Investment Fundamentals Flashcards
CAPM assumptions
▶ investors are risk averse
▶ investors care only about mean and standard deviation or security returns are normally distributed
▶ NEW: investors have the same estimates of expected returns, volatilities and correlations (“homogeneous expectations”)
▶ NEW: investors can borrow/lend at the same riskfree rate
Security Market Line (SML)
- plot of the linear relationship between expected return and beta given by the CAPM
- SML is plotted in beta/return space ▶ as opposed to being plotted in standard deviation/return space (as in the CML)
the size effect
While small stocks do tend to have high beta, their returns are still high even after accounting for their higher beta
Regressions with a size factor (here Fama French’s SMB factor) as a second explanatory variable to beta - formula:
ri − rf = ai + bi * (rm − rf ) + si * rSMB + εi
Sort stocks on book-to-market (B/M):
High B/M → value stocks
Low B/M → growth stocks
value-minus-growth effect:
- Value stocks tend to have positive alphas
- growth stocks tend to have low or negative alpha
Fama-French 3-Factor Model:
- E (ri) = rf + bi * (E (rm) − rf ) + si * E (rSMB) + hi * E (rHML)
- rSMB - is the return on the ’size’ factor (return on a small cap portfolio minus return on a large cap portfolio)
- rHML - is the return on a ’value-minus-growth’ factor (return on high B/M portfolio minus return on a low B/M portfolio
- Also often written: E (ri) = rf + bi * E(MKT RF) + si * E(SMB) + hi * E(HML)
Empirical tests of the CAPM are difficult - CAPM could ’fail’ because:
▶ Behavioral biases
▶ No relation between return and marginal risk
▶ Frictionless market assumption in CAPM invalid: Non-tradable risks?
CAPM vs Multifactor Models
- CAPM: Expected returns are a function of covariance with the market portfolio
- Multifactor Models: ▶ Expected returns are a function of covariance with one or more “risk factors”. ▶ Assume that we have identified portfolios that we can combine to form an efficient portfolio; we call these portfolios factor portfolios
a one factor model
Typical factor portfolios:
Examples: Market Factor, Value Factor, Size Factor, Momentum Factor, High dividend, Quality, Volatility, …
Taking a factor/index model approach greatly simplifies the estimation of inputs, specifically covariances, into portfolio construction. The covariance between two stocks 𝑖 and 𝑗 can be simplified using their betas and the market variance - formula:
The covariance is then: σ2i,j = bi * bj * σ2m
Factor Models and Asset Allocation - we can simplify the variances/covariances of stocks for factor models, particularly index models, when estimating the inputs required for portfolio construction - moreover, firm-specific risks are uncorrelated across individual securities. For K risk factors we have:
Efficient Market Hypothesis – 3 versions
- Weak-form EMH: Stock prices already reflect all information contained in history of trading
- Semistrong-form EMH: Stock prices already reflect all public information
- Strong-form EMH: Stock prices already reflect all relevant information, including inside information
Post–earnings-announcement drift (PEAD)
tendency for stock prices to drift in the direction of an earnings surprise for several weeks following an earnings announcement
CAPM - how to calculate the beta
βi = σi,m/σ2m = ρi,m * σi/σm
CAPM – insights
- Developed by William Sharpe (published 1964, Nobel 1990) et al.
- In equilibrium, the tangency portfolio is the market portfolio
- (1) Leads to a relationship between the expected return on a security and its risk as measured by its beta
- (2) beta is the systematic risk of the security
The tangent portfolio is the market
portfolio - conditions
- Same beliefs ⇒ agree on tangency portfolio (PT )
- Everybody holds risky assets in same proportions ▶ Because risky holdings are proportional to the tangency portfolio (from two-fund separation) ▶ I hold 20% risk-free bond, 80% tangency port ▶ You hold -30% risk-free bond, 130% tangency port
- Thus, sum of everybody’s holdings has these properties: ▶ Proportional to tangent portfolio (by two-fund sep.) ▶ Equals the market portfolio (by definition).
The Capital Market Line (CML)
simply the CAL that combines investments in the riskfree asset and the market portfolio: E[ri] = rf + (E[m] − rf) / σm * σi
Contribution to excess expected return of market portfolio by individual stock - formula:
wi(E[ri] − rf )
Contribution to RISK of market portfolio by individual stock - formula:
- Not volatility (standard deviation), as firm-specific risk is diversified away, only market risk is priced
- Instead, the appropriate measure will be a function of how much that security contributes to the risk of the market portfolio, which in turn depends on the covariance of the security’s returns with the market portfolio
contribution to excess expected return of market portfolio for security i:
(E [ri] − rf) / σim –> In equilibrium this has to be the same for all securities (otherwise you can imporove the risk-return properties of the market portfolio) and it has to equal the ratio for the market as a whole:
What is beta in the CAPM equal to
βi = σi,m / σ2m = ρi,m * σi/σm
Capital Market Line (CML) - formula
rp = rf + σp * (rmkt − rf)/σmkt
What is the relation between β and standard deviation for efficient portfolios? Does this relation hold for inefficient portfolios?
- The relation between β and standard deviation for efficient portfolios is β = σp/σmkt
- Under this relation the CML agrees with the CAPM. This relation does not hold for inefficient portfolios
Formula for βi
βi = cov (ri, rm) / σ2m
OLS regression analysis for estimating beta
ri − rf = ai + bi (rm − rf ) + εi
If you assume a constant riskfree rate then, by the properties of OLS, specifically corr(rm, ϵi)=0, risk can be split into the following parts:
Sharpe Ratio formula
Sharpe Ratio = (Rp - Rf) / σm
Security Market Line (SML) - formula
ri = rf + βi * (rmkt − rf )
A security i’s alpha is defined as:
αi = E[Ri] − Rf − βi * (E[RM] − Rf )
Information Ratio (IR) - formula
- IR = αp / σ(ϵp)
- also known as the Appraisal ratio
- measures the extra return we can obtain
from security analysis
Treynor Ratio - formula
Tp = (E(rp)−rf) / βp
Constant (Gordon) Growth Model (simplified DDM) - P0 = …
P0 = D1 / (r − g)
Definition Return on equity (ROE) is defined as:
ROE = Earnings / Book value of equity
How is the reinvestment ratio of earnings also called?
The plowback ratio (or retention ratio) “b”
What is the payout ratio expressed as the plowback ratio?
1-b
Formula for “firm’s sustainable growth rate”:
g = Earnings_t+1 / Earnings_t − 1 = b ∗ ROE
P/E ratio expressed with r and b:
P/EPS = (1 − b) / (r − b × ROE)
Price-to-earnings ratio and future stock
returns - chart
low long-run growth rate (LLTG) vs high long-run growth rate (HLTG) stocks - performance - observation
- LLTG stocks appear undervalued and HLTG overvalued Observation
- LLTG portfolio earns a compounded average return of 15% in the year after formation, while the HLTG portfolio earns only 3%
- HLTG portfolio has higher market beta than the LLTG portfolio
Analyst biases/anomalies could also result from information frictions, such as:
- Sticky Information: It is costly to update forecasts frequently.
- Noisy Information: The available information contains significant noise, making it difficult to extract a clear signal about the forecasted variable.
- Rational Learning: Investors are rational but are still learning about the processes that dictate the variables being forecasted, and may not fully understand the underlying model yet
standard deviation of a return - referred to as..
volatility –> σ
Alternatives to standard deviation that focus more on downside risk (more advanced topics):
- semi-standard deviation
- (conditional) value-at-risk
- expected tail loss (the expected loss in the worst x% of outcomes)
annualise monthly variance
σ2 (annual) = 12 × σ2 (monthly)
Covariance formula
Cov(ri, rj) = σi,j = 1 / (N − 1) “Sum N, i = 1” (ri − r¯i) * (rj − r¯j )
With 2 securities, the variance of portfolio return is:
six useful rules that can be derived from the
respective definitions for portfolio variance, covariance, and correlation:
Markowitz’s Modern Portfolio Theory
- Markowitz’s Modern Portfolio Theory
- Start with individual securities whose investment properties are summarized by: ▶ Expected return, eg estimated as the historical average return ▶ Risk or volatility
- MPT assumes that either: 1. Investors do not care about higher moments (skewness, etc) 2. Security returns are normally distributed
- Frictionless markets
Are individual securities in general dominated by the
efficient frontier? What does adding securities do to the frontier?
- Yes
- Shifts it to the left
Risk in Equally-Weighted Portfolios: - does variance of portfolio return converge to average covariance of returns?
Yes
Classifications of Sources of Risk
- Part that cannot be diversified away: ▶ “covariance risk,” “systematic risk” or “non-diversifiable risk.” ▶ E.g. market risk, macroeconomic risk, industry risk
- Part that can be diversified away: ▶ “idiosyncratic risk,” “non-systematic risk,” “diversifiable risk” or
“unique risk” ▶ E.g. individual company news.
Why does the simplification σi,j =bi * bj * σm2 occur for factor models, particularly index models, when estimating the inputs required for portfolio construction, specifically covariances and variances of assets?
- because the factors are uncorrelated with each other and with firm-specific risks (the variance-covariance matrix is diagonal)
- Moreover, firm-specific risks are uncorrelated across individual securities
When you have two assets in a portfolio, 𝐴 and 𝐵, the variance of the portfolio return, 𝜎𝑃2, is calculated using:
What is the standard deviation of a portfolio of a risky and a risk free asset? What is the weight w?
- σP = square root(w2*σ2i) = |w|σi
- w = σP / σi
Capital Allocation Line - formula/derivation
Portfolio that combined with the risk free asset has the CAL with the highest slope or Sharpe Ratio - how is this portfolio known:
This portfolio is the one for which the CAL is tangent to the efficient frontier - known as the tangency or mean-variance efficient (MVE) portfolio
What term is given by the slope of the Capital Allocation Line?
The Sharpe ratio (named after William Sharpe, 1990 Nobel Laureate):
The most effective way to determine if portfolio weights align with two-fund separation theorem is to eliminate the influence of individual risk preferences. This can be achieved by:
This ratio should remain constant across different levels of risk aversion as it is independent of γi (y = risk preference)
Diversified portfolio - different approaches - some alternative diversification schemes:
PV, FV, R, T are tied by Definition - formulas
Perpetuity and growing perpetuity - formulas
PV = C / r and PV = C / (r-g)
Annuity - formula
simplest fixed income asset:
- the zero coupon bond, or simply a “zero”
- created: Treasuries issued in primary markets, coupons “stripped” and resold separately
- A multiple payment bond is just a collection of zero
coupon bonds
Growing Annuity - formula
Date of final cash flow of bonds is called
Maturity or Redemption Date
Cash flow at maturity is called
Notional, Face Value or Principal
T-Bills vs T-Notes vs T-Bonds vs TIPS
What is the shape of bond prices vs interest rates?
- convex shape, this property is called “convexity”
- means that an increase in interest rates results in a price decline that is smaller than the price increase resulting from an interest rate fall of equal magnitude.
- the increase in price as r drops from 2% to 1% is greater than the decrease in prices as r goes up from 2% to 3%
How is it called backing out interest rates from coupon paying bonds?
“bootstrap”
Bond trades above/at/below par if …
Coupon rate is above/equal to/below the interest rate
How are spot rates obtained from the coupon bond prices in practice?
through a procedure known as bootstrapping
What is the yield-to-maturity (or yield for short) of a bond?
- The single discount rate, y, that sets the PV of a bond’s promised cashflows equal to the observed price
- Is a “blend” of all the interest rates that apply to the dates of the bond’s cash flows
- The YTM is equal to the bond’s maturity interest rate only if all spot rates are identical
- The YTM on a coupon paying bond is equal to the compound return if held to maturity and the coupons can be reinvested at the original yield
The total return earned on the bond over a period is called what?
“Sometimes” called the holding period return (HPR), consists of: capital gain/loss and return due to cash income (coupon)
Can fairly priced coupon paying bonds with the same
maturity have different YTMs?
Yes, if interest rates differ by maturity (“coupon effect”)
If the investment horizon differs from the bond maturity then:
- If the investment horizon is longer than the original bond maturity, then all bonds (coupon paying or not) will face reinvestment risk on the principal
- If the investment horizon is shorter than the bond maturity, then all bonds (coupon paying or not) will face “price risk” in that the value of the bonds at the end of the horizon will not be known ex-ante (as they depend on interest rates at that future date)
What is an interest rate forward contract?
An interest rate forward contract is a contract today that fixes the interest rate for a loan in the future.
formula for the forward rate:
Where/how are forward rates traded?
Forward rates are also traded directly:
▶ FRA’s: Forward Rate Agreements.
▶ Eurocurrency Interest Rate Futures.
▶ Bond Futures.
Settled daily by the exchange to minimize counter-party risk.
different shapes of the yield curve
What is the Expectations Hypothesis?
- Under the Expectations Hypothesis long-term interest rates are determined by expectations of future short-term interest rates
- Under E.H. the long term rate is a geometric average of current and expected future short rates
Role of Risk Premium in the Expectations Hypothesis
- The EH assumes that risk premiums are negligible and that investors are indifferent to bonds with different maturities, relying solely on expectations of future interest rate movements to make decisions.
- Investors in longer maturity bonds need to earn a risk premium to compensate them for greater interest rate risk. All else equal, this should result in an upward sloping yield curve
Preferred habitat or market segmentation hypothesis (vs expectation hypothesis)
- Curve is shaped by supply and demand at various maturities
- This can lead to any shape of curve – e.g. if most investors prefer short-run investing then this could also explain the traditional upward sloping shape of the yield curve
- Commonly accepted that high demand from pension funds and insurance companies for long dated bonds depresses yields at the long end
The duration D of a bond is defined as
- minus the elasticity of its price P0 with respect to (1 plus YTM)
- The duration D is equal to the average of the times at which the cash-flows occur, weighted by their respective contribution to the present value of the bond.
The relative price response to a yield change is
What is the duration of a 4-year coupon bond with a face
value of $1,000 and a coupon rate of 8%? Current YTM
on the bond is 10%.
The duration of a portfolio is…
The weighted average of the durations of the constituents
What is the duration of a perpetuity?
What happens to the duration of a coupon bond if, all else equal, the coupon rate increases?
Duration shortens as cash-flows more concentrated in the near term
What happens to the duration of the bond if, all else equal, the YTM increases?
Duration shortens as cash-flows more concentrated in the near term
What can you say about the duration of a negative coupon rate bond?
Duration longer than maturity
The convexity of a bond is the curvature of its price-yield relationship:
What does a duration-matching balance sheet mean?
- the equity value does not change with interest rates
- Interest rate changes makes the values of assets and liabilities change by the same amount: the portfolio is immunized
Problems with Immunization - what does it assume?
- It is an approximation that assumes:
- Flat term structure of interest
- Only risk of changes in the level of interest rates; not in the slope of the term-structure or other types of shape changes.
- Small interest rate changes—can improve duration matching by also matching convexity
The fund wants to invest in 1-year and 30-year zero-coupon bonds. ▶ What fraction should it invest in 1- and 30-year bonds in order to be immunized against interest rate risk?
Two common type of options
▶ European: can be exercised only at the expiration date
▶ American: can also be exercised before expiration
Two types of payoff diagrams for options:
▶ Gross payoff diagrams: final payoff not including any premium paid (if buying the option) or received (if selling the options)
▶ Net payoff (P&L) diagrams: final payoff net of any premium paid or received
Collar options strategy:
Straddle options strategy
Bear Spread options strategy:
Other option strategies:
Lower Bounds on Option Prices
Put-Call Parity for European Options
The Binomial Option Pricing Model:
▶ A binomial model of share prices
▶ Pricing options based on replicating portfolios
▶ Pricing options using “risk-neutral” valuation
