Diversification and Portfolio Management (Ch. 8)

1. Diversification and Portfolio Management (Ch. 8) 05/10/06

2. How investors view risk and return • Investors like return. They seek to maximize return. • But investors dislike risk. They seek to avoid or minimize risk. Why? • Because human beings possess the psychological trait of “risk aversion” which is a dislike for taking risks.

3. Implications of risk aversion • The “risk-return tradeoff” - Risk averse investors require higher rates of return to induce them to invest in higher risk securities. • The higher a security’s risk, the higher the return investors demand. Thus, the less they are willing to pay for the investment, i.e. as risk increase, P0 decreases. • Risk averse investors will diversify their investments in order to reduce risk.

4. Diversification • Definition - An investment strategy designed to reduce risk by spreading the funds invested across many securities. • It is holding a broad portfolio of securities so as “not to have all your eggs in one basket.” • Since people hold diversified portfolios of securities, they are not very concerned about the risk and return of a single security. They are more concerned about the risk and return of their entire portfolio.

5. Two components of an asset’s risk (standard deviation) • Unique Risk - Also called “diversifiable risk” and “unsystematic risk.” The part of a security’s risk associated with random outcomes generated by events specific to the firm. This risk can be eliminated by proper diversification. • Market Risk – Also called “systematic risk.” The part of a security’s risk that cannot be eliminated by diversification because it is associated with economic or market factors that systematically affect most firms. Market risk reflects economy-wide sources of risk that affect most firms and, hence, the overall stock market.

6. The expected return on a portfolio of stocks • Assume N stocks are held in the portfolio. • Stock i is held in the proportion, wi • Then the expected return on the portfolio of stocks is the weighted average of the individual stock expected returns:

7. The standard deviation of returns for a portfolio of stocks • The standard deviation of returns for the portfolio of stocks is given by: where returni is the return of the portfolio in state i and n represents the number of states of the economy.

8. The standard deviation of returns for a portfolio of stocks • We can also calculate the standard deviation of returns for the portfolio of stocks as: where ρij= the correlation coefficient for stocks i and j

9. Correlation coefficient • The “Correlation Coefficient” is a measure of the extent that two variables move or vary together. • It ranges between –1.0 and +1.0 • Positive correlation: a high value on one variable is likely to be associated with a high value on the other. • Negative correlation: a high value on one variable is likely to be associated with a low value on the other. • No correlation: values of each are independent of the other

10. Correlation coefficient • It is denoted by the Greek letter, “rho”: ρ • If ρ = +1.0, perfect positive correlation • If ρ = -1.0, perfect negative correlation • If ρ = 0, uncorrelated or independent

11. How diversification reduces risk • Combining stocks into a portfolio reduces the variability of possible returns as long as the returns on the individual stocks are not perfectly correlated, i.e. as long as their correlation coefficients are less than +1.0.

12. Portfolio risk falls as you add securities

13. You can’t eliminate “market risk”

14. This pattern occurs because of the two components of a stock’s risk • Total Risk = Market risk + unique risk • The unique risk is “diversified away” when individual stocks are combined in a portfolio. • Only market risk remains. • The amount of the market risk is determined by the market risk of the individual stocks in the portfolio.

15. How should we measure portfolio risk now? • Since diversification eliminates unique risk and leaves market or non-diversifiable risk, the latter is the only relevant risk for a diversified investor. • Therefore, the relevant measure of risk for a portfolio is a measure of the sensitivity of the portfolio’s returns to changes in the return on the “market portfolio”. • This is known as the portfolio’s beta (β) • By definition, the market portfolio has a beta of 1 and the risk-free asset has a beta of 0.

16. How to interpret a beta • If βi > 1, returns to stock i are amplified relative to the market. • If βiis between 0 and 1.0, returns to stock i tend to move in the same direction as the market but not as far. • If βi < 1(very rare), returns to stock i tend to move in the opposite direction as the market.

17. How to interpret a beta-cont’d • A stock with β = 1 has average market risk. • A well-diversified portfolio of such stocks tends to move by the same percentage as the overall market moves. • A stock with β = +.5 has below average market risk. • A well-diversified portfolio of these stocks tends to move half as far as the overall market moves.

18. Measuring individual security betas • Security betas are estimated by running a regression between the historical returns on the security and the historical returns on the market portfolio over the same period of time. • Typically, betas are estimated using 5 years of historical monthly returns. • The slope of the regression represents the beta

19. General comments about risk • Most stocks are positively correlated with the market (ρi,m 0.65). • σ 0.35 for an average stock. • Combining stocks in a portfolio generally lowers risk.

20. Calculating portfolio betas • Assume N stocks are held in the portfolio. • Stock i is held in the proportion, wi • Then the portfolio beta is the weighted average of the individual stock betas:

21. Risk-return trade-off revisited • For a diversified investor whose only concern is non-diversifiable risk, measured by beta, this investor will now want higher return for a security with a higher beta. • This linear relationship between a security’s expected return and beta is formalized by the Security Market Line (SML).

22. The Security Market Line E(r) SML E(rm) rf BETA 1 where rf is the return on the risk-free security and E(rm) is the expected return on the market portfolio

23. The Security Market Line E(r) SML E(rA) E(rm) rf BETA 1 βA

24. Reward-to-risk ratio • The reward-to-risk ratio is calculated as the ratio of the excess return (beyond that of the risk-free return) that is required or expected for a particular security given its level of risk: Reward to risk ratio • This excess return (E(rA) – rf) is referred to as the asset’s risk premium, which is the return investors require beyond that of the risk free rate for security A

25. Capital Asset Pricing Model (CAPM) • The CAPM assumes that the reward-to-risk ratio of all securities are equal….giving us the following model to estimate the expected or required return on a stock: where • rf is the return on the risk-free security and is often proxied by the 3-month U.S. Treasury Bill or Treasury Bond rate; • E(rm) is the expected return on the market portfolio and is often proxied by the return on the S&P 500 index and; • E(rm) - rf represents the market risk premium

26. Jensen’s alpha (α) • Using the CAPM, and assuming that securities are priced based on this model, one can measure whether a particular security performed better or worse than expected by this model, i.e., did the security provide a return greater than that required and expected by investors? • This performance measure is called Jensen’s alpha and is calculated as follows: where kArepresents the actual return achieved by security A and E(kA) represents the expected return based on CAPM.

27. The Security Market Line and Jensen’s alpha E(r) SML E(rA) E(rm) Jensen’s alpha rA rf BETA 1 βA

28. CAPM assumptions • The CAPM relies on historical data to calculate its inputs, thus it implicitly assumes that the past is a good measure of the future • The model assumes no transaction costs, identical and complete information, rational investors and that securities that are mispriced will self-adjust. These are all efficient market assumptions.

29. CAPM limitations • Theoretically, the market portfolio should consist of all assets. • Studies have shown that additional explanatory (risk) factors must be considered in explaining security returns.