return of asset i.
– Alternatively, active return can be written as:
“RA = T∑_i=1 (Δwi * RAi)”
— RAi = Ri - RB: Active return of asset i relative to the benchmark.
Key Takeaways
– Active return measures the value added by an actively managed portfolio compared to a benchmark.
– It can result from asset selection, weighting differences, or both, providing insights into portfolio performance.
[Decomposition of Value Added]
1- Overview of Decomposition of Value Added:
– Performance attribution analyzes the sources of value added by separating the contribution of asset allocation decisions and security selection.
2- Key Components:
– Asset Allocation:
— Adds value by overweighting high-return asset classes or underweighting poor-performing asset classes relative to the benchmark.
– Security Selection:
— Adds value by selecting individual securities within asset classes that outperform others in the same class.
3- Formula for Total Value Added:
– Total value added (RA) can be expressed as:
“RA = T∑_j=1 (Δwj * RB,j) + T∑_j=1 (wp,j * RA,j)”
— Δwj: Active weight for asset class j.
— RB,j: Benchmark return for asset class j.
— wp,j: Portfolio weight for asset class j.
— RA,j: Active return for securities within asset class j.
Key Takeaways
– Decomposition highlights the distinct contributions of asset allocation and security selection to active return.
– By attributing value added to these sources, managers can assess the effectiveness of their strategies and refine their approaches.
[Sharpe Ratio]
1- Overview of Sharpe Ratio:
– The Sharpe ratio measures the excess portfolio return over the risk-free rate, relative to the portfolio’s return standard deviation.
2- Formula for Sharpe Ratio:
– Formula: “SRP = (RP - RF) / STD(RP)”
— RP: Portfolio return.
— RF: Risk-free rate.
— STD(RP): Standard deviation of portfolio returns.
3- Key Characteristics:
– The Sharpe ratio is a risk-return trade-off metric and can be calculated on an ex-ante (forward-looking) or ex-post (historical) basis.
– Typically, returns and risks are annualized for comparison purposes.
– When comparing funds, the same time period must be used to ensure consistency.
– The Sharpe ratio remains unaffected by portfolio leverage or the addition of cash since both the numerator and denominator are proportionally adjusted.
Key Takeaways
– The Sharpe ratio provides a standardized way to evaluate the performance of a portfolio relative to its risk.
– It helps investors compare different funds or strategies based on their risk-adjusted returns.
[Information Ratio]
1- Overview of Information Ratio:
– The information ratio (IR) measures a portfolio’s active return relative to its active risk, where active risk represents the standard deviation of the portfolio’s excess return over the benchmark.
2- Formula for Information Ratio:
– Formula: “IR = (RP - RB) / STD(RP - RB)” or equivalently “IR = RA / STD(RA)”
— RP: Portfolio return.
— RB: Benchmark return.
— RA: Active return (RP - RB).
— STD(RA): Standard deviation of active return, representing active risk.
3- Key Characteristics:
– The measure assumes the portfolio has a beta of 1 relative to the benchmark, and residual risk is considered a beta-adjusted measure of active risk.
– The information ratio evaluates the consistency of active returns and is typically annualized.
– It can be applied on an ex-ante (forward-looking) or ex-post (historical) basis.
4- Differences from the Sharpe Ratio:
– The information ratio is sensitive to changes in portfolio weights. For example, doubling all active weights leaves the ratio unchanged because both active return and active risk scale proportionately.
– Unlike the Sharpe ratio, the IR reflects the impact of leverage and cash additions on the portfolio.
Key Takeaways
– The information ratio is a critical tool for assessing the performance of active management by evaluating how consistently a portfolio outperforms its benchmark on a risk-adjusted basis.
– High information ratios indicate effective portfolio management in delivering superior risk-adjusted returns relative to a benchmark.
If cash is added to a portfolio of risky assets information ratio will typically decrease. However, adding cash to such a portfolio will reduce the expected excess return and return volatility in the same proportion, leaving the Sharpe ratio unaffected.
Unlike the Sharpe ratio, the information ratio is affected by the addition of both cash and leverage.
[Constructing Optimal Portfolios]
1- Overview of Optimal Risky Portfolios:
– The optimal risky portfolio on the efficient frontier is the one that maximizes the Sharpe ratio, regardless of an investor’s risk aversion.
– The objective is to combine the actively managed portfolio and the benchmark portfolio to achieve the best risk-return outcome.
2- Formula for Portfolio Sharpe Ratio:
– Formula: “SRP^2 = SRB^2 + IR^2”
— SRP: Sharpe ratio of the portfolio.
— SRB: Sharpe ratio of the benchmark.
— IR: Information ratio of the active portfolio.
– This equation shows that the portfolio with the highest squared information ratio will also have the highest squared Sharpe ratio.
3- Determining the Optimal Active Risk:
– Formula: “STD(RA) = (IR / SRB) * STD(RB)”
— STD(RA): Standard deviation of active return (active risk).
— STD(RB): Standard deviation of benchmark return.
– This formula calculates the optimal amount of active risk to allocate to an actively managed portfolio.
4- Overall Portfolio Risk:
– Formula: “STD(RP)^2 = STD(RB)^2 + STD(RA)^2”
— STD(RP): Total standard deviation of the portfolio.
5- Example of Allocation:
– If an actively managed fund has an active risk of 5% and the optimal active risk is calculated as 3%, then the allocation would be 60% to the active portfolio (i.e., 3% / 5%) and 40% to the benchmark portfolio.
Key Takeaways
– Investors should select active managers with the highest expected skill as measured by the information ratio.
– The goal of constructing optimal portfolios is to achieve the highest risk-adjusted return by balancing active and benchmark components efficiently.
[Maximum Sharpe Ratio with Combination of Manager C and Benchmark
1- Overview of the Concept
– The maximum Sharpe ratio of a portfolio combining an active manager and a benchmark is derived from both the Sharpe ratio of the benchmark and the manager’s information ratio.
– This assumes optimal mixing of the active and passive components.
2- Formula Used
– Maximum portfolio Sharpe ratio:
— “SRp^2 = SRb^2 + IR^2”
— Then take the square root: “SRp = √(SRb^2 + IR^2)”
— Where:
—- SRp: Sharpe ratio of the combined portfolio.
—- SRb: Sharpe ratio of the benchmark.
—- IR: Information ratio of the active manager (Manager C in this case).
3- Input Values
– Benchmark expected return = 13.15%
– Risk-free rate = 2.55%
– Benchmark standard deviation = 19.27%
– Manager C information ratio = 0.54
4- Step-by-Step Calculation
– Step 1: Calculate Sharpe ratio of benchmark:
— SRb = (13.15% – 2.55%) ÷ 19.27% = 0.55
– Step 2: Apply the formula:
— SRp^2 = (0.55)^2 + (0.54)^2
— SRp^2 = 0.3025 + 0.2916 = 0.5941
– Step 3: Take the square root:
— SRp = √0.5941 = 0.7708
5- Final Answer
– Maximum Sharpe ratio achievable: 0.77
[Ex Ante vs. Ex Post Measures]
1- Ex Ante Measure:
– Calculated in advance based on expected return and risk.
– Makes the measure subjective as it depends on assumptions about the future.
2- Ex Post Measure:
– Calculated after a specified time period using actual return and risk over that time.
– Portfolio return and portfolio risk are annualized:
— Monthly returns are multiplied by 12.
— Monthly standard deviations are multiplied by sqrt(12).
Key Takeaways
– When comparing the Sharpe ratio between funds, always use data from the same time period for consistency.
[Information Ratio and Active Weights]
1- Unconstrained Portfolio:
– An unconstrained portfolio has no limits on positive or negative active weights.
– The Information Ratio (IR) of an unconstrained portfolio is not affected by changes in the degree of active weights.
2- Adding Cash to a Portfolio:
– Adding cash to a portfolio of risky assets reduces the information ratio of the combined portfolio.
3- Mathematical Explanation:
– The sum of active returns (2∑Δwi·RAi) equals 2RA.
– The sum of standard deviations (2∑Δwi·STD(RAi)) equals 2STD(RA).
– New IR = 2RA / 2STD(RA) = IR.
Key Takeaways
– The IR of an unconstrained portfolio remains constant despite changes in active weights or doubling the portfolio size.
[Active Security Returns]
1- Overview of Active Security Returns:
– Active returns are generated by an investor’s ability to outperform the benchmark return (RB).
– The goal is to maximize active return while managing active risk.
2- Formula for Active Return:
– Active return for a security: RAi = Ri - RB, where:
— RAi: Active return for security i.
— Ri: Return on security i.
— RB: Benchmark return.
– Expected active return: μi = E[RAi].
3- Multifactor Model for Active Return:
– Active return can also be expressed using a multifactor statistical model:
— RAi = Ri - Σj=1K βj,iRj, where:
—- βj,i: Sensitivity of security i to factor j.
—- Rj: Return of factor j.
—- K: Number of factors in the model.
4- Key Parameters in Active Management:
– Forecasted active returns (μi).
– Active weights (Δwi).
– Realized active returns (RAi).
5- Value Addition in Active Management:
– Active weights are determined by forecasted active returns.
– Value is created only if forecasted returns (μi) align with realized returns (RAi).
Key Takeaways
– Active management’s success depends on accurately forecasting active returns and aligning them with realized performance.
– Multifactor models are essential tools for estimating active security returns.
[Optimization of Active Security Weights]
1- Scaling of Active Return Forecasts:
– Active return forecasts are adjusted before optimization, using the formula:
— μi = IC * σi * Si, where:
—- μi: Forecasted active return for security i.
—- IC: Information coefficient (correlation between forecasted and realized active returns).
—- σi: Forecasted volatility of the active return for security i.
—- Si: Standardized forecast of expected returns (often scores that sum to zero).
2- Determining Optimal Active Weights:
– The mean-variance-optimal active security weights (Δwi) are calculated using:
— Δwi = (μi / σi²) * (σA / (IC * √BR)), where:
—- Δwi*: Optimal active weight for security i.
—- μi: Forecasted active return for security i.
—- σi²: Variance of active return for security i.
—- σA: Active portfolio risk.
—- IC: Information coefficient.
—- BR: Breadth, representing the number of independent investment decisions.
3- Key Variable Explanations:
– σA: Represents the total active risk of the portfolio.
– σi: Measures the expected variability in the active return of security i.
– IC: Reflects the quality of forecasts; higher IC indicates better predictive ability.
– BR: Captures diversification through the number of independent investment opportunities.
Key Takeaways
– Optimization ensures that active weights maximize the portfolio’s risk-adjusted active return.
– The relationship between IC and BR highlights the importance of accurate forecasts and diversification.
[Information Coefficient (IC) in Active Management]
1- Overview of the Concept
– The Information Coefficient (IC) measures the quality of an active manager’s forecasts, indicating their ability to predict active returns accurately.
– It represents the correlation between forecasted active returns (μi) and realized active returns (RAi).
2- Range and Interpretation of IC
– Formula: “-1 ≤ IC ≤ 1”.
— A value of 1 indicates perfect positive correlation, implying highly effective forecasts.
— A value of -1 indicates perfect negative correlation, meaning the manager’s forecasts are consistently incorrect.
— A value near 0 indicates no forecasting ability.
3- Typical Values of IC
– In practice, IC typically falls within the range “0 ≤ IC < 0.2”.
— An IC closer to 0.2 suggests moderate forecasting skill.
— An IC close to 0 reflects poor or no predictive ability.
4- Investor Preferences
– Investors ideally aim for an IC as close to +1 as possible.
— A high IC indicates the manager is effective at forecasting active returns, enhancing the likelihood of adding value through active management.
Key Takeaways
– The IC is a critical measure of the skill and accuracy of an active manager’s forecasts.
– Typical IC values are modest but still provide a meaningful contribution to portfolio performance when combined with other factors such as breadth and the transfer coefficient.
[Breadth (BR) in Active Management]
1- Overview of Breadth (BR)
– Breadth (BR) represents the number of independent decisions or forecasts made in the investment process. It is a key component in the Fundamental Law of Active Management, which links breadth, skill (IC), and portfolio performance.
2- Single-Factor Risk Model Assumptions
– In a single-factor model:
— There is only one common market factor.
— Portfolio adjustments are made annually.
— Formula: “BR = Number of securities in the portfolio”.
— Assumes each active return is independent of the others.
3- Multi-Factor Risk Models
– Most risk models include additional factors such as sectors or industries.
— In these cases, “BR < Number of securities” because securities within the same sector are assumed to be positively correlated.
4- Impact of Portfolio Rebalancing
– When rebalancing occurs more frequently, breadth increases.
— Formula: “BR = Number of securities × Number of forecasts per year”.
— Example: For monthly forecasts and 12 rebalancing events per year, BR increases proportionally.
Key Takeaways
– Breadth measures the diversification of independent forecasts in the portfolio.
– Greater BR enhances the portfolio’s potential for adding value, assuming high forecast quality (IC).
– Multi-factor models and correlations between securities can reduce effective breadth.
– Frequent rebalancing increases BR and provides more opportunities for active management.
[Basic Fundamental Law]
1- Overview of the Concept
– The Basic Fundamental Law states that the expected active portfolio return is the sum product of active security weights and expected security returns.
2- Formula for Expected Active Portfolio Return
– Formula: “E[RA] = N∑_i=1 (Δwi * μi)”
— E[RA]: Expected active portfolio return.
— N: Number of securities.
— Δwi: Active weight of security i (difference from benchmark weight).
— μi: Expected return of security i.
3- Expected Active Portfolio Return with Optimal Weights
– Formula: “E[RA]* = IC * √BR * σA”
— E[RA]*: Expected active return for the optimal portfolio.
— IC: Information coefficient (measure of manager skill).
— BR: Breadth, representing the number of securities.
— σA: Active risk (portfolio tracking error).
4- Breadth (BR) and Number of Securities
– Breadth (BR) is equal to the number of securities in the portfolio.
— Formula: “BR = N”.
5- Information Ratio of the Unconstrained Optimal Portfolio
– Formula: “E[RA]* / σA = IR* = IC * √BR”
— E[RA]* / σA: Information ratio of the portfolio.
— IR*: Optimal information ratio.
Key Takeaways
– The Basic Fundamental Law links portfolio performance to the skill of the manager (IC), the number of securities (BR), and portfolio tracking error (σA).
– Optimal performance is achieved by maximizing IC and BR while managing active risk (σA).
[Full Fundamental Law]
1- Overview of the Concept
– The Full Fundamental Law accounts for constraints that prevent investors from using optimal active security weights (e.g., no short selling).
– It introduces the transfer coefficient (TC), which measures the correlation between forecasted active security returns and actual active weights, ranging between 0 and 1.
2- Formula for Expected Active Portfolio Return (Constrained)
– Formula: “E[RA] = (TC)(IC) * √BR * σA”
— E[RA]: Expected active return.
— TC: Transfer coefficient (degree to which constraints impact active weights).
— IC: Information coefficient (manager skill).
— BR: Breadth (number of securities).
— σA: Active risk.
3- Information Ratio for Constrained Optimal Portfolio
– Formula: “E[RA] / σA = IR = (TC)(IC) * √BR”
— IR: Information ratio for constrained portfolio.
4- Active Risk for Constrained Portfolio
– Formula: “σA = TC * (IR* / SRB) * σB”
— σA: Active risk.
— TC: Transfer coefficient.
— IR*: Information ratio of the unconstrained portfolio.
— SRB: Sharpe ratio of the benchmark.
— σB: Standard deviation of the benchmark.
5- Squared Sharpe Ratio for Constrained Portfolio
– Formula: “SRP² = SRB² + (TC)²(IR)²”
— SRP²: Squared Sharpe ratio of constrained portfolio.
— SRB²: Squared Sharpe ratio of benchmark.
— (TC)²(IR)²: Incremental contribution from active management with constraints.
Key Takeaways
– The transfer coefficient adjusts performance expectations based on constraints affecting active weights.
– A higher TC indicates fewer constraints and better alignment with forecasted active returns.
– The Full Fundamental Law connects the impact of constraints (via TC) with manager skill (IC), breadth (BR), and portfolio risk (σA).
[Transfer Coefficient (TC)]
1- Overview of the Transfer Coefficient (TC)
– The transfer coefficient (TC) measures the efficiency of implementing forecasted active returns in a portfolio, considering constraints.
– It reflects the risk-weighted correlation between forecasted returns, optimal active weights, and actual active weights.
2- Definitions of TC
– 1- TC is the risk-weighted correlation between forecasted active security returns and actual active weights.
– 2- Alternatively, TC is the risk-weighted correlation between the optimal active weights and the actual active weights.
3- Range and Typical Values
– Formula: “-1 ≤ TC ≤ 1”.
— A TC of 1 indicates no constraints, allowing full alignment of weights with forecasts.
— A TC near 0 or negative implies severe constraints, limiting the portfolio manager’s ability to act on forecasts.
– Typical range: “0.2 ≤ TC < 0.9”.
4- Implications of TC
– A low TC implies significant portfolio constraints (e.g., restrictions on short-selling or turnover limits).
– A TC of 1 means no binding constraints, and the full expected value can be realized through active management.
Key Takeaways
– The TC is a critical factor in determining how effectively a manager can translate skill (IC) into realized active returns.
– High TC indicates fewer implementation constraints, enhancing the portfolio’s performance potential.
– Typical values for TC suggest that some level of constraints is common in practice.
[Calculating Information Ratios for Sub-Portfolios]
1- Formula for the Information Ratio (IR)
– The information ratio measures a portfolio’s active return relative to its active risk, calculated as follows:
— Formula: “IR = (TC) × (IC) × √BR”
— Where:
—- TC: Transfer coefficient (portfolio’s alignment with active signals).
—- IC: Information coefficient (accuracy of active signals).
—- BR: Breadth (number of independent investment decisions).
[Ex Post Performance Measurement]
1- Overview of the Concept
– Ex post performance measurement evaluates actual portfolio performance, which may deviate from expectations due to constraints and noise.
– Actual performance is influenced by the realized information coefficient (ICR) and portfolio constraints.
2- Formula for Expected Value Added Conditional on Realized IC
– Formula: “E[RA | ICR] = (TC)(ICR) * √BR * σA”
— E[RA | ICR]: Expected active return given the realized IC.
— TC: Transfer coefficient.
— ICR: Realized information coefficient.
— BR: Breadth (number of securities).
— σA: Active risk.
3- Formula for Actual Active Return
– Formula: “RA = E[RA | ICR] + Noise”
— RA: Actual active return.
— E[RA | ICR]: Conditional expected active return.
— Noise: Random deviation caused by constraints or unforeseen factors.
4- Portfolio Active Return Variance Decomposition
– The portfolio’s active return variance can be divided into:
— 1- Variation due to realized information coefficient, “TC²”.
— 2- Variation due to constraint-induced noise, “1 - TC²”.
Key Takeaways
– Realized performance depends on both the skill of the investor (captured by ICR) and the impact of constraints (captured by TC).
– Noise represents factors beyond the investor’s control that influence actual returns.
– Decomposing variance helps identify the effects of constraints and skill on portfolio performance.
[The Correlation Triangle]
1- Overview of the Concept
– The correlation triangle highlights the relationships among forecasted active returns (μi), active weights (wi), and realized active returns (RAi).
– It emphasizes the roles of portfolio construction (transfer coefficient, TC) and signal quality (information coefficient, IC) in determining portfolio performance.
2- Key Relationships in the Correlation Triangle
– Forecasted Active Returns (μi): Represent the expected returns based on forecasts.
– Active Weights (wi): Reflect the positions taken in the portfolio based on forecasted returns.
— Value is added by overweighting securities that perform well and underweighting those that perform poorly.
– Realized Active Returns (RAi): Actual returns derived from the active portfolio.
3- Key Metrics
– Transfer Coefficient (TC): Measures the effectiveness of portfolio construction in translating forecasts into actual portfolio weights.
— A higher TC indicates fewer constraints and greater confidence in forecasts, leading to larger active weights.
– Information Coefficient (IC): Measures the quality and accuracy of the forecasts.
— Higher IC reflects better signal quality, increasing the correlation between forecasted and realized returns.
4- Active Management Requirements
– Active management success depends on the manager’s ability to:
— Accurately predict realized active returns (high IC).
— Construct a portfolio that effectively reflects forecasts despite constraints (high TC).
Key Takeaways
– Value from active management can only be achieved to the extent that active positions are taken, measured by the active weights (wi).
– Confidence in forecasts (TC) and the quality of those forecasts (IC) are critical in achieving better realized returns.
– Active weights, forecasted returns, and realized returns form a feedback loop for evaluating and improving portfolio management.
Active return Forecast are scaled first ! With the Grinold Rule
[Grinold Rule for Forecasted Active Returns]
1- Overview of the Grinold Rule
– The Grinold Rule is used to calculate scaled forecasted active returns (μi) based on the information coefficient (IC), security volatility, and a security’s score.
2- Formula
– Formula: “μi = IC * σi * Si”
— μi: Forecasted active return for security i.
— IC: Information coefficient, reflecting the manager’s forecasting skill.
— σi: Volatility of security i.
— Si: Score of security i (e.g., alpha ranking or predicted performance).
3- Explanation of Components
– IC: Represents the quality of the manager’s forecasts, with higher IC indicating greater skill.
– σi: Adjusts the forecasted return based on the risk level of the security.
– Si: Serves as a measure of the relative attractiveness or ranking of the security.
[Global Equity and Fixed-Income Strategies]
1- Global Equity Strategy
– This strategy evaluates active return performance relative to the MSCI All Country World Index (ACWI).
– Active return is driven by deviations in weights from the 24 assets in the benchmark.
– Active weights are determined based on return forecasts and optimized to maximize active return while adhering to constraints on active risk.
– Common constraints:
— 1- Active weights must sum to zero to avoid risk-free cash and leverage, slightly reducing the transfer coefficient (TC < 1.0).
— 2- Positions are restricted to being long-only, prohibiting short selling. Active weights cannot be less than the opposite of the benchmark weight.
– Constrained portfolios reduce the aggressiveness of the strategy.
2- Fixed-Income Strategies
– The Fundamental Law of Active Management also applies to fixed-income portfolios.
– Breadth (BR) is determined by the frequency of independent decisions:
— If rebalancing occurs quarterly, BR = 4, assuming independent active returns.
— Increasing the frequency of rebalancing to daily could raise BR to 250, provided decisions remain independent (an unlikely assumption).
– Limited breadth restricts the information ratio, as the number of independent decisions directly impacts performance potential.
Key Takeaways
– Constrained portfolios (e.g., long-only positions, sum-to-zero constraints) limit the aggressiveness of equity strategies and reduce TC.
– In fixed-income portfolios, increasing decision frequency improves breadth but is constrained by the assumption of independence.
– Both strategies highlight the balance between constraints, breadth, and portfolio performance potential.
[Ex Ante Measurement of Skill and Independence of Investment Decisions]
1- Ex Ante Measurement of Skill
– The information coefficient (IC) measures the correlation between a portfolio investor’s forecasts and actual outcomes, serving as a proxy for the investor’s skill.
– Challenges:
— IC is uncertain and subject to change over time, leading to variability in performance assessments.
— This uncertainty often results in actual information ratios being lower than those predicted by the Fundamental Law of Active Management.
2- Independence of Investment Decisions
– The number of individual assets (N) is not always a reliable measure of breadth (BR), particularly in these cases:
— 1- Active returns between assets are correlated.
— 2- Forecasts are not independent across time periods.
– Adjusted Formula for Breadth (BR):
— Formula: “BR = N / [1 + (N - 1)ρ]”
— ρ: Average correlation between active returns of the assets.
– Implications of Non-Independence:
— Many investment decisions, especially in fixed-income portfolios, lack independence due to factors like shared duration and credit risk.
— Time-series correlation can limit breadth; for example, a stock expected to outperform in one month is likely to have a similar forecast in subsequent months.
Key Takeaways
– IC provides an ex ante measure of skill but is inherently uncertain and variable over time.
– Breadth (BR) must account for correlations (ρ) between assets and across time to reflect the true number of independent investment decisions.
– Non-independence reduces the effective breadth and diminishes the potential impact of active management.
9.3 Exchange-Traded Funds: Mechanics & Applications
– Explain the creation/redemption process of ETFs and the function of authorized participants.
– Describe how ETFs are traded in secondary markets.
– Describe sources of tracking error for ETFs.
– Describe factors affecting ETF bid–ask spreads.
– Describe sources of ETF premiums and discounts to NAV.
– Describe costs of owning an ETF.
– Describe types of ETF risk.
– Identify and describe portfolio uses of ETFs.
[Exchange-Traded Funds (ETFs)]
1- Overview of ETFs
– Exchange-traded funds (ETFs) were introduced in the 1990s and have grown rapidly due to their numerous advantages.
2- Key Features of ETFs
– Low cost: ETFs typically have lower expense ratios compared to mutual funds.
– Exchange access: ETFs are traded on stock exchanges, providing liquidity and ease of access.
– Transparency: Holdings are disclosed frequently, offering clarity to investors.
– Asset class variety: ETFs offer exposure to a wide range of asset classes.
– Index-based investing: Increased use of passive, index-based strategies has contributed to ETF popularity.
3- Regulatory and Tax Treatment
– ETFs are regulated similarly to mutual funds but differ in tax efficiency and trading mechanisms.
[ETF Share Creation and Redemption Process]
1- Key Differences from Mutual Funds
– ETFs differ from mutual funds in their share creation and redemption process:
— ETF shares are created and redeemed in kind (shares-for-shares swap), allowing them to trade throughout the trading day.
— Mutual fund shares are bought and sold at the end of each trading day with the fund manager, based on the closing NAV.
– Advantages of the ETF process:
— Lower costs and higher tax efficiency compared to mutual funds.
— ETFs generally trade close to their net asset value (NAV) throughout the day, while mutual funds only transact at the daily closing NAV.
2- ETF Market Mechanics
– ETF transactions occur in two interconnected markets:
— Primary Market: Over-the-counter (OTC) transactions between authorized participants (APs) and the ETF manager (sponsor).
—- APs are institutional brokers or dealers.
—- APs deliver a basket of securities to the sponsor in exchange for ETF shares (in-kind trade).
— Secondary Market: ETF shares are then traded by investors on exchanges, where supply and demand determine prices.
Key Takeaways
– The creation and redemption mechanism is central to ETF efficiency, enabling continuous trading, tax advantages, and tight tracking of NAV.
– Authorized participants (APs) play a crucial role in facilitating ETF liquidity and price alignment with the underlying assets.
[The Creation/Redemption Process for ETFs]
1- Overview
– ETFs are bought and sold by individual investors through brokerage accounts, similar to stocks.
– In the secondary market, ETF trades occur between investors, without any direct transaction with the ETF manager.
2- Role of Authorized Participants (APs)
– Only APs, large brokers or market makers, have the authority to create or redeem ETF shares.
– APs transfer securities to the ETF sponsor in exchange for new ETF shares or redeem ETF shares for securities.
3- Creation Process
– The ETF sponsor provides a daily list of required in-kind securities, known as the creation basket, which determines the fund’s NAV.
— APs deliver the securities from their inventory or purchase them in the secondary market.
— These in-kind transactions occur in large blocks called creation units, typically consisting of 50,000 shares, but ranging from 10,000 to 600,000 shares.
4- Redemption Process
– APs redeem ETF shares for a specified basket of securities, known as the redemption basket.
— The composition of the redemption basket may differ from the creation basket, depending on tax, compliance, or investment objectives.
5- Key Features of In-Kind Transactions
– In-kind transactions between APs and ETF sponsors typically occur outside regular trading hours.
– APs can sell ETF shares in the secondary market while purchasing an equivalent amount of securities from the creation basket.
– These transactions are economically neutral for APs because the values of the ETF shares and the securities in the basket are identical.
Key Takeaways
– The creation/redemption process ensures efficient ETF liquidity and alignment with the fund’s NAV.
– APs are critical intermediaries, facilitating both the creation of new shares and the redemption of existing ones without impacting market dynamics.
– In-kind transactions contribute to ETF tax efficiency and cost-effectiveness.
[The Creation/Redemption Process]
1- Economic Incentives for APs
– Authorized participants (APs) are incentivized to participate in the creation/redemption process due to arbitrage opportunities:
— Example: If ETF shares trade at $10.10 but the underlying assets are worth $10.00, APs can sell ETF shares at $10.10 and purchase the underlying basket for $10.00, earning a profit.
— This arbitrage keeps ETF share prices closely aligned with their NAV.
2- Factors Affecting the Arbitrage Gap
– APs monitor ETF and underlying security prices but act only if the price discrepancy (arbitrage gap) exceeds transaction costs.
— Liquidity of the underlying securities and transaction costs incurred by APs influence whether they engage in creation/redemption.
— Delays in trading the underlying securities can widen the arbitrage gap, especially in foreign equity ETFs where market frictions exist.
3- AP Costs and Fees
– APs pay processing fees to ETF sponsors for creating or redeeming shares, ranging from $50 to over $25,000.
– Settlement costs, taxes, and other expenses incurred by APs are reflected in bid-ask spreads for ETF shares.
— Advantage for ETFs: These costs are borne only by trading investors, unlike mutual funds where all shareholders incur these costs.
4- Customized Redemption Baskets
– ETF sponsors can offer customized redemption baskets to improve tax efficiency and reduce creation costs.
— For example, ETFs holding illiquid securities (e.g., high-yield bonds or OTC assets) may allow APs to deliver equivalent-value baskets of cash or liquid securities instead of illiquid ones.
— These customizations are common in specialized ETFs, such as those for commodities, leveraged, or inverse strategies.
Key Takeaways
– The creation/redemption process enables efficient price alignment and tax efficiency for ETFs.
– AP activity is driven by arbitrage opportunities, constrained by transaction costs and market frictions.
– Customized baskets enhance flexibility for sponsors and reduce costs in illiquid or specialized ETF markets.
[Trading and Settlement]
1- US Settlement
– In the US, all ETF trades are settled through the National Securities Clearing Corporation (NSCC), which guarantees transactions against financial and operational risks.
— Buyers gain beneficial ownership as of the trade execution, even if the seller defaults.
— The NSCC, a subsidiary of the Depository Trust Company (DTC), maintains the book of accounts at the firm level, while firms track individual client ownership.
– Settlement Details:
— Most trades settle within two days of the trade date (T+2).
— Market makers have up to six days to settle, allowing flexibility to maintain liquidity in the ETF market.
2- European Trading and Settlement
– In Europe, most ETF trades occur between institutions and are typically negotiated over-the-counter (OTC).
— OTC trades do not appear as bid/ask orders before execution but are reported after clearing.
– European Market Characteristics:
— The market is highly fragmented, with most ETFs cross-listed on multiple exchanges.
— Different share classes may vary by factors like dividend distribution or currency hedging.
— ETF trades can be cleared in one of 29 central securities depositories.
– Rather than having centralized clearing (as in the United States), European ETF trades are settled by one of 29 different securities depositories.
– Implications for Investors:
— The complexity of the European system may lead to wider spreads and higher transaction costs but has no direct impact on the trading process for individual investors.
Key Takeaways
– US settlement is standardized, with the NSCC providing strong guarantees and efficient trade processing (T+2).
– European ETF markets are less centralized and more fragmented, with cross-listings and OTC trades leading to higher complexity and costs.
– Despite differences, both systems ensure smooth ETF trading and settlement for investors.
[Ownership and Trading of European ETFs]
1- ETF Ownership in Europe:
– The majority of ETF owners in Europe are institutional investors, such as asset managers and pension funds.
– Retail investors have a relatively smaller presence compared to institutional participants.
2- ETF Trading Mechanisms:
– Most ETF trading in Europe occurs in the over-the-counter (OTC) market rather than on public exchanges.
– Institutional investors typically trade ETFs directly with market makers or authorized participants, avoiding exchange-based trading.
[ETF Arbitrage Mechanism]
1- Arbitrage When ETF Shares Trade at a Premium
– Condition: Market price of ETF shares > Net Asset Value (NAV).
– Arbitrage process for authorized participants (APs):
— 1- Sell the ETF share in the market at the inflated price.
— 2- Buy the underlying securities at their NAV.
— 3- Transfer the underlying securities to the ETF manager in exchange for newly created ETF shares.
2- Arbitrage When ETF Shares Trade at a Discount
– Condition: Market price of ETF shares < Net Asset Value (NAV).
– Arbitrage process for APs:
— 1- Buy the ETF share in the market at the discounted price.
— 2- Sell the underlying securities at their NAV.
— 3- Transfer the ETF shares to the ETF manager in exchange for the underlying securities (redemption).
Key Takeaways
– The arbitrage mechanism ensures that ETF prices stay closely aligned with their NAV.
– APs capitalize on price discrepancies by creating or redeeming ETF shares.
– This process enhances market efficiency, liquidity, and price stability in ETFs.
[Evaluating ETF Performance]
1- Key Expectations for an ETF
– An ETF should:
— Closely track its benchmark index.
— Provide accurate and transparent information on fund structure and composition.
— Maintain low and predictable investment costs.
— Minimize tax exposure for investors.
2- Areas to Consider in ETF Evaluation
– Expense Ratios: Assess management fees and other costs relative to similar ETFs.
– Index Tracking: Evaluate tracking error to ensure the ETF follows its benchmark efficiently.
– Tax Treatment: Consider capital gains distributions, dividend taxation, and in-kind redemption benefits.
– Trading Costs: Look at bid-ask spreads and market liquidity.
– Ownership Costs: Factor in brokerage commissions and additional fees.
– Risks: Assess liquidity risk, market volatility, and structural risks of the ETF.
Key Takeaways
– ETFs should offer cost-efficient exposure to an index while maintaining accurate tracking and transparency.
– Investors should review key metrics like tracking error, fees, and tax efficiency to determine if an ETF meets expectations.
– Trading and ownership costs can impact long-term returns and should be considered in ETF selection.
[ETF Tracking and Expense Ratios]
1- ETF Tracking and Cost Efficiency
– A well-managed ETF closely follows its benchmark index, ensuring transparency in fund composition and performance.
– ETFs are designed to offer low, predictable fees while minimizing tax exposure.
2- Expense Ratios of ETFs
– ETFs generally have lower expense ratios than mutual funds due to passive index tracking, eliminating the need for active management and alpha research.
– Factors affecting ETF fees:
— 1- Number of securities held.
— 2- Frequency of rebalancing to maintain index weights.
– Expense Ratio Trends:
— 1- Lowest for domestic equity and fixed-income ETFs.
— 2- Highest for leveraged and inverse ETFs.
— 3- Very low for broad-based, capitalization-weighted index ETFs.
Key Takeaways
– ETFs offer cost-effective investing due to passive management and lower expense ratios.
– Expense ratios vary depending on complexity, asset class, and rebalancing frequency.
– Broad-based index ETFs tend to have the lowest fees, making them an attractive option for long-term investors.
[Index Tracking and Tracking Error]
1- Overview of Index Tracking
– Most ETFs (98% in the US) are designed to track an index, but their returns may not perfectly match the index due to various factors.
– Tracking error measures the dispersion of return differentials between the ETF and its benchmark, calculated as the annualized standard deviation of these differences.
– Tracking error does not indicate outperformance or underperformance but rather the consistency of tracking.
2- Sources of Tracking Error
– Fees and expenses: Management fees create a drag on ETF returns relative to the index.
– Representative sampling/optimization: Instead of full replication, ETFs may use sampling techniques, especially for illiquid assets, leading to deviations in performance.
– Depository receipts and ETFs: Holding assets outside the index, such as shares in other ETFs or depository receipts, increases tracking risk.
– Index changes: ETFs must adjust holdings when the index changes; exact replication may not always be possible, leading to potential errors.
– Fund accounting practices: Differences in valuation timing for bonds, foreign currencies, or commodities can introduce discrepancies.
– Regulatory and tax requirements: Withholding taxes on dividends and other regulatory factors can affect ETF returns relative to the index.
– Asset manager operations: Securities lending and foreign dividend recapture may act as “negative costs,” reducing expense ratios and affecting tracking performance.
Key Takeaways
– Tracking error measures consistency in ETF performance relative to its index but does not indicate relative outperformance.
– Fees, sampling methods, index rebalancing, and regulatory factors all contribute to tracking error.
– Investors should use rolling tracking differences to evaluate an ETF’s effectiveness in tracking its benchmark over time.
[Understanding Tracking Error in ETFs]
1- Definition of Tracking Error
– Tracking error is the standard deviation of the differences between ETF returns and index returns.
– It measures the consistency of an ETF’s performance relative to its benchmark but does not indicate outperformance or underperformance.
2- Limitations of Tracking Error
– Does not show how often daily returns differ:
— An ETF might track its index closely for most of the year but deviate significantly in a short period.
– Does not show the distribution of errors:
— An ETF may consistently outperform or underperform its index in small amounts, which tracking error alone does not capture.
3- Best Practices for Evaluating Tracking Accuracy
– Monitor mean or median return differences: This helps assess systematic biases rather than just variability.
– Use rolling period graphs: These help identify trends over time and fluctuations in tracking error.
– Long-run expectation:
— Formula: “Expected ETF Return = Index Return - Fund’s Expense Ratio”
— If this relationship holds, the ETF is effectively tracking its benchmark.
Key Takeaways
– Tracking error is a useful measure of dispersion but should be complemented with additional metrics.
– Investors should monitor return distributions and rolling trends to assess ETF tracking quality.
– A well-tracking ETF should have an expected return close to the index return minus the fund’s expense ratio.
[Tax Issues in ETFs]
1- Capital Gains Distribution
– ETFs typically distribute any realized capital gains at the end of the year, though quarterly distributions may also occur.
– Capital gains distributions are generally lower for ETFs compared to mutual funds due to tax-efficient structures.
– Tax Efficiency of ETFs:
— ETFs are tax fair, meaning one shareholder’s trades do not create tax liabilities for others, unlike mutual funds.
— Mutual fund redemptions require managers to sell assets, triggering taxable gains distributed among shareholders.
— In contrast, ETF shares trade in the secondary market, and redemptions through authorized participants (APs) using in-kind transactions do not generate taxable events.
— ETF managers can minimize capital gains by swapping out low-cost basis securities in the redemption process.
— However, capital gains can still be realized when securities are dropped from an index, particularly in funds that rely on cash redemptions (e.g., bond ETFs).
2- Other Distributions
– ETFs distribute dividends in many jurisdictions, potentially triggering tax liabilities based on investor tax status and regional tax laws.
3- Taxes on Sale
– ETFs are taxed based on their underlying holdings.
– Some ETFs receive preferential tax treatment compared to others:
— Example: In the US, exchange-traded notes (ETNs) tracking commodity indexes may have more favorable tax treatment than commodity futures-based ETFs.
Key Takeaways
– ETFs generally have lower capital gains distributions than mutual funds due to in-kind redemptions.
– Secondary market trading prevents tax burdens from being passed to other investors.
– Investors should consider tax implications on dividend distributions and ETF sales based on jurisdiction and asset class.
[Tax Advantages of ETFs]
1- ETFs and Capital Gains Distributions
– ETFs generally distribute fewer capital gains compared to mutual funds due to their tax-efficient structure.
2- Tax Fairness of ETFs
– Shareholders are not affected when others sell their ETF shares and realize capital gains.
– The ETF manager does not directly handle investor transactions, as trades occur in the secondary market.
– Redemptions between authorized participants (APs) and the ETF fund do not trigger taxable events since in-kind securities are exchanged instead of cash.
– In contrast, mutual funds must sell securities when an investor redeems shares, generating taxable gains distributed among remaining shareholders.
3- Tax Efficiency of ETFs
– ETF managers can strategically manage tax liabilities through the redemption process.
– They can select share lots with the lowest cost basis when fulfilling redemption requests, reducing unrealized capital gains.
4- Situations Where Capital Gains Can Still Occur
– If an index removes a constituent, the ETF must sell the security, potentially triggering capital gains.
– Index rebalancing can lead to capital gains realizations.
– Fixed-income ETFs are more susceptible to capital gains since bonds bought at a discount generate taxable gains upon sale or maturity.
Key Takeaways
– ETFs are tax fair because individual investor sales do not affect others.
– ETFs are tax efficient because managers can manage redemptions strategically to minimize realized capital gains.
– While ETFs generally distribute fewer capital gains than mutual funds, rebalancing and security removals can still create taxable events.
[ETF Trading Costs and Bid-Ask Spreads]
1- ETF Trading Costs
– ETFs can be traded throughout market hours, but their trading costs are influenced by bid-ask spreads, order size, and market maker activity.
– The ETF’s closing price may reflect a premium or discount based on supply and demand dynamics.
2- ETF Bid-Ask Spreads
– The liquidity of the underlying assets is the primary factor affecting an ETF’s bid-ask spread.
– Less liquid assets lead to wider bid-ask spreads as authorized participants (APs) require compensation for acquiring such assets.
– Example: A corporate bond ETF typically has a wider spread than a large-cap equity ETF.
– APs also face risks when ETFs hold foreign assets, as these markets may be closed during ETF trading hours.
– Even if underlying assets are liquid, bid-ask spreads can widen if:
— The ETF itself trades infrequently.
— The creation/redemption basket is too large, preventing APs from efficiently adjusting positions.
Key Takeaways
– Trading costs for ETFs depend on bid-ask spreads, order execution, and market conditions.
– Liquidity of the ETF’s underlying assets significantly impacts trading costs.
– Wider bid-ask spreads can occur in ETFs with illiquid assets, infrequent trading, or large creation/redemption units.
[ETF Bid-Ask Spreads]
1- Factors Affecting ETF Bid-Ask Spreads
– Bid-ask spreads for ETFs depend on costs and risks incurred by authorized participants (APs).
– The main factors include:
— Bid-ask spreads of the underlying securities.
— The market maker’s profit spread, which decreases as competition increases.
— Compensation for the market maker’s risk of holding positions, which is reduced if hedging instruments are available.
— Direct trading costs such as brokerage fees, exchange fees, and creation/redemption fees.
2- Characteristics of ETF Bid-Ask Spreads
– ETF spreads are typically less than or equal to the sum of these cost factors.
– Highly liquid ETFs may have tighter bid-ask spreads than their underlying securities, especially for small orders.
– ETFs used for long-term strategies, such as alternatives and asset allocation, tend to have wider spreads due to:
— Lower average daily trading volume.
— Greater reliance on market makers for liquidity.
– ETF spreads tend to widen during periods of market volatility or when new information about the underlying assets becomes available.
Key Takeaways
– Bid-ask spreads reflect APs’ costs and risks when facilitating ETF trading.
– Liquidity, market maker competition, and hedging ability significantly impact spreads.
– ETFs with lower liquidity or increased market volatility tend to have wider spreads.
[Premiums and Discounts]
1- Definition
– The premium or discount for an ETF represents the difference between the ETF’s market price and its net asset value (NAV) per share.
2- Types of Premium/Discount Measures
– End-of-day premium/discount:
— Formula: “(ETF Price - NAV per share) ÷ NAV per share”.
– Intraday premium/discount:
— Formula: “(ETF Price - iNAV per share) ÷ iNAV per share”.
— iNAV (indicated NAV) provides an estimated fair value for the ETF based on its creation basket composition during the trading day.
3- Causes of Premiums and Discounts
– Timing differences:
— NAV is calculated based on the closing prices of underlying securities, while ETF prices fluctuate throughout the trading day.
– Stale pricing:
— Some underlying securities may not trade frequently, causing temporary mispricing.
– Currency fluctuations:
— ETFs tracking foreign markets may show discrepancies when the domestic market remains open while the foreign market is closed.
Key Takeaways
– Premiums and discounts occur due to market inefficiencies, pricing lags, and liquidity variations.
– ETFs trading on exchanges with different trading hours than their underlying assets may experience persistent deviations.
– iNAV provides intraday price guidance, while end-of-day NAV is used for official valuation.
[Stale Pricing and ETF Premiums/Discounts]
1- Stale Pricing Impact on ETFs
– ETFs that trade infrequently:
— Can lead to large premiums or discounts to NAV.
— ETFs not trading in the last few hours of the market session may reflect outdated prices.
2- Underlying Securities Considerations
– Market closures for the underlying securities can cause discrepancies.
– Illiquid and infrequently traded securities may distort ETF pricing relative to NAV.
Key Takeaways
– ETF prices may serve as a better estimate of intrinsic value than NAV when pricing delays or illiquidity affect NAV calculations.
[Premiums and Discounts]
1- Factors Affecting Premiums and Discounts
– NAV calculation inputs:
— Bid prices used in NAV calculations can be adjusted down, leading to a premium.
— Bond ETFs may show discounts if NAV models fail to incorporate market turbulence.
– Market liquidity:
— Less frequently traded ETFs experience greater premiums or discounts.
— US bond ETFs tend to have higher deviations than domestic equity ETFs.
– Market timing:
— ETFs tracking foreign markets may see discrepancies when the underlying market is closed.
– Volatility:
— Premiums and discounts widen during periods of increased market uncertainty.
Key Takeaways
– ETF prices may better reflect intrinsic value than NAV during illiquid market conditions.
– NAV estimates vary based on trade pricing methodologies, potentially distorting premium/discount measures.
– Investors should account for liquidity, market timing, and bid-ask spreads when evaluating ETF pricing.
[Comparison of Bond ETFs and Equity ETFs]
1- Discounts and Premiums to NAV:
– Bond ETFs often trade at greater discounts or premiums relative to their net asset value (NAV) compared to equity ETFs.
– This occurs because bond markets do not have continuous pricing like equity markets, leading to valuation discrepancies.
2- Capital Gains Realization:
– Bond ETFs are more likely to realize capital gains when selling bonds that were originally purchased at a discount.
– Equity ETFs, by contrast, have more flexibility in managing capital gains by strategically trading stocks with different cost bases.
[Total Costs of ETF Ownership]
1- Explicit Costs
– Management fees: Charged as a percentage of AUM, typically lower for ETFs than mutual funds.
– Commissions: Costs incurred when trading ETF shares, though some brokers waive them.
– Taxes: Applied when assets are sold for a profit, but ETFs often have lower tax liabilities due to in-kind trading.
2- Implicit Costs
– Tracking error: Cost due to deviation from the index; can be negative if the ETF outperforms.
– Bid-ask spreads: Costs incurred by APs in the ETF creation/redemption process, affecting trading prices.
– Premiums to NAV: Cost when ETF shares trade above NAV; conversely, discounts to NAV can be a benefitc (Negative cost [double negative]).
– Portfolio turnover: Incurred due to rebalancing, though lower for ETFs than mutual funds.
– Security lending: A negative cost (income) that benefits ETF investors over longer holding periods.
Key Takeaways
– ETFs generally have lower explicit costs than mutual funds but still incur implicit costs.
– Investors should evaluate tracking error, bid-ask spreads, and NAV premiums when assessing total costs.
– Security lending provides a cost advantage to ETFs over longer investment horizons.
[Explicit Costs of ETF Ownership]
1- Management Fees
– Charged as a percentage of assets under management (AUM).
– Impact on total returns increases as the holding period lengthens.
– Typically lower for ETFs compared to mutual funds.
2- Commissions
– Paid when trading ETF shares, similar to stocks.
– Some brokers waive ETF commissions to compete with mutual funds.
– Mutual funds do not incur commissions when trading.
3- Taxable Gains and Losses
– Taxes are incurred when ETFs are sold at a gain.
– Selling at a loss may lead to negative taxes.
– ETF managers can adjust the cost basis by selecting specific tax lots when redeeming shares.
– Taxes tend to be lower for ETFs than mutual funds due to tax-efficient structures.
Key Takeaways
– ETFs generally have lower management fees and taxes but may involve commissions.
– Mutual funds benefit from no trading commissions but tend to have higher management fees and taxable distributions.
[Implicit Costs of ETF Ownership]
1- Tracking Error
– Occurs when an ETF does not fully replicate the index.
– Can be a cost if the ETF underperforms the index or a negative cost if it outperforms.
– Greater impact over longer holding periods.
– Lower for ETFs than for index-tracking mutual funds.
2- Bid-Ask Spread
– Charged by authorized participants as compensation for liquidity risks.
– Only paid when an investor buys or sells the ETF.
– Not affected by the holding period.
– Exists for ETFs but not for mutual funds.
3- Premium or Discount to NAV
– Premiums occur when ETF price > NAV, representing an implicit cost.
– Discounts occur when ETF price < NAV, representing a negative implicit cost.
– Often results from timing differences in NAV calculation.
– Mutual funds always trade at NAV, avoiding this cost.
4- Portfolio Turnover
– Results from fund managers buying and selling securities to maintain fund strategy.
– Higher turnover reduces long-term returns.
– ETFs generally have lower turnover than mutual funds due to passive index tracking.
5- Security Lending
– ETFs generate income by lending securities.
– This creates an implicit negative cost that benefits investors.
– More common for ETFs than mutual funds.
Key Takeaways
– Implicit costs impact ETF performance based on liquidity, trading activity, and market structure.
– ETFs may have tracking error and bid-ask spreads but benefit from lower portfolio turnover and income from security lending.
– Mutual funds avoid bid-ask spreads and NAV deviations but typically have higher turnover and less security lending income.
Key Takeaways
ETFs generally have lower costs due to lower management fees, lower taxable distributions, and passive strategies.
Mutual funds always trade at NAV, while ETFs can experience bid-ask spreads and NAV deviations.
ETFs provide tax efficiency through in-kind redemptions, whereas mutual funds distribute realized capital gains to investors.
ETFs have lower turnover, leading to lower implicit costs compared to mutual funds.
Security lending is more prevalent in ETFs, providing an additional revenue stream that benefits investors over time.
[Holding Period Cost Analysis]
1- Formula for Holding Period Cost
– “Holding period cost = Round-trip trading cost + Management fee”
– Round-trip trading cost: “(One-way commission × 2) + (0.5 × Bid-ask spread × 2)”
– Management fee: “Annual management fee × (n ÷ 12)”, where n is the number of months in the holding period.
2- Impact of Holding Period on Costs
– Example: If commissions are 0.10%, management fees are 0.25%, and the bid-ask spread is 0.20%:
— 6-month holding period: “[(0.10% × 2) + 0.20%] + (0.25% × 6 ÷ 12) = 0.525%”
— 2-year holding period: “[(0.10% × 2) + 0.20%] + (0.25% × 24 ÷ 12) = 0.90%”
– The round-trip cost remains constant at 0.40%, but its proportion of total costs is higher for shorter holding periods. Greater impact on the performance of short-term investments, while longer-term investors will give greater consideration to ongoing costs such as management fees
Key Takeaways
– Transaction costs (commissions and bid-ask spreads) significantly impact short-term investments.
– Long-term investors are more affected by ongoing costs such as management fees.
[Leveraged ETF Swap Exposure Adjustment Calculation]
1- Initial Swap Exposure Calculation:
– A leveraged ETF targeting 3X exposure to an index starts with a notional swap exposure equal to three times its initial NAV.
– If the initial NAV is $100, then the swap exposure is:
— “Initial Swap Exposure = 100 × 3 = 300”
2- Impact of a Daily Index Increase:
– Suppose the index rises by 2% in a single day. The NAV of the fund will adjust accordingly:
— “New NAV = 100 × (1 + 3 × 0.02) = 100 × 1.06 = 106”
– The total notional exposure at the end of the day, before any adjustments, is:
— “Unadjusted Swap Exposure = 300 × (1 + 0.02) = 300 × 1.02 = 306”
3- Required Exposure Reset:
– To maintain the 3X leverage, the fund must adjust its swap exposure based on the updated NAV:
— “Required Swap Exposure = 106 × 3 = 318”
4- Necessary Adjustment to Swap Exposure:
– The difference between the new required exposure and the unadjusted exposure determines the adjustment:
— “Swap Exposure Adjustment = 318 - 306 = 12”
Key Takeaways
– The fund needs to increase its swap exposure by $12 per share after Day 1 to maintain 3X leverage.
– Leveraged ETFs reset daily, meaning compounding effects can cause deviations from expected simple multiples over time.
[NAV Calculation for Leveraged and Inverse ETFs]
1- Initial Setup:
– Fund A provides +200% exposure to the index, meaning it moves twice the index’s daily return.
– Fund B provides -200% exposure, meaning it moves in the opposite direction of twice the index’s return.
– Both funds start with an initial NAV of $100.
2- Daily Returns and NAV Adjustments:
– The index moves as follows over three days:
— Day 1: +2%
— Day 2: -4%
— Day 3: +2%
– Fund A’s daily returns (2X the index):
— Day 1: +4%
— Day 2: -8%
— Day 3: +4%
– Fund B’s daily returns (-2X the index):
— Day 1: -4%
— Day 2: +8%
— Day 3: -4%
3- NAV Calculations After Three Days:
– Fund A’s NAV:
— “NAV_A = 100 × (1 + 0.04) × (1 - 0.08) × (1 + 0.04)”
— “NAV_A = 100 × 1.04 × 0.92 × 1.04 = 99.51”
– Fund B’s NAV:
— “NAV_B = 100 × (1 - 0.04) × (1 + 0.08) × (1 - 0.04)”
— “NAV_B = 100 × 0.96 × 1.08 × 0.96 = 99.53”
Key Takeaways
– Fund B, the inverse leveraged ETF, ends with a slightly higher NAV than Fund A.
– Compounding effects make daily leveraged and inverse ETFs deviate from expected simple multiples over time.
[Proportion of Holding-Period Costs Attributable to Management Fees]
1- Total Holding-Period Cost Components:
– Costs include two commissions, the full bid-ask spread, and the periodic management fee.
– Given data:
— Commission per trade: 0.08%
— Bid-ask spread: 0.32%
— Management fee (annualized): 0.85%
— Holding period: 3 months (3/12 of a year)
2- Total Holding-Period Cost Calculation:
– Formula:
— “Total Cost = (Commission × 2) + Bid-Ask Spread + (Management Fee × Holding Period)”
— “Total Cost = (0.08% × 2) + 0.32% + (3/12 × 0.85%)”
— “Total Cost = 0.16% + 0.32% + 0.2125%”
— “Total Cost = 0.6925%”
3- Proportion of Total Cost Attributable to Management Fees:
– Formula:
— “Proportion = (Management Fee Contribution ÷ Total Cost)”
— “Proportion = 0.2125% ÷ 0.6925%”
— “Proportion ≈ 30.7%”
[Counterparty Risk]
1- Definition: Counterparty risk refers to the potential loss investors face if the counterparty defaults on its obligations.
2- Counterparty Risk in ETNs:
– Exchange-traded notes (ETNs) are unsecured, unsubordinated debt instruments, meaning investors bear full exposure to the issuer’s default risk.
– The risk can be measured using the issuer’s credit default swap (CDS) spread.
3- Counterparty Risk in ETFs:
– Some ETFs hold foreign currency deposits in offshore banks, which are subject to default risk.
– ETFs that use over-the-counter (OTC) derivatives for exposure to certain assets face settlement risk, which is typically managed through frequent settlements.
– ETF providers disclose holdings but may not fully disclose counterparty details.
4- Securities Lending:
– ETFs lend securities to short sellers to generate additional income.
– Loans are collateralized (102% for domestic short sellers, 105% for international).
– Lending income can lower an ETF’s expense ratio but carries counterparty risk.
Key Takeaways
– ETNs have higher counterparty risk than ETFs due to their unsecured structure.
– ETFs mitigate counterparty risk using collateralized lending and frequent derivative settlements.
– Securities lending adds revenue for ETFs but introduces exposure to borrowers’ default risk.
[Fund Closures]
1- Reasons for Fund Closure:
– Regulatory requirements may force funds to close or convert, as seen in Israel’s 2018 mandate for ETNs to transition to ETFs.
– Competitive pressures can lead to insufficient assets under management (AUM) and low trading volumes, prompting closure.
– Mergers and acquisitions between sponsors may result in fund consolidation or termination.
2- Soft Closures:
– A soft closure occurs when a fund halts new unit creation or changes its investment strategy without fully shutting down.
– ETNs may stop issuing new shares to avoid increasing debt, which disrupts arbitrage mechanisms and can lead to significant premiums to NAV.
– ETFs can also implement creation halts, though this is more common with ETNs.
3- Investment Strategy Changes:
– Instead of closing, a fund may repurpose an existing ETF, which is often simpler than launching a new one.
– These changes are usually minor but can significantly alter the fund’s structure.
Key Takeaways
– Fund closures can be driven by regulation, competition, or corporate actions.
– Soft closures limit new share creation but do not liquidate the fund.
– ETFs may change strategies rather than close, depending on market conditions.
[Exchange-Traded Notes (ETNs)]
1- Characteristics of ETNs:
– Trade on exchanges like ETFs.
– Have a similar creation and redemption process.
– Not true funds, as they do not hold underlying securities.
– Represent unsecured debt obligations of the issuer.
– Registered under the Securities Act of 1933 in the U.S.
– Have the highest counterparty risk among exchange-traded products due to lack of asset backing.
2- Counterparty Risk in ETNs:
– ETN issuers promise to deliver returns based on an index.
– Issuers are responsible for hedging the risk.
– Investors fully rely on the issuer’s ability to meet obligations.
– If the issuer defaults, investors can lose 100% of their investment.
– Example: Lehman Brothers’ ETNs in 2008 became worthless after the firm’s collapse.
3- Risk Measurement and Comparison with Deposit-Based ETFs:
– ETN default risk can be assessed using the issuing bank’s credit default swap (CDS) pricing.
– Deposit-based ETFs also have counterparty risk, but they are typically backed by assets.
Key Takeaways
– ETNs are unsecured debt instruments that trade like ETFs but do not hold underlying assets.
– Investors face high counterparty risk and must assess the issuer’s creditworthiness.
– Unlike ETFs, ETNs depend entirely on the financial health of the issuing institution.
[Investor-Related Risk]
1- Leveraged and Inverse ETFs:
– Leveraged ETFs aim to provide returns that are a multiple of an index’s return, while inverse ETFs provide returns opposite to the index.
– These funds are typically used for short-term tactical positioning, often with a holding period of one month or less.
2- Maintaining Leverage with Swaps:
– To maintain exposure, leveraged ETF managers use swaps to adjust their notional exposure daily.
– Example: A 2X leveraged ETF on the S&P 500 with a NAV of $100 will rise to $110 if the index gains 5%, since NAV = $100(1 + 2 × 5%).
– To maintain the leverage for the next day, swaps must be adjusted to match the updated NAV.
3- Impact of Compounding on Leveraged ETFs:
– Compounding causes leveraged ETFs to deviate from their expected multiple of index returns over time.
– Example: A 2X ETF and a traditional ETF both start at $100, but with daily returns of {-1.88%, +7.52%, -5.21%}, the traditional ETF ends at $100, while the 2X ETF ends at $99.17.
– The difference arises because leveraged ETFs are affected by daily resets and compounding effects.
Key Takeaways
– Leveraged ETFs require daily adjustments to maintain target exposure.
– Compounding leads to deviations from expected returns over multiple periods.
– These funds are best suited for short-term strategies rather than long-term investments.
[ETF Strategies]
1- Purpose of ETFs in Investment Strategies:
– ETFs provide exposure to various asset classes, including stocks, bonds, and commodities.
– Can be used for both top-down macro allocation and bottom-up security selection.
– Serve purposes such as asset allocation, rebalancing, and risk management.
2- Strategic and Tactical Uses:
– ETFs facilitate strategic asset allocation based on long-term market expectations.
– Can be used for tactical tilts and short-term positioning in response to market conditions.
– Dynamic allocations are more aligned with capital market expectations rather than tactical shifts.
– Certain investment strategies, such as illiquid securities or non-transparent (“black box”) strategies, may not be well suited for ETFs.
3- Passive vs. Active Use of ETFs:
– Although often considered passive investments, ETFs require some level of active decision-making.
– Using ETFs for short-term deviations from a strategic asset allocation is an active management decision.
– ETFs themselves can be actively managed through rule-based strategies or with limited discretionary management.
Key Takeaways
– ETFs support a range of investment strategies, from passive long-term allocations to active tactical positioning.
– Their flexibility allows them to be used for rebalancing, risk management, and dynamic market exposure.
– While typically seen as passive investments, ETFs can play an active role in portfolio management.
[ETF Strategies]
1- Objectives of ETF Strategies:
– Strategic: Long-term asset allocation based on market expectations.
– Tactical: Short-term positioning to exploit market inefficiencies.
– Dynamic: Adjusting allocations based on changing market conditions.
– Portfolio Efficiency: Enhancing diversification and liquidity management.
2- Limitations of ETFs for Certain Strategies:
– Illiquidity: Some underlying assets may be too illiquid for ETF structures.
– Analytical Complexity: Strategies that cannot be easily defined analytically may not be viable as ETFs.
– Confidentiality Concerns: Active managers may avoid ETFs to prevent disclosure of proprietary investment strategies.
Key Takeaways
– ETFs can support strategic, tactical, and dynamic investment approaches.
– Certain strategies, particularly those involving illiquid assets or proprietary methods, may not be well-suited for ETFs.
[Efficient Portfolio Management]
1- Cash Flow (Liquidity) Management:
– ETFs help institutional investors manage liquidity by investing excess cash balances.
– Cash drag from idle cash can lead to portfolio underperformance relative to benchmarks.
– A cash equitization strategy involves investing excess cash in ETFs for better returns while maintaining liquidity.
2- Portfolio Rebalancing:
– ETFs allow portfolio managers to efficiently adjust asset allocations to maintain target weights.
– The ability to sell ETF shares short enhances their usefulness in rebalancing strategies.
– ETFs can complement bottom-up security selection by filling exposure gaps in a portfolio.
3- Managerial Transitions:
– ETFs offer a transition tool for investors shifting between fund managers while maintaining benchmark exposure.
– Large asset owners may prefer separately managed accounts (SMAs) over ETFs for lower costs and customization.
– Large ETF positions might require disclosure, which can complicate and increase the cost of liquidating positions.
Key Takeaways
– ETFs enhance portfolio efficiency through cash flow management, rebalancing, and transition strategies.
– Cash equitization helps mitigate cash drag, improving portfolio performance.
– SMAs can be an alternative to ETFs for institutional investors seeking lower costs and greater customization.
[Portfolio Efficiency]
1- Uses of ETFs for Portfolio Efficiency:
– Portfolio liquidity management: Ensuring sufficient cash flow for withdrawals and investments.
– Portfolio rebalancing: Adjusting asset allocations to maintain target weights.
– Portfolio completion strategies: Filling gaps in asset class exposures.
– Transition management: Facilitating shifts in investment strategies or managers.
[Portfolio Completion Strategies]
1- Situations Leading to Portfolio Gaps:
– External manager changes: When asset management responsibilities shift.
– Strategy shifts by existing managers: Changes in investment approaches may create imbalances.
2- Role of ETFs in Portfolio Completion:
– ETFs can adjust exposures while keeping the same manager, ensuring portfolio continuity.
Key Takeaways
– ETFs enhance portfolio efficiency by managing liquidity, rebalancing, and supporting strategy transitions.
– Portfolio completion strategies address asset gaps due to external or internal management changes.
[Asset Class Exposure Management]
1- Core Asset Class Exposure:
– ETFs provide access to core asset classes such as fixed income, offering greater liquidity and efficiency than the underlying assets.
2- Tactical Asset Allocation (TAA):
– ETFs are well-suited for short-term tactical allocation decisions.
– The ability to short-sell ETFs makes them more flexible than mutual funds.
3- Thematic ETFs:
– Thematic ETFs target niche asset classes, such as cybersecurity, which may be difficult to access through traditional investment vehicles.
– Given the short-term nature of tactical asset allocation, trading costs and liquidity take precedence over management fees.
– Trading volume is a key factor for investors using ETFs in tactical strategies.
Key Takeaways
– ETFs enhance portfolio flexibility by providing liquid exposure to core, tactical, and thematic asset classes.
– The ability to short-sell ETFs improves their adaptability in dynamic asset allocation.
– Liquidity and trading costs are crucial considerations for tactical ETF investors.
[Asset Class Exposure Management]
1- Strategic Uses of ETFs for Asset Class Exposure:
– Core exposure: ETFs provide access to broad asset classes or sub-asset classes.
– Tactical strategies: Used for short-term positioning and adjustments.
2- Types of Exposure Through ETFs:
– Broad categories: ETFs enable investment in global equities and other diversified markets.
– Targeted exposures: Investors can gain specific asset exposure, such as commodities.
Key Takeaways
– ETFs facilitate easy access to broad and specific asset classes.
– They support both long-term strategic allocations and short-term tactical adjustments.
[Asset Class Exposure Management]
1- Tactical Strategies in ETFs:
– ETFs are useful for tactical strategies aimed at short-term market adjustments.
2- Key Considerations for Tactical Trading:
– High liquidity is essential for efficient trading execution.
– Trading costs and liquidity outweigh management fees in importance.
Key Takeaways
– Tactical traders should prioritize very liquid ETFs.
– Short holding periods make trading costs a crucial factor in ETF selection.
[Active and Factor Investing]
1- Factor (Smart Beta) ETFs:
– These ETFs follow index benchmarks with specific rules for weighting constituents.
– They represent approximately 20% of the ETF market by AUM.
– Used for long-term exposure to rewarded factors, such as overweighting stocks with value characteristics.
– Multifactor approaches reduce factor timing risk while maintaining lower volatility than single-factor strategies.
2- ETFs as Risk Management Tools:
– ETFs help manage portfolio volatility by adjusting exposure to various risk factors.
– International ETFs can mitigate currency risk.
– Fixed-income ETFs offer a means to manage interest rate risk.
3- Alternatively-Weighted ETFs:
– These ETFs use weighting methodologies other than market capitalization.
– Equal-weighted and fundamentally-weighted ETFs allow investors to express specific market views, such as targeting high-dividend stocks.
Key Takeaways
– ETFs can go beyond passive indexing to incorporate factor-based and rules-based investing.
– Smart beta ETFs optimize exposure to factors while maintaining diversification.
– Risk management strategies using ETFs can help mitigate currency and interest rate risks.
[Active and Factor Investing]
1- Discretionary Active Management:
– Some ETFs incorporate active management, focusing mainly on fixed-income markets.
– Equity ETFs with active strategies often target niche sectors, such as technology.
2- Liquid Alternative ETFs:
– These ETFs provide exposure to alternative assets, aiming for higher returns and risk diversification.
– They may track alternative indexes or use hedge fund-like strategies, including long-short positions, managed futures, and merger arbitrage.
3- Dynamic and Top-Down Strategies:
– Increased ETF availability has led to strategies based on return and risk forecasts.
– These strategies often employ swaps and futures to enhance efficiency in trading.
Key Takeaways
– Active ETFs can incorporate discretionary management, often in fixed income and niche equity markets.
– Liquid alternative ETFs replicate hedge fund strategies, offering diversification and alternative exposures.
– Dynamic ETFs use top-down forecasting methods with derivatives to enhance portfolio efficiency.
9.4 Using Multifactor Models
– Describe arbitrage pricing theory (APT), including its underlying assumptions and its relation to multifactor models.
– Define arbitrage opportunity and determine whether an arbitrage opportunity exists.
– Calculate the expected return on an asset given an asset’s factor sensitivities and the factor risk premiums.
– Describe and compare macroeconomic factor models, fundamental factor models, and statistical factor models.
– Describe uses of multifactor models and interpret the output of analyses based on multifactor models.
– Describe the potential benefits for investors in considering multiple risk dimensions when modeling asset returns.
– Explain sources of active risk and interpret tracking risk and the information ratio.
[Multifactor Models and Portfolio Theory]
1- Factors and Asset Returns:
– Factors are variables correlated with asset returns, providing deeper insights than single-factor models.
– Multifactor models enhance risk understanding and portfolio diversification.
2- Development of Modern Portfolio Theory (MPT):
– Introduced by Harry Markowitz in 1952.
– Focuses on portfolio contributions rather than isolated investments.
– Demonstrates risk reduction through diversification.
3- Capital Asset Pricing Model (CAPM):
– Introduced by William Sharpe in 1964.
– Models asset returns based on a single factor—market risk.
– Investors are compensated for bearing non-diversifiable systematic risk but not for diversifiable risk.
4- Limitations of CAPM:
– Fails to account for multiple sources of systematic risk.
– Multifactor models are necessary to properly capture these risks.
Key Takeaways
– Multifactor models improve risk assessment and return predictions.
– Diversification is essential for risk management beyond CAPM’s assumptions.
[Multifactor Models]
1- Definition and Usage:
– Used by asset owners, managers, investment consultants, and risk managers.
– Factors are variables correlated with individual asset returns.
– Incorporating multiple factors enhances explanatory power.
2- Common Uses:
– Construct portfolios that replicate an index with modifications.
– Establish exposure to one or more risk factors.
– Perform risk and return attribution for actively managed portfolios.
– Assess risk exposures across equity, fixed income, and other asset classes.
– Identify active investment decisions relative to a benchmark.
– Ensure portfolios meet active risk and return objectives.
[Arbitrage Pricing Theory]
1- Overview of the Concept
– The arbitrage pricing theory (APT) is an alternative to the capital asset pricing model (CAPM) that explains asset returns as a linear function of multiple risk factors.
– It accounts for systematic risk through various factor exposures rather than a single market risk factor.
2- Formula
– Arbitrage pricing theory equation:
— “Ri = ai + b1I1 + b2I2 + … + bkIk + εi”
3- Explanation of Variables
– Ri: Expected return of asset i.
– ai: Expected return of asset i when all factors are zero.
– bk: Sensitivity of the asset to return factor k.
– Ik: Systematic return factor k.
– εi: Error term representing unsystematic risk.
Key Takeaways
– APT is a multifactor model that extends beyond CAPM by incorporating multiple systematic risk factors.
– It allows for different sensitivities to various economic, financial, or macroeconomic factors.
– Unlike CAPM, APT does not require a specific market portfolio and can be empirically tested with factor analysis.
[Arbitrage Pricing Theory (APT)]
1- Overview of the Concept
– APT is a multi-factor asset pricing model that explains expected returns based on multiple risk factors.
– Unlike CAPM, APT does not specify the number or type of risk factors affecting asset returns.
2- Formula
– Expected return on a portfolio:
— E(Rp) = RF + λ1βp,1 + … + λKβp,K
3- Explanation of Variables
– E(Rp): Expected return on the portfolio.
– RF: Risk-free rate.
– λ: Expected reward for bearing the risk factor.
– β: Sensitivity of the portfolio to a risk factor.
– K: Number of factors in the model.
4- Assumptions of APT
– 1- A factor model describes the asset returns (number not specified).
– 2- A well-diversified portfolio can be created to eliminate asset-specific risk.
– 3- No arbitrage opportunities exist in well-diversified portfolios.
Key Takeaways
– APT allows for multiple factors influencing asset returns, unlike CAPM, which relies on a single market factor.
– The absence of arbitrage ensures that asset prices reflect fair compensation for risk.
– The model is widely used for explaining returns based on macroeconomic and firm-specific risks.
[Carhart Four-Factor Model]
1- Overview of the Concept
– The Carhart model extends the CAPM by incorporating size, value, and momentum factors to explain asset returns.
– It is a multifactor model used to describe expected portfolio returns based on systematic risk factors.
2- Formula
– Expected return on a portfolio:
— E(Rp) = RF + βp1RMRF + βp2SMB + βp3HML + βp4WML
3- Explanation of Variables
– E(Rp): Expected return on the portfolio.
– RF: Risk-free rate.
– RMRF: Return on a value-weighted equity index in excess of the risk-free rate.
– SMB: Small-minus-big factor, representing the return difference between small-cap and large-cap portfolios.
– HML: High-minus-low factor, representing the return difference between high and low book-to-market portfolios.
– WML: Winners-minus-losers factor, representing momentum returns (past winners minus past losers).
– β: Sensitivity of the portfolio to each risk factor.
4- Factors in the Carhart Model
– 1- Market Risk: Captured by RMRF, representing the equity premium.
– 2- Size Factor: SMB accounts for the tendency of small-cap stocks to outperform large-cap stocks.
– 3- Value Factor: HML reflects the historical outperformance of value stocks over growth stocks.
– 4- Momentum Factor: WML captures the excess returns of stocks with strong past performance.
Key Takeaways
– The Carhart model builds on the Fama-French three-factor model by adding momentum as a fourth factor.
– It helps explain cross-sectional differences in stock returns beyond CAPM.
– The model assumes that these factors represent systematic risks that investors must be compensated for.
[Factors and Types of Multifactor Models]
1- Macroeconomic Factor Model
– Factors are macroeconomic surprises affecting assets via cash flows or discount rates.
– Common examples include interest rates, inflation risk, and credit spreads.
2- Fundamental Factor Model
– Factors are financial metrics like book-to-price and price-to-earnings ratios.
– These are crucial for explaining cross-sectional differences in stock prices.
3- Statistical Factor Model
– Factors are derived from statistical analysis of historical return data.
– Two main types:
– 1- Factor Analysis Models: Identify factors that explain/reproduce historical return covariance.
– 2- Principal Components Models: Seek factors that best explain/reproduce historical return variance.
– These models require minimal assumptions but can be difficult to interpret.
Key Takeaways
– Multifactor models help explain asset returns using different risk factor categories.
– Macroeconomic models use economic variables, fundamental models use financial ratios, and statistical models derive factors from data.
– Statistical models provide flexibility but lack intuitive economic interpretation.
A macroeconomic factor model, which means that the expected return on an asset is given by its intercept. The expected portfolio return is calculated as the weighted average of the expected return on each of its components.
[Contribution to Portfolio Return in a Macroeconomic Factor Model]
1- Overview of the Contribution Calculation:
– The portfolio’s total return is determined by the weighted contributions of each asset based on their expected returns and portfolio allocations.
– Each asset’s return is derived using a macroeconomic factor model incorporating GDP growth and inflation surprises.
2- Expected Return Calculation for Each Asset in 20X6:
– Formula: “Expected Return = Intercept + (GDP Sensitivity × GDP Surprise) + (Inflation Sensitivity × Inflation Surprise) + Error Term”
– Applying the formula:
— Asset A: “7.6% + (-1.0 × 0.3%) + (1.2 × -0.2%) + 1.2%” = 8.26%
— Asset B: “8.1% + (0.3 × 0.3%) + (1.8 × -0.2%) + 0.8%” = 8.63%
— Asset C: “6.9% + (-0.8 × 0.3%) + (0.6 × -0.2%) + 1.8%” = 8.34%
3- Contribution to Portfolio Return:
– Contribution Formula: “Asset Return × Portfolio Weight”
— Asset A: “8.26% × 40%” = 3.30%
— Asset B: “8.63% × 25%” = 2.16%
— Asset C: “8.34% × 35%” = 2.92%
– Total Portfolio Return: 8.38%
Key Takeaways
– Asset A had the largest contribution (3.30%) to the total portfolio return of 8.38%, due to its higher portfolio weight of 40%.
– While Asset B had the highest individual return (8.63%), its lower weight (25%) led to a smaller contribution than Asset A.
– Asset C contributed moderately, aligning closely with Asset B.
[The Structure of Fundamental Factor Models]
1- Overview
– Fundamental factor models use the same basic equation as macroeconomic factor models:
— Ri = ai + b_i1 F1 + b_i2 F2 + … + b_ik FK + ε_i
– Factors are stated as returns, not surprises, so the intercept does not represent the expected return.
2- Factor Sensitivities
– Sensitivities are attributes of securities, such as the price-to-earnings (P/E) ratio.
– Sensitivity to a factor is expressed as a standardized beta, scaled by standard deviation.
– Sensitivity is 0 if the asset has an average value for that attribute.
— Formula: b_ik = (Value of attribute k for asset i - Average value of attribute k) / σ(Values of attribute k)
3- Process of Model Estimation
– Fundamental factor models first specify factor sensitivities.
– Factor returns are then estimated using regression.
– This contrasts with macroeconomic factor models, where factors are developed as surprises before estimating sensitivities.
Key Takeaways
– Fundamental factor models explain asset returns using firm-specific attributes rather than macroeconomic variables.
– Factor sensitivities are predefined, allowing for regression-based estimation of factor returns.
– Unlike macroeconomic models, fundamental factor models do not treat factors as unexpected economic shocks.
[Types of Fundamental Factor Models]
1- Company Fundamental Factors
– Reflect internal company performance.
– Examples: Earnings growth, earnings momentum, earnings variability, financial leverage.
2- Company Share-Related Factors
– Related to valuation measures.
– Examples: Dividend yield, earnings yield, book-to-market ratio.
3- Macroeconomic Factors
– Measure systematic risk and broader economic influences.
– Examples: CAPM beta, yield curve sensitivity, sector or industry membership factors.
Key Takeaways
– Fundamental factor models classify factors into company-specific and macroeconomic categories.
– These models help in performance attribution and risk analysis.
– Macroeconomic factors capture systematic risk, while company-related factors focus on valuation and internal performance.
Types of fixed-income multifactor models:
Macroeconomic multifactor models
Fundamental multifactor models
Risk and style multifactor models
[Macroeconomic Factor Models]
1- Definition
– Asset returns are correlated with surprises in macroeconomic factors, such as inflation.
– Example: If expected inflation is 3% but actual inflation is 4%, the inflation surprise variable is 1%.
2- Formula
– General model: Ri = ai + b1iF1 + b2iF2 + … + bKiFK + εi
– Two-factor model using inflation and GDP growth: Ri = ai + b1iFINFL + b2iFGDP + εi
3- Interpretation
– Factor sensitivities (betas) measure how assets respond to macroeconomic shocks.
– The intercept ai represents expected return assuming no surprises.
4- Asset Class Performance Under Different Macroeconomic Conditions
– Low inflation, low growth: Cash, government bonds.
– Low inflation, high growth: Equities, corporate bonds.
– High inflation, low growth: Inflation-linked bonds, commodities, infrastructure.
– High inflation, high growth: Real estate, farmland, timberland, energy.
Key Takeaways
– Macroeconomic factor models analyze the impact of economic surprises on asset returns.
– Different asset classes perform better under varying combinations of inflation and GDP growth.
– Regression analysis is used to estimate factor sensitivities and asset class responses.
[Macroeconomic Factor Models]
1- Definition
– These models assume that asset returns are correlated with surprises in macroeconomic factors.
– A surprise is defined as actual value minus expected value.
2- Formula
– General model: Ri = ai + b1iF1 + b2iF2 + … + bKiFK + εi
– Two-factor model using inflation and GDP growth: Ri = ai + b1iFINFL + b2iFGDP + εi
3- Explanation of Components
– Ri: Return on asset i.
– ai: Expected return of asset i, assuming no surprises.
– Fk: Surprise in factor k.
– bik: Sensitivity of the return to factor k.
– εi: Error term, representing the part of return not explained by the model.
4- Interpretation
– Factors are defined in terms of surprises, with their predicted values embedded in ai.
– The return on an asset is broken into three parts:
— Expected return assuming no surprises.
— Unexpected return from new information (factor surprises).
— An error term.
Key Takeaways
– Macroeconomic factor models decompose asset returns into expected components, surprises, and error terms.
– These models help analyze the impact of macroeconomic shocks like inflation and GDP growth.
– Factor sensitivities (betas) measure how asset returns react to macroeconomic surprises.
[Comparison of Macroeconomic and Fundamental Factor Models]
1- Macroeconomic Multifactor Models
– Begin by developing the factor series based on macroeconomic variables.
– Use regression analysis to estimate factor sensitivities (betas).
2- Fundamental Factor Models
– Take the opposite approach by specifying factor sensitivities first.
– Use regression analysis to derive the factor returns.
Key Takeaways
– Macroeconomic models focus on estimating how asset returns react to macroeconomic surprises.
– Fundamental models start by defining factor sensitivities and then derive their returns.
– Both models rely on regression analysis but differ in their approach to defining factors.
[Groups of Fundamental Factor Models for Equities]
1- Company Fundamental Factors
– Reflect internal performance aspects of a company.
– Examples:
— Earnings growth.
— Earnings variability.
— Earnings momentum.
— Financial leverage.
2- Share-Related Factors
– Include valuation measures and factors based on investor expectations.
– Examples:
— Dividend yield.
— Earnings yield.
— Book-to-market ratio.
3- Macroeconomic Factors
– Cover sector and industry factors along with systematic risk measures.
– Examples:
— Yield curve level sensitivity.
[Multifactor Models and Applications]
1- Macroeconomic Factor Models
– Equity application: Surprises in macroeconomic variables.
– Fixed-income application: Surprises for inflation and GDP growth.
2- Fundamental Factor Models
– Equity application: Attributes of stocks or companies.
– Fixed-income application: Different duration factors.
3- Statistical Factor Models
– Factors are deduced from historical data.
– Can be most easily applied to various asset classes.
[Factor Models in Return Attribution]
1- Purpose of Factor Models
– Help analysts understand the sources of a manager’s returns relative to a benchmark.
– Fundamental multifactor models are preferred for return attribution over statistical and macroeconomic factor models.
2- Active Return Calculation
– Active return is the difference between a portfolio’s return and the return on the manager’s benchmark.
— Formula: “Active return = Rp - RB”.
— Rp: Portfolio return.
— RB: Benchmark return.
3- Components of Active Return
– Factor tilts: Overweighting or underweighting factor sensitivities relative to the benchmark.
– Security selection: Holding securities different from those in the benchmark.
4- Decomposition of Active Returns
– Formula: “T∑_k=1 [(Portfolio sensitivity)k - (Benchmark sensitivity)k] × (Factor return)k + Security selection”.
[Factor Models in Return Attribution]
1- Role of Multifactor Models
– These models help analysts understand the sources of a manager’s returns relative to a benchmark.
2- Preference for Fundamental Multifactor Models
– Fundamental models are often preferred over statistical models.
– They are easier to explain and understand for clients.
3- Return Attribution Process
– Begins with understanding the active manager’s objectives.
4- Active Managers’ Portfolio Weighting
– They weight security holdings differently than the benchmark.
– 1- The goal is to add value to the portfolio relative to a passive benchmark.
– 2- Weighting decisions are based on the manager’s expectations.
Key Takeaways
– Multifactor models provide insights into portfolio performance attribution.
– Fundamental factor models are favored due to their interpretability.
– Active managers use factor exposures to enhance returns beyond passive benchmarks.
[Factor Models in Risk Attribution]
1- Definition of Active Risk
– Active risk, also known as tracking error (TE), is the standard deviation of active returns.
— Formula: “TE = s(Rp - RB)”.
— Rp: Portfolio return.
— RB: Benchmark return.
— s: Standard deviation function.
2- Information Ratio (IR)
– Measures active return relative to active risk.
— Formula: “IR = (R̅p - R̅B) / s(Rp - RB)”.
— R̅p: Average portfolio return.
— R̅B: Average benchmark return.
3- Decomposition of Active Risk
– Active risk squared is used to analyze risk variances.
— Formula: “Active risk squared = s^2(Rp - RB)”.
– Active risk consists of two components:
— Active factor risk: Risk from factor exposures differing from the benchmark.
— Active specific risk: Risk from security selection.
4- Active Specific Risk Calculation
– Formula: “Active specific risk = n∑_i=1 (wi^a)^2 σεi^2”.
— wi^a: Active weight of the i-th asset.
— σεi^2: Residual risk of the i-th asset.
Example Calculation
– Given data over the past 10 years:
— Mean portfolio return = 8%.
— Mean benchmark return = 7.5%.
— Portfolio’s tracking error = 4%.
– Calculation: “IR = (8% - 7.5%) ÷ 4% = 0.125”.
Key Takeaways
– A higher IR indicates better risk-adjusted active returns.
– IR helps evaluate a manager’s skill in generating excess returns relative to risk.
– It is useful for comparing different investment strategies and portfolio managers.
[Active Risk Comparison of Managers A, B, and C
1- Overview of the Concept
– Active risk (also called tracking error) measures the standard deviation of active returns.
– It can be derived from the Information Ratio (IR) formula:
— IR = RA ÷ STD(RA)
— Rearranged: STD(RA) = RA ÷ IR
2- Formula Used
– Information Ratio formula (rearranged for active risk):
— “ActiveRisk = ActiveReturn ÷ InformationRatio”
— Where:
—- ActiveReturn = Fund expected return – Benchmark return
—- InformationRatio = IR (provided in the table)
—- ActiveRisk = STD(RA)
3- Inputs from Case
– Benchmark return = 13.15%
– Manager A:
— Fund return = 14.39%
— ActiveReturn = 14.39% – 13.15% = 1.24%
— IR = 0.52
— ActiveRisk = 1.24% ÷ 0.52 = 2.38%
– Manager B:
— Fund return = 14.91%
— ActiveReturn = 14.91% – 13.15% = 1.76%
— IR = 0.63
— ActiveRisk = 1.76% ÷ 0.63 = 2.79%
– Manager C:
— Fund return = 14.76%
— ActiveReturn = 14.76% – 13.15% = 1.61%
— IR = 0.54
— ActiveRisk = 1.61% ÷ 0.54 = 2.98%
4- Final Answer
– Manager C has the highest active risk at 2.98%.
[Factor Models in Portfolio Construction]
1- Role of Multifactor Models in Portfolio Construction
– Multifactor models help construct portfolios by enabling managers to make focused bets or control risk.
– These models are applicable to both passive and active portfolio management.
– Rules-based alternative indexes can be created to capture systemic exposure to managerial skill.
2- Pure Factor Portfolio
– A portfolio can be designed to have sensitivity to only one risk factor.
– A “pure factor portfolio” has a sensitivity of 1 to the relevant risk factor and 0 to all other factors.
Key Takeaways
– Multifactor models assist in portfolio risk control and strategy implementation.
– They can be used in both active and passive management approaches.
– Pure factor portfolios isolate exposure to a single risk factor for targeted investment strategies.
[Factor Models in Portfolio Construction]
1- Application of Multifactor Models
– Used to construct portfolios and analyze active returns and risks.
– Applicable in both passive and active management strategies.
2- Portfolio Management Strategies and Their Applications
– Passive management: Selecting index securities that match factor exposures.
– Active management: Predicting alpha or relative returns and establishing risk profiles.
– Rules-based active management: Creating factor and style biases relative to an index.
3- Key Concepts
– Alpha: Measures excess risk-adjusted returns.
– Relative return: Compares the return of one asset to another.
Key Takeaways
– Multifactor models help optimize portfolio construction and risk assessment.
– They provide insights for both passive index replication and active return prediction.
– Understanding alpha and relative return is crucial for evaluating active strategies.
[Factor Models in Strategic Portfolio Decisions]
1- Role of Multifactor Models in Strategic Decisions
– These models help investors focus on risks where they have a comparative advantage.
– Investors can avoid unnecessary risks while optimizing portfolio exposure.
2- Example: University Endowments
– Endowments have longer investment horizons than most investors.
– They may have a comparative advantage in business cycle risk.
3- Multifactor Models vs. CAPM
– Multifactor models provide more detail and granularity than the CAPM framework.
– This can lead to better diversification and potentially more efficient portfolios.
Key Takeaways
– Multifactor models allow investors to align risk exposure with their strengths.
– They offer improved portfolio diversification compared to CAPM.
– Longer-term investors, such as endowments, can benefit from their ability to manage business cycle risk.
[Factor Considerations in Strategic Portfolio Decisions]
1- Role of Multifactor Models
– Help investors make informed investment decisions based on comparative advantages.
– Provide more detailed insights than the CAPM framework.
2- Investment Decision Framework
– According to CAPM, investors choose a mix of the market portfolio and the risk-free rate.
– Investors may deviate from the market portfolio based on specific comparative advantages or disadvantages.
3- Customization for Investors
– Multifactor models assist in constructing optimal portfolios tailored to individual investor circumstances.
Key Takeaways
– Multifactor models offer deeper insights than CAPM by incorporating investor-specific factors.
– Portfolio decisions can be adjusted based on comparative advantages.
– Customization using multifactor models helps align investments with unique investor circumstances.
Quiz - [Arbitrage Opportunity in Factor-Based Investing]
1- Understanding the Arbitrage Opportunity:
– Arbitrage occurs when two portfolios have identical factor sensitivities but different expected returns.
– If one portfolio has a higher expected return than another with the same risk, an arbitrage strategy can exploit this mispricing.
2- Expected Returns and Factor Sensitivities Calculation:
– Portfolio 60/40 consists of 60% allocation to Fund A and 40% allocation to Fund B.
– Expected return calculation:
— “Expected Return of Portfolio 60/40 = (60% × 2%) + (40% × 4%)”
— “Expected Return = 1.2% + 1.6%” = 2.8%
– Factor sensitivity calculation:
— “Factor Sensitivity of Portfolio 60/40 = (60% × 0.5) + (40% × 1.5)”
— “Factor Sensitivity = 0.3 + 0.6” = 0.9
3- Comparing Portfolio 60/40 to Fund C:
– Fund C has a factor sensitivity of 0.9, identical to Portfolio 60/40.
– However, Fund C’s expected return is 3.0%, which is higher than Portfolio 60/40’s return of 2.8%.
– This mismatch suggests an arbitrage opportunity:
— Long Fund C (higher expected return, same factor risk).
— Short Portfolio 60/40 (lower expected return, same factor risk).
4- Optimal Arbitrage Strategy:
– To exploit this mispricing, Strategy 3 is correct:
— Sell short $60,000 of Fund A and $40,000 of Fund B.
— Buy $100,000 of Fund C.
Key Takeaways
– Factor-based arbitrage arises when two investments have the same factor exposures but different expected returns.
– Portfolio 60/40 and Fund C have identical factor risks, but Fund C offers a higher return.
– Going long Fund C and short Portfolio 60/40 generates a risk-free arbitrage profit.
[Exercise: Determining Parameters in a One-Factor APT Model]
1- Overview of the Concept
– The Arbitrage Pricing Theory (APT) assumes that asset returns are determined by factor sensitivities.
– The goal is to determine the risk-free rate (RF) and factor risk premium (λ1) using given portfolio data.
2- Formula
– The expected return in a one-factor APT model follows the equation:
— Expected return formula: E(Rp) = RF + λ1 * βp,1
3- Given Data
– Portfolio A: Expected return = 0.070, Factor sensitivity = 0.5
– Portfolio B: Expected return = 0.105, Factor sensitivity = 1.2
4- Solving for Parameters
– Step 1: Express RF in terms of λ1 using Portfolio A:
— 0.07 = RF + λ1(0.5)
— RF = -λ1(0.5) + 0.07
– Step 2: Substitute RF into Portfolio B’s equation:
— 0.105 = RF + λ1(1.2)
— 0.105 - 0.07 = -λ1(0.5) + λ1(1.2)
— 0.035 = 0.7λ1
— λ1 = 0.05
– Step 3: Solve for RF:
— RF = 0.07 - 0.05(0.5)
— RF = 0.045
[Exercise: Arbitrage Opportunity in APT]
1- Overview of the Concept
– Arbitrage arises when an asset’s expected return deviates from the return predicted by the APT model.
– Portfolio C has an expected return of 9.0% and a factor sensitivity of 0.85, requiring verification of arbitrage.
2- Formula
– Expected return in a one-factor APT model:
— E(Rp) = RF + λ1 * βp,1
3- Calculation of Expected Return for Portfolio C
– Using previously determined values:
— E(Rc) = 0.045 + 0.05(0.85)
— E(Rc) = 0.0875 (8.75%)
4- Identifying Arbitrage
– Portfolio C’s quoted expected return of 9.0% exceeds 8.75%, indicating an arbitrage opportunity.
– Construct an arbitrage portfolio by offsetting risk via shorting a blended portfolio of A and B.
5- Solving for Portfolio Weights
– Solve for weight (w) to match Portfolio C’s factor sensitivity:
— w(0.5) + (1 - w)(1.2) = 0.85
— 0.5w + 1.2 - 1.2w = 0.85
— -0.7w = -0.35
— w = 0.5
6- Constructing the Arbitrage Portfolio
– Sell short a blended portfolio: 50% A and 50% B (expected return = 8.75%).
– Use proceeds to buy Portfolio C (expected return = 9.0%).
– Risk-free arbitrage profit = 9.0% - 8.75% = 0.25%.
Quiz - [Calculating the Return for Portfolio AC Using the Two-Factor Model]
1- Macroeconomic Two-Factor Model Formula:
– “Ri = αi + βi1FINF + βi2FGDP + εi”
– Where:
— “Ri” = Return for asset “i”.
— “αi” = Expected return.
— “βi1, βi2” = Sensitivities to inflation (FINF) and GDP growth (FGDP).
— “εi” = Error term (assumed to be zero).
2- Given Data from Exhibits:
– Surprises in Economic Factors:
— Surprise in inflation = 2.2% - 2.0% = 0.2%.
— Surprise in GDP growth = 1.0% - 1.5% = -0.5%.
– Factor Sensitivities & Expected Returns:
— Fund A: αA = 2.0%, βA1 = 0.5, βA2 = 1.0.
— Fund C: αC = 3.0%, βC1 = 1.0, βC2 = 1.1.
3- Calculating the Returns for Fund A and Fund C:
– “RA = 0.02 + (0.5 × 0.002) + (1.0 × -0.005)”
– “RA = 0.02 + 0.001 - 0.005 = 1.6%”
– “RC = 0.03 + (1.0 × 0.002) + (1.1 × -0.005)”
– “RC = 0.03 + 0.002 - 0.0055 = 2.45%”
4- Portfolio AC Return Calculation:
– Portfolio AC consists of 60% Fund A and 40% Fund C:
– “RAC = (0.6 × 1.6%) + (0.4 × 2.45%)”
– “RAC = 0.96% + 0.98%”
– “RAC = 2.02%”
Key Takeaways
– Portfolio AC’s return is 2.02%, confirming answer choice A.
– The two-factor model incorporates inflation and GDP growth surprises to refine return estimations.
– Weighting the returns of Fund A and Fund C appropriately leads to the final portfolio return.
Quiz - [Impact of Economic Surprises on Fund Returns]
1- Definition of Surprise in a Macroeconomic Model:
– “Surprise = Actual Factor - Predicted Factor”.
– The inflation surprise is 2.2% - 2.0% = 0.2%.
– The GDP growth surprise is 1.0% - 1.5% = -0.5%.
– The impact on fund returns is calculated by multiplying the surprise by the fund’s factor sensitivity.
2- Effect of Inflation and GDP Growth Surprises on Fund Returns:
– Fund A: The impact of the inflation surprise is 0.5 × 0.2% = 0.10%, and the impact of the GDP growth surprise is 1.0 × (-0.5%) = -0.50%.
– Fund B: The impact of the inflation surprise is 1.6 × 0.2% = 0.32%, while the impact of the GDP growth surprise is 0.0 × (-0.5%) = 0.00%.
– Fund C: The impact of the inflation surprise is 1.0 × 0.2% = 0.20%, and the impact of the GDP growth surprise is 1.1 × (-0.5%) = -0.55%.
3- Conclusion:
– The GDP growth surprise on Fund C (-0.55%) had the greatest effect on fund returns.
– Fund C had the highest sensitivity (1.1) to GDP growth, amplifying the negative surprise of -0.5%.
– The impact was larger than the inflation effects across all funds, confirming answer C as correct.
9.5 Measuring & Managing Market Risk
– Explain the use of value at risk (VaR) in measuring portfolio risk.
– Compare the parametric (variance–covariance), historical simulation, and Monte Carlo simulation methods for estimating VaR.
– Estimate and interpret VaR under the parametric, historical simulation, and Monte Carlo simulation methods.
– Describe advantages and limitations of VaR.
– Describe extensions of VaR.
– Describe sensitivity risk measures and scenario risk measures and compare these measures to VaR.
– Demonstrate how equity, fixed-income, and options exposure measures may be used in measuring and managing market risk and volatility risk.
– Describe the use of sensitivity risk measures and scenario risk measures.
– Describe advantages and limitations of sensitivity risk measures and scenario risk measures.
– Explain constraints used in managing market risks, including risk budgeting, position limits, scenario limits, and stop-loss limits.
– Explain how risk measures may be used in capital allocation decisions.
– Describe risk measures used by banks, asset managers, pension funds, and insurers.
Market risk is the risk from movements in the stock market, interest rates, exchange rates, and commodity prices. Risk management is the process of identifying and measuring risk. Various financial models are used to identify and measure the risk. Judgment and experience are needed to correctly interpret the outputs of these models.
[Value at Risk (VaR): Formal Definition]
1- Overview of the Concept
– Value at Risk (VaR) estimates the minimum loss expected over a given time period at a specified confidence level.
– It can be expressed in currency units or as a percentage of portfolio value.
– Example: A 5% VaR of £1.4 million means the portfolio is expected to lose at least £1.4 million on 5% of the days.
2- Confidence Level and Interpretation
– A 5% VaR corresponds to a 95% confidence level, meaning there is a 95% probability that losses will not exceed this amount on any given day.
– VaR focuses on both the frequency and magnitude of potential losses.
3- Statistical Interpretation
– VaR measures the left tail of the return distribution.
– The number of standard deviations below the mean depends on the confidence level:
— 5% VaR = 1.65 standard deviations below the mean.
— 1% VaR = 2.33 standard deviations below the mean.
— 16% VaR = 1 standard deviation below the mean.
Key Takeaways
– VaR quantifies potential downside risk in a portfolio.
– It is widely used in risk management to set capital requirements and assess potential losses.
– The confidence level determines the likelihood that actual losses will not exceed the estimated VaR.
[Estimating Value at Risk (VaR)]
1- Methods for Estimating VaR
– Three main approaches are used to estimate VaR:
— Parametric method.
— Historical simulation method.
— Monte Carlo simulation method.
2- Key Steps in VaR Estimation
– 1- Define risk factors with risk decomposition (e.g., equity risk, currency risk, interest rate risk).
– 2- Gather historical data for each risk factor, including return, standard deviation, and correlation.
– 3- Use the collected data to estimate VaR using the selected methodology.
3- Example of VaR Estimation
– A portfolio with two assets is used for illustration:
— An S&P 500 index ETF (SPY) represents equity exposure.
— A long-term corporate bond fund (LWC) measures bond exposure.
– The estimation process remains the same for larger portfolios but increases in complexity.
Key Takeaways
– VaR estimation requires defining risk factors, collecting historical data, and applying an appropriate methodology.
– The historical data used includes return, standard deviation, and correlation over different time periods.
– Judgment is necessary in determining the lookback period for historical data analysis.
[Parametric Method of VaR Estimation]
1- Overview of the Parametric Method
– Also known as the analytical or variance-covariance method.
– Assumes returns are normally distributed, requiring only the expected value and standard deviation.
– Other distributions can be used but may require additional parameters.
2- Standardization of Returns
– A normally distributed return R can be converted to a z-score:
— Formula: “z = (R - μ) / σ”
— Where:
—- μ: Mean (expected return).
—- σ: Standard deviation.
3- Portfolio Return and Volatility Calculation
– For a portfolio with two assets (SPY and LWC), expected return is:
— Formula: “E[Rp] = wSPY * E[RSPY] + wLWC * E[RLWC]”
– Portfolio standard deviation (volatility) is:
— Formula: “σp = sqrt[wSPY^2 * σSPY^2 + wLWC^2 * σLWC^2 + 2 * wSPY * wLWC * ρSPY,LWC * σSPY * σLWC]”
4- Conversion to Daily Values
– Assumes 250 trading days per year.
– Daily expected return: “Annual expected return ÷ 250”.
– Daily volatility: “Annual volatility ÷ sqrt(250)”.
5- Computing the 5% VaR
– Formula: “{ (E[Rp] - 1.65 * σp) * (-1) } * (Portfolio value)”.
– The 5% VaR represents the minimum expected loss at a 95% confidence level.
Key Takeaways
– The parametric VaR method assumes normal return distributions and uses expected return and standard deviation.
– Portfolio risk is determined using variance and correlation between asset returns.
– VaR calculations can be converted to daily values for short-term risk assessment.
The major advantage of the parametric method is its simplicity. It works best when the normal distribution assumption is reasonable and the parameter estimates are reliable. By contrast, the parametric method does not work well for portfolios that contain options because the returns on options are not normally distributed.
[The Historical Simulation Method of VaR Estimation]
1- Overview of the Concept
– The historical simulation method estimates VaR by analyzing portfolio returns over a lookback period.
– Returns are sorted from the largest loss to the largest gain, and the VaR is chosen based on the desired confidence level.
2- Key Assumptions and Characteristics
– Unlike the parametric method, it does not assume returns are normally distributed.
– The standard method assumes all observations are equally weighted, but adjustments can be made to weight observations differently.
– The chosen lookback period should be representative of future conditions for accuracy.
3- Application and Suitability
– This method is particularly useful for portfolios containing options, as it can handle non-normal return distributions.
Key Takeaways
– Historical simulation calculates VaR based on actual past returns rather than statistical assumptions.
– It does not rely on normal distribution assumptions, making it effective for portfolios with derivatives.
– The quality of results depends on selecting a representative lookback period.
[The Monte Carlo Simulation Method of VaR Estimation]
1- Overview of the Concept
– The Monte Carlo simulation method estimates VaR by generating random portfolio return outcomes based on statistical assumptions about distributions.
– This method is flexible and can be used for portfolios with many assets without requiring a normal distribution assumption.
2- Process of Monte Carlo Simulation for VaR
– Random values are generated for each unknown variable, such as asset returns, using assumed statistical distributions.
– Asset correlations must be incorporated into the model to reflect realistic portfolio behavior.
– The model generates multiple possible portfolio return scenarios.
– Returns are then sorted from worst to best, and VaR is determined similarly to the historical simulation method.
3- Key Considerations and Limitations
– There is no fixed standard for the number of scenarios required, but more simulations improve the reliability of VaR estimates at the cost of higher computation time.
– The Monte Carlo method does not impose normality on the final returns, making it more flexible than the parametric method.
– If the input variables follow a normal distribution, Monte Carlo estimates will closely resemble those from the parametric method.
Key Takeaways
– Monte Carlo simulation provides a flexible approach to estimating VaR, accommodating complex portfolios.
– It accounts for asset correlations and does not require normal return distributions.
– The accuracy of VaR depends on the number of scenarios generated, balancing precision with computational efficiency.
VaR does not take into account asset liquidity
VaR does not take into account asset liquidity
[Advantages of VaR]
1- Simplicity and Communication
– VaR is easy to understand, even for those without a technical background.
– The concept can be effectively communicated using a single number.
2- Risk Measurement and Comparison
– VaR serves as a standardized measure for comparing risks across asset classes and business units.
– It helps in assessing and managing financial risks systematically.
3- Capital Allocation and Performance Evaluation
– VaR assists in determining appropriate capital allocation across different trading units.
– It allows for performance evaluation by adjusting returns based on risk before making comparisons.
4- Verification and Regulatory Acceptance
– Historical data can be used to verify the reliability of VaR estimates by checking whether actual losses align with expected values.
– VaR is widely accepted by regulators such as the SEC and global banking authorities.
Key Takeaways
– VaR is a simple yet effective risk measurement tool that facilitates risk comparison and capital allocation.
– It helps in performance evaluation by incorporating risk adjustments.
– The method is widely accepted and used by financial regulators for risk assessment.
[Limitations of VaR]
1- Subjectivity in Assumptions
– VaR results depend on the choice of percentile, time horizon, and estimation method.
– Different input choices can lead to varying risk estimates.
2- Underestimation of Extreme Events
– Assuming a normal distribution often fails to capture the frequency of left-tail events.
– Market crashes and rare events may be underestimated.
3- Liquidity and Correlation Risks
– VaR may not account for the additional losses from liquidity constraints during market stress.
– Correlations tend to increase during downturns, reducing diversification benefits.
4- Sensitivity to Market Conditions
– Portfolios can lose significantly over time without hitting the VaR threshold, masking cumulative risks.
– Volatility regimes can shift, making past estimates unreliable.
5- Misinterpretation and Oversimplification
– VaR is sometimes misunderstood as the worst-case scenario or maximum loss.
– Despite its simplicity, risk is complex and cannot always be reduced to a single number.
6- Incomplete Risk Assessment
– VaR focuses on left-tail risk and ignores potential upside risks (right-tail events).
– A full distribution analysis is needed for a complete risk profile.
Key Takeaways
– VaR relies on assumptions that can lead to underestimating extreme risks.
– Liquidity, correlation shifts, and volatility changes can impact its accuracy.
– It provides a simplified risk metric but should be complemented with other risk measures.
[Extensions of VaR]
1- Conditional VaR (CVaR)
– Also known as expected shortfall or expected tail loss, CVaR measures the expected loss beyond the VaR threshold.
– Unlike standard VaR, CVaR provides insight into the magnitude of extreme losses.
– It is commonly calculated using historical simulation or Monte Carlo methods.
2- Incremental VaR (IVaR)
– IVaR quantifies the impact of adjusting a position’s size within a portfolio.
– Used to measure risk changes when adding or removing assets.
– Helps in optimizing portfolio allocation by assessing risk contributions.
3- Marginal VaR (MVaR)
– Similar to IVaR, MVaR measures the change in VaR for a small change in position size.
– It is derived using calculus to assess each asset’s contribution to total portfolio risk.
– Useful for risk budgeting and position sizing.
4- Relative VaR
– Also known as ex-ante tracking error, it measures a portfolio’s deviation from its benchmark.
– Calculated using the difference between portfolio weights and benchmark weights.
– A higher relative VaR indicates greater divergence from the benchmark.
Key Takeaways
– CVaR captures tail risk beyond the VaR threshold, providing additional insights into extreme losses.
– IVaR and MVaR assess risk sensitivity to position changes, aiding in portfolio optimization.
– Relative VaR measures tracking error relative to a benchmark, useful for performance evaluation.
Quiz - [Assessing George’s Comments on Extensions of VaR]
1- Misinterpretation of Relative VaR (Statement B)
– Relative VaR, or ex-ante tracking error, measures the risk of deviating from a benchmark.
– George incorrectly claims that if a portfolio outperforms, relative VaR might be negative.
– This is incorrect because tracking error is based on squared deviations, making it always positive.
2- Misuse of Marginal VaR (Statement C)
– Marginal VaR estimates how a small position change affects total portfolio risk.
– George incorrectly applies it to a large allocation shift (e.g., 30% reallocation).
– Marginal VaR is only valid for small incremental changes, not major rebalancing.
Key Takeaways
– Relative VaR (tracking error) is always positive, regardless of portfolio outperformance.
– Marginal VaR applies only to small position changes, not large allocation shifts.
– George’s statements contain errors, making Answer A the correct choice.
[Sensitivity Risk Measures]
1- Equity Exposure Measures
– Beta (β) is the key measure used to assess exposure to equity risk in the Capital Asset Pricing Model (CAPM).
– The expected return on a portfolio is given by the CAPM formula:
— Formula: “E[Ri] = RF + βi (E[RM] - RF)”
– Where:
— E[Ri]: Expected return on portfolio i.
— RF: Risk-free rate.
— E[RM]: Expected market return.
— βi: Risk measure (beta).
2- Interpretation of Beta
– Beta quantifies the portfolio’s sensitivity to the equity risk premium.
– It is computed as the covariance between the asset return and market return, divided by the variance of the market return.
– A higher beta indicates greater exposure to market risk.
Key Takeaways
– Beta is essential for measuring systematic risk in equity investments.
– CAPM helps estimate expected returns based on market risk exposure.
– Understanding beta allows investors to align portfolios with their risk tolerance.
[Fixed-Income Exposure Measures]
1- Duration and Convexity
– Duration measures the sensitivity of a bond’s price to changes in yield, approximating percentage price changes.
– Convexity accounts for non-linear price-yield relationships, improving accuracy for larger yield changes.
2- Duration Formula
– Formula: “ΔB / B ≈ -D * (Δy / (1 + y))”
– Where:
— ΔB / B: Percentage change in bond price.
— D: Duration.
— Δy: Change in yield.
— y: Initial yield.
3- Convexity Adjustment
– The impact of convexity refines the duration estimate for larger yield shifts.
– Formula: “ΔB / B ≈ -D * (Δy / (1 + y)) + (1/2) * C * (Δy^2 / (1 + y)^2)”
– Where:
— C: Convexity.
Key Takeaways
– Duration provides a linear estimate of bond price changes due to yield shifts.
– Convexity improves the estimate by accounting for non-linearity in price changes.
– These measures help investors manage interest rate risk in fixed-income portfolios.
Duration is the weighted average life of a bond’s cash flows. It is used to approximate the percentage change in a bond price for a given change in yield.
[Options Risk Measures]
1- Delta: Measures an option’s price sensitivity to changes in the underlying asset price.
– Formula: “Δ ≈ (Change in value of option) / (Change in value of underlying)”
– Call options have delta values between 0 and 1, while put options have delta values between 0 and -1.
– Formula for a call option: “Δc ≈ Δc * ΔS”
— Δc: Change in option price.
— Δc: Option delta.
— ΔS: Change in underlying price.
2- Gamma: Measures the rate of change of delta with respect to changes in the underlying asset price.
– Formula: “Γ ≈ (Change in delta) / (Change in value of underlying)”
– Accounts for the non-linearity in an option’s price change.
– Adjusted price change formula: “Δc ≈ Δc * ΔS + (1/2) * Γc * (ΔS)^2”
3- Vega: Measures an option’s price sensitivity to changes in the volatility of the underlying asset.
– Formula: “Vega ≈ (Change in value of option) / (Change in volatility of underlying)”
– Adjusted price change formula with vega: “Δc ≈ Δc * ΔS + (1/2) * Γc * (ΔS)^2 + vega * Δσ”
— Δσ: Change in volatility.
Key Takeaways
– Delta provides a linear approximation of option price sensitivity to underlying price changes.
– Gamma adjusts for non-linearity, improving accuracy for larger price movements.
– Vega accounts for the impact of volatility changes on option prices.
[Scenario Risk Measures]
1- Overview: Scenario risk measures estimate portfolio returns under specific market conditions, which can be either hypothetical or historical.
– Unlike sensitivity measures, scenario analysis considers multiple factors changing simultaneously.
– These measures focus on extreme outcomes, with stress tests emphasizing severe negative events.
2- Historical Scenarios: Measures portfolio returns if past market events were to recur.
– Common examples:
— 1- Currency crisis of 1997-1998.
— 2- Technology bubble of 2001.
– Price histories are used for equity positions, while fixed-income securities and derivatives are repriced based on historical conditions.
– Outputs can be:
— 1- Total return.
— 2- Return relative to a benchmark.
— 3- Return relative to a liability.
3- Variations and Considerations:
– Some approaches assume managers adjust portfolios as scenarios unfold, potentially underestimating risks.
– All relevant factors should be included, such as foreign exchange rate changes.
– Historical simulation is effective when using rate and price changes but may not be accurate for extreme market shifts.
– Some securities may not have existed during the historical period, requiring adjustments through factor models.
Key Takeaways
– Scenario risk measures capture multiple risk factors and assess extreme conditions.
– Historical scenarios provide insights but require careful assumptions about market conditions.
– Stress testing helps evaluate a portfolio’s resilience to severe negative market events.
[Scenario Risk Measures]
1- Overview: Scenario risk measures assess portfolio returns under specific market conditions.
– Unlike historical scenarios, hypothetical scenarios help managers prepare for events that have not yet occurred.
– These scenarios do not assume a normal distribution and can account for multiple factors changing simultaneously.
2- Hypothetical Scenarios:
– Designed to identify portfolio exposures to potential market events.
– Used for stress testing, including reverse stress testing, which estimates losses from multiple exposures in a crisis.
– Geopolitical scenarios measure the impact of crises spreading across countries.
– Correlations often increase during market stress, which can be incorporated into scenario testing.
3- Applications and Limitations:
– Helps identify vulnerabilities in a portfolio, leading to adjustments in risk exposure.
– Risks cannot be fully eliminated without affecting expected returns.
– Best used as a final screen rather than an initial risk measure. Initial screens should include exposure limits and VaR.
Key Takeaways
– Hypothetical scenarios provide insight into market risks beyond historical events.
– Reverse stress testing is useful for evaluating crisis exposure.
– Scenario testing is most effective when used alongside other risk measures like VaR.
[Sensitivity and Scenario Risk Measures and VaR]
1- Overview: These risk measures evaluate asset value changes under different risk conditions.
– VaR estimates the probability of large losses but does not specify which risk factors drive them.
– Sensitivity measures assess how asset values change with other risk factors but do not determine probabilities.
– Scenario risk measures analyze outcomes when multiple factors change, making them complementary to VaR.
2- Advantages of Sensitivity and Scenario Risk Measures:
– Do not rely on historical volatility or correlation, making them useful beyond lookback periods.
– No assumption of normal distribution is required.
– Can identify key portfolio exposures, such as concentrated positions.
3- Limitations of Scenario Analysis and Stress Testing:
– Hypothetical scenarios are needed when historical data is insufficient.
– They may incorrectly specify asset correlations or fail to adjust for factors like liquidity.
– Maintaining hypothetical scenarios is difficult due to uncertainty in tested ranges.
– Establishing limits based on scenario analysis is challenging.
– Extreme scenarios may not be taken seriously by management, while limiting scenarios to plausible cases may restrict risk assessment.
Key Takeaways
– Combining VaR, sensitivity measures, and scenario analysis provides a more comprehensive risk assessment.
– Sensitivity measures help quantify exposure but do not indicate probability.
– Scenario analysis helps identify extreme risks but is challenging to implement reliably.
[Quiz - Limitation of Historical Scenario Analysis in Risk Measurement]
1- Purpose of Historical Scenario Analysis:
– Ming requested an analysis using a historical financial crisis to estimate its impact on Flask’s portfolio.
– This approach assumes that past market conditions will repeat in the same way, affecting assets similarly.
2- Key Limitation of This Approach:
– It assumes no deviation from historical market events.
– Market crises rarely unfold in exactly the same manner, so relying on historical data alone can be misleading.
– Additional measures are needed to address potential differences between past and future financial events.
3- Explanation of Incorrect Choices:
– Omitting asset correlations (A) is incorrect because historical scenarios do capture past correlations between asset classes.
– Precluding portfolio manager actions (B) is incorrect since historical scenario analysis does not necessarily ignore the possibility of active management but focuses primarily on past conditions.
Key Takeaways:
– Historical scenario analysis is useful but should not be relied on exclusively.
– Market crises evolve differently each time, so risk models should incorporate flexibility beyond historical patterns.
[Market Participants and the Different Risk Measures They Use]
1- Factors Influencing Risk Measures:
– Different market participants use various risk measures based on their exposure and objectives.
2- Key Factors Affecting Risk Measure Selection:
– Degree of leverage: Highly leveraged participants focus on extreme loss scenarios to avoid bankruptcy, while those with low leverage aim to minimize underperformance.
– Mix of risk factors: Fixed-income portfolios prioritize interest rate and credit risk, while equity portfolios assess broad market and industry risks.
– Accounting or regulatory requirements: VaR, economic capital, duration, and beta are preferred for fair-value portfolios, whereas book-value portfolios rely on asset/liability gap models.
Key Takeaways
– The choice of risk measures depends on leverage, asset type, and regulatory requirements.
– Fixed-income and equity portfolios require different risk metrics due to differing exposures.
– Regulatory and accounting considerations shape the risk assessment framework.
[Banks and Risk Management]
1- Risk Management Considerations for Banks:
– Banks must balance regulatory, investor, and depositor expectations while managing risk effectively.
2- Key Risk Factors Banks Address:
– Liquidity gap: Monitoring the mismatch between assets and liabilities.
– VaR application: Used for the fair value portion of the balance sheet.
– Leverage assessment: Assigning higher weights to riskier assets.
– Sensitivity analysis: Applied to the held-for-sale portion of the balance sheet.
– Economic capital: Covers market, credit, and operational risks.
– Scenario analysis and stress testing: Ensuring capital adequacy under adverse conditions.
Key Takeaways
– Banks use various risk measures to manage liquidity, capital adequacy, and regulatory compliance.
– Scenario analysis and stress testing are crucial in determining capital sufficiency.
– Different balance sheet components require distinct risk measurement approaches.
[Asset Managers and Risk Measures]
1- Risk Considerations for Asset Managers:
– Asset managers focus on volatility, probability of loss, and risk of underperformance rather than capital standards.
2- Traditional Asset Manager Risk Measures:
– Position limits: Constraints on country, currency, sector, and asset class exposures.
– Sensitivity measures: Duration for fixed-income securities and Greeks for options.
– Beta sensitivity: Used for equity accounts.
– Liquidity assessment: Evaluates how long it takes to liquidate a security.
– Scenario analysis: Tests portfolio responses to stressed market conditions.
– Active share: Measures how much a portfolio differs from its benchmark.
– Redemption risk: Assesses the proportion of the portfolio that could be redeemed during peak times.
– Tracking error: Ex post measures historical deviations, while ex ante assesses risk in the current portfolio.
– VaR: Less commonly used compared to tracking error but still relevant in some strategies.
3- Hedge Fund Risk Measures:
– Sensitivities: Used to gauge various market exposures.
– Gross exposure: The absolute sum of long and short positions.
– Leverage: Monitored due to its impact on risk and returns.
– VaR: Typically applied for short time horizons and high confidence levels.
– Scenario-specific risks: Used to assess potential stress events.
– Maximum drawdown: Identifies worst-performing periods, crucial for non-normal return distributions.
Key Takeaways
– Traditional asset managers emphasize tracking error, liquidity, and active share.
– Hedge funds require additional monitoring of leverage, margin calls, and credit risks.
– VaR is more frequently used in hedge fund risk management than in traditional asset management.
[Pension Fund Risk Measures]
1- Overview of Pension Fund Risks:
– Defined benefit pension plans must manage assets to meet retiree obligations while avoiding overfunding or underfunding.
2- Key Risk Measures:
– Interest rate and curve risk: Cash flows are grouped by maturity and currency, with liabilities expressed as negative cash flows.
– Surplus at risk: Uses VaR to assess the potential for asset underperformance relative to liabilities. Higher risk arises when assets are more volatile or have low correlation with liabilities.
– Glide path: A strategy that gradually adjusts asset allocation to align the pension plan with its target funding level.
– Liability hedging vs. return generation: Some assets hedge liabilities, while others seek excess returns.
Key Takeaways
– Pension funds must balance asset growth with liability management.
– Interest rate risks and surplus risk are primary concerns.
– A glide path helps maintain appropriate funding levels.
[Risk Measures for Insurers]
1- Risk Management in Property and Casualty Insurance: Insurers use various risk measures to manage liabilities and asset allocations.
– Sensitivities and exposures: Asset allocations are designed within specified exposure limits.
– Economic capital and VaR: Capital is required to cover expected payouts and absorb adverse deviations.
– Scenario analysis: Market and insurance risks are analyzed together under stress scenarios.
2- Risk Management in Life Insurance and Annuities: These products have longer-term liabilities and align more closely with financial market risks.
– Sensitivities: Risk exposures are continuously measured and monitored.
– Asset and liability matching: While a perfect match is not required, life insurance companies align assets and liabilities more closely than property and casualty insurers.
– Scenario analysis: Stress tests jointly assess market and non-market risks, such as changes in cash flows due to declining asset values.
Key Takeaways
– Insurers use sensitivities, scenario analysis, and economic capital to manage liabilities.
– Property and casualty insurers focus more on exposure limits, while life insurers emphasize asset-liability management.]
[Risk Limits and Budgeting]
1- Establishing Risk Limits: Proper risk limits balance return potential with loss mitigation.
– Limits can be applied at the business unit level and the aggregate level.
– Correlation among business units should be considered when setting limits.
2- Risk Budgeting: Allocating risk exposure across business units or asset classes.
– Risk budgets are often measured using VaR or ex-ante tracking error.
– Pension funds may use surplus at risk to assess expected changes in funded status.
3- Position Limits: Preventing excessive concentration risk.
– Limits are set on exposure to issuers, countries, sectors, or investment strategies.
– Can be defined by market value of securities or notional principal of derivatives.
4- Scenario Limits: Setting loss thresholds for specific stress scenarios.
– Addresses shortcomings of VaR by ensuring losses stay within acceptable ranges.
– Requires corrective action if exceeded.
Key Takeaways
– Risk limits must balance flexibility with loss control.
– Position and scenario limits complement VaR-based risk budgeting.
Quiz - [Sector Allocation and Position Limits]
1- Sector Weighting and Position Limits: Used to manage sector-specific risk by capping sector exposure.
– Example: DLB fund caps sector allocation at 20%, despite the benchmark having 35% in Sector 1.
– This indicates compliance with a position limit rather than a lack of investment opportunities.
2- Liquidity Considerations: Not the primary factor behind underweighting Sector 1.
– Sector 1 consists of large-cap stocks, which are generally more liquid.
– DLB overweights Sectors 4 and 5, which have smaller, less liquid stocks.
– Market capitalization comparisons:
— Sector 1: 1.84% of total market capitalization.
— Sector 4: 0.60%, Sector 5: 0.28%.
3- Active Share and Benchmark Deviation:
– Active share measures deviation from the benchmark.
– DLB underweights Sector 1 significantly, increasing active share rather than minimizing it.
– This suggests the sector allocation is due to position limits rather than tracking the benchmark.
Key Takeaways
– Position limits are the primary reason for underweighting Sector 1.
– Liquidity was not a major concern, as smaller, less liquid stocks were overweighted.
– Active share percentage increased due to sector allocation differences.
[Stop-Loss Limits and Capital Allocation]
1- Stop-Loss Limits: Mechanism to mitigate prolonged losses.
– Triggered when losses exceed a predefined threshold within a specific period.
– Helps prevent losses that remain just below VaR limits.
– May require hedging or portfolio adjustments instead of liquidation.
2- Risk Measures and Capital Allocation: Ensuring efficient capital use.
– Economic capital estimates potential portfolio loss in adverse conditions.
– Company capital should exceed economic capital to maintain stability.
– Capital allocation prioritizes risk-adjusted returns and accounts for tail risk, particularly in portfolios with options exposure.
Key Takeaways
– Stop-loss limits act as a safeguard against prolonged drawdowns.
– Capital allocation strategies should optimize risk-adjusted returns while maintaining adequate reserves.
[Economic Capital and Risk Budgeting]
1- Economic Capital: Measures the capital required to cover potential portfolio losses.
– Larger for portfolios with sizable tail risk (e.g., options sellers).
– Allocated to maximize return per unit of capital, aligning with shareholder objectives.
2- Capital Allocation by Business Unit: Ensures risk-adjusted capital distribution.
– More conservative approach.
– Uses a very high confidence level.
3- Risk Budgeting: Distributes risk across business units and investment strategies.
– Often focuses on losses at one standard deviation.
Key Takeaways
– Actual capital should exceed economic capital to provide a risk cushion.
– Risk budgeting helps manage losses, while economic capital ensures financial stability.
[VaR Constraints and Stop-Loss Limits]
1- VaR Constraint: Defines risk tolerance using Value at Risk.
– Example: A portfolio has a 5-day, 1% VaR limit of $10 million.
2- Stop-Loss Limit: Triggers liquidation if losses exceed a threshold.
– Example: A portfolio will be liquidated if monthly losses exceed $15 million.
3- Risk Management Actions: Implemented when limits are breached.
– Liquidating positions.
– Using hedges.
– Applying drawdown control or portfolio insurance.
Key Takeaways
– VaR constraints help monitor risk exposure, while stop-loss limits enforce capital protection.
[Using Constraints in Market Risk Management]
1- Role of Risk Measures: Used to set operational limits for companies.
– Constraints define acceptable risk exposure and guide decision-making.
2- Tight Constraints: Lower risk but limit return potential.
– Companies may struggle to meet return objectives.
– Asset managers may fail to generate sufficient returns for investors.
3- Loose Constraints: Higher return potential but greater risk.
– Expected returns may increase.
– The probability of catastrophic losses rises.
Key Takeaways
– Effective risk management requires balancing constraints to optimize returns while mitigating excessive risk.
Quiz - [Use of Historical Bond Yields for Crisis Simulation]
1- Purpose of Using Historical Data for Analysis 1:
– McKee aims to evaluate the impact of a repeat of the last financial crisis on Flask’s bond portfolio.
– The appropriate data to use should best capture how bond values and risks responded to that past event.
2- Why Historical Yields Are the Best Choice:
– Bond yields drive bond pricing, making them the most important factor in risk assessment.
– Historical prices may not be available or relevant if the current bonds differ in maturity or characteristics from those traded during the crisis.
– Durations change over time, making them unreliable for accurately simulating past events.
3- Explanation of Incorrect Choices:
– Historical prices (A) are misleading, as price movements alone do not reflect the true yield environment or credit conditions.
– Historical durations (C) are not relevant, since durations change as bonds age, making them unsuitable for comparing past and present risks.
Key Takeaways:
– Yields should be used because they directly influence bond valuation and risk measures.
– Bond prices and durations are not reliable indicators for stress-testing current holdings under past crisis conditions.
Quiz - [Estimating 1% Annual VaR for GEF-B]
1- Understanding Value at Risk (VaR) Calculation
– VaR estimates potential portfolio loss under normal market conditions over a specified period.
– The 1% VaR represents the loss threshold that will be exceeded with a 1% probability in a year.
– Key inputs:
— Expected portfolio return (E(Rp))
— Portfolio standard deviation (σRp)
— Z-score for 1% confidence level (-2.33)
2- Given Portfolio Composition and Inputs
– GEF-B portfolio allocation:
— 30% IEF (International Equity Fund)
— 70% DHB (Domestic High Beta Equity Fund)
– Expected return: E(Rp) = 12.62%
– Portfolio standard deviation: σRp = 22.25%
3- Applying the VaR Formula
– Formula: “VaR = E(Rp) - Z × σRp”
– Substituting values:
— “VaR = 12.62% - (2.33 × 22.25%)”
— “VaR = 12.62% - 51.62%”
— “VaR = -39.22%”
– Interpretation: In 1% of years, losses of at least 39.2% are expected.
Key Takeaways
– VaR estimates loss at a specified confidence level using return and volatility.
– Portfolio risk assessment must account for weighted asset contributions and correlations.
– GEF-B’s 1% annual VaR is -39.2%, meaning a 1% chance of a loss of 39.2% or worse in a given year.
9.6 Backtesting and Simulation
– Describe objectives in backtesting an investment strategy.
– Describe and contrast steps and procedures in backtesting an investment strategy.
– Interpret metrics and visuals reported in a backtest of an investment strategy.
– Identify problems in a backtest of an investment strategy.
– Evaluate and interpret a historical scenario analysis.
– Contrast Monte Carlo and historical simulation approaches.
– Explain inputs and decisions in simulation and interpret a simulation.
– Demonstrate the use of sensitivity analysis.
[Methods for Evaluating Investment Strategies]
1- Backtesting
– Assesses how a strategy would have performed if implemented in the past.
2- Historical (Scenario Analysis) Stress Testing
– Evaluates strategy performance under specific historical conditions.
3- Simulation
– Provides insights into strategy performance in hypothetical scenarios rather than historical ones.
4- Sensitivity Analysis
– Measures the impact of changing assumptions on a strategy’s performance.
[Objectives of Backtesting]
1- Purpose of Backtesting: Used by investment managers to evaluate how strategies would have performed using historical data.
2- Key Assumption: It assumes that past market conditions will resemble future conditions, which may not always be accurate.
3- Role of Randomness: To improve reliability, randomness should be incorporated to account for future uncertainty.
4- Complementary Techniques: Backtesting is most effective when combined with other evaluation methods.
[Backtesting Process]
1- Strategy Design: Establish the investment strategy, including asset selection and trading rules.
2- Historical Investment Simulation: Apply the strategy to past market data to assess its performance.
3- Analysis of Backtesting: Evaluate results, identify biases, and refine the strategy based on findings.
Key Takeaways
– Backtesting helps assess how an investment strategy would have performed using historical data.
– The process involves designing the strategy, running simulations, and analyzing outcomes.
– Proper validation ensures that results are not driven by overfitting or unrealistic assumptions.
[Strategy Design]
1- Investment Hypothesis and Goals: Establishing a clear investment objective, typically to outperform a benchmark on a risk-adjusted basis.
2- Investment Rules and Processes: Defining strategies for overweighting or underweighting stocks based on valuation metrics, such as identifying undervalued stocks.
3- Key Parameters Selection: Choosing relevant metrics like price-to-book ratio to construct a portfolio, potentially incorporating short positions in overvalued stocks.
4- Investment Universe: Determining the range of securities considered, often represented by a broad market index like the TSX for Canadian equity managers.
Key Takeaways
– Strategy design is crucial for active managers aiming to outperform benchmarks.
– Investment rules and processes help ensure disciplined decision-making.
– Portfolio construction is guided by pre-determined financial metrics and investment universes.
[Strategy Design]
1- Return Definition: Determining whether returns should be measured as price return, total return, or adjusted for fees. Consideration is given to currency conversion for multinational portfolios and hedging foreign exchange exposure.
2- Rebalancing Frequency and Transaction Costs: Evaluating the impact of rebalancing on strategy performance, particularly in portfolios using valuation-based approaches like low P/E strategies. Frequent rebalancing increases transaction costs, affecting net returns.
3- Start and End Date: Selecting an appropriate historical period for backtesting to ensure the results are meaningful. A longer period provides more reliability, but non-stationarity in financial data necessitates analyzing different economic conditions, such as inflationary periods.
Key Takeaways
– Return definitions influence portfolio performance assessment, particularly for international investments.
– Rebalancing frequency and trading costs must be factored into backtesting to avoid overstating returns.
– Historical data selection should consider different market conditions to ensure strategy robustness.
Historical Investment Simulation
1- Purpose of Historical Investment Simulation
– The second step in backtesting involves constructing a portfolio based on predefined rules and simulating its historical performance.
– This process includes rebalancing at specified intervals and accounting for transaction costs.
2- Consideration of Portfolio Constraints
– Portfolio constraints such as size, liquidity, geographic limitations, and short-selling ability must be considered.
– The backtested portfolio should reflect the constraints applicable to future trading conditions.
3- Testing with Out-of-Sample Data
– Once a portfolio is established, it is tested using out-of-sample data to assess its performance.
– Example: A portfolio of the 30 lowest P/E stocks from January to December 2016 can be tested using January 2017 data.
— January 2017 data initially serves as out-of-sample data.
— It later becomes in-sample data for the next test, which uses February 2017 data as the new out-of-sample period.
4- Rolling Window Backtesting
– This iterative process, known as rolling window backtesting, allows multiple testing periods by continuously shifting the in-sample and out-of-sample data.
Key Takeaways
– Historical investment simulation validates portfolio strategies by using historical data.
– Portfolio constraints must be incorporated for realistic results.
– Rolling window backtesting enables multiple testing periods, improving robustness.
[Analysis of Backtesting Output]
1- Importance of Risk-Adjusted Metrics
– Beyond portfolio returns, analysts should consider risk-adjusted measures such as the Sharpe ratio and the Sortino ratio.
– Maximum drawdown analysis helps assess the worst loss a portfolio experiences after reaching its peak value.
– Skewness and kurtosis of return distributions should also be evaluated.
2- Use of Cumulative Performance Graphs
– Analysts should use cumulative performance graphs over extended periods to identify downside risk and performance decay.
– Backtested portfolios may initially outperform benchmarks before experiencing performance deterioration.
3- Identifying Structural Breaks
– Cumulative performance graphs help detect structural breaks due to economic crises, geopolitical events, monetary policy shifts, and technological advances.
– A logarithmic scale for the y-axis is recommended to ensure equal percentage changes have consistent vertical representation.
Key Takeaways
– Risk-adjusted metrics provide deeper insights into portfolio performance beyond returns.
– Cumulative performance graphs help analyze risk, decay, and structural shifts in performance.
– Using a logarithmic scale improves the clarity of performance trends over time.
While it is possible to backtest a portfolio based on a single parameter (e.g., earnings yield), backtested portfolios usually involve a combination of multiple factors and rules. A multifactor portfolio may give equal weight to each of its parameters. Alternatively, it may be preferable to use a risk-based weighting scheme that includes factors according to their contribution to the portfolio’s total risk.
[Strategy Design]
1- Multifactor Portfolio Composition
– The multifactor portfolio consists of eight long-short sub-portfolios.
– Example: The defensive value factor sub-portfolio goes long on the top-quintile stocks and shorts the lowest-quintile stocks based on the factor metric.
2- Portfolio Weighting Schemes
– Two portfolios are created using different weighting approaches:
— Benchmark (BM) portfolio: Assigns equal weights to all sub-portfolios.
— Risk parity (RP) portfolio: Assigns weights to equalize each factor’s risk contribution.
– More volatile sub-portfolios receive lower weights in the RP portfolio.
– While sub-portfolios take short positions, the benchmark portfolio remains long-only.
[Historical Investment Simulation]
1- Rolling Window Backtesting for Multifactor Portfolios
– The same rolling window backtesting approach used for single-factor portfolios applies to multifactor portfolios.
– The RP portfolio requires monthly re-estimation of the covariance matrix to optimize risk weights.
[Analysis of Backtesting Output]
1- Performance Comparison of BM and RP Portfolios
– The equally weighted BM portfolio achieves higher absolute returns.
– The RP portfolio exhibits lower volatility and a higher Sharpe ratio.
2- Skewness and Kurtosis Differences
– The BM portfolio is negatively skewed, while the RP portfolio is slightly positively skewed with lower kurtosis.
– The RP portfolio’s lower volatility relative to individual factors highlights diversification benefits.
Key Takeaways
– Multifactor portfolios consist of long-short sub-portfolios, with weighting schemes affecting risk-return characteristics.
– Rolling window backtesting applies to multifactor portfolios, with RP portfolios requiring covariance re-estimation.
– The RP portfolio offers better risk-adjusted returns through diversification, despite lower absolute returns.
[Survivorship Bias]
1- Definition and Impact
– Over time, companies are removed from indexes, and new ones are added.
– Example: More than 2600 stocks were removed from the Russell 3000 index over 20 years, leaving only around 400 survivors.
– Focusing only on surviving stocks ignores the uncertainty of which companies would remain in the index.
2- Avoiding Survivorship Bias
– A point-in-time approach is recommended to include stocks that met criteria at the time, even if they later dropped out.
– Backtesting a strategy using only surviving stocks distorts results and is not recommended.
– Many analysts fail to properly account for the impact of survivorship bias.
3- Misleading Conclusions from Survivorship Bias
– Data from surviving firms often reflects only the most successful and innovative companies.
– A point-in-time approach shows that low-volatility stocks outperform high-volatility stocks in the long run.
– However, analyzing only surviving firms falsely suggests that high-volatility stocks performed better.
Key Takeaways
– Survivorship bias skews historical analysis by ignoring failed companies.
– A point-in-time approach ensures backtests include all eligible stocks, not just survivors.
– Misinterpreting survivorship-biased data can lead to incorrect investment conclusions.
[Impact of Survivorship Bias on Backtesting]
1- Backtesting Using Only Survivors
– When analyzing only surviving stocks, high-volatility stocks appear to outperform low-volatility stocks.
– The highest-volatility quintile (Survivor Universe) shows significant cumulative wealth growth compared to the lowest-volatility quintile.
– This misleading result suggests greater rewards for taking on risk.
2- Backtesting Using Point-in-Time Data
– A more accurate approach includes all stocks available at the time, not just survivors.
– The highest-volatility quintile (Russell 3000) underperforms and experiences sharp declines, while the lowest-volatility quintile shows superior long-term performance.
– This reveals that many high-volatility stocks do not survive, leading to survivorship bias in the first approach.
Key Takeaways
– Backtesting with only survivors overstates the performance of high-volatility stocks.
– A point-in-time approach shows that low-volatility stocks tend to outperform over the long run.
– Survivorship bias can lead to incorrect investment conclusions by excluding failed stocks from analysis.
[Look-Ahead Bias]
1- Definition and Causes
– Look-ahead bias occurs when backtesting relies on information that was unknowable at the time.
– Example: Using January returns for companies with low year-end P/E ratios, despite earnings being reported in February, creates look-ahead bias due to reporting lags.
– Survivorship bias is a specific form of look-ahead bias.
2- Sources of Look-Ahead Bias
– Data revisions: Vendors update macroeconomic and company-level data after revisions, introducing hindsight bias.
– Backfilled data: New companies added to indexes or databases may include historical data that was unavailable during the original period.
– Although accurate databases are useful, backtesting should only use information available at the time investment decisions would have been made.
3- Mitigating Look-Ahead Bias
– Analysts can extend reporting lags to ensure only contemporaneous information is used.
– Example: Adjusting P/E metrics to exclude data unavailable at the time of backtesting.
– However, applying a uniform reporting lag can result in stale data, requiring a balance between accuracy and timeliness.
Key Takeaways
– Look-ahead bias occurs when backtests use future or unavailable information.
– Data revisions and backfilled data are major contributors to this bias.
– Extending reporting lags can help, but excessive lags may reduce data relevance.
[Common Forms of Look-Ahead Bias]
1- Reporting Lags
– Occurs when backtesting uses financial statement data that was not yet available at the time.
– Example: A portfolio is rebalanced in February using prior-year financial data, but the data was only publicly released in March.
– Solution: Adjust reporting lags to reflect actual data availability.
2- Data Revisions
– Macroeconomic data and financial statements are often revised multiple times.
– Many databases only keep the latest figures, leading to the use of updated (but unavailable) data in backtesting.
– Example: Backtests relying on restated financials introduce bias since original figures were different.
– Solution: Use point-in-time data to ensure only contemporaneous information is included.
3- Index Additions
– New companies added to an index may have historical data backfilled into databases.
– Using this data in backtesting implies knowledge of companies that were not in the index at the time.
– Solution: Exclude backfilled data for companies not in the index during the original period.
Key Takeaways
– Look-ahead bias occurs when backtesting incorporates information unavailable at the time.
– Reporting lags, data revisions, and index additions are primary sources of this bias.
– Using point-in-time data and adjusting for reporting delays helps mitigate look-ahead bias.
[Data Snooping]
1- Definition and Causes
– Also known as p-hacking, data snooping occurs when analysts select strategies based on statistical significance rather than prior inferences.
– It is a form of selection bias that distorts backtesting results.
2- Common Data Snooping Practices
– Waiting to review interim results before deciding whether to continue data collection.
– Excluding outliers only after completing the analysis.
– Using many independent variables without theoretical justification and selecting them based on observed results.
3- Methods to Combat Data Snooping
– Using a higher t-value to determine statistical significance for independent variables.
– Cross-validation with out-of-sample data to confirm results; rolling window backtesting serves as a cross-validation method.
– Testing strategies across different geographic markets to improve robustness.
Key Takeaways
– Data snooping introduces bias by selecting strategies after observing statistical outcomes.
– Common mistakes include selective outlier exclusion and overfitting to historical data.
– Cross-validation and robust statistical thresholds help mitigate data snooping risks.
[Data Snooping]
1- Definition and Process
– Data snooping occurs when analysts backtest multiple strategies and select the best-performing one without prior inference.
– This replaces sound portfolio construction with a purely statistical selection approach.
– Example: Making an inference only after analyzing statistical results instead of testing a prior hypothesis.
2- Problem with Data Snooping
– The ultimate results often produce false positives.
– A strategy may appear successful in backtesting, but its performance could be due to random chance rather than genuine predictive power.
3- Solution to Data Snooping
– Strategies should be tested based on prior inferences rather than selecting models based purely on historical performance.
– Cross-validation and robustness testing help mitigate the risks of overfitting to past data.
[Structural Breaks in Economic Conditions]
1- Key Structural Breaks to Consider
– Expansions vs. Recessions: Changes in the economic cycle that can significantly impact market performance.
– High-volatility vs. Low-volatility Regimes: Periods of increased or decreased market volatility affect portfolio returns.
2- Scenario Analysis and Its Importance
– Historical scenario analysis reveals deeper insights into return distributions through probability density plots.
– These plots provide more information than stand-alone metrics like the Sharpe ratio.
3- Portfolio Performance During Structural Breaks
– The RP portfolio maintains a stable Sharpe ratio during both recessions and economic expansions, as well as during periods of high and low volatility.
– The BM portfolio’s Sharpe ratio significantly decreases during recessions and, to a lesser extent, during low-volatility periods.
– BM portfolio returns are more volatile, negatively skewed, and show higher kurtosis during recessions.
Key Takeaways
– Structural breaks, such as economic expansions and recessions, heavily influence portfolio performance.
– Scenario analysis using probability density plots provides more insight than traditional metrics.
– The RP portfolio proves more resilient across different economic and volatility regimes compared to the BM portfolio.
[Historical Simulation and Monte Carlo Simulation]
1- Historical Simulation
– A non-deterministic form of rolling window backtesting commonly used by banks.
– Relies on historical data but fails to adjust for changing market conditions.
2- Monte Carlo Simulation
– Introduces randomness to key variables to address limitations of historical data.
– Allows incorporation of factors such as non-normality, excess kurtosis, and tail dependence.
– More flexible but computationally intensive.
3- Steps in a Properly Designed Simulation
– 1- Determine the target variable, typically the portfolio return.
– 2- Specify key decision variables, such as asset returns and portfolio weights.
– 3- Define the number of trials, usually between 1,000 and 10,000, for stability.
– 4- Establish the distributional properties of key variables (e.g., normal, lognormal, binomial).
– 5- Generate random numbers for each key decision variable.
– 6- Compute the target variable for each simulated set.
– 7- Repeat Steps 5 and 6 until all trials are complete.
– 8- Calculate performance metrics such as mean return, volatility, Sharpe ratio, VaR, and CVaR.
Key Takeaways
– Historical simulation is simple but does not adapt to changing market conditions.
– Monte Carlo simulation provides flexibility but requires significant computational power.
– Proper simulation design follows structured steps to improve reliability.
[Historical Simulation]
1- Overview of Historical Simulation
– Unlike backtesting, historical simulation is non-deterministic, incorporating randomness in selecting historical returns.
– A key methodological choice is whether to use bootstrapping (sampling with replacement) or sampling without replacement.
– Bootstrapping is common when the number of simulations exceeds the number of historical observations.
2- Steps for BM and RP Portfolios
– 1- Define target variables: Returns for the BM and RP portfolios.
– 2- Identify key decision variables: Returns for the 8 underlying portfolios (with predefined weights).
– 3- Set the number of trials (N = 1,000).
– 4- Assign probabilities to each of the 374 historical months and use a random number generator to select which month’s return will be used for each sub-portfolio.
– 5- If a random number corresponds to a specific month, that month’s returns are used for all sub-portfolios.
– 6- Calculate portfolio returns:
— BM portfolio: Uses equally weighted returns from the 8 sub-portfolios.
— RP portfolio: Uses weighted returns based on final-month factor allocations.
– 7- Repeat Steps 5 and 6 until 1,000 trials are completed.
– 8- Use the 1,000 generated returns to calculate performance metrics.
3- Key Findings
– Results align with rolling window backtesting.
– The RP portfolio has a higher Sharpe ratio and lower downside risk than the BM portfolio.
Key Takeaways
– Historical simulation introduces randomness, improving robustness over traditional backtesting.
– Bootstrapping ensures sufficient sample size when historical data is limited.
– The RP portfolio exhibits superior risk-adjusted returns compared to the BM portfolio.
[Rolling Window Backtesting: How It Works]
1- Definition and Purpose
– Rolling window backtesting is a deterministic method that evaluates investment strategies by applying them to historical data in a structured, time-sequential manner.
– It ensures that only information available at a given point in time is used, making it more realistic than traditional full-period backtesting.
2- How the Rolling Window Method Works
– Step 1: Define a fixed-length window (e.g., 12 months, 3 years) for testing the strategy.
– Step 2: Apply the strategy to the initial window and record the performance.
– Step 3: Shift the window forward by a fixed period (e.g., one month, one quarter).
– Step 4: Repeat steps 2 and 3 until the dataset is fully covered.
3- Key Characteristics
– Sequential Analysis: The window moves forward in time while maintaining chronological order.
– Overlapping Data: Depending on the step size, previous data may be included in multiple test periods.
– Consistency in Testing: Helps evaluate strategy performance across different market conditions.
4- Example of Rolling Window Backtesting
– Suppose an investor wants to backtest a stock-selection strategy using a 12-month rolling window:
— The strategy is tested on data from January 2010 – December 2010.
— The window shifts forward one month, now covering February 2010 – January 2011.
— This process repeats until the dataset is fully tested, ensuring multiple independent performance evaluations.
5- Advantages and Limitations
– Advantages:
— Reduces look-ahead bias by maintaining a time-consistent approach.
— Provides multiple performance snapshots instead of a single backtest.
— Helps identify performance stability across different periods.
– Limitations:
— May not work well for short datasets with limited historical data.
— Computationally intensive when using small step sizes.
Key Takeaways
– Rolling window backtesting maintains a deterministic, structured approach to strategy evaluation.
– It systematically shifts a fixed-length period forward, ensuring realistic historical testing.
– This method is especially useful for assessing strategy consistency and robustness over time.
[Deterministic vs. Non-Deterministic Methods]
1- Deterministic Methods
– Follow a fixed, structured process where outcomes are entirely determined by initial conditions.
– Given the same inputs, they always produce the same results.
– Example: Rolling-window backtesting assumes asset returns follow their historical order, maintaining a sequential structure.
2- Non-Deterministic Methods
– Incorporate randomness or variability in their process, allowing for different possible outcomes.
– Can produce different results even with the same initial inputs.
– Example: Historical simulation randomly selects past data points to create future scenarios.
Quiz - [Backtesting Methods for Lukowich’s Investment Hypothesis]
1- Criteria for Selecting a Backtesting Method
– Lukowich prefers a non-deterministic method, assuming that past asset returns are indicative of future returns.
– Both historical simulation and rolling-window backtesting rely on past data but differ in their approach.
2- Key Differences Between Historical Simulation and Rolling-Window Backtesting
– Historical Simulation (Non-Deterministic):
— Randomly selects past historical data points to construct future return scenarios.
— Introduces variability, making each simulation potentially different.
– Rolling-Window Backtesting (Deterministic):
— Uses a strict chronological sequence of returns, maintaining their historical order.
— Results remain the same if inputs are unchanged.
3- Most Appropriate Method for Lukowich
– Historical simulation aligns best with Lukowich’s preference for a non-deterministic approach.
– Rolling-window backtesting, while useful, is deterministic and does not allow for randomness in return selection.
Key Takeaways
– Historical simulation is non-deterministic and allows for random selection of past returns, making it better suited for Lukowich’s approach.
– Rolling-window backtesting is deterministic, maintaining a strict sequence of historical returns.
[Historical Scenario Analysis]
1- Purpose of Historical Scenario Analysis
– Evaluates the performance and risk of an investment strategy under different structural breaks.
– Provides insights beyond traditional backtesting by considering real-world economic shifts.
2- Common Structural Breaks Considered
– 1- Transitions from economic expansions to recessions.
– 2- Shifts from high-volatility to low-volatility regimes.
3- Comparison with Other Evaluation Methods
– Complements backtesting, simulation, and sensitivity analysis by identifying potential weaknesses in a strategy.
– Helps assess how strategies perform under extreme market conditions.
Key Takeaways
– Historical scenario analysis tests investment strategies under major economic shifts.
– It identifies risks and weaknesses that may not be evident in standard backtesting.
– Structural breaks such as recessions and volatility changes are crucial considerations for robust strategy evaluation.
Quiz- [Best Method to Communicate Historical Scenario Analysis Results]
1- Moore’s Hypothesis and Methodology
– Moore hypothesizes that her four-factor model will generate positive excess risk-adjusted returns during periods of low inflation.
– She conducts historical scenario analysis to evaluate portfolio performance across different inflation regimes.
2- Why Overlapping Probability Density Plots?
– Reveals distribution of returns: Enables direct comparison of return distributions between high- and low-inflation periods.
– Highlights return characteristics: Shows differences in mean returns, volatility, and skewness, which are crucial for risk-adjusted performance assessment.
– Effectively communicates historical scenario results: Provides a clear statistical visualization of return variations across different time periods.
3- Why Other Graphs Are Less Effective?
– Logarithmic Scale for Returns (Incorrect Choice): Helps track performance decay over time but does not facilitate comparison across different market regimes.
– Cumulative Performance Graph (Incorrect Choice): Useful for identifying performance trends but does not offer insights into return distributions across inflation regimes.
Key Takeaways
– Overlapping probability density plots are the most effective tool for visualizing historical scenario analysis results.
– These plots highlight critical statistical properties of returns under different economic conditions.
– While cumulative performance graphs and logarithmic scales have their uses, they are not ideal for comparing return distributions across macroeconomic environments.
[Monte Carlo Simulation]
1- Overview of Monte Carlo Simulation
– Similar to historical simulation but differs in specifying a functional form for statistical distributions.
– Focuses on capturing correlations between key decision variables rather than treating them in isolation.
2- Importance of Distribution Assumptions
– The effectiveness of Monte Carlo simulation depends on how well the statistical distribution represents actual data.
– Analysts must analyze metrics such as mean, standard deviation, skewness, kurtosis, and tail dependence to refine model assumptions.
3- Handling Multiple Decision Variables
– A multivariate normal distribution is typically used to model correlations between decision variables.
– Example: A portfolio with K assets requires K means, K standard deviations, and (K × (K - 1)) ÷ 2 correlations.
– However, this approach may not capture negative skewness, excess kurtosis, or tail dependence in financial data.
4- Model Complexity and the Bias-Variance Trade-Off
– Complex models have low specification errors but high estimation errors.
– Overly simplistic models reduce estimation errors but may not fit real data well.
– This aligns with the bias-variance trade-off concept in quantitative methods.
5- Monte Carlo Simulation Steps
– 1- Define target variables: Returns for BM and RP portfolios.
– 2- Use returns of the 8 sub-portfolios as key decision variables.
– 3- Set number of simulations (N = 1,000).
– 4- Specify the multivariate normal distribution as the functional form, calibrated with 8 mean returns, 8 standard deviations, and 28 covariance values.
– 5- Simulate factor returns by generating random numbers mapped onto a joint cumulative probability function.
– 6- Use the same factor weightings for BM and RP portfolios as in historical simulation.
– 7- Repeat Steps 5 and 6 until 1,000 trials are completed.
– 8- Compute and summarize simulation results.
6- Key Observations from Monte Carlo Simulation
– Results align with backtesting and historical simulation for the Sharpe ratio.
– The multivariate normal assumption in Monte Carlo may understate the BM portfolio’s downside risk.
Key Takeaways
– Monte Carlo simulation enhances analysis by modeling correlations between variables.
– Proper distributional assumptions are critical for accuracy.
– While flexible, Monte Carlo may fail to capture tail risk and non-normal characteristics in financial data.
[Simulation Design – Addressing Factor Correlation in Monte Carlo Modeling]
1- Context of Concern 1
– Concern 1: Returns from six of the nine factors are correlated.
– Approach 2: Monte Carlo simulation is being used to model investment performance.
2- Proper Simulation Design Adjustment
– In Monte Carlo simulations, if returns from multiple factors are correlated, specifying a multivariate distribution is required.
– Modeling each factor independently ignores the correlations, leading to invalid simulations.
3- Correct Methodological Step
– To address the correlation in factor returns, Yuen should specify a multivariate distribution rather than modeling each factor on a standalone basis.
4- Why Other Options Are Incorrect
– A is incorrect: Modeling each factor separately violates the assumption of joint return distribution and omits correlation structure.
– B is incorrect: Only computing 15 covariance terms is insufficient; the full covariance matrix for all 9 factors (including variances and all covariances) requires 36 elements.
Key Takeaways
– A multivariate distribution accounts for the interdependencies across factors.
– Correct calibration requires means, variances, and full covariance structure of all factors.
Quiz - [Assessing Eichel’s Statements on Monte Carlo Simulation]
1- Importance of Exploratory Data Analysis in Monte Carlo Simulation
Eichel’s Statement 1 is incorrect because exploratory data analysis is essential before performing a Monte Carlo simulation.
– Monte Carlo methods rely on assumptions about how key variables are distributed.
– If a variable exhibits significant skewness or non-normality, running a Monte Carlo simulation without adjusting for this can lead to misleading results.
2- Monte Carlo Simulation Requires Distributional Assumptions
Eichel’s Statement 2 is also incorrect because Monte Carlo simulation does not eliminate the need for distributional assumptions.
– True data distributions cannot be directly observed; instead, Monte Carlo requires specifying functional forms for these distributions.
– Errors in these assumptions can produce misleading or biased results.
Key Takeaways
– Exploratory data analysis is critical before Monte Carlo simulation to determine the appropriate distributional assumptions.
– Monte Carlo does require assumptions about data distributions, meaning errors in these assumptions can distort results.
[Sensitivity Analysis]
1- Purpose of Sensitivity Analysis
– Estimates how changes in input variables affect the target variable.
– Helps assess whether a model calibrated to a specific distribution, such as a multivariate normal, underestimates risk.
2- Importance of Distribution Assumptions
– The multivariate normal distribution is commonly used but does not capture negative skew, excess kurtosis, or tail dependence.
– A Student’s t-distribution accounts for these factors but requires estimating more parameters.
– Proper model calibration should consider non-normality when appropriate.
3- Application in Portfolio Analysis
– Sensitivity analysis follows the same steps as simulation analysis but assumes a multivariate skewed Student’s t-distribution.
– The Sharpe ratio calculations appear robust to the choice of distribution.
– However, sensitivity analysis, like simulations, suggests the BM portfolio’s downside risk may be understated.
– Additional sensitivity tests with alternative functional forms are recommended.
Key Takeaways
– Sensitivity analysis helps determine how model assumptions impact results.
– Accounting for non-normality improves the accuracy of risk assessment.
– Testing multiple distributional assumptions ensures robustness in portfolio evaluation.
– Historical scenario analysis is justified when datasets experience structural breaks, making past relationships unreliable.
– Point-in-time data and return distribution characteristics do not determine the need for scenario analysis.
[Why Not Just Use Normal Distributions?]
1- Limitations of Normal Distributions in Finance
– Many models assume financial data follows a normal distribution (bell curve).
– However, real-world data includes extreme events (e.g., market crashes) that normal distributions fail to capture.
2- Advantages of Student’s t-Distribution
– Accounts for fat tails, meaning extreme events occur more frequently.
– Considers asymmetry, capturing skewness in financial data.
– More complex but provides a better representation of risk.
3- Key Findings from Sensitivity Analysis
– Switching to a different distribution did not impact Sharpe ratios.
– Sensitivity analysis revealed that risk was still being underestimated.
– Suggests that testing multiple models improves risk assessment.
Key Takeaways
– Normal distributions may underestimate financial risk due to their inability to capture extreme events.
– Student’s t-distribution provides a more realistic representation of financial data.
– Using multiple models helps refine risk estimation and improve decision-making.
[Simulation Testing Method – Sensitivity Analysis of Factor Return Distribution]
1- Context of the Scenario
– Ruckey proposes using a multivariate skewed Student’s t-distribution to simulate 1,000 factor return scenarios for Factor 1.
– The goal is to assess how the distributional characteristics of Factor 1 affect the performance of Portfolios A and B under Approach 2 (Monte Carlo simulation).
2- Definition of Sensitivity Analysis
– Sensitivity analysis is used to assess how changes in key input variables (e.g., distributional assumptions) impact model outputs (e.g., portfolio return distributions).
– In this context, it tests the impact of non-normal return characteristics such as skewness and excess kurtosis on the simulation outcomes.
3- Why This is Sensitivity Analysis
– The use of an alternative distribution (multivariate skewed Student’s t-distribution) is intended to isolate the effect of a single factor’s distribution shape.
– It helps determine whether the assumption of multivariate normality significantly affects the results of the simulation.
4- Incorrect Answer Rationales
– A is incorrect: Data snooping refers to repeated testing or mining of historical data to find patterns, often leading to overfitting—not applicable here.
– C is incorrect: Inverse transformation refers to a statistical technique for returning data to its original scale—not related to the process being applied.
Quiz - [Justification for Using Historical Scenario Analysis]
1- Why Historical Scenario Analysis?
– Useful for evaluating investment strategies across different structural regimes, such as high and low inflation periods.
– Commonly applied when datasets contain structural breaks, such as changes in central bank monetary policy.
2- Moore’s Dataset and Non-Stationarity
– Spans 20 years and includes a structural break where the central bank reduced its inflation target from 3.5% to 2.0%.
– Structural breaks cause past relationships between variables to change, leading to non-stationary datasets.
– Non-stationarity makes traditional time-series models less reliable.
3- Misinterpretation of Other Factors
– Point-in-time data (Incorrect Justification): Ensures data integrity but does not address structural breaks.
– Skewed Student’s t-distribution (Incorrect Justification): Affects return modeling but does not justify scenario analysis.
Key Takeaways
– Historical scenario analysis is justified when datasets experience structural breaks, making past relationships unreliable.
– Point-in-time data and return distribution characteristics do not determine the need for scenario analysis.
– Best used for evaluating strategy performance under changing macroeconomic conditions.
[Methods to Communicate Results and Their Applications]
1- Overlapping Probability Density Plots
– Purpose: Compare return distributions across different scenarios.
– Best for: Historical Scenario Analysis.
– Key Insights: Highlights differences in mean returns, volatility, skewness, and kurtosis.
– Example Use Case: Comparing returns under high- and low-inflation regimes.
2- Cumulative Performance Graphs
– Purpose: Track portfolio performance over time.
– Best for: Backtesting and Rolling Window Backtesting.
– Key Insights: Shows performance trends, drawdowns, and long-term growth.
– Example Use Case: Evaluating strategy consistency across different market periods.
3- Logarithmic Scale Graphs
– Purpose: Maintain proportionality of returns for better visualization.
– Best for: Long-term Backtesting and Performance Decay Analysis.
– Key Insights: Ensures equal percentage returns appear equidistant, making trends clearer.
– Example Use Case: Tracking performance decay and compounding effects.
4- Box Plots
– Purpose: Summarize return distribution characteristics (median, quartiles, outliers).
– Best for: Monte Carlo Simulation and Sensitivity Analysis.
– Key Insights: Displays the spread of simulated returns and highlights extreme outcomes.
– Example Use Case: Analyzing tail risk in Monte Carlo-generated return distributions.
5- Scenario Tables
– Purpose: Present numerical summaries of key performance metrics.
– Best for: Historical Scenario Analysis and Sensitivity Analysis.
– Key Insights: Provides structured comparisons of risk-adjusted returns, drawdowns, and Sharpe ratios.
– Example Use Case: Showing portfolio performance under different volatility or inflation regimes.
Key Takeaways
– Different result communication methods are suited for specific analytical approaches.
– Historical Scenario Analysis benefits most from probability density plots and scenario tables.
– Monte Carlo and Sensitivity Analysis require visualization of return distributions, making box plots effective.
– Long-term backtesting and performance analysis are best visualized using cumulative performance graphs and logarithmic scales.
Quiz - [Backtesting Methodology for Lukowich’s Investment Hypothesis]
1- Choosing the Appropriate Investment Universe
When backtesting an investment strategy, the investment universe refers to the range of securities that can be included in the analysis. For Lukowich’s hypothesis—that Canadian value stocks outperform their peers—a broad market index of Canadian equities is the most appropriate universe because:
– It includes both value and non-value stocks, providing a basis for comparison.
– It ensures that results are not biased by an overly narrow selection of stocks.
2- Currency Considerations
Lukowich’s fund hedges foreign currency risk, meaning returns should be measured in local currency terms (Canadian dollars) rather than in EUR.
– Since currency risk is eliminated through hedging, defining returns in EUR terms would not be relevant.
– If the fund did not hedge currency risk, then returns should be expressed in EUR to reflect potential exchange rate impacts.
3- Transaction Cost Adjustments
While quarterly rebalancing instead of monthly rebalancing may reduce transaction costs, this is a secondary concern. Lukowich’s objective is to account for transaction costs, not necessarily minimize them through less frequent trading.
– The key is including transaction costs in the backtest, whether using monthly or quarterly data.
Key Takeaways
– A broad market index of Canadian equities is the best investment universe for testing Lukowich’s hypothesis.
– Returns should be measured in CAD (local currency) because currency risk is fully hedged.
– Accounting for transaction costs is important, but the frequency of rebalancing is not the primary concern.
