Investment Analysis and Portfolio Management

1. Investment Analysis and Portfolio Management Lecture 7 Gareth Myles

2. The Capital Asset Pricing Model (CAPM) • The CAPM is a model of equilibrium in the market for securities. • Previous lectures have addressed the question of how investors should choose assets given the observed structure of returns. • Now the question is changed to: • If investors follow these strategies, how will returns be determined in equilibrium?

3. The Capital Asset Pricing Model (CAPM) • The simplest and most fundamental model of equilibrium in the security market • Builds on the Markowitz model of portfolio choice • Aggregates the choices of individual investors • Trading ensures an equilibrium where returns adjust so that the demand and supply of assets are equal • Many modifications/extensions can be made • But basic insights always extend

4. Assumptions • The CAPM is built on a set of assumptions • Individual investors • Investors evaluate portfolios by the mean and variance of returns over a one period horizon • Preferences satisfy non-satiation • Investors are risk averse • Trading conditions • Assets are infinitely divisible • Borrowing and lending can be undertaken at the risk-free rate of return • There are no taxes or transactions costs

5. Assumptions • The risk-free rate is the same for all • Information flows perfectly • The set of investors • All investors have the same time horizon • Investors have identical expectations

6. Assumptions • The first six assumptions are the Markowitz model • The seventh and eighth assumptions add a perfect capital market and perfect information • The final two assumptions make all investors identical except for their degree of risk aversion

7. Direct Implications • All investors face the same efficient set of portfolios

8. Direct Implications • All investors choose a location on the efficient frontier • The location depends on the degree of risk aversion • The chosen portfolio mixes the risk-free asset and portfolio M of risky assets

9. Separation Theorem • The optimal combination of risky assets is determined without knowledge of preferences • All choose portfolio M • This is the Separation Theorem • M must be the market portfolio of risky assets • All investors hold it to a greater or lesser extent • No other portfolio of risky assets is held • There is a question about the interpretation of this portfolio

10. Equilibrium • The only assets that need to be marketed are: • The risk-free asset • A mutual fund representing the market portfolio • No other assets are required • In equilibrium there can be no short sales of the risky assets • All investors buy the same risky assets • No-one can be short since all would be short • If all are short the market is not in equilibrium

11. Equilibrium • Equilibrium occurs when the demand for assets matches the supply • This also applies to the risk-free • Borrowing must equal lending • This is achieved by the adjustment of asset prices • As prices change so do the returns on the assets • This process generates an equilibrium structure of returns

12. The Capital Market Line • All efficient portfolios must lie on this line • Slope = • Equation of the line

13. Interpretation • rf is the reward for "time" • Patience is rewarded • Investment delays consumption • is the reward for accepting "risk" • The market price of risk • Judged to be equilibrium reward • Obtained by matching demand to supply

14. Security Market Line • Now consider the implications for individual assets • Graph covariance against return • The risk on the market portfolio is • The covariance of the risk-free asset is zero • The covariance of the market with the market is

15. Security Market Line • Can mix M and the risk-free asset along the line • If there was a portfolio above the line all investors would buy it • No investor would hold one below • The equation of the line is M

16. Security Market Line • Define • The equation of the line becomes • This is the security market line (SML)

17. Security Market Line • There is a linear trade-off between risk measured by and return • In equilibrium all assets and portfolios must have risk-return combinations that lie on this line

18. Market Model and CAPM • Market model uses • CAPM uses • is derived from an assumption about the determination of returns • it is derived from a statistical model • the index is chosen not specified by any underlying analysis • is derived from an equilibrium theory

19. Market Model and CAPM • In addition: • I is usually assumed to be the market index, but in principal could be any index • M is always the market portfolio • There is a difference between these • But they are often used interchangeably • The market index is taken as an approximation of the market portfolio

20. Estimation of CAPM • Use the regression equation • Take the expected value • The security market line implies • It also shows

21. CAPM and Pricing • CAPM also implies the equilibrium asset prices • The security market line is • But where pi(0) is the value of the asset at time 0 and pi(1) is the value at time 1

22. CAPM and Pricing • So the security market line gives • This can be rearranged to find • The price today is related to the expected value at the end of the holding period

23. CAPM and Project Appraisal • Consider an investment project • It requires an investment of p(0) today • It provides a payment of p(1) in a year • Should the project be undertaken? • The answer is yes if the present discounted value (PDV) of the project is positive

24. CAPM and Project Appraisal • If both p(0) and p(1) are certain then the risk-free interest rate is used to discount • The PDV is • The decision is to accept project if

25. CAPM and Project Appraisal • Now assume p(1) is uncertain • Cannot simply discount at risk-free rate if investors are risk averse • For example using will over-value the project • With risk aversion the project is worth less than its expected return

26. CAPM and Project Appraisal • One method to obtain the correct value is to adjust the rate of discount to reflect risk • But by how much? • The CAPM pricing rule gives the answer • The correct PDV of the project is