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

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

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

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

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

Figure 6.2 The Indifference Curve

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

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)

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

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%

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   

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

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

Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates (0.15-0.09)/0.22 (0.15-0.07)/0.22

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

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

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

Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves

 y* = (E(rp) – rf ) / (0.01*A* p2

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

Table 6.7 Expected Returns on Four Indifference Curves and the CAL

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