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1. Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory

2. 2 Capital Asset Pricing Model Equilibrium model that underlies modern financial theory Developed by Sharpe, Lintner and Mossin Useful benchmark for the expected return of an asset �Expected rate of return (or risk premium) is determined by a security�s risk as measured by beta� Builds on Markowitz portfolio theory Each investor is assumed to diversify his or her portfolio according to the Markowitz model In this case, each asset�s risk can be measured by its contribution to the risk (�beta�) of the market portfolio

3. 3 Assumptions Individual investors can borrow or lend money at the risk-free rate of return Investors are rational mean-variance optimizers Homogeneous expectations Single-period investment horizon Investments are limited to traded financial assets Information is costless and available to all investors No taxes, and transaction costs

4. 4 Resulting Equilibrium Conditions In equilibrium, market portfolio containing all assets in proportion to its market value will be on the efficient frontier, and at the tangent point of the capital market line Therefore, it is optimal among all risky assets, and everyone will find it optimal to hold the market portfolio since it gives the highest reward-to-risk Simply buying an index mutual fund, e.g., a passive strategy, would be optimal: �Mutual Fund Theorem�

5. 5 Risk premium on an individual security is proportional to the risk premium on the market portfolio and the beta coefficient E(ri) � rf = ?i [ E(rM) � rf ] Risk premium on the market portfolio depends on the average risk aversion of all market participants E(rM) � rf = A�?M2 Resulting Equilibrium Conditions (cont.)

6. 6 Capital Market Line (CML)

7. 7 Market Risk Premium

8. 8 Risk Premium on Individual Securities Due to the diversification effect, investors want to be compensated only for bearing systematic risk Individual security�s risk premium is a function of the individual security�s contribution to the risk of the market portfolio, which is measured by beta Thus, the ratio of risk premium to beta must be the same for all securities in equilibrium Comparing this ratio with the market portfolio (?m = 1), [ E(ri) � rf ] / ?i = [ E(rM) � rf ] / 1 ?E(ri) = rf + ?i [ E(rM) � rf ] ? Security Market Line (SML) Cf. CML equation only applies to markets (M) in equilibrium and efficient portfolios

9. 9 Security Market Line (SML)

10. 10 Sample Calculations for SML E(rm) - rf = 0.08 and rf = 0.03 If bx = 1.25 E(rx) = 0.03 + 1.25(0.08) = 0.13 or 13% If bx = 1.0 E(rx) = 0.03 + 1.0(0.08) = 0.11 or 11% If by = 0.6 E(ry) = 0.03 + 0.6(0.08) = 0.078 or 7.8%

11. 11 Graph of Sample Calculations

12. 12 Disequilibrium Example

13. 13 Disequilibrium Example :Underpriced Suppose a security with a b of 1.25 is offering expected return of 15% According to SML, it should be 13% That is, it is underpriced, because it is offering too high of a rate of return for its level of risk

14. 14 CAPM and Index Models CAPM relies on a hypothetical market portfolio, and deals with expected returns as opposed to actual returns. For implementation, we cast it in the form of an index model, where actual index portfolio are used as a proxy for the market portfolio: ri � rf = ?i + ?i (rm � rf) + ?i

15. 15 Estimating an index model

16. 16 Estimation Example

17. 17 Estimation Results

18. 18 Decomposition of total variance Taking variance on both sides of the index model gives the following: ri � rf = ?i + ?i (rm � rf) + ?i ?i2 = ?i2 ?m2 + ??2 = Systematic + unsystematic components

19. 19 Arbitrage Pricing Theory Based on the Law of One Price Since two otherwise identical assets cannot sell at different prices, equilibrium prices adjust to eliminate all arbitrage opportunities Arbitrage opportunity arises if an investor can construct a zero investment portfolio with no risk, but with a positive profit Since no investment is required, an investor can create large positions in long and short to secure large levels of profits In an efficient market, profitable arbitrage opportunities will quickly disappear

20. 20 Arbitrage Example Current Expected Standard Stock Price$ Return% Dev.% A 10 25.0 29.58 B 10 20.0 33.91 C 10 32.5 48.15 D 10 22.5 8.58

21. 21 Arbitrage Example (Cont.) Consider an equally-weighted portfolio of the first three stocks Mean Std. Correlation Return Dev. Of Returns EW Portfolio of A,B,C 25.83 6.40 0.94 D 22.25 8.58 This portfolio yields higher returns and lower risk (std. dev.) than those of the asset D alone How to exploit this opportunity?

22. 22 Arbitrage Example (Cont.)

23. 23 APT Model APT assumes returns generated by a factor model Factor Characteristics Each risk factor must have a pervasive influence on stock returns Risk factors must have nonzero prices Risk factors must be unpredictable to the market The expected return-risk relationship for the APT: E(Ri) = RF + bi1 (risk premium for factor 1) + bi2 (risk premium for factor 2) + ... + bin (risk premium for factor n)

24. 24 APT and CAPM Compared APT applies to well diversified portfolios, and not necessarily to individual stocks With APT, it is possible for some individual stocks to be mispriced - not lie on the SML APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio Unlike CAPM, APT does not assume mean-variance decisions, riskless borrowing or lending, and existence of a market portfolio APT can be extended to multifactor models

25. 25 Problems with APT Factors are not well specified ex ante To implement the APT model, we need the factors that account for the differences among security returns This is a similar problem with the CAPM, which defines the unobservable market portfolio as a single factor Neither CAPM or APT has been proven superior Both rely on unobservable expectations