Risk, Return, Portfolio Theory and CAPM

1. Risk, Return, Portfolio Theory and CAPM TIP If you do not understand anything, ask me! Where does the discount rate (for stock valuation) come from?

2. So far, • We have taken the discount rate as given. • We have also learned the factors that determine the discount rate in bond valuation. • What is the appropriate discount rate in stock valuation?

3. Topics • Risk and returns • 75 Years of Capital Market History • How to measuring risk • Individual security risk • Portfolio risk • Diversification • Unique risk • Systematic risk or market risk • Measure market risk: beta • CAPM

4. The Value of an Investment of $1 in 1926 6402 2587 64.1 48.9 16.6 Index 1 Year End Source: Ibbotson Associates

5. The Value of an Investment of $1 in 1926 Real returns 660 267 6.6 5.0 1.7 Index 1 Year End Source: Ibbotson Associates

6. Rates of Return 1926-2000 Percentage Return Year Source: Ibbotson Associates

7. Selected Realized Returns, 1926 – 2001 Average Standard ReturnDeviation Small-company stocks 17.3% 33.2% Large-company stocks 12.7 20.2 L-T corporate bonds 6.1 8.6 L-T government bonds 5.7 9.4 U.S. Treasury bills 3.9 3.2 Source: Ibbotson Associates.

8. Risk premium • Risk aversion – in Finance we assume investors dislike risk, so when they invest in risky securities, they require a higher expected rate of return to encourage them to bear the risk. • The risk premium is the difference between the expected rate of return on a risky security and the expected rate of return on a risk-free security, e.g., T-bills. • Over the last century, the average risk premium is about 7% for stocks.

9. Measuring Risks • How to measure the risk of a security? • Stand-alone risk: when the return is analyzed in isolation. This provides a starting point. • Portfolio risk: when the return is analyzed in a portfolio. This is what matters in reality when people hold portfolios.

10. PART I: Standard alone risk • The risk an investor would face if s/he held only one asset. • Investment risk is related to the probability of earning a low or negative actual return. • The greater the chance of lower than expected or negative returns, the riskier the investment. • The greater the range of possible events that can occur, the greater the risk

11. Firm X Firm Y Rate of Return (%) -70 0 15 100 Expected Rate of Return Probability distributions Which firm is more likely to have a return closer to its expected value? • A listing of all possible outcomes, and the probability of each occurrence. • Can be shown graphically.

12. Measuring Risk In financial markets, we use the volatility of a security return to measure its risk. Variance – Weighted average value of squared deviations from mean. Standard Deviation – Squared root of variance.

13. Some basic concepts • Some basic formula for Expectation and Variance • Let X be a return of a security in the next period. Then we have

14. Investment alternatives

15. Return: Calculating the expected return for each alternative

16. Risk: Calculating the standard deviation for each alternative

17. Standard deviation calculation

18. Comparing risk and return

19. Comments on standard deviation as a measure of risk • Standard deviation (σi) measures “total”, or stand-alone, risk. • The larger the σi , the lower the probability that actual returns will be closer to expected returns, that is : the larger the stand-alone risk.

20. PART II: Risk in a portfolio context • Portfolio risk is more important because in reality no one holds just one single asset. • The risk & return of an individual security should be analyzed in terms of how this asset contributes the risk and return of the whole portfolio being held.

21. Portfolio • A portfolio is a set of securities and can be regarded as a security. • If you invest W dollars in a portfolio of n securities, let Wi be the money invested in security i, then the portfolio weight on stock i is , with property

22. Example 1 • Suppose that you want to invest $1,000 in a portfolio of IBM and GE. You spend $200 on IBM and the other $800 on GE. What is the portfolio weight on each stock?

23. Portfolio return and risk of two stocks

24. In a portfolio… • In English, we say: • The expected return = weighted average of each stock’s expected return. • But the portfolio standard deviation is <= the weighted average of each stock’s standard deviation. • I Mathematics, we say:

25. (Not required) Suppose a portfolio is made up of x1 shares of stock 1 and x2 shares of stock 2.

26. Suppose you invest 50% in HT and 50% in Coll. What are the expected returns and standard deviation for the 2-stock portfolio?

27. Calculating portfolio expected return

28. An alternative method for determining portfolio expected return

29. Calculating portfolio standard deviation

30. Comments on portfolio risk measures • σp = 3.3% is lower than the weighted average of HT and Coll.’s σ (16.7%). This is true so long as the two stocks’ returns are not perfectly positively correlated. • Perfect correlation means the returns of two stocks will move exactly in same rhythm. • Portfolio provides average return of component stocks, but lower than average risk. • The more negatively correlated the two stocks, the more dramatic reduction in portfolio standard deviation.

31. Stock W Stock M Portfolio WM 25 15 0 0 0 -10 -10 Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0) 25 25 15 15 -10

32. Stock M’ Portfolio MM’ Stock M 25 25 25 15 15 15 0 0 0 -10 -10 -10 Returns distribution for two perfectly positively correlated stocks (ρ = 1.0)

33. General comments about risk • Most stocks are positively correlated with the market (ρk,m 0.65). • σ 35% for an average stock. • Combining stocks in a portfolio generally lowers risk.

34. Total Risk • A stock’s realized return is often different from its expected return. • Total return= expected return + unexpected return • Unexpected return=systematic portion + unsystematic portion • Thus: Total return= expected return +systematic portion + unsystematic portion • Total risk (stand-alone risk)= systematic portion + unsystematic portion

35. Systematic Risk • Total risk (stand-alone risk)= systematic portion + unsystematic portion • The systematic portion will be affected by factors such as changes in GDP, inflation, interest rates, etc. • This portion is not diversifiable because the factor will affect all stocks in the market. • Such risk factors affect a large number of stocks. Also called Market risk, non-diversifiable risk, beta risk.

36. Unsystematic Risk • Total risk (stand-alone risk)= systematic portion + unsystematic portion • This portion is affected by factors such as labor strikes, part shortages, etc, that will only affect a specific firm, or a small number of firms. • Also called diversifiable risk, firm specific risk.

37. Diversification • Portfolio diversification is the investment in several different classes or sectors of stocks. • Diversification is not just holding a lot of stocks. • For example, if you hold 50 internet stocks, you are not well diversified.

38. Creating a portfolio:Beginning with one stock and adding randomly selected stocks to portfolio • σp decreases in general as stocks added. • Expected return of the portfolio would remain relatively constant. • Diversification can substantially reduce the variability of returns with out an equivalent reduction in expected returns. • Eventually the diversification benefits of adding more stocks dissipates (after about 10 stocks), and for large stock portfolios, σp tends to converge to  20%.

39. sp (%) Company-Specific Risk 35 Stand-Alone Risk, sp 20 0 Market Risk 10 20 30 40 2,000+ # Stocks in Portfolio Illustrating diversification effects of a stock portfolio

40. Breaking down sources of total risk Total risk= systematic portion (market risk) + unsystematic portion (firm-specific risk) • Market risk (systematic risk, non-diversifiable risk, beta risk) – portion of a security’s stand-alone risk that cannot be eliminated through diversification. It is affected by economy-wide sources of risk that affect the overall stock market. • Firm-specific risk (unsystematic risk, diversifiable risk, idiosyncratic risk) – portion of a security’s stand-alone risk that can be eliminated through proper diversification. • If a portfolio is well diversified, unsystematic is very small. Rational, risk-averse investors are just concerned with portfolio standard deviation σp, which is based upon market risk. That is: investor care little about a stock’s firm–specific risk.

41. How do we measure a tock’s systematic (market) risk? Beta • Measures a stock’s market risk, and shows a stock’s volatility relative to the market (i.e., degree of co-movement with the market return.) • Indicates how risky a stock is if the stock is held in a well-diversified portfolio.

42. Calculating betas • Run a regression of past returns of a security against past market returns. • Market return is the return of market portfolio. • Market Portfolio - Portfolio of all assets in the economy. In practice, a broad stock market index, such as the S&P Composite, is used to represent the market. • The slope of the regression line (called the security’s characteristic line) is defined as the beta for the security.

43. _ ki . 20 15 10 5 . Year kM ki 1 15% 18% 2 -5 -10 3 12 16 _ -5 0 5 10 15 20 kM Regression line: ki = -2.59 + 1.44 kM . -5 -10 ^ ^ Illustrating the calculation of beta (security’s characteristic line)

44. Security Character Line • What does the slope of SCL mean? • Beta • What variable is in the horizontal line? • Market return. • The steeper the line, the more sensitive the stock’s return relative to the market return, that is, the greater the beta.

45. Comments on beta • A stock with a Beta of 0 has no systematic risk • A stock with a Beta of 1 has systematic risk equal to the “typical” stock in the marketplace • A stock with a Beta greater than 1 has systematic risk greater than the “typical” stock in the marketplace • A stock with a Beta less than 1 has systematic risk less than the “typical” stock in the marketplace • The market return has a beta=1 (why?). • Most stocks have betas in the range of 0.5 to 1.5.

46. Can the beta of a security be negative? • Yes, if the correlation between Stock i and the market is negative. • If the correlation is negative, the regression line would slope downward, and the beta would be negative. • However, a negative beta is very rare. • A stock with negative beta will give your higher return in recession and hence is more valuable to investors, thus required rate of return is lower.

47. _ ki HT: β = 1.30 40 20 T-bills: β = 0 _ kM -20 0 20 40 Coll: β = -0.87 -20 Beta coefficients for HT, Coll, and T-Bills

48. Comparing expected return and beta coefficients SecurityExp. Ret. Beta HT 17.4% 1.30 Market 15.0 1.00 USR 13.8 0.89 T-Bills 8.0 0.00 Coll. 1.7 -0.87 Riskier securities have higher returns, so the rank order is OK.

49. Until now... • We argued that well-diversified investors only cares about a stock’s systematic risk (measured by beta). • The higher the systematic risk, the higher the rate of return investors will require to compensate them for bearing the risk. • This extra return above risk free rate that investors require for bearing the non-diversifiable risk of a stock is called risk premium.

50. Beta and risk premium • That is: the higher the systematic risk (measured by beta), the greater the reward (measured by risk premium). • risk premium =expected return - risk free rate. • In equilibrium, all stocks must have the same reward to systematic risk ratio. • For any stock i and stock m: • (ri-rf)/beta(i)=(rm-rf)/beta(m)