Risk and Risk Aversion Chapter 6 Mc GrawHillIrwin

Risk and Risk Aversion Chapter 6 Mc. Graw-Hill/Irwin Copyright © 2005 by The Mc. Graw-Hill Companies, Inc. All rights reserved.

Risk - Uncertain Outcomes p =. 6 W 1 = 150 Profit = 50 W = 100 1 -p =. 4 W 2 = 80 Profit = -20 E(W) = p. W 1 + (1 -p)W 2 = 6 (150) +. 4(80) = 122 2 = p[W 1 - E(W)]2 + (1 -p) [W 2 - E(W)]2 =. 6 (150 -122)2 +. 4(80=122)2 = 1, 176, 000 = 34. 293 6 -2

Risky Investments with Risk-Free Risky Inv. 100 p =. 6 1 -p =. 4 Risk Free T-bills W 1 = 150 Profit = 50 W 2 = 80 Profit = -20 Profit = 5 Risk Premium = 17 6 -3

Risk Aversion & Utility Investor’s view of risk Risk Averse Risk Neutral Risk Seeking Utility Function U = E ( r ) -. 005 A 2 A measures the degree of risk aversion 6 -4

Risk Aversion and Value: U = E ( r ) -. 005 A 2 =. 22 -. 005 A (34%) 2 Risk Aversion A Value High 5 -6. 90 3 4. 66 Low 1 16. 22 T-bill = 5% 6 -5

Dominance Principle Expected Return 4 2 3 1 Variance or Standard Deviation • 2 dominates 1; has a higher return • 2 dominates 3; has a lower risk • 4 dominates 3; has a higher return 6 -6

Utility and Indifference Curves Represent an investor’s willingness to trade-off return and risk. Example Exp Ret 10 15 20 25 St Deviation U=E ( r ) -. 005 A 2 20. 0 2 25. 5 2 30. 0 2 33. 9 2 6 -7

Indifference Curves Expected Return Increasing Utility Standard Deviation 6 -8

Expected Return Rule 1 : The return for an asset is the probability weighted average return in all scenarios. 6 -9

Variance of Return Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return. 6 -10

Return on a Portfolio Rule 3: The rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions as weights. rp = W 1 r 1 + W 2 r 2 W 1 = Proportion of funds in Security 1 W 2 = Proportion of funds in Security 2 r 1 = Expected return on Security 1 r 2 = Expected return on Security 2 6 -11

Portfolio Risk with Risk-Free Asset Rule 4: When a risky asset is combined with a risk-free asset, the portfolio standard deviation equals the risky asset’s standard deviation multiplied by the portfolio proportion invested in the risky asset. 6 -12

Portfolio Risk Rule 5: When two risky assets with variances 12 and 22, respectively, are combined into a portfolio with portfolio weights w 1 and w 2, respectively, the portfolio variance is given by: p 2 = w 12 12 + w 22 22 + 2 W 1 W 2 Cov(r 1 r 2) = Covariance of returns for Security 1 and Security 2 6 -13