Fixed Income Flashcards
The discount function and spot yield curve (or spot curve) represent the discount factors and spot rates for a range of maturities. The same information can be derived from each.
The spot rate represents the annualized return on a zero-coupon bond that has no default risk and no embedded options. Because a zero-coupon bond does not produce any cash flows prior to maturity, there is no reinvestment risk.
Forward Rates and Forward Pricing Model
1- Forward Rates:
– Forward rates are interest rates agreed upon today for loans or deposits that will occur in the future. The set of these rates across different time horizons is called the forward curve.
– A forward rate for a deposit made at time “A” and maturing at time “B” is denoted as “f_A,B-A.”
– Example: If an investor plans to deposit money in 2 years for a period of 3 years, the forward rate is denoted as “f_2,3.”
2- Forward Pricing Model:
– The forward pricing model ensures that there is no arbitrage. It equates the return of an investor making a single deposit for a longer term to the return of rolling over shorter-term deposits.
– The forward price for a one-unit deposit made at time “A” and maturing at time “B” is derived using the discount factors “DF_A” and “DF_B”:
“F_A,B-A = DF_B / DF_A.”
– Example: The forward price of a one-year deposit starting in 4 years can be calculated using the ratio of discount factors for years 4 and 5.
DFn = [1 / (1 + Zn)^n]
Zn : Spot rate
n : Maturity
The Forward Rate Model
1- Forward Price Determination:
– The forward price “F_A,B-A” is determined using the forward rate “f_A,B-A”.
2- Formula (using forward rate):
– “F_A,B-A = 1 / (1 + f_A,B-A)^(B-A)”
3- Relation to Spot Rates (Forward Rate Model):
– The forward rate can also be defined by spot rates:
– “(1 + z_B)^B = (1 + z_A)^A * (1 + f_A,B-A)^(B-A)”
Forward Rate Model and Spot Rate Relationships
1- T-Year Spot Rate Function:
– The T-year spot rate, “z_T,” is a function of the one-year spot rate and successive one-year forward rates:
– “(1 + z_T)^T = (1 + z_1) * (1 + f_1,1) * (1 + f_2,1) * … * (1 + f_T-1,1)”
2- Geometric Mean Interpretation:
– The formula shows that the T-year spot rate is the geometric mean of the one-year spot rate and forward rates:
– “z_T = [(1 + z_1) * (1 + f_1,1) * (1 + f_2,1) * … * (1 + f_T-1,1)]^(1/T) - 1”
3- Spot and Forward Curve Relationship:
– Positive Slope (Normal Curve):
— A positively sloped spot curve means the forward curve will lie above the spot curve.
– Negative Slope (Inverted Curve):
— A negatively sloped spot curve will lie below the forward curve.
– Equal Rates:
— The spot and forward curves match only when rates are identical across all maturities.
4- Par Curve Representation:
– The par curve represents the yield to maturity on coupon-paying government bonds priced at par.
– Par curves are based on recently issued (“on-the-run”) bonds.
5- Bootstrapping Method:
– Bootstrapping is a technique used to derive the spot curve from the par curve.
Explanation of the Spot Curve, Forward Curve, and Par Curve
1- Spot Curve:
– The spot curve represents the yields to maturity of zero-coupon bonds (bonds with no interim coupon payments) at various maturities.
– Uses:
— It is used to calculate present values of cash flows by discounting them using the specific spot rate for the cash flow’s maturity.
— Helps determine the price of any fixed-income security through discounting.
– Key Characteristics:
— Each point on the curve reflects the yield for a zero-coupon bond of a specific maturity.
— Derived using bootstrapping from the par curve when zero-coupon bonds are not available in the market.
2- Forward Curve:
– The forward curve represents the expected future interest rates for specific time periods, as implied by today’s spot rates.
– Uses:
— Helps investors predict short-term rates and guide expectations for future monetary policy.
— Useful for pricing derivatives, such as forward rate agreements (FRAs) and swaps.
– Key Characteristics:
— Calculated based on the spot curve using the no-arbitrage principle.
— Shows implied future yields, not necessarily the rates that will actually occur.
— Example: A one-year forward rate two years from now represents the interest rate expected for the period starting two years from now and ending three years from now.
3- Par Curve:
– The par curve represents the yields to maturity of coupon-paying bonds that are priced at par.
– Uses:
— Used as a reference for constructing the spot curve through bootstrapping.
— Reflects yields of “on-the-run” bonds, which are newly issued and actively traded government securities.
– Key Characteristics:
— Each point on the curve reflects the yield for a par bond (coupon-paying bond priced at face value) of a specific maturity.
— Easier to observe in the market compared to spot and forward curves, making it a common starting point for deriving other curves.
Practical Insights for Investors:
Spot Curve: Provides the purest measure of interest rates for different maturities, crucial for bond valuation.
Forward Curve: Key for making expectations about future rates and market sentiment.
Par Curve: Offers real-world yields that are observable in the market and serves as the basis for deriving the spot curve.
Yield to maturity (YTM) is a commonly used pricing concept in bond markets. It can be thought of as a weighted average of the spot rates that are used to value a bond’s expected cash flows.
Assumes the curve is flat (dose not change)
Understanding Yield to Maturity (YTM)
1- Definition of YTM:
– Yield to maturity is the internal rate of return (IRR) for a bond, calculated by equating the bond’s current price with the present value of its expected future cash flows (coupon payments and principal repayment).
– It is essentially the rate of return an investor can expect if they:
— Hold the bond until maturity.
— Receive all payments (coupons and principal) on time.
— Reinvest all interim cash flows at the same rate as the YTM.
2- Key Assumptions Behind YTM:
– For investors to earn the YTM, three conditions must hold:
— The bond must be held to maturity—selling it before maturity introduces reinvestment and price risk.
— The bond issuer must pay all cash flows (coupons and principal) on time and in full—default risk would affect actual returns.
— All interim cash flows must be reinvested at the YTM—deviations in reinvestment rates can alter actual returns.
3- When YTM is a Poor Estimate of Expected Return:
– Interest Rate Volatility:
— If interest rates fluctuate, reinvestment at the original YTM becomes unrealistic.
– Steep Yield Curve:
— When the yield curve is sharply sloped (positively or negatively), reinvestment assumptions implied by YTM are inaccurate.
– High Default Risk:
— If there’s a significant chance of default, YTM overestimates expected return because it assumes full payment of all cash flows.
– Embedded Options in the Bond:
— Callable bonds, putable bonds, or other options can alter cash flows, making YTM an unreliable measure of return.
4- Limitations of YTM:
– YTM implicitly assumes a flat yield curve, meaning the same interest rate applies across all maturities. This is rarely the case in real markets where yield curves can be positively sloped (normal), negatively sloped (inverted), or humped.
5- Practical Implications for Investors:
– YTM is a useful measure for comparing bonds under stable market conditions, but investors should not rely solely on it when:
— Rates are expected to change.
— There is credit risk or embedded options.
– In such cases, alternative measures like the yield to call (YTC), option-adjusted yield (OAS), or scenario analysis may provide better insights.
Yield Curve Movement and the Forward Curve
If future spot rates evolve as predicted in today’s forward curve, forward contract prices will not change. In this scenario, all risk-free bonds will earn the current one-year spot rate over a one-year holding period, regardless of their maturity.
Forward prices change when the spot curve deviates from what is predicted in the current forward curve. Active bond managers can try to anticipate changes in interest rates relative to those implied by current forward rates. For example, a manager who expects that future spot rates will be less than what is implied by the current forward curve should buy the forward contract because it will increase in price.
How Forward Prices Change with the Spot Curve
1- Relationship Between Forward Prices and Spot Rates:
– Forward prices are based on current spot rates and the forward curve, which predicts future interest rates.
– If actual future spot rates deviate from the predictions in the current forward curve, forward prices will adjust to reflect the new expectations.
2- Example of Forward Price Adjustment:
– Suppose the current forward curve predicts that future spot rates will rise to 5%.
– If, in reality, future spot rates are expected to rise less than 5% (say, to 4%), the forward price of a bond will increase because the implied discount rate (based on the lower spot rates) decreases, raising the bond’s value.
3- How Active Bond Managers Use This Information:
– Active bond managers analyze forward curves to anticipate whether actual future spot rates will differ from the rates implied in the forward curve.
– For Example:
— If a manager expects future spot rates to be lower than what the forward curve predicts, they will buy the forward contract now.
— As forward prices increase due to the unexpected drop in spot rates, the manager can profit from the price appreciation.
4- Summary of Strategy:
– The key idea is that if actual future interest rates (spot rates) are lower than those implied by the forward curve, forward contracts become more valuable.
– Conversely, if spot rates are higher than expected, forward contracts decrease in value.
– Managers leverage these deviations to make profitable trades based on their predictions of future rate movements.
Components of Total Return for Fixed-Rate, Option-Free Bonds:
1- Receipt of Coupons:
– The periodic coupon payments that bondholders receive.
2- Return of Principal:
– The face value of the bond, repaid at maturity.
3- Reinvestment of Coupons:
– The additional returns earned by reinvesting the coupon payments.
4- Capital Gains/Losses on Sales Before Maturity:
– If the bond is sold before maturity, the price difference from its purchase price results in a gain or loss.
Carry Trade:
1- General Definition:
– Borrowing in a low-yielding currency (or market) and investing in a higher-yielding currency (or market) to capture the yield difference.
2- Maturity Spread Carry Trade in Bond Markets:
– A type of carry trade based on the expectation that future interest rates will remain stable or rise less than implied by the spot curve.
3- How It Works:
– Borrow short-term at lower interest rates.
– Invest in longer-term bonds with higher yields.
4- Profitability Conditions:
– The strategy is profitable as long as short-term interest rates do not rise sharply.
5- Risk of Maturity Spread Carry Trade:
– Vulnerable to a spike in short-term interest rates, which can increase borrowing costs and lead to losses.
Riding the Yield Curve:
1- Definition:
– A trading strategy used when the yield curve is positively sloped.
2- Mechanism:
– Traders buy bonds with longer maturities and hold them as they “roll down the yield curve.”
– As time passes, the bond’s remaining maturity shortens, and its price rises because shorter-maturity bonds typically have lower yields (higher prices).
3- Profit Opportunity:
– The trader earns returns from the price increase (capital gains) as the bond rolls down the curve.
– The strategy is profitable when the yield curve maintains its current shape and slope.
4- Best Conditions for This Strategy:
– A positively sloped yield curve where the forward rates exceed spot rates.
Key Insights:
Riding the Yield Curve: Profits from holding bonds and benefiting from price increases due to the yield curve’s slope.
Carry Trade: Profits from the yield differential between borrowing short-term and investing long-term, but with greater sensitivity to interest rate changes.
Bond Strategy Regarding the Yield Curve
1- Key Components of the Formula:
– Left Side:
— Represents the accumulated value of a zero-coupon bond maturing at time B:
(1 + zB)^B
– Right Side:
— Combines two components:
—- The accumulated value of a zero-coupon bond maturing at time A:
(1 + zA)^A
—- The accumulated value of a (B - A)-year zero-coupon bond starting at time A, calculated using the forward rate:
(1 + fA,B-A)^(B - A)
– Formula:
(1 + zB)^B = (1 + zA)^A * (1 + fA,B-A)^(B - A)
2- Interpretation of the Formula:
– This equation demonstrates the relationship between spot rates (z) and forward rates (f).
– It shows that the forward rate reflects the implied rate for reinvesting from the shorter maturity bond to the longer maturity bond.
3- Strategy Based on Spot and Forward Rates:
– Compare the expected future spot rate with the implied forward rate:
– Case 1:
Expected future spot rate is lower than the forward rate:
— Implication: The bond is undervalued.
— Action: Buy the bond now.
– Case 2:
Expected future spot rate is higher than the forward rate:
— Implication: The bond is overvalued.
— Action: Sell the bond now.
4- Practical Application for Bond Traders:
– Evaluate the current forward curve and predict future spot rates.
– If rates deviate from the forward curve as expected, traders can exploit price changes for profit.
– This strategy requires accurate forecasting of future interest rate movements.
5- Related Formula for Comparing Spot Rates Over Time:
(1 + zB)^B / (1 + fB-1,1)^(B - 1) = (1 + z1)
– This formula shows how forward rates are linked to one-year spot rates and (B - 1)-year bonds.
Summary:
– The forward curve provides expectations of future interest rates.
– A mismatch between the expected future spot rate and the implied forward rate creates trading opportunities.
– Buy undervalued bonds when the expected spot rate is below the forward rate.
– Sell overvalued bonds when the expected spot rate is above the forward rate.
Riding the Yield Curve / Rolling Down the Yield Curve Strategy
This is a popular yield curve trading strategy designed to take advantage of the positive slope of the yield curve. Below are the key points and steps involved:
1- Assumptions Underlying the Strategy: – The yield curve has a positive slope, meaning longer-term bonds have higher yields compared to shorter-term bonds.
– The forward curve is always above the spot curve because the forward curve incorporates expectations of future rates.
– The yield curve remains stable over the investment horizon, meaning the shape and slope of the curve do not change.
– Future spot rates are expected to be less than forward rates, which creates an opportunity for capital gains.
2- How the Strategy Works: – Traders purchase bonds with longer maturities to earn higher yields associated with longer-term instruments.
– As the bond approaches its maturity date, its yield declines (rolls down the yield curve), and the bond’s price increases. This generates capital gains in addition to coupon payments.
– By holding the bond and selling it before maturity, traders can realize returns greater than those earned by reinvesting in a series of shorter-term bonds.
3- Key Advantage: – Traders can earn an extra return by investing in longer-maturity bonds compared to continually reinvesting in shorter-term bonds.
— Return on bonds with longer maturity > Return on continually reinvesting in shorter-maturity bonds.
Practical Implications:
This strategy is effective when:
– Interest rates remain stable or decline during the holding period.
– Future spot rates are indeed lower than the forward rates implied by the current yield curve.
However, the strategy carries interest rate risk because an unexpected rise in interest rates would reduce bond prices, leading to potential capital losses.
Swap Rate Curve
1- Definition of Swaps and Their Function:
– A swap is a derivative contract in which counterparties exchange fixed-rate interest payments for floating-rate interest payments.
– The size of the payments is based on the swap rates (for the fixed leg), the floating reference rate (e.g., LIBOR or SOFR), and the notional principal amount.
– Swaps are used for speculation and risk management by hedging interest rate exposure.
2- Key Features of Swap Rates:
– The fixed-rate leg of the swap is called the swap rate and is denoted as sT, where T is the maturity of the swap.
– The swap rate is set such that the swap has a zero value at inception, meaning the present value (PV) of fixed-rate payments equals the PV of floating-rate payments.
– The floating leg is tied to a short-term reference rate, such as LIBOR or SOFR, and resets periodically.
3- The Swap Curve:
– The swap curve represents the yield curve for swap rates across different maturities. It provides a benchmark for interest rates in swap markets.
– The swap curve differs from government bond yield curves because it is derived from swap rates, not bond yields.
– For countries with illiquid long-term government bond markets, the swap curve often serves as a more reliable benchmark for interest rates.
4- Advantages of Swaps and Their Market:
– Liquidity:
— The swap market is highly liquid because it does not require matching borrowers and lenders; it only needs counterparties willing to exchange cash flows.
— Swaps are widely used for hedging interest rate risks, which increases their liquidity.
– Benchmark Role:
— In countries with less liquid government bond markets, the swap curve serves as an essential benchmark curve for valuing fixed-income securities.
— Swap rates and government bond yields are often used together for comprehensive valuation.
Summary:
– Swaps exchange fixed-rate and floating-rate interest payments for speculation and risk management.
– The swap curve reflects the yield curve for swap rates across maturities and is a vital benchmark, especially where long-term government bonds are less liquid.
– Swap markets are highly liquid due to their effectiveness in hedging interest rate risk and the simplicity of counterparties agreeing to exchange cash flows.
Why Do Market Participants Use Swap Rates When Valuing Bonds?
Government spot curves and swap rate curves are the primary choices for fixed-income valuation. The choice is based on many factors, including the relative liquidity of the markets. In the US, wholesale banks tend to use the swap curve for valuation, while retail banks prefer the government spot curve.
How Do Market Participants Use the Swap Curve in Valuation?
Swaps are customized over-the-counter contracts, where cash flows for the fixed payment portion (based on the swap rate) are exchanged for floating-rate payments (based on a reference rate such as LIBOR). The valuation of swaps involves the following considerations:
1- Initial Value and Zero Net Value at Inception:
– At inception, the swap has a value of zero.
– The present value of the fixed-rate payments equals the present value of the expected floating-rate payments.
2- Swap Rate Calculation:
– The swap rate for a term of T years is denoted as “s_T”.
– The formula for calculating the swap rate is:
“T∑_t=1 [s_T ÷ (1 + z_t)^t] + [1 ÷ (1 + z_T)^T] = 1”
— Where:
—- s_T: Swap rate for term T.
—- z_t: Spot rate for period t.
—- T: Total term in years.
3- Alternate Formula Using Discount Factors:
– The formula can also be expressed using discount factors (DF_t):
“T∑_t=1 [s_T × DF_t] + DF_T = 1”
— Where:
—- DF_t: Discount factor for period t.
—- DF_T: Discount factor for the final period (T).
4- Application in Swap Valuation:
– Swaps are valued based on the forward curve, as the floating leg’s payment dates align with expected forward rates.
– Pricing is straightforward for the fixed-rate leg but more complex for the floating-rate leg because cash flows depend on future interest rates.
5- Use of the Swap Curve:
– The swap curve, representing the yield curve of swap rates, serves as a benchmark for interest rates in markets where government bond markets may be less liquid.
– It is widely used for valuing fixed-income securities and managing interest rate risk.
Summary:
– The swap curve is crucial for valuing swaps and fixed-income securities.
– Swap rates are determined such that the present value of fixed and floating payments is equal at inception.
– Alternate formulas using spot rates or discount factors simplify calculations.
– The floating leg depends on forward rates, making its valuation more complex than the fixed leg.
Importance of the Swap Curve
1- Countries Without Liquid Government Bond Markets:
– For countries where government bond markets lack liquidity for maturities beyond one year, the swap curve serves as a benchmark for interest rates.
— This provides market participants with a reliable reference for pricing and valuing financial instruments.
2- Countries with Larger Private Sectors:
– In countries where the private sector is larger than government debt markets, the swap curve is used as a measure of the time value of money.
— It reflects the market-driven cost of borrowing and investing across different maturities.
Government Bond Market vs. Swap Market in Valuation
1- Both Markets Are Very Liquid:
– In countries like the U.S., where both the government bond market and the swap market are highly liquid, the choice of benchmark depends on the nature of the business operations.
2- Wholesale Banks:
– Risk Hedging: These banks often engage in hedging activities to manage risk.
– Preference for Swap Curve: Likely to value fixed-income securities using the swap curve due to its relevance in derivative and hedging activities.
3- Retail Banks:
– Limited Swap Exposure: Retail banks typically have limited involvement in swaps.
– Preference for Government Spot Curve: They tend to use the government spot curve as a benchmark, reflecting their focus on traditional banking products like loans and deposits.
Why the Swap Rate is Slightly Less Than the Spot Rate
1- Weighted Average of Spot Rates:
– The swap rate represents a weighted average of the spot rates.
— Weights Depend on Cash Flows: The weights are determined by the cash flow structure of the swap.
2- Concentration on Key Spot Rates:
– Most of the weight will align with the spot rate corresponding to the timing of the notional amount.
— Example: For a three-year swap, the majority of the weight is on the three-year spot rate, reflecting the critical cash flow timing.
Key Concepts on Swap Spread, I-Spread, and Z-Spread
1- Swap Spread:
– Definition: The swap spread is the difference between the swap rate and the yield of the on-the-run government bond with the same maturity.
– Interpretation:
— Indicates credit spreads and liquidity spreads in the market.
— Highlights the risk premium of firms compared to risk-free government securities.
2- I-Spread (Interpolated Spread):
– Definition: The I-spread measures the difference between a bond’s yield and the swap rate for the same maturity.
– Comparison with Swap Spread:
— The I-spread focuses on the difference between a specific bond and the swap rate, while the swap spread compares the swap rate to a government bond yield.
– Bonds yield - [Spot rate + Swap Spread]
– [Spot rate + Swap Spread] = Swap Rate
3- Libor/Swap Curve:
– Most widely used interest rate curve due to its association with the credit risk of firms rated A1/A+ (typical for commercial banks).
– Influences on Swap Rates:
— Default risk of firms.
— Supply and demand conditions in government bond markets.
4- Advantages of Swap Markets:
– Unregulated by governments, providing cross-country comparability.
– Offers more maturities compared to government bond markets, increasing flexibility for pricing and hedging.
5- Z-Spread (Zero-Volatility Spread):
– Definition: The Z-spread is a constant basis point spread added to the implied spot yield curve so that discounted future cash flows equal the bond’s current market price.
– Key Characteristics:
— More accurate than a linearly interpolated yield, particularly when the yield curve is steep.
— Reflects credit and liquidity risks better than simple spreads.
Summary:
– Swap Spread: Measures credit and liquidity spreads by comparing the swap rate to government bond yields of the same maturity.
– I-Spread: Focuses on the difference between a bond’s yield and the swap rate for the same maturity.
– Z-Spread: Captures credit and liquidity risks by adjusting spot yields to equate discounted cash flows to the bond’s market price.
– The Libor/swap curve is preferred due to its association with A1/A+ firms and its broad applicability across maturities and countries.
Key Concepts on Spreads as a Price Quotation Convention
1- Treasury Rates vs. Swap Rates:
– Treasury rates and swap rates differ due to the following reasons:
— Default Risk: Swap rates reflect counterparty credit risk, while Treasury yields are considered risk-free.
— Liquidity Differences: Treasury bonds are generally more liquid, but liquidity varies by maturity for both markets.
— Arbitrage Inefficiency: Arbitrage trades between the two markets cannot be perfectly executed due to market frictions.
2- Swap Spread:
– Definition: The swap spread is the difference between the swap rate and the yield on a government bond with the same maturity.
— Formula: Swap Spread = Swap Rate − Government Bond Yield.
— Characteristics:
— Generally positive due to compensation for counterparty risk.
— Rarely negative, but can occur briefly under certain market conditions.
— Uses on-the-run government bonds for comparison.
3- TED Spread (Treasury-Eurodollar Spread):
– Definition: The TED spread is the difference between Libor and the yield on a T-bill with matching maturity.
— Formula: TED Spread = Libor − T-Bill Yield.
— Indicator of Credit Risk:
— A higher TED spread signals increased perceived default risk in interbank loans.
— Reflects counterparty credit risk and overall risk in the banking system.
4- Libor-OIS Spread:
– Definition: The Libor-OIS spread measures the difference between Libor and the Overnight Indexed Swap (OIS) rate.
— Characteristics:
— The OIS rate is the geometric average of a floating overnight rate during the payment period (e.g., federal funds rate for USD).
— Indicates the risk and liquidity of money market securities.
— A higher spread implies increased risk or reduced liquidity in the money markets.
Summary:
– Treasury rates and swap rates differ due to default risk, liquidity variations, and arbitrage limitations.
– The Swap Spread reflects compensation for counterparty credit risk, typically positive but occasionally negative.
– The TED Spread captures counterparty credit risk in the banking system, rising during times of increased interbank default concerns.
– The Libor-OIS Spread highlights risk and liquidity conditions in money markets, with wider spreads indicating heightened financial stress.
Key Spreads in the Fixed-Income World
1- I-Spread:
– Purpose: Indicates the difference between a bond’s yield and the swap rate for the same maturity. It measures credit risk relative to swaps rather than government bonds.
– Advantages:
— Provides a better measure of credit risk when government bond yields are distorted (e.g., due to high liquidity or central bank interventions).
— Useful for comparing bonds across different issuers.
– Disadvantages:
— Does not account for liquidity risk or changes in interest rate volatility.
2- TED Spread:
– Purpose: Measures the perceived credit risk in the banking system. Defined as the difference between Libor and the yield on a T-bill of matching maturity.
– Advantages:
— Provides a direct indicator of interbank lending risk.
— Widely used during periods of economic uncertainty or financial crises to gauge market stress.
– Disadvantages:
— Heavily influenced by central bank interventions in the money market.
— May not accurately reflect longer-term credit risks.
3- Libor-OIS Spread:
– Purpose: Indicates the credit risk and liquidity in money markets by comparing Libor to the overnight indexed swap (OIS) rate.
– Advantages:
— Offers insight into short-term funding stresses in the interbank market.
— Less influenced by sovereign bond yields compared to the TED spread.
– Disadvantages:
— Can become less relevant as markets transition from Libor to alternative benchmarks (e.g., SOFR).
— May be volatile during periods of changing interest rate expectations.
4- SOFR (Secured Overnight Financing Rate):
– Purpose: Serves as a risk-free benchmark rate for the US dollar-denominated derivatives and loan markets. Based on transactions in the overnight repo market.
– Advantages:
— Reflects actual transactions and is less prone to manipulation compared to Libor.
— Backed by secured lending, making it a better representation of funding costs.
– Disadvantages:
— May not fully reflect unsecured interbank lending risks.
— Limited historical data compared to Libor.
5- Swap Spread:
– Purpose: Measures the credit and liquidity risk by comparing the swap rate to government bond yields of the same maturity.
– Advantages:
— Useful for hedging and pricing fixed-income instruments.
— Reflects market conditions when government bond markets are illiquid.
– Disadvantages:
— Can be distorted by supply and demand in the swap or government bond markets.
— May not fully capture default risk in certain market conditions.
6- Z-Spread (Zero-Volatility Spread):
– Purpose: Measures the credit and liquidity risk by adding a constant spread to the spot yield curve to equate the bond’s present value with its market price.
– Advantages:
— Provides a more precise measure of credit risk than I-spreads or swap spreads.
— Accounts for term structure differences across cash flows.
– Disadvantages:
— Requires an accurate spot curve, which can be difficult to construct.
— Sensitive to changes in market conditions, making it harder to interpret during periods of high volatility.
Summary:
– Swap Spread: Adds a single spread over the government bond yield to account for credit and liquidity risk, providing a simplified discount rate.
– Z-Spread: Offers a more accurate measure by adjusting each cash flow for its timing, aligning the bond’s price with the implied spot curve.
Expectations Theory
1- Unbiased Expectations Theory (Pure Expectations Theory):
– Purpose: States that forward rates are unbiased predictors of future spot rates.
– Key Concept: Bonds of any maturity are perfect substitutes over any holding period. For example, the expected return of holding a seven-year bond for three years is equal to the expected return of holding a five-year bond for the same three years.
– Assumption: Investors are risk-neutral (indifferent to risk).
– Criticism: Not consistent with observed risk aversion, as most investors demand risk premiums for longer maturities.
2- Local Expectations Theory:
– Purpose: A modified version of the pure expectations theory that focuses on short time periods.
– Key Concept: All bonds, whether risk-free or risky, are expected to earn the risk-free rate of return over short holding periods.
— Over longer periods, risk premiums may exist, allowing for risk-aversion effects.
– Advantage: Applicable to both risk-free and risky bonds, making it more flexible than the pure expectations theory.
– Inconsistency in Practice:
— Empirical evidence shows that longer-dated bonds often produce higher returns than shorter-maturity bonds over short holding periods.
— This suggests investors require compensation for the illiquidity of longer-term bonds and the challenges in hedging risks for these securities.
Summary:
– The Unbiased Expectations Theory assumes forward rates are accurate predictors of future spot rates and that investors are risk-neutral.
– The Local Expectations Theory adjusts for short-term neutrality but allows for risk premiums over longer periods, making it more applicable in practice.
– Empirical evidence suggests that both theories struggle to fully explain observed market behavior, particularly for longer-maturity bonds.
Liquidity Preference Theory
1- Purpose:
– Explains the shape of the yield curve by incorporating the idea of a risk premium (liquidity premium) for longer-term bonds.
– Suggests that investors demand additional compensation for the interest rate risk and reduced liquidity associated with lending over longer periods.
2- Key Features:
– Risk Premium: Longer maturities have higher interest rate risk, so investors require higher returns (liquidity premiums) to compensate for holding them.
– Yield Curve Implications:
— Typically leads to an upward-sloping yield curve because the risk premium increases with maturity.
— Can still produce a downward-sloping or hump-shaped yield curve if deflationary expectations dominate the market.
3- Implications for Forward Rates:
– Forward rates derived from the yield curve are upwardly biased estimators of future spot rates because they include the liquidity premium.
– This distinguishes the liquidity preference theory from the pure expectations theory, which assumes no risk premium.
Summary:
– The Liquidity Preference Theory accounts for the interest rate risk and liquidity concerns associated with longer maturities, leading to the inclusion of a liquidity premium in yields.
– This theory generally explains the upward-sloping yield curve but allows for other shapes in unique scenarios like deflation.
– Forward rates under this theory are biased upwards due to the embedded liquidity premium.
Segmented Markets Theory
1- Purpose:
– Explains the shape of the yield curve based on supply and demand dynamics across different maturities.
– Suggests that lenders and borrowers operate in distinct maturity segments based on their specific needs and preferences, leading to independently determined interest rates for each segment.
2- Key Features:
– No Arbitrage Between Segments: Bonds of different maturities are not perfect substitutes, meaning investors are unwilling to move between segments to exploit arbitrage opportunities.
– Influence of Market Participants:
— Life insurers and pension funds drive demand for long-term bonds to match their long-term liabilities.
— Money market funds dominate demand for short-term bonds due to their liquidity requirements.
– Yield Curve Variability: The shape of the yield curve reflects imbalances in supply and demand within specific maturity segments.
3- Implications for the Yield Curve:
– High demand or limited supply in a segment reduces yields in that maturity range, while low demand or high supply increases yields.
– The theory does not inherently predict an upward or downward slope for the yield curve but provides insight into anomalies in specific sections of the curve.
Summary:
– The Segmented Markets Theory explains yield curve shapes by focusing on supply and demand dynamics within distinct maturity segments.
– It assumes that bonds of different maturities are not substitutes, so market participants’ preferences dominate pricing for each segment.
– The shape of the yield curve varies depending on which segments experience imbalances in demand or supply.
Preferred Habitat Theory
1- Purpose:
– Expands on Segmented Markets Theory by allowing for investor flexibility.
– Explains yield curve shapes based on investors’ maturity preferences and the compensation required to leave their preferred maturity range (or “habitat”).
2- Key Features:
– Preferred Segments: Investors and institutions have a “natural” maturity range (e.g., long-term bonds for pension funds).
– Inducement to Switch: Investors may purchase bonds outside their preferred range if they receive additional compensation (e.g., higher yields).
– Market Dynamics:
— If demand for a specific segment is high and supply is low, yields in that segment will decrease.
— Investors may then move to adjacent segments if they are adequately compensated, affecting yields in those segments as well.
3- Implications for the Yield Curve:
– The theory can explain both positively sloped yield curves (investors demand a premium for long-term bonds) and negatively sloped curves (if sufficient demand exists for long-term bonds or short-term yields spike).
Summary:
– The Preferred Habitat Theory assumes that investors have natural maturity preferences but will move to other segments for sufficient compensation.
– Unlike Segmented Markets Theory, it explains how demand for specific maturities can spill over into other segments.
– The theory provides a framework for understanding both upward and downward-sloping yield curves.
Liquidity Preference Theory (Expanded Explanation)
1- Purpose of the Theory:
– Explains why the yield curve is typically upward-sloping by incorporating a liquidity premium to account for interest rate risk.
– Recognizes that investors prefer shorter maturities due to lower interest rate risk and require additional compensation (liquidity premium) to hold longer-term bonds.
2- Key Insights:
– Liquidity Premium Explained:
— Represents compensation for interest rate risk (not liquidity risk) associated with holding longer-term bonds.
— Longer-term bonds experience greater price sensitivity to interest rate changes.
– Forward Rate as a Biased Estimate:
— Forward rates are upwardly biased predictors of future spot rates because of the liquidity premium.
— Even if spot rates are expected to remain constant, the liquidity premium causes forward rates to exceed expected future spot rates.
– Yield Curve Implications:
— An upward-sloping yield curve is expected because the liquidity premium increases with maturity.
— A downward-sloping yield curve can still occur if future spot rates are expected to decline more than the liquidity premium.
3- Applications in Practice:
– Investors require enticement (via higher yields) to purchase bonds with maturities beyond their investment horizons.
– The liquidity premium becomes a key factor in the pricing of longer-term bonds.
Summary:
– Liquidity Preference Theory predicts a positively sloped yield curve, even if short-term spot rates are not expected to change, due to the liquidity premium required for interest rate risk.
– Forward rates are biased estimates of future spot rates because they include this liquidity premium.
– While typically explaining upward-sloping curves, the theory accommodates downward-sloping yield curves when future interest rate declines outweigh the premium.
A Bond’s Exposure to Yield Curve Movement
Shaping risk is the sensitivity of a bond’s price to the changing shape of the yield curve. Yield curves rarely shift in a purely parallel fashion, so the shape is constantly changing. Such changes are particularly important for the valuation of bonds with embedded options.
Factors Affecting the Shape of the Yield Curve
Yield curve factor models aim to explain historical interest rate movements by identifying the key factors that influence the curve’s shape. The Litterman and Scheinkman three-factor model highlights the following factors:
1- Level (Parallel Shifts):
– Description: Represents parallel upward or downward shifts in the entire yield curve.
– Impact:
— Negative for all bonds because of the inverse relationship between yields and bond prices.
— Example: A level coefficient of -0.5 means the bond’s price will decline by 0.5% for a 1 standard deviation increase in the yield curve level.
– Importance: The level factor is the dominant component of yield curve movements, explaining most variations.
2- Steepness (Nonparallel Shifts):
– Description: Captures differences in changes between short-term and long-term rates.
— Example: Short-term rates rise more than long-term rates, causing the curve to become steeper.
– Impact: Nonparallel shifts are generally less significant compared to parallel shifts (level factor).
– Relevance: Changes in steepness often reflect monetary policy actions or expectations for future growth and inflation.
3- Curvature (Twists):
– Description: Reflects changes where short-term rates increase while long-term rates fall, creating a twist in the curve.
– Impact:
— Negative for intermediate maturities (e.g., 5- to 10-year bonds).
— Positive for short-term and long-term maturities.
– Relevance: Often linked to changes in market sentiment or unique economic conditions (e.g., flight to safety in long-term bonds).
Summary:
– Level: Describes parallel shifts and has the greatest explanatory power.
– Steepness: Captures nonparallel shifts (e.g., short-term rates rising faster than long-term rates).
– Curvature: Represents twists in the yield curve, typically with intermediate maturities falling while short and long-term rates rise.
– Together, these factors offer a comprehensive understanding of how and why the yield curve changes over time.
Yield Volatility
Interest rate volatility plays a significant role in fixed-income securities valuation and risk management, particularly for securities with embedded options. Below are the key aspects:
1- Definition and Impact:
– Description: Yield volatility reflects fluctuations in interest rates, influencing the value of bonds and their embedded options (e.g., callable or putable bonds).
– Relevance:
— Higher volatility increases the value of embedded options, as uncertainty raises the potential benefit of exercising options.
— Critical in risk management for forecasting potential changes in bond values due to rate movements.
2- Volatility Curve (Term Structure):
– Definition: The volatility curve shows how interest rate volatility varies across maturities.
– Shape: Typically downward sloping, indicating:
— Short-term rates: Tend to be more volatile due to higher sensitivity to monetary policy changes.
— Long-term rates: Are relatively stable because they are influenced by slower-moving factors such as long-term growth and inflation expectations.
3- Factors Driving Volatility:
– Short-term rates: Primarily influenced by monetary policy uncertainty.
— Example: Market speculation on central bank interest rate changes increases short-term rate volatility.
– Long-term rates: Driven by uncertainty about real economic growth and inflation expectations.
— Example: Unpredictable long-term inflation trends can cause volatility in long-term rates.
4- Duration Effect on Prices:
– Despite the relative stability of long-term rates, prices of long-term bonds are generally more volatile than short-term bonds due to the duration effect (greater sensitivity to interest rate changes).
— Example: A small change in long-term interest rates can significantly affect the price of a long-duration bond.
Summary:
– Yield volatility is crucial for understanding the behavior of fixed-income securities, particularly those with embedded options.
– The volatility curve is typically downward sloping, as short-term rates are more volatile than long-term rates.
– Short-term volatility reflects monetary policy uncertainty, while long-term volatility is driven by economic growth and inflation expectations.
– Longer-term bonds, while less volatile in rates, exhibit higher price volatility due to the duration effect.
Managing Yield Curve Risks Using Key Rate Duration
Yield curve risk refers to the potential impact of unanticipated changes in the yield curve on a portfolio’s value. Key rate durations and factor models can be used to manage this risk.
1- Key Rate Duration:
– Measures the sensitivity of a bond’s value to changes in specific points on the yield curve.
– The bond’s effective duration is calculated as the sum of its key rate durations across all maturities.
– For a portfolio of bonds with maturities at 1-year, 5-year, and 10-year, the expected percentage change in value can be approximated as:
ΔP ÷ P ≈ -D1 Δr1 - D5 Δr5 - D10 Δr10
Where:
— ΔP ÷ P: Percentage change in portfolio value.
— D1, D5, D10: Key rate durations for 1-year, 5-year, and 10-year maturities.
— Δr1, Δr5, Δr10: Changes in interest rates for 1-year, 5-year, and 10-year maturities.
2- Factor Models for Yield Curve Movements:
– Yield curve movements can be decomposed into:
— Level: Parallel shifts.
— Steepness: Nonparallel changes where short-term rates increase more than long-term rates.
— Curvature: “Twists” where short-term rates and long-term rates move in opposite directions.
For a portfolio with maturities at 1-year, 5-year, and 10-year:
— D_Level: D1 + D5 + D10
— D_Steepness: -D1 + D10
— D_Curvature: D1 - D5 + D10
Summary:
– Key rate durations measure sensitivity to specific maturities, enabling precise management of yield curve risk.
– Factor models break yield curve changes into components (level, steepness, curvature) for additional insights.
– These tools help portfolio managers identify risks and make informed decisions to mitigate exposure.
List of Duration Measures
Below is a detailed breakdown of key duration measures, their uses, advantages, and disadvantages.
1- Effective Duration:
– Usage:
— Measures the sensitivity of a bond’s price to a small parallel shift in the yield curve.
— Commonly used for bonds with embedded options.
– Advantages:
— Captures the effects of changes in interest rates and option-related convexity.
— Applicable to option-embedded securities.
– Disadvantages:
— Assumes parallel yield curve shifts, which may not always hold true.
— Requires complex option-adjusted calculations.
2- Key Rate Duration:
– Usage:
— Measures the sensitivity of a bond’s price to changes in specific maturities (key rates) on the yield curve.
— Useful for managing yield curve risk.
– Advantages:
— Identifies vulnerabilities to non-parallel shifts in the yield curve.
— Provides precision in managing risk for portfolios with varying maturities.
– Disadvantages:
— Complex and requires detailed yield curve data.
— Not useful for securities that are evenly impacted by yield curve shifts.
3- Modified Duration:
– Usage:
— Measures the percentage change in a bond’s price for a 1% change in yield.
— Often used for plain vanilla fixed-income securities.
– Advantages:
— Straightforward and widely understood.
— Applicable to bonds without embedded options.
– Disadvantages:
— Assumes constant cash flows, so unsuitable for callable/option-embedded bonds.
— Does not capture convexity effects.
4- Macaulay Duration:
– Usage:
— Measures the weighted-average time to receive a bond’s cash flows.
— Primarily used for immunization strategies in portfolio management.
– Advantages:
— Provides a time-weighted measure of bond maturity.
— Useful for matching duration to liability timelines.
– Disadvantages:
— Assumes a flat yield curve and constant cash flows.
— Less relevant for actively managed portfolios.
5- Portfolio Duration:
– Usage:
— Measures the weighted-average duration of a portfolio of bonds.
— Useful for assessing overall interest rate risk of a portfolio.
– Advantages:
— Aggregates interest rate sensitivity across multiple bonds.
— Useful for diversified portfolios.
– Disadvantages:
— Ignores interaction effects between bonds.
— Assumes parallel shifts in the yield curve.
6- Convexity-Adjusted Duration:
– Usage:
— Adjusts effective or modified duration to account for convexity (nonlinear price changes).
— Used when interest rate changes are large.
– Advantages:
— Improves accuracy for non-linear price-yield relationships.
— Essential for large interest rate shifts.
– Disadvantages:
— More complex to calculate.
— May overstate effects for smaller interest rate changes.
Key Insights:
– Effective Duration is ideal for option-embedded bonds.
– Key Rate Duration is best for assessing yield curve risk.
– Modified Duration is the simplest measure for vanilla bonds.
– Convexity Adjustment is crucial for large rate shifts.
Factors Affecting Spot and Forward Rate Curves
1- Short- and Intermediate-Term Rates:
– Primary Influences:
— Monetary Policy: Central banks’ decisions on interest rates directly impact short- and intermediate-term rates.
— Macroeconomic Factors: GDP growth and inflation also play a role, but to a lesser extent.
2- Long-Term Rates:
– Primary Influences:
— Inflation: Approximately two-thirds of the variation in long-term rates is driven by inflation expectations.
— Monetary Policy: Residual variation in long-term rates is attributed to central bank actions and monetary policies.
3- Additional Factors Influencing Bond Yields:
– Fiscal Policy:
— Governments issuing more debt to finance budget deficits cause bond yields to rise as supply increases.
— Restrictive fiscal policy reduces new government debt issuance, lowering yields.
– Maturity Structure:
— An increase in the supply of longer-dated bonds raises yields and enhances excess returns for longer maturities.
– Investor Demand:
— Domestic demand (e.g., from pension funds or insurance companies) lowers yields, particularly in the long-term segment.
— Foreign investors influence yields indirectly through exchange rate management and international transactions.
Summary:
– Short- and intermediate-term rates are mainly influenced by monetary policy, with macroeconomic factors like GDP growth and inflation having secondary effects.
– Long-term rates are primarily driven by inflation expectations and, to a lesser extent, monetary policy.
– Fiscal policy, maturity structure, and investor demand also play critical roles in shaping bond yields and rate curves.
Active Bond Investment Strategies and Yield Curve Dynamics
1- Passive vs. Active Investors:
– Passive Investors: Accept the forward rates implied by the yield curve and simply roll bonds over as they mature.
– Active Investors: Develop independent views of interest rate movements and market conditions, seeking to profit from discrepancies between their expectations and market consensus.
— Derivative Usage: Active investors can use derivatives, such as futures contracts, to express their views while minimizing portfolio turnover.
2- Bond Risk Premium (Term Premium):
– Definition: The additional yield required to hold a long-term, default-free bond above the short-term risk-free rate.
– Characteristics:
— It is forward-looking and based on government bond yields.
— It excludes illiquidity or credit risk premiums.
3- Strategies Based on Interest Rate Expectations:
– Parallel Shifts in Rates:
— Downward Shift: Increase portfolio duration to benefit from rising bond prices.
— Upward Shift: Decrease portfolio duration to minimize losses from falling bond prices.
— Key Note: Parallel shifts are rare; more nuanced strategies are typically required.
– Non-Parallel Shifts:
— Steepening Yield Curve: Focus on shorter maturities as long-term yields rise faster than short-term yields.
— Flattening Yield Curve: Extend portfolio duration as long-term yields fall relative to short-term yields.
Summary:
– Passive investors accept forward rates and manage portfolios with minimal turnover.
– Active investors seek to profit from discrepancies between market consensus and their own expectations, often using derivatives to express their views.
– The bond risk premium explains why yields are typically upward-sloping, reflecting compensation for holding longer-maturity bonds.
– Portfolio strategies vary based on expected interest rate movements, with duration adjustments playing a critical role.
Yield Curve Movements and Investment Strategies
1- Bullish Steepening
– Definition: Short-term rates fall faster than long-term rates.
– Cause: Central banks loosen monetary policy to stimulate an economy operating below potential.
– Investment Strategy:
— Position: Short long-term bonds and buy short-term bonds.
— Objective: Profit from the widening spread between short- and long-term yields.
2- Bearish Steepening
– Definition: Long-term rates rise faster than short-term rates.
– Cause: Often occurs when inflation expectations increase, pushing long-term rates higher relative to short-term rates.
– Investment Strategy:
— Position: Buy long-term bonds and short short-term bonds.
— Objective: Benefit from the widening gap as long-term rates increase more than short-term rates.
3- Bullish Flattening
– Definition: Long-term rates fall faster than short-term rates.
– Cause: Observed after market turmoil due to a “flight to quality” where investors move into safe government bonds.
– Investment Strategy:
— Position: Buy long-term bonds and short short-term bonds.
— Alternative: For investors unable to short, shift from a bullet portfolio (concentrated on a single maturity) to a barbell portfolio (combining short- and long-dated bonds).
— Objective: Profit from narrowing spreads between short- and long-term yields.
4- Bearish Flattening
– Definition: Short-term rates rise faster than long-term rates.
– Cause: Typically occurs during an economic expansion when central banks tighten monetary policy to control inflation.
– Investment Strategy:
— Position: Short short-term bonds and hold long-term bonds.
— Objective: Take advantage of narrowing spreads between short- and long-term rates.
5- Bullish Parallel Shift
– Definition: Rates across all maturities decline simultaneously.
– Cause: A decrease in inflation expectations or significant monetary policy easing.
– Investment Strategy:
— Position: Increase portfolio duration by buying long-term bonds.
— Objective: Maximize gains from falling rates as longer-duration bonds are more sensitive to interest rate changes.
6- Bearish Parallel Shift
– Definition: Rates across all maturities increase simultaneously.
– Cause: A rise in inflation expectations or significant monetary policy tightening.
– Investment Strategy:
— Position: Reduce portfolio duration by holding short-term bonds.
— Objective: Minimize losses from rising rates as shorter-duration bonds are less sensitive to interest rate changes.
7- Bullish Twist
– Definition: Short-term rates fall while long-term rates rise.
– Cause: Often associated with monetary policy easing accompanied by rising inflation expectations in the long run.
– Investment Strategy:
— Position: Buy short-term bonds and short long-term bonds.
— Objective: Benefit from declining short-term rates and rising long-term rates.
8- Bearish Twist
– Definition: Short-term rates rise while long-term rates fall.
– Cause: Often observed during stagflation or when the economy faces both weak growth and inflationary pressures.
– Investment Strategy:
— Position: Short short-term bonds and buy long-term bonds.
— Objective: Profit from short-term rates increasing more than long-term rates declining.
Duration-Neutral Positioning
– Definition: Structuring a portfolio to eliminate exposure to changes in the overall level of the yield curve while focusing solely on steepening or flattening effects.
– Benefit: Isolates yield curve slope changes from shifts in the curve’s level, providing a clearer risk-return profile.
Yield Curve Movement Strategies
1- Overall Interest Rates Expected to Fall
– Action: Extend portfolio duration.
– Reasoning: Longer-duration bonds are more sensitive to interest rate changes, allowing investors to benefit more from falling rates.
– Objective: Maximize gains from a declining interest rate environment.
2- Overall Interest Rates Expected to Rise
– Action: Reduce portfolio duration.
– Reasoning: Shorter-duration bonds are less sensitive to rising interest rates, minimizing potential losses.
– Objective: Protect the portfolio from adverse price movements due to rising rates.
3- Yield Curve Expected to Steepen
– Action:
— Sell: Long-term bonds.
— Buy: Short-term bonds.
– Reasoning: Steepening implies long-term rates increase relative to short-term rates. Selling long-term bonds minimizes exposure to rate increases, while buying short-term bonds captures relative stability.
– Objective: Profit from the divergence between short- and long-term rates.
4- Yield Curve Expected to Flatten
– Action:
— Sell: Short-term bonds.
— Buy: Long-term bonds.
– Reasoning: Flattening implies short-term rates rise or long-term rates fall. Selling short-term bonds avoids potential losses from rising rates, while buying long-term bonds captures gains from falling rates.
– Objective: Benefit from the convergence of short- and long-term yields.
- Banks
Preferred Rate: Swap rates.
Reason: Swap rates reflect the credit risk of A-rated financial institutions, aligning better with the risk profile of banks. They are also more liquid and globally comparable, making them a suitable benchmark for bank-issued bonds. - Corporate Issuers
Preferred Rate: Interpolated spreads (I-spreads) over swap rates.
Reason: I-spreads reflect credit risk relative to swaps, making them more accurate for pricing corporate bonds with varying levels of credit risk. - Municipal Bond Issuers
Preferred Rate: Municipal bond indices or taxable municipal spreads.
Reason: These issuers often rely on tax-advantaged pricing benchmarks, such as municipal yield curves, to reflect tax-adjusted returns for investors. - Emerging Market Governments
Preferred Rate: Sovereign credit spreads over swaps or global bond indices.
Reason: Emerging market governments often issue bonds with higher credit risk than developed market governments, so swap-based spreads or sovereign-specific indices better reflect the associated risk. - Supranational Organizations (e.g., World Bank)
Preferred Rate: LIBOR or SOFR-based curves.
Reason: Supranational entities often have AAA ratings, and they issue bonds in highly liquid markets. They may prefer floating rates or other benchmark curves that align with global funding conventions. - High-Yield Bond Issuers
Preferred Rate: High-yield bond indices or spreads relative to corporate indices.
Reason: These issuers often require benchmarks that reflect their lower credit ratings and higher default risks, which are not captured by government or swap rates. - Securitized Debt Issuers (e.g., Mortgage-Backed Securities)
Preferred Rate: Asset-specific benchmarks such as OAS (option-adjusted spread).
Reason: Securitized debt pricing depends on cash flow structures and embedded options, requiring benchmarks that adjust for prepayment or extension risk.
6.2 The Arbitrage Free Valuation Framework
– Explain what is meant by arbitrage-free valuation of a fixed-income instrument.
– Calculate the arbitrage-free value of an option-free, fixed-rate coupon bond.
– Describe a binomial interest rate tree framework.
– Describe the process of calibrating a binomial interest rate tree to match a specific term structure.
– Describe the backward induction valuation methodology and calculate the value of a fixed-income instrument given its cash flow at each node.
– Compare pricing using the zero-coupon yield curve with pricing using an arbitrage-free binomial lattice.
– Describe pathwise valuation in a binomial interest rate framework and calculate the value of a fixed-income instrument given its cash flows along each path.
– Describe a Monte Carlo forward-rate simulation and its application.
– Describe term structure models and how they are used.
Arbitrage-Free Valuation: Key Points
– Definition: Arbitrage-free valuation calculates the value of securities under the assumption that no arbitrage opportunities exist, ensuring the market prices reflect all available information.
– Principle of No Arbitrage: In efficient markets, arbitrage opportunities may arise temporarily but are eliminated quickly as prices adjust. Arbitrage allows investors to earn riskless profits with no net investment, which violates market efficiency.
– Valuation Framework: — The value of a financial asset equals the present value of its cash flows. — Risk-free assets: Discount cash flows at the risk-free rate. — Riskier assets: Use higher discount rates reflecting the risk premium.
– Yield Curve Impact: — If the yield curve is flat, all cash flows are discounted at the same rate. — If the yield curve is not flat, each cash flow is discounted using the appropriate spot rate, treating cash flows as individual zero-coupon bonds. — This ensures consistency with arbitrage-free pricing principles by reflecting the time value of money and differing rates for various maturities.
Law of One Price
The law of one price states two goods that are perfect substitutes should have the same price. If different prices existed, the trader could buy the cheaper one and sell the more expensive one to lock in a risk-free profit based on the price differential. However, for two assets to be perfect substitutes, it is not sufficient that they offer identical cash flows. They must also be equal in terms of risk.
Arbitrage Opportunity:
– Definition: Arbitrage opportunities arise when an investor can achieve a riskless profit with zero net investment.
– Types of Arbitrage:
— 1. Violation of Value Additivity:
—- Occurs when the value of the whole differs from the sum of its parts.
—- Example: If a bond’s individual cash flows (valued as separate zero-coupon bonds) sum to $1,000 but the bond is priced at $950, an arbitrage opportunity exists.
— 2. Violation of Dominance:
—- Occurs when a risk-free future payoff is available at a zero or negative price today.
—- Example: If a security guaranteeing a $1,000 risk-free payment is priced at $950, investors can arbitrage by purchasing the security and receiving a certain profit at maturity.
– Market Impact: Investors will exploit these arbitrage opportunities until prices adjust, restoring efficiency in well-functioning markets.
Implications of Arbitrage-Free Valuation for Fixed-Income Securities
– Concept: Fixed-income securities can be treated as packages of zero-coupon bonds, with each cash flow representing a separate bond.
– Processes:
— 1. Stripping:
—- Dealers separate a bond’s cash flows into individual zero-coupon bonds.
—- Example: A 5-year annual coupon bond is split into 6 zero-coupon bonds—5 for the annual coupons and 1 for the principal repayment.
— 2. Reconstitution:
—- Dealers recombine zero-coupon bonds to replicate the cash flows of a coupon-paying bond.
—- This process ensures the reconstructed bond aligns with the original bond’s price, maintaining arbitrage-free pricing.
– Market Significance:
— Arbitrage-free valuation ensures that the value of the bond equals the combined present value of its individual cash flows, preventing arbitrage opportunities.
Valuing Bonds with Embedded Options Using Lattice Models
When a bond includes embedded options (e.g., callable or putable), its cash flows are uncertain because their timing and amount depend on future interest rate changes. Lattice models address this uncertainty by modeling possible future interest rate environments.
Framework and Types of Models:
1- Interest Rate Models:
– Some models, like one-factor models, project changes in a single interest rate (e.g., the one-year rate).
– Multi-factor models account for behavior across multiple rates (e.g., one-year and ten-year rates).
2- Binomial Lattice Framework:
– Composed of nodes that represent distinct time intervals.
– At each node, the interest rate can rise or fall, creating a network of potential paths for future rates.
Conditions for Consistency:
The future interest rate paths generated must satisfy the following:
– The current benchmark yield curve (e.g., spot or forward rate curves).
– The assumed interest rate volatility (representing market uncertainty).
– The stochastic process that governs the random behavior of interest rates, such as changes in short-term or long-term rates.
Key Concepts:
– Uncertain Cash Flows: Interest rate changes affect the bond’s cash flows, making valuation more complex.
– Lattice Models: Used to create interest rate paths based on an assumed level of volatility.
Binomial Interest Rate Tree
A binomial interest rate tree is a framework used to value bonds with embedded options. It models the possible interest rate paths over time while incorporating assumptions about volatility and the current yield curve.
Key Features:
1- Starting Point:
– The tree begins with the benchmark par curve as the base.
– The par, spot, and forward curves are related:
— If the curves are upward sloping, the forward curve will lie above the spot curve, which will lie above the par curve.
— All three curves converge only if yields are flat for all maturities.
2- Structure of the Tree:
– Nodes represent possible interest rates at distinct time intervals (e.g., years).
– At each node, the interest rate can:
— Rise to i_t,H (higher rate).
— Fall to i_t,L (lower rate).
– The tree progresses from Time 0 to subsequent time intervals (Time 1, Time 2, etc.).
Conditions for Consistency:
– The tree is calibrated to reflect:
— The current benchmark yield curve (par, spot, or forward).
— The assumed volatility of interest rates.
— A stochastic interest rate model governing how rates evolve over time.
Example of Node Representation:
– For one-year intervals:
— i0 represents the current one-year rate at Time 0.
— Moving to Time 1, the rate will either increase to i1,H or decrease to i1,L.
— At Time 2, i2,HL represents a scenario where the rate rose once and fell once.
Summary:
This framework is essential for valuing bonds with embedded options, as it accounts for the range of possible interest rate environments. The binomial tree structure provides a systematic way to calculate the timing, magnitude, and discounting of cash flows.
Lognormal Random Walk Model in a Binomial Interest Rate Tree
When building a binomial interest rate tree, a lognormal random walk model is often employed to model the evolution of interest rates.
Key Characteristics of the Lognormal Random Walk Model:
1- Prevention of Negative Rates:
– Rates cannot be negative, which aligns with real-world conditions.
2- Higher Volatility for Higher Rates:
– As interest rates increase, their volatility also increases.
Relationship Between Adjacent Nodes:
Under the lognormal model, the relationship between rates at adjacent nodes in the same time period is determined by the assumed interest rate volatility (σ).
1- Upper Node Rate:
– The rate at the upper node at Time t+1 can be calculated using:
– i_t,H = i_t,L * e^(2σ).
2- Lower Node Rate:
– Given the upper node rate, the lower node rate can be calculated as:
– i_t,L = i_t,H * e^(-2σ).
Key Observation:
– The rates at any two adjacent nodes will be two standard deviations apart.
Example Calculation:
Assume:
– Rate at Time t = 3%.
– Volatility (σ) = 20%.
At Time t+1:
– Upper Node Rate: i1,H = 0.03 * e^(20.2) ≈ 4.48%.
– Lower Node Rate: i1,L = 0.03 * e^(-20.2) ≈ 2.01%.
At Time t+2 (for i2,HL):
– The rates i2,HH and i2,LL will be four standard deviations apart:
— i2,HH = i2,LL * e^(4*σ).
Substituting:
– i2,HH = 2.01% * e^(4*0.2) = 4.48%.
Summary:
This lognormal random walk model ensures that the binomial tree appropriately reflects interest rate volatility and prevents unrealistic outcomes, such as negative rates.
Lognormal Model and Approximation of Interest Rates
The lognormal random walk model provides an efficient way to approximate future interest rates by utilizing implied forward rates. This model ensures the interest rate tree aligns with the forward rates implied by the benchmark yield curve.
Key Concepts:
1- Midpoint Approximation in Nodes:
– The one-year forward rate for a given period serves as the best estimate for the midpoint between adjacent nodes in that period.
– Example:
— At Time 1, the forward rate approximates the midpoint between i1,H (upper node rate) and i1,L (lower node rate).
— Similarly, at Time 2, the forward rate approximates the midpoint for i2,HL.
– However, actual calibrated rates may not match the midpoint exactly.
2- Simplified Notation for the Binomial Lattice:
– Rates at each node can be represented in terms of the one-year forward rate at a given time (i_T).
– Each node’s interest rate is adjusted for the number of standard deviations (σ) away from the midpoint.
– Formula:
— For an upward movement: i_T * e^(nσ).
— For a downward movement: i_T * e^(-nσ).
Restated Binomial Lattice (Simplified):
Using the notation i_T to represent the one-year forward rate at time T, the tree can be expressed as:
Time 0:
– i_0.
Time 1:
– Upper Node: i_1 * e^(σ).
– Lower Node: i_1 * e^(-σ).
Time 2:
– Upper Node: i_2 * e^(2σ).
– Lower Node: i_2 * e^(-2σ).
Time 3:
– Upper Node: i_3 * e^(3σ).
– Lower Node: i_3 * e^(-3σ).
This restated lattice maintains the flexibility of the binomial tree while simplifying the interpretation of each node’s rates in terms of the forward rate and its deviations.
Key Insight:
The lognormal model ensures consistency with the implied forward curve while accounting for volatility through the standard deviation factor (σ). This ensures a robust framework for valuing bonds with embedded options.
Steps for Demonstrating the Binomial Valuation Method
1- Start with the Benchmark Par Curve:
– Use the par curve as the initial reference for deriving other yield curves.
2- Derive the Spot Rates and Forward Rates:
– Spot rates and one-year forward rates are derived from the par curve using the bootstrapping method.
– If yields are identical across maturities, the par curve, spot curve, and forward curve will be the same (indicative of a flat yield curve).
Key Points:
– The par rates, spot rates, and forward rates can all be used to discount cash flows of the benchmark bonds.
– The present value (PV) of the benchmark bonds will match their market prices, ensuring consistency across the curves.
With the lognormal distribution, the standard deviation of the one-year rate is the product of the current one-year rate and the assumed level of volatility. This means interest rate movements are larger when interest rates are high. This also means negative interest rates are not possible.
Volatility can be measured with two methods. One method measures historical interest rate volatility. Another method calculates the volatility that is implied by the current market prices of interest rate derivatives.
Binomial Interest Rate Tree Using the Lognormal Model
1- Relationship Between Higher and Lower Interest Rates:
– The lognormal model relates the higher interest rate i1,H and the lower interest rate i1,L as follows:
i1,H = i1,L × e^(2σ),
where:
— σ: Assumed volatility of the one-year rate.
2- One-Year Forward Rate as the Average:
– The one-year forward rate is the midpoint between i1,H (higher rate) and i1,L (lower rate).
– This ensures that the tree is calibrated to the forward curve.
3- Tree Structure Representation:
– The binomial interest rate tree shows the evolution of rates over time:
— i1: One-year forward rate at time 1.
— i2: One-year forward rate at time 2.
— i3: One-year forward rate at time 3.
4- Calculation of Rates at Each Node:
– Higher and lower rates at each node are calculated using:
— iT,H = iT,L × e^(2σ).
– The rates at each node reflect potential upward or downward movements in rates.
This lognormal framework ensures the tree aligns with the current benchmark yield curve and captures interest rate volatility.
Determining the Value of a Bond at a Node
1- Process Overview:
– To value a bond in the binomial tree, the backward induction valuation method is used.
– This involves starting at the bond’s known cash flows at maturity and working backward to determine its value at each node.
2- Node Value Representation:
– At each node:
— VH represents the bond’s value if the forward rate increases.
— VL represents the bond’s value if the forward rate decreases.
– The model assumes equal probabilities of an increase or decrease in rates (50% probability for each outcome).
3- Calculation at Each Node:
– The value at any node is determined by calculating the present value of the bond’s probability-weighted future values using the appropriate one-period rates.
– Formula for node valuation:
NodeValue = 0.5 * [ [( VH + C) ÷ (1 + i)] + [ (VL + C) ÷ (1 + i)] ]. or = [C + 0.5 * (VH + VL) / 1+ i ]
4- Purpose:
– This backward-looking process ensures that the bond’s present value is consistent with the binomial interest rate tree, reflecting possible future rate paths.
Example Explanation of “i”:
If the node is at Time 0 and the interest rate i at this node is 2%, we discount the bond values from Time 1 (both VH and VL) and the coupon payment at Time 1 back to Time 0 using this rate.
Recall that a binomial interest rate tree reflects the current benchmark yield curve, an assumed level of interest rate volatility, and a stochastic process that determines the interest rate movements. If market prices are assumed to be correct (i.e., no arbitrage opportunities exist), then a binomial interest rate tree is calibrated by choosing rates that are consistent with current benchmark yields and bond prices.
Explanation: Calibration of a Binomial Interest Rate Tree
1- Key Components of the Binomial Interest Rate Tree:
– The binomial interest rate tree models interest rate movements based on:
— The current benchmark yield curve: Reflects observed market interest rates for different maturities.
— Assumed interest rate volatility: Captures the uncertainty in future interest rate changes.
— A stochastic process: Governs the random movement of rates (upward or downward at each node).
2- Calibration Process:
– The tree is calibrated under the assumption of no arbitrage, meaning that market prices are correct and free from arbitrage opportunities.
– Rates are chosen such that:
— The tree matches the current benchmark yields (to reflect observed market interest rates).
— Bond prices calculated using the tree align with market prices of those bonds.
3- Purpose of Calibration:
– Ensures consistency between the binomial tree, observed market conditions, and no-arbitrage pricing.
– Enables the accurate valuation of bonds and other fixed-income securities based on the tree.
Calibration of a Binomial Interest Rate Tree
1- Purpose of Calibration:
– Calibration ensures that the binomial tree reflects the current market conditions.
– This involves matching the tree to:
— The current par curve (i.e., bond yields observed in the market).
— The market prices of bonds (ensuring no arbitrage opportunities).
2- Iterative Calibration Process:
– The process starts with a guess for the interest rate at one node in the next time period (e.g., at Time 1).
– Adjustments are made iteratively until:
— Rates at each node are consistent with the current par curve.
— The calculated bond price matches its face value or market price.
3- Lognormal Model Relationship:
– In a lognormal random walk model, rates at adjacent nodes are determined by:
— The volatility (σ) of interest rates.
— The relationship:
“Rate at the upper node (i1_H) = Rate at the lower node (i1_L) × e^(2σ).”
4- Application Example:
– For a two-year par rate bond:
— Start with the current one-year rate at Time 0 (i0).
— Guess the lower rate at Time 1 (i1_L).
— Calculate the upper rate (i1_H) using the lognormal formula.
— Refine i1_L until Time 1 rates are consistent with the par curve and bond price equals its face value.
Building and Calibrating a Three-Year Binomial Tree
1- Three-Year Tree Calibration Process:
– Begin with the calibrated rates for the 2-year par rate bond at nodes i1_L and i1_H.
– Use an iterative process to find the rates for the nodes at Time 2 that will bring the price of a 3-year par bond to its face value.
— Start with an estimated value for i2, which will determine the rates for the nodes above and below it.
— Adjust the rates until the 3-year bond price equals par.
2- Key Considerations for Calibration:
– The 1-year forward rate in 2 years, implied by the spot curve, serves as a good initial estimate for i2. However, the actual rate may vary slightly once the tree is fully calibrated.
– Once rates at Time 2 are determined, they can be used to calibrate longer-term trees (e.g., 4-year or 5-year trees) based on the same par curve.
3- Lognormal Random Walk Model:
– Rates at adjacent nodes are separated by two standard deviations of the assumed volatility (σ) at that time period.
— Example: At Time 1, the lower node rate (i1_L) is two standard deviations less than the upper node rate (i1_H):
“i1_L = i1_H × e^(-2σ).”
— Rates at each node reflect the upward or downward movements from previous nodes.
4- Illustrative Example:
– Assume par rates of 2.5%, 2.75%, and 3.0% for maturities of 1, 2, and 3 years, respectively, with a 20% volatility assumption.
— At Time 0, the one-year rate is 2.500%.
— At Time 1, rates are 3.605% (i1_H) and 2.417% (i1_L).
— At Time 2, rates are 5.074%, 3.401%, and 2.280% for the respective nodes.
5- Consistency with the Forward Curve:
– The binomial tree rates closely approximate the one-period forward rates implied by the par curve, but slight differences may exist.
— Example:
— One-year forward rate in one year = 3.0075% (implied by the par curve).
— Middle node rate at Time 2 = 3.401% (close to the implied forward rate of 3.5254%).
6- Key Insights:
– A properly calibrated binomial tree reflects:
— The stochastic interest rate process (e.g., 50% probability of upward or downward movement).
— The assumed volatility level (σ).
— The current par curve and bond prices.
– Higher volatility assumptions widen the range of rates, which is critical for valuing bonds with embedded options.
Steps to Calibrate a Two-Period Arbitrage-Free Binomial Tree
1- Start with i0:
– Use the current one-year rate from the benchmark par curve.
2- Make a Guess for i1_L:
– Begin with an initial guess for the lower node rate at Time 1 (i1_L).
3- Calculate i1_H:
– Determine the upper node rate at Time 1 (i1_H) using the lognormal relationship:
“i1_H = i1_L × e^(2σ).”
4- Calculate the Value of the Two-Year Bond:
– Use the binomial tree to compute the value of the two-year benchmark bond that pays a coupon matching the two-year par rate.
— If the bond’s value is less than the market price, reduce the guess for i1_L to increase the bond value.
— If the bond’s value is greater than the market price, increase the guess for i1_L to lower the bond value.
5- Adjust i1_L:
– Modify the initial guess for i1_L until the value of the two-year bond equals its market price.
6- Make a Guess for i2_HL:
– Use a similar iterative approach to estimate i2_HL at Time 2.
7- Calculate i2_LL and i2_HH:
– Apply the lognormal relationships to compute the lower node (i2_LL) and upper node (i2_HH) rates at Time 2:
– “i2_LL = i2_HL × e^(-2σ).”
– “i2_HH = i2_HL × e^(2σ).”
8- Use the Expanded Tree to Value the Three-Year Bond:
– Utilize the expanded tree to calculate the value of the three-year benchmark bond.
9- Adjust i2_LL:
– Refine i2_LL until the value of the three-year bond equals its market price.
To construct a binomial interest rate tree, two key assumptions are required: the process to generate interest rates and the assumed volatility. Volatility is particularly important when valuing bonds with embedded options, as it influences the range of possible interest rate outcomes in the tree.
If a greater volatility is assumed:
– The interest rates in the tree will be more spread out, meaning lower rates will be lower, and higher rates will be higher.
Valuing Option-Free Bonds Using a Binomial Interest Rate Tree
When valuing option-free bonds, the binomial interest rate tree provides a structured approach to calculating their arbitrage-free value. Here’s how the theory applies:
- Key Process Overview:
– The binomial tree is constructed using calibrated interest rates that are consistent with the current benchmark yield curve.
– The tree ensures that cash flows are discounted at the correct rates at each node. This results in a valuation that matches arbitrage-free prices.
– Working backward from the bond’s known future cash flows, values are calculated at each preceding node until the value at Time 0 is determined. - Backward Induction:
Backward induction is the primary method for valuing bonds within the binomial tree framework.
– Step 1: Start with the terminal cash flows at the bond’s maturity (e.g., principal repayment and the final coupon payment).
– Step 2: Use the appropriate interest rate at each node to discount these terminal cash flows back to the preceding nodes.
– Step 3: At intermediate nodes, combine the discounted values from both the upward and downward paths in the tree.
Formula for the value at any node:
V = [C + 0.5(V_H + V_L)] ÷ (1 + i),
where:
— V = Bond value at the current node.
— C = Coupon payment.
— V_H = Bond value in the upward rate path.
— V_L = Bond value in the downward rate path.
— i = One-year interest rate at the node.
- Spot Rate Consistency:
The calibration of the binomial tree ensures that it aligns with the current benchmark spot rates. These spot rates are derived from par rates using the bootstrapping method.
For example:
– The 1-year, 2-year, and 3-year spot rates may be 2.50%, 2.7534%, and 3.0101%, respectively.
– The values generated by the tree must match the present value of the bond’s cash flows when discounted using these spot rates.
This consistency confirms that the binomial tree is accurately calibrated.
- Validation Using Arbitrage-Free Pricing:
To validate the tree’s accuracy:
– Discount the bond’s cash flows directly using spot rates to calculate its arbitrage-free value.
– Compare this value with the bond’s calculated value at Time 0 in the tree.
For example:
A bond with a 3.2% annual coupon and a $1,000 par value generates cash flows of $32 at the end of Years 1 and 2 and $1,032 at the end of Year 3.
Using spot rates (e.g., 2.50%, 2.7534%, 3.0101%), the present value of these cash flows sums to approximately $1,005.68, matching the tree’s value at Time 0.
Pathwise Valuation for Option-Free Bonds Using a Binomial Tree (Correctly Following The Rules)
Steps for Pathwise Valuation:
Specify All Paths Through the Tree:
Each path represents a sequence of interest rate movements over the bond’s life. In a three-time-period tree, there are 4 paths:
– Path 1: Up to 3.605%, up to 5.074%.
– Path 2: Up to 3.605%, down to 3.401%.
– Path 3: Down to 2.417%, up to 3.401%.
– Path 4: Down to 2.417%, down to 2.280%.
Calculating Present Value for Path 1:
– Step 1: Discount the cash flow of 1,032 at Year 3 using the forward rate of 5.074%:
PresentValue = 1,032 ÷ (1 + 0.05074) = 982.16
– Step 2: Add the Year 2 coupon of 32, and discount the sum using the forward rate of 3.605%:
PresentValue = (982.16 + 32) ÷ (1 + 0.03605) = 978.88
– Step 3: Add the Year 1 coupon of 32, and discount the sum using the forward rate of 2.500%:
PresentValue = (978.88 + 32) ÷ (1 + 0.025) = 986.22
Averaging Results Across All Paths:
– Repeat the above process for all four paths.
– The average of all present values matches the bond’s arbitrage-free value of 1,005.68.
Monte Carlo Method for Valuation
1- Definition and Purpose:
– The Monte Carlo method simulates numerous possible interest rate paths by selecting them randomly.
– It is particularly useful for valuing securities with path-dependent cash flows, such as mortgage-backed securities, where prepayments depend on the sequence of interest rates rather than the current rate alone.
2- Key Features:
– Arbitrage-Free Calibration:
— Like binomial trees, the Monte Carlo method becomes arbitrage-free by:
—- Making an interest rate volatility assumption.
—- Calibrating the paths to the current benchmark term structure.
— A drift-adjusted model adds a constant drift term to short-term rates, ensuring the benchmark bond’s value aligns with its market price.
– Path-Dependence:
— Captures cash flow behaviors that depend on the sequence of interest rate changes, not just the terminal value at a specific node.
– Mean Reversion:
— Can simulate mean-reverting interest rates by setting upper and lower boundaries.
— Produces rates that cluster around the implied forward rates derived from the current yield curve.
3- Statistical Accuracy vs. Input Quality:
– Increasing Modeled Paths:
— Increases statistical accuracy, reducing sampling errors in simulated outputs.
— Does not guarantee closer alignment with the security’s intrinsic value unless inputs are accurate and realistic.
— Quality of the model and assumptions (e.g., volatility, drift, and yield curve structure) significantly impacts output reliability.
4- Applications:
– Mortgage-Backed Securities:
— Models prepayment behavior, which depends on the path of interest rate movements.
– Complex Derivatives:
— Useful for pricing derivatives with path-dependent payoffs (e.g., Asian options).
Interest Rate Factors
The simplest term structure models rely entirely on the one-period rate. This single factor determines the entire term structure. These one-factor models assume that all rates move in the same direction over a given interval, although they may not necessarily move by the same amount. More complex multi-factor models allow analysts to incorporate additional assumptions about, for example, the slope of the yield curve.
Interest Rate Process
1- Definition:
– Interest rate processes are stochastic models used to describe the dynamics of interest rate movements, assuming rates evolve randomly over time.
– These processes are continuous and can be applied to frameworks like binomial lattice models.
2- General Formula for the Process:
– The process consists of two components:
— A drift term that defines the path assuming zero volatility.
— A dispersion term that introduces volatility into the model.
– General formula:
“dr = θ_t * dt + σ_t * dZ”
— Where:
—- dr: Change in the short-term interest rate (𝒾).
—- θ_t * dt: Drift term, which can be constant or mean-reverting.
—- σ_t * dZ: Dispersion term, incorporating randomness through a Weiner process (Z).
3- Explanation of Terms:
– The drift term (θ_t * dt):
— Represents the expected deterministic component of rate movements.
— May vary based on assumptions like mean reversion.
– The dispersion term (σ_t * dZ):
— Adds randomness and reflects the volatility of interest rates.
— Z: A normally distributed Weiner process allowing for the possibility of negative interest rates.
4- Applications:
– Used for pricing fixed-income securities with embedded options (e.g., callable/putable bonds).
– Also applied in valuing derivatives tied to interest rates.
5- Model Preferences:
– Analysts may select models based on objectives:
— Equilibrium models: Focus on theoretical relationships (e.g., term structure of interest rates).
— Arbitrage-free models: Ensure consistency with current market data (e.g., yield curves).
– Differences in models stem from how they structure the drift and volatility components.
Class of Models for Interest Rate Dynamics
1- Overview:
– Interest rate models are categorized into arbitrage-free models and equilibrium models, each with distinct features, assumptions, and use cases.
2- Arbitrage-Free Models:
– Definition:
— Assume that bond prices and the term structure implied by those prices are correct, meaning no arbitrage opportunities exist.
– Key Features:
— Parameterized to align with current prices and rates.
— Grounded in market prices, making them suitable for applications requiring consistency with current market conditions (e.g., hedging).
– Advantages:
— Provide accurate pricing for derivatives and securities based on current market data.
– Disadvantages:
— Computationally intensive due to the higher number of parameters required.
— Outputs rely on risk-neutral probabilities, which may not align with real-world probabilities.
3- Equilibrium Models:
– Definition:
— Use fundamental economic variables to model the term structure of interest rates.
— Derive equilibrium prices for bonds and interest rate options based on these variables.
– Key Features:
— Parameter values may deviate from current market prices, making them less suitable for static decisions (e.g., hedging).
— Allow for multiple possible future interest rate paths, making them ideal for dynamic applications.
– Well-Known Examples:
— Cox-Ingersoll-Ross (CIR) Model:
—- Incorporates mean reversion and avoids negative interest rates.
— Vasicek Model:
—- Allows for negative interest rates, but also assumes mean reversion.
– Advantages:
— Simpler than arbitrage-free models.
— Suitable for dynamic analyses where future uncertainty matters.
– Disadvantages:
— Not parameterized to current market prices, reducing their accuracy for pricing based on existing data.
Equilibrium Models: Cox-Ingersoll-Ross (CIR) Model
1- Overview:
– The Cox-Ingersoll-Ross (CIR) model is a one-factor equilibrium model used to describe the dynamics of the short-term interest rate.
– The model assumes that short-term rates are mean-reverting to a long-term average and incorporates stochastic elements to account for randomness.
2- Components of the CIR Model:
– Deterministic Component (Drift Term):
— Ensures that interest rates revert to a long-run mean value (θ).
— Described as: k(θ - rt), where:
—- θ = Long-run mean of the short-term rate.
—- rt = Current short-term rate.
—- k = Speed of mean reversion (how quickly rt moves toward θ).
– Stochastic Component (Random Risk Term):
— Accounts for random interest rate movements.
— Described as: σ√rt dZ, where:
—- σ = Volatility of the short-term rate.
—- √rt = Ensures volatility is proportional to the square root of the short-term rate, avoiding negative rates.
—- dZ = A normally distributed random variable.
3- General Formula:
– CIR Formula:
“dr_t = k(θ - r_t) dt + σ√r_t dZ”
Where:
— dr_t = Expected change in the short-term rate in the next period.
— dt = Number of periods over which the rate change occurs.
4- Key Assumptions:
– Interest rates are mean-reverting, meaning they eventually return to the long-term average (θ).
– Volatility is positively correlated with the level of the short-term rate, making large rate swings less likely when rates are low.
– No negative interest rates: Since the volatility term is proportional to √rt, rates cannot drop below zero.
5- Advantages:
– Realistic modeling of interest rates by incorporating mean reversion.
– Avoids negative rates, making it suitable for many practical applications.
6- Disadvantages:
– Complexity: Requires multiple parameters (k, θ, σ) that can make calibration challenging.
– Assumes a single-factor process, which may not fully explain multi-dimensional changes in the yield curve.
7- Application:
– The CIR model is widely used in pricing interest rate derivatives and fixed-income securities due to its theoretical rigor and practical assumptions.
Equilibrium Models: Vasicek Model
1- Overview:
– The Vasicek model is another one-factor equilibrium model used to describe the dynamics of short-term interest rates.
– Like the CIR model, it assumes that interest rates are mean-reverting to a long-term average.
– However, the Vasicek model differs by assuming constant volatility, which can theoretically allow for negative interest rates.
2- Formula:
– Vasicek Formula:
“dr_t = k(θ - r_t) dt + σ dZ”
Where:
— dr_t = Expected change (differential) in the short-term rate in the next period.
— θ = Long-term mean of the short-term rate.
— r_t = Current short-term rate.
— k = Speed of mean reversion (rate at which r_t moves toward θ).
— σ = Constant volatility of the short-term rate.
— dZ = Normally distributed random variable representing stochastic changes.
— dt = Time period over which the rate change occurs.
3- Key Differences Compared to CIR Model:
– Volatility: In the Vasicek model, volatility (σ) is constant, while the CIR model links volatility to the square root of the short-term rate.
– Negative Rates: Unlike the CIR model, the Vasicek model can theoretically allow for negative interest rates due to its linear volatility assumption.
4- Assumptions:
– Interest rates are mean-reverting, eventually moving back toward the long-term average (θ).
– Volatility is constant and does not depend on the level of interest rates.
5- Advantages:
– Simplicity: The Vasicek model’s constant volatility makes it mathematically simpler and easier to implement.
– Useful for theoretical insights into interest rate dynamics.
6- Disadvantages:
– The possibility of negative interest rates is unrealistic in many practical scenarios.
– May not accurately reflect the real-world relationship between volatility and the level of interest rates.
7- Applications:
– The Vasicek model is commonly used in academic research and theoretical modeling.
– It provides a foundation for more complex models, including those that adjust for its limitations (e.g., allowing for positive-only rates).
Arbitrage-Free Models: Ho-Lee Model
1- Overview:
– The Ho-Lee model is an arbitrage-free model that uses a binomial lattice framework to simulate changes in the short-term interest rate.
– At each node in the lattice, the short-term interest rate can either increase or decrease by equal probabilities.
– This model ensures that the generated term structure is consistent with current market prices of bonds and the observed term structure.
2- Formula:
– Ho-Lee Formula:
“dr_t = θ_t dt + σ dZ”
Where:
— dr_t = Expected change (differential) in the short-term rate in the next period.
— θ_t = Time-dependent drift term ensuring arbitrage-free valuation.
— σ = Constant volatility of the short-term rate.
— dZ = Normally distributed random variable representing stochastic changes.
— dt = Time period over which the rate change occurs.
3- Key Features:
– Arbitrage-Free: Ensures consistency with the observed market prices of bonds and the term structure.
– Time-Dependent Drift: The drift term (θ_t) adjusts over time to ensure that the model generates correct term structures and avoids arbitrage opportunities.
– Constant Volatility: Assumes σ is constant, simplifying calculations.
4- Advantages:
– Ensures that interest rate simulations match current bond prices and the yield curve.
– Simple implementation using a binomial tree structure.
– Provides a normal symmetrical distribution of possible future interest rates.
5- Disadvantages:
– The assumption of constant volatility may not reflect real-world conditions where interest rate volatility varies with the level of rates.
– Negative Interest Rates: The model can produce negative rates due to its linear drift and volatility assumptions.
6- Applications:
– The Ho-Lee model is primarily used for pricing and valuing fixed-income securities, including bonds and derivatives, in markets where arbitrage-free valuation is critical.
– It serves as a foundation for more complex arbitrage-free models.
Arbitrage-Free Models: Kalotay-Williams-Fabozzi (KWF) Model
1- Overview:
– The KWF model, like the Ho-Lee model, is an arbitrage-free model assuming constant volatility and no mean reversion in interest rates.
– However, the KWF model focuses on the logarithm of the short-term interest rate rather than the rate itself.
– This ensures that the short-term interest rate is lognormally distributed, which prevents the generation of negative interest rates.
2- Formula:
– KWF Formula:
“ dln(r_t) = θ_t dt + σ dZ “
Where:
— dln(r_t) = Change in the natural logarithm of the short-term rate.
— r_t = Short-term rate, assumed to be lognormally distributed.
— θ_t = Time-dependent drift term ensuring arbitrage-free valuation.
— σ = Constant volatility of the short-term rate.
— dZ = Normally distributed random variable representing stochastic changes.
— dt = Time period over which the rate change occurs.
3- Key Features:
– Lognormal Distribution: The logarithm of the short-term rate (ln(r_t)) is modeled, ensuring that the rate remains positive and avoids unrealistic negative rates.
– Arbitrage-Free: The drift term (θ_t) adjusts to match current bond prices and the term structure.
– Constant Volatility: Assumes that volatility (σ) is constant over time.
– No Mean Reversion: Rates are not expected to revert to a long-term average value.
4- Advantages:
– The model avoids generating negative interest rates due to its lognormal distribution assumption.
– Ensures arbitrage-free valuation consistent with current market prices and yield curves.
– Suitable for applications where rates are expected to remain positive.
5- Disadvantages:
– The assumption of constant volatility may not align with real-world interest rate dynamics.
– No Mean Reversion: This limitation makes the model less applicable in environments where rates are expected to revert to historical averages.
6- Applications:
– Used for pricing and valuation of fixed-income securities in arbitrage-free contexts.
– Suitable for scenarios where maintaining positive interest rates is a critical assumption.
Modern Models
The models that have been covered to this point use a single factor, the short-term interest rate, to model the entire term structure. More recent multi-factor models are more sophisticated, using data from observed market rates as well as the volatilities implied by option prices. The Gauss+ model, for example, uses short-, medium-, and long-term rates to produce a humped-shaped volatility curve. Changes in short-term rates are not assumed to be random, which is consistent with central banks manipulating rates at this end of the curve to achieve their monetary policy objectives. Medium- and long-term rates are more volatile in comparison.
Models Allowing Negative Rates
1- Ho-Lee Model (AB Model):
– The model assumes constant volatility and a normal distribution for future rates, meaning interest rates can fall below zero.
2- Vasicek Model:
– Due to its assumption of constant volatility and lack of restrictions on rate levels, the model allows for the possibility of negative interest rates.
Models That Do Not Allow Negative Rates
1- Cox-Ingersoll-Ross (CIR) Model:
– Rates are mean-reverting with volatility proportional to the square root of the current rate. This ensures that as rates approach zero, volatility decreases, preventing negative rates.
2- Kalotay-Williams-Fabozzi (KWF) Model (AB Model):
– Models the logarithm of the short-term rate (ln(r_t)), ensuring rates remain lognormally distributed and always positive.
Written Solution:
The standard deviation of the one-year rate is calculated as the product of the volatility and the one-year rate.
Formula: Standard deviation = Volatility × One-year rate
Calculation: 0.15 × 0.0135 = 0.002025
This value can be expressed as either:
0.2025%
20.25 basis points (bps)
