1. Investments, 8 th edition Bodie, Kane and Marcus Slides by Susan Hine McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved. CHAPTER 6 Risk Aversion and Capital Allocation to Risky Assets

2. 6-2 Risk and Risk Aversion Speculation –Considerable risk Sufficient to affect the decision –Commensurate gain: big enough risk premium Gamble –Bet or wager on an uncertain outcome for pure enjoyment of risk –e.g. fair game=investment with 0 risk premium

3. 6-3 Risk Aversion and Utility Values Risk averse investors reject investment portfolios that are fair games or worse These investors are willing to consider only risk-free or speculative prospects with positive risk premiums Intuitively one would rank those portfolios as more attractive with higher expected returns But if the risks increase along with the expected returns?

4. 6-4 Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%)

5. 6-5 Utility Function U=E(r)-(1/2)A   Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion   = variance of returns Think what the equation signifies. Compare utility from a risk free portolio

6. 6-6 Utility Function Risk avert Risk neutral Risk lover

7. 6-7 Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion Calculate these with respect to the above utility function.

8. 6-8 Figure 6.1 The Trade-off Between Risk and Returns of a Potential Investment Portfolio, P

9. 6-9 Mean-variance criterion Portfolio A is preferred to portfolio B if E(r A )  E(r B ) and     Called mean-variance criterion

10. 6-10 Figure 6.2 The Indifference Curve How's the indif. curve of a less risk averse person look like?

11. 6-11 Table 6.3 Utility Values of Possible Portfolios for an Investor with Risk Aversion, A = 4

12. 6-12 Estimating Risk Aversion Observe individuals’ decisions when confronted with risk Observe how much people are willing to pay to avoid risk –Insurance against large losses

13. 6-13 Table 6.4 Investor’s Willingness to Pay for Catastrophe Insurance

14. 6-14 Capital Allocation Across Risky and Risk- Free Portfolios Control risk –Asset allocation choice Fraction of the portfolio invested in Treasury bills or other safe money market securities Here we consider only proportion between risk-free assets and risky assets

15. 6-15 The Risky Asset Example Total portfolio value = $300,000 Risk-free value = 90,000 Risky (Vanguard & Fidelity) = 210,000 Vanguard (V) = 54% Fidelity (F) = 46%

16. 6-16 The Risky Asset Example Continued Vanguard 113,400/300,000 = 0.378 Fidelity 96,600/300,000 = 0.322 Portfolio P 210,000/300,000 = 0.700 Risk-Free Assets F 90,000/300,000 = 0.300 Portfolio C 300,000/300,000 = 1.000

17. 6-17 The Risk-Free Asset Only the government can issue default-free bonds –Guaranteed real rate only if the duration of the bond is identical to the investor’s desired holding period T-bills viewed as the risk-free asset –Less sensitive to interest rate fluctuations, inflation

18. 6-18 Figure 6.3 Spread Between 3-Month CD and T-bill Rates

19. 6-19 It’s possible to split investment funds between safe and risky assets. Risk free asset F: proxy; T-bills;rate of return r f Risky asset P: stock (or a portfolio), rate of return r p, variance σ p 2 The total portfolio C with proportion y invested in P and (1-y) in F; rate of return of C, r c Portfolios of One Risky Asset and a Risk- Free Asset

20. 6-20 r f = 7%  f = 0% E(r p ) = 15%  p = 22% y = % in P(1-y) = % in F Example Using Chapter 6.4 Numbers

21. 6-21 r c = complete or combined portfolio E(r c )=E(yr p +(1-y)r f )=r f +y(E(r p )-r f ) For example, y =.75 E(r c ) =.75(.15) +.25(.07) =.13 or 13% Expected Returns for Combinations

22. 6-22 c =.75(.22) =.165 or 16.5% If y =.75, then c = 1(.22) =.22 or 22% If y = 1 c = (.22) =.00 or 0% If y = 0    Combinations Without Leverage

23. 6-23 Borrow at the Risk-Free Rate and invest in stock: y>1 and 1-y<0. Using 50% Leverage, r c = (-.5) (.07) + (1.5) (.15) =.19  c = (1.5) (.22) =.33 Capital Allocation Line with Leverage

24. 6-24 Figure 6.4 The Investment Opportunity Set with a Risky Asset and a Risk-free Asset in the Expected Return-Standard Deviation Plane

25. 6-25 Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates

26. 6-26 Risk Tolerance and Asset Allocation The investor must choose one optimal portfolio, C, from the set of feasible choices –Trade-off between risk and return –Expected return of the complete portfolio is given by: –Variance is:

27. 6-27 Risk tolerance and... An inverstor chooses the best y to maximize his utility. Max y U=E(r c )-(1/2)Aσ c 2 =r f +y(E(r p )-r f )- (1/2)Ay 2 σ p 2 →y * =(E(r p )-r f )/(Aσ p 2 ) optimal position

28. 6-28 Table 6.5 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4

29. 6-29 Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y

30. 6-30 Table 6.6 Spreadsheet Calculations of Indifference Curves

31. 6-31 Figure 6.7 Indifference Curves for U =.05 and U =.09 with A = 2 and A = 4

32. 6-32 Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves

33. 6-33 Table 6.7 Expected Returns on Four Indifference Curves and the CAL

34. 6-34 Passive Strategies: The Capital Market Line Passive strategy involves a decision that avoids any direct or indirect security analysis Supply and demand forces may make such a strategy a reasonable choice for many investors

35. 6-35 Passive Strategies: The Capital Market Line Continued A natural candidate for a passively held risky asset would be a well-diversified portfolio of common stocks Because a passive strategy requires devoting no resources to acquiring information on any individual stock or group we must follow a “neutral” diversification strategy

36. 6-36 Table 6.8 Average Annual Return on Stocks and 1-Month T-bills; Standard Deviation and Reward- to-Variability Ratio of Stocks Over Time