1. Risk Aversion and Capital Allocation to Risky Assets
CHAPTER 6
Risk Aversion and Capital Allocation to Risky Assets

2. Three Steps in Investment Decisions – Top-down Approach
I. Capital Allocation Decision
Allocate funds between risky and risk-free assets
Made at higher organization levels 
II. Asset Allocation Decision
Distribute risk investments across asset classes – small-cap stocks, large-cap stocks, bonds, & foreign assets
 III. Security Selection Decision
Select particular securities within each asset class
Made at lower organization levels

3. Risk and Risk Aversion Speculation Gamble Considerable risk
Sufficient to affect the decision
Commensurate gain
Gamble
Bet or wager on an uncertain outcome

4. 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

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

7. Utility Function Where U = utility
E ( r ) = expected return on the asset or portfolio
A = coefficient of risk aversion
s2 = variance of returns

8. Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion

9. 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

10. Figure 6.2 The Indifference Curve10

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

12. 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

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

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

15. 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 desire holding period
T-bills viewed as the risk-free asset
Less sensitive to interest rate fluctuations

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

17. Portfolios of One Risky Asset and a Risk-Free Asset
It’s possible to split investment funds between safe and risky assets.
Risk free asset: proxy; T-bills
Risky asset: stock (or a portfolio)

18. Example Using Chapter 6.4 Numbers
rf = 7%
rf = 0%
E(rp) = 15%
p = 22%
y = % in p
(1-y) = % in rf

19. Expected Returns for Combinations
rc = complete or combined portfolio
For example, y = .75
E(rc) = .75(.15) + .25(.07)
= .13 or 13%

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

21. Capital Allocation Line (CAL)
E(rc) = yE(rp) + (1 – y)rf
= rf +[(E(rp) – rf)]y (1)
σc = yσp → y = σc/σp (2)
From (1) and (2)
E(rc) = rf +[(E(rp) - rf)/σp]σc (CAL)

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

23. Capital Allocation Line with Leverage
Borrow at the Risk-Free Rate and invest in stock.
Using 50% Leverage,
rc = (-.5) (.07) + (1.5) (.15) = .19
c = (1.5) (.22) = .33

24. Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates

25. 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:

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

27. Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y

28. Analytical Solution
U = E(rc) – (1/2)Aσc2 (1) where E(rc) = yE(rp) + (1-y)rf (2) σc = yσp (3) Substituting (2) and (3) into (1), we obtain U = yE(rp) + (1-y)rf – (1/2)A(yσp)2 From dU/dy = E(rp) – rf – Ayσp2 = 0, y* = (E(rp) – rf)/Aσp2

29. y* = (E(rp) – rf)/Aσp2 = (0.15 – 0.07)/4*(0.22)2 = 0.413

30. Indifference curve
We can trace combinations of E(rc) and σc for given values of U and A. From U = E(rc) – (1/2)Aσc2 E(rc) = U + (1/2)Aσc2 Example: E(rc) = (1/2)(2)σc2

31. Table 6.6 Spreadsheet Calculations of Indifference Curves

32. Figure 6. 7 Indifference Curves for U =. 05 and U =
Figure 6.7 Indifference Curves for U = .05 and U = .09 with A = 2 and A = 4

33. Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves

34. Passive Strategies: The Capital Market Line
E(rc) = rf +[(E(rM) - rf)/σM]σc
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. 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