1. Risk Aversion and Capital Allocation to Risky Assets
Investments
Cover image
CHAPTER 6 & 7
Risk Aversion and Capital Allocation to Risky Assets
Slides by
Richard D. Johnson
McGraw-Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved

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

3. Utility Function
U = E ( r ) – 1/2 A s2
Where
U = utility
E ( r ) = expected return on the asset or portfolio
A = coefficient of risk aversion
s2 = variance of returns

4. Table 6. 2 Utility Scores of Alter
Table 6.2 Utility Scores of Alter. Portfolios for Investors with Varying Risk Aversion

5. Figure 6.1 The Trade-off Between Risk and Returns of a Potential Investment Portfolio

6. Figure 6.2 The Indifference Curve

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

8. Allocating Capital: Risky & Risk Free Assets
It’s possible to split investment funds between safe and risky assets.
Risk free asset: proxy; T-bills
Risky asset: stock (or a portfolio)

9. rf = 7% rf = 0% E(rp) = 15% p = 22% y = % in p (1-y) = % in rf
Example
rf = 7%
rf = 0%
E(rp) = 15%
p = 22%
y = % in p
(1-y) = % in rf

10. Expected Returns for Combinations
E(rc) = yE(rp) + (1 - y)rf
rc = complete or combined portfolio
For example, y = .75
E(rc) = .75(.15) + .25(.07)
= .13 or 13%

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

12. 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

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

14. Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates
( )/0.22
( )/0.22

15. Table 6.5 Utility Levels for Positions in Risky Assets for an Investor with Risk Aversion A = 4

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

17. 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

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

19. 
y* = (E(rp) – rf ) / (0.01*A* p2

20. Capital Market Line
Passive strategy: portfolio decision that avoids any direct or indirect security analysis
CML: investment opportunity set that generated by a passive strategy

21. Table 6.7 Expected Returns on Four Indifference Curves and the CAL

22. Table 6. 8 Average Annual Return on Stocks and 1-Month T-bills; S. Dev
Table 6.8 Average Annual Return on Stocks and 1-Month T-bills; S. Dev. and Reward to Variability of Stocks Over Time