Lecture 10 The Capital Asset Pricing Model

Expectation, variance, standard error (deviation), covariance, and correlation of returns may be based on (i) fundamental analysis (ii) historical data Preliminaries Fundamental or Theoretical Analysis S possible states  s probability of state s = 1,2,…,S R s likely return is state s Notation

4 business cycle states (boom, normal, recession, depression) 3 industry demand states 2 firm demand share states 3 firm cost states Then, there are 4*3*2*3 = 72 possible states (or situations) Example: Suppose there are Expectation (mean) Variance Standard error

returns on stock A R As s = 1,…,S returns on stock B R Bs s = 1,…,S Covariance measures how two random variables are related Correlation is a normalized covariance Note !

Example:Suppose we have a theoretical model that predicts the following returns on stocks A and B in 3 states. Expected returns Variances

Standard errors Covariance Correlation Returns on stocks A and B are perfectly negatively correlated. Stocks A can be used as a hedge against the risk in holding stock B

Historical Data Based Approach From historical data, calculate the percentage returns R 1, R 2, …, R T Sample standard deviation (or standard deviation) Sample average percentage return Sample Variance

Historical Data Based Approach (continued) Sample covariance of returns on stocks A and B, calculated from the historical samples of R A and R B R A = (R A1, …, R AT ) ; R B = (R B1, …, R BT ) Sample correlation of R A and R B

Expected Return and Variance of Returns on Portfolios A portfolio is an investment in stocks. Let be the proportion invested in stock n. Then If the return on stock n is R n, then the return on the portfolio is and the expected return on the portfolio is

The variance of the returns on the portfolio is given by Expected Return and Variance of Returns on Portfolios (continued)

Diversification 1. Variances are diversified away 2. Average covariance converges to covariance from economy- wide shocks affecting all stocks Consider a special case with for each. Then - In a diversified portfolio, only systematic risk affects returns. - Diversifiable or unsystematic (idiosyncratic) risk is irrelevant to returns.

Diversification (continued) Recall ; Suppose you invest $100 in stock A and $200 in stock B. Returns on investment in assets A and B The mean return on the portfolio is 10%.

Diversification (continued) The standard deviation of the return on the portfolio is zero. No risk! The mean return on the portfolio is a weighted average of and Recall that the correlation between the returns on A and B is -1. This implies that the variation in returns on either asset can be completely offset by holding the right proportion of the other asset.

Deriving an appropriate discount rate for risky cash flows 1. The opportunity set for two assets 3. The efficient set with a riskless asset 2. The opportunity set and efficient set with many securities 4. The CAPM (capital asset pricing model) equation 5. A risk-return separation theorem

The opportunity set for two assets Suppose there are two assets A and B in proportions and. Then, since.

The opportunity set for two assets (continued) From, we have. Then we have Using the above equation, we can trace a feasible (or opportunity) set of attainable and for given

Example We are given the following parameter values, For these values, the above equation becomes approximately which looks like the following in space.

Example (continued) Opportunity set for assets A and B Portfolio MV (minimum variance) has the lowest risk obtainable with assets A and B. Between B and MV, replacement of B by A increases and reduces. This always happens if and may happen for. When, a riskless portfolio can be obtained by holding A and B in right proportions.

The opportunity set and efficient set with many securities Each pair of securities ((A,B),(A,C),(B,C)) gives an opportunity set Except for portfolios close to MV, the efficient set is very close to a straight line. Also as the variance of the MV portfolio decreases, the efficient set gets closer to a straight line. Suppose we add asset C, to the previous example, with the parameter values Linear combination of portfolios in any of these opportunity set will lead to additional curve in s space. It can be shown that the opportunity set for assets is an area bounded by a rectangular hyperbola.

The efficient set with a riskless asset If one asset is riskless, the variance of returns on that asset, and the covariance with returns on all other assets will be zero. In equilibrium, the riskless rate < return on MV. Hence, the opportunity set will be the tangent line from the riskless asset to the efficient set. In the two security case discussed earlier, suppose B is riskless, I.e.,. Then from the above equation, we have

The efficient set with a riskless asset (continued) Homogeneous expectations assumption All investors have the same estimates on expectations, variances and covariances. Under homogeneous expectations, all investors would hold the portfolio of risky assets represented by the tangency portfolio. It is a market-valued weighted portfolio of all existing securities, I.e. market portfolio. A proxy commonly used is S&P 500. What is the tangency portfolio? Use of such a broad-based index as a proxy is justified since most investors hold diversified portfolios.

Formula for beta covariance between the return on asset i and the return on market variance of market portfolio The efficient set with a riskless asset (continued) The best measure of the risk of a security in a large portfolio is the beta of the security, which measures the responsiveness of the security to the movements in the market portfolio.

Example

Example (continued) The beta coefficient for this firm is Returns on this firm’s stock magnify market returns.

The CAPM equation Relationship between risk and expected return If there is a riskless asset with return r, there is a straight line trade off between risk and expected return for a security. is the contribution of this security to the portfolio risk. If the tangency portfolio is the market portfolio with expected return and standard deviation, then

The CAPM equation (continued) Equilibrium expected return on asset j : It can be shown that Then we have CAPM equation

The CAPM equation (continued) (Expected return on a security) = (current risk free interest rate) + (beta coefficient of the security)*(historical market risk premium) CAPM equation Finally, we established a way of determining appropriate discount rate for risky cash flows. We first measure its risk by its beta coefficient, and then obtain the required return from the CAPM equation.

The CAPM equation (continued) Interpretation Recall that the variance of return on a diversified portfolio is basically the “average covariance”. The beta coefficient for asset j can be considered as the share of overall market risk contributed by asset j. Then CAPM equation says that an asset shares the market excess return to the extent that it contributes to the total market risk. In practice, we usually estimate using linear regression using historical returns data on and Regression

The CAPM equation (continued) statistical (least squares) estimator for

The Security Market Line (SML) When The Security Market Line (SML) below graphs expected return against beta, using the CAPM equation. Slope of the SML is the risk premium. For the S&P500 and US treasury bills, the risk premium is about 8.5%. (The book uses 9.2%, which is based on Ibbotson et. al study). This estimate is often used as a forecast for the risk premium on stocks in the future.

SML (continued) The SML applies to portfolios as well as individual securities. For a portfolio with of A and of B, with beta coefficients and the expected return on the portfolio is Note that implying Hence, the portfolio also will be on the SML. The SML should not be confused with the efficient set.

A Risk-Return Separation Theorem An investment will be worth taking only if it is at least as desirable as what is already available in the financial markets. A new investment will be worthwhile if and only if it is outside (above) the efficient set (or the risk-return budget constraint). No matter where individual would choose to be on the efficient set, an investment can only make them better off if it is above the efficient set. If the two financial separation theorems did not hold, then the firms would need to know the inter-temporal and risk-return preferences of each owner to decide desirable investments.

There are 3 securities in the market with the following payoffs: What are expected returns and standard deviations of the returns? Problem from the text

What are covariances and correlations between the returns? For j = A,B,C and k = A,B,C Problem from the text (continued)

What are expected returns and standard deviations of the portfolios?

Problem from the text Suppose you have invested $30,000 in the following 4 stocks The risk free rate is 4% and the expected return on the market portfolio is 15%. Based on the CAPM, what is the expected return on the above portfolio? Let denote the proportion invested in stock i (I=A,B,C,D) and the beta coefficient of the stock i.

Problem from the text (continued) There are two ways to answer the question. 1. Calculate the beta coefficient for the portfolio, and get the expected return on the portfolio directly from CAPM equation. 2. Calculate the expected return individually for I = A,B,C,D and obtain the expected return on the portfolio as