Capital Asset Pricing Model and Arbitrage Theory

1. Capital Asset Pricing Model and Arbitrage Theory Riccardo Colacito

2. Capital Asset Pricing Model (CAPM) • Equilibrium model that underlies all modern financial theory • Derived using principles of diversification with simplified assumptions • Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development

3. Link to Factor Models • What risk should be priced? • Idiosyncratic risk: no • Aggregate risk: yes • Only aggregate/macro risk commands a premium

4. Why? Because: • idiosyncratic risk can be diversified away • Macro risk affects all assets and cannot be diversified Remember our example using factor models?

5. Example: two assets

6. What is the portfolio variance? Systematic Risk Idiosyncratic Risk

7. Example: three assets

8. What is the portfolio variance? Systematic Risk Idiosyncratic Risk • Systematic risk: unchanged • Idiosyncratic risk: decreased • Can you guess what would happen if we had an infinite number of assets?

9. Infinite assets • For a well diversified portfolio • That is: we got rid of any idiosyncratic shock and we are left only with systematic risk

10. What lesson did we learn? • The only source of risk that we are entitled to ask a compensation for is aggregate risk. • Idiosyncratic risk does not entitle to any compensation because it can be diversified away.

11. Risk compensation

12. The fundamental equation of the Capital Asset Pricing Model (CAPM) Any asset entitles to a risk premium that is proportional to the risk premium of the market portfolio. The b of the asset is the coefficient of proportionality.

13. Questions • What assumptions need to be satisfied for the CAPM to hold? • Why the market portfolio? What is it anyway? • Why b? What is b anyway? • What are the empirical predictions of the CAPM? • Can we measure it from the data? • If so, is it empirically accepted?

14. Assumptions • Individual investors are price takers • Single-period investment horizon • Investments are limited to traded financial assets • No taxes, and transaction costs • Lending and borrowing can be done at same rate • Information is costless and available to all investors • Investors are rational mean-variance optimizers • Homogeneous expectations

15. Assumptions… in other words! • Everybody knows how to solve the problem that we discussed during the last three classes • Everybody is forecasting returns, variances and correlations in the same way • … if this is the case we are all going to end up with same optimal risky portfolio!

16. The Efficient Frontier and the Capital Market Line

17. Some re-labeling • We call the optimal CAL, the Capital Market Line • We call the optimal risky portfolio, the Market Portfolio • Remember the separation property?

18. Separation property • Aka mutual fund theorem • All investors desire the same portfolio of risky assets: the optimal risky portfolio or market portfolio.

19. This answers: Why the market portfolio? • If everybody is holding this portfolio, then it is an excellent candidate to proxy for aggregate market risk. • We are now left with the question of why b is the coefficient of proportionality!

20. First things first: what is b? • Remember factor models • Remember how we computed b

21. b in words • The more correlated the asset is with market the larger b is. • What does it mean in terms of the CAPM?

22. b and the risk premium • It means that the higher the correlation with the market, the higher the risk premium that an asset commands. • Why?

23. b and the risk premium: intuition Which asset is more appealing: • An asset whose return goes up when the market goes up • An asset whose return goes down when the market goes up

24. Answer • Everything else equal you prefer an asset that co-varies negatively with the market, because it gives you an insurance against bad states of the world. • If an asset is perceived as a good asset because it provides an hedge against bad states, its demand will go up, increasing the price and decreasing the expected return. • How about an example?

25. Example • Two assets, the market portfolio, a risk free rate. Two states of the world. • Let’s compute b’s and required risk premia!

26. Results • Some computations: • Hence

27. What did we learn? • Asset 2 doesn’t do too much to protect us against market fluctuations: therefore it must `promise’ a higher return to convince us to buy it • Asset 1 can protect us against market fluctuations and therefore we are willing to buy it even if its return is low in expectation.

28. What happens to the price of the first security? • The CAPM says that the expected return of asset 1 should be 0 • However at the current prices, asset 1 has an expected return of 1 • Looks like a great deal! • Investors will want to buy more of asset 1 • But if its current price goes up, its return will go down in expectation • When does the price increase stop? • When the expected return at the current prices equals the CAPM expected return • What happens to the current price of security 2? • Nothing! This security is price correctly!

29. Empirical predictions of the CAPM • Remember the one factor model: • The intercept should be equal to zero!

30. The Security Market Line and Positive Alpha Stock

31. Alpha (a) • The abnormal rate of return on a security in excess of what would be predicted by an equilibrium model such as the CAPM • Can we test this prediction?

32. Estimating the Index Model • Using historical data on T-bills, S&P 500 and individual securities • Regress risk premiums for individual stocks against the risk premiums for the S&P 500 • Slope is the beta for the individual stock

33. Characteristic Line for GM

34. Statistical analysis • In this example we cannot reject that a is equal to zero • However in many cases a is significantly different from zero: what does it mean?

35. Rejecting the CAPM? • Not necessarily! • We may have chosen an imprecise proxy for the market risk: after all the Market Portfolio is not directly observable. • We may be omitting some risk factors: research shows that this is possible.

36. Fama-French Research • Returns are related to factors other than market returns • Size • Book value relative to market value • Three factor model better describes returns

37. Why are these factors supposed to help? • Firms with high ratios of book to market value are more likely to be in financial distress • Small firms are more sensitive to changes in business conditions

38. Fama-French applied • A Fama-French three factors regression for GM • where

39. Regression Statistics for the Single-index and FF Three-factor Model

40. Resulting Equilibrium Conditions • All investors will hold the same portfolio for risky assets – market portfolio • Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value • Risk premium on an individual security is a function of its covariance with the market

41. Arbitrage Pricing Theory (APT) Arbitrage - arises if an investor can construct a zero beta investment portfolio with a return greater than the risk-free rate • If two portfolios are mispriced, the investor could buy the low-priced portfolio and sell the high-priced portfolio • In efficient markets, profitable arbitrage opportunities will quickly disappear

42. APT and well diversified portfolios • A well diversified portfolio has no exposure to idiosyncratic risk • Claim: aP must equal zero. Why?

43. aP=0 to avoid arbitrage opportunities • Take two well-diversified portfolios • Pick the following shares of investment • The resulting portfolio is not subject to any risk and provides a non-zero return • Hence a’s must be equal to zero

44. Conclusion • The equilibrium condition is the same as the CAPM • Only assumption needed is the absence of arbitrage opportunities • Can be extended beyond well diversified portfolios