and Risk Risk Aversion
Chapter 6
Risk - Uncertain Outcomes
p = .6
1
W = 150 [Profit = 50] W = 100 1-p = .4
1 2 2
W = 80 [Profit = -20]
E(W) = pW + (1-p)W = .6(150) + .4(80) = 122
2 1 2 2 2
= p[W - E(W)] + (1-p) [W - E(W)] = .6 (150-122)2 + .4(80=122)2 = 1,176,000 = 3 4 .2 9 3
Risky Investments with RiskFree
Risky Inv.
p = .6
1
W = 150 Profit = 50 1-p = .4
100
2
Risk Free T-bills
W = 80 Profit = -20 Profit = 5
Risk Premium = 17
Risk Aversion & Utility
Investor's view of risk
Risk Averse Risk Neutral Risk Seeking
Utility Utility Function
U = E ( r ) - .005 A
2
A measures the degree of risk aversion
Risk Aversion and Value:
U = E ( r ) - .005 A 2 = 22 - .005 A (34) 2 Risk Aversion A High 5 3 Low 1 Value -6.90 4.66 16.22
T-bill = 5%
Dominance Principle
Expected Return 4 2 1 Variance or Standard Deviation 3
2 dominates 1; has a higher return 2 dominates 3; has a lower risk 4 dominates 3; has a higher return
Dominance Principle
Expected Return 4 2 1 Variance or Standard Deviation 3
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 ) - .005A2 20.0 2 25.5 2 30.0 2 33.9 2
Indifference Curves
Expected Return
Increasing Utility
Standard Deviation
Expected Return
Rule 1 : The return for an asset the is probability weighted average return in all scenarios.
E (r ) = P( s)r ( s)
s
Variance of Return
Rule 2: The variance of an asset's return is the expected value of the squared deviations from the expected return.
= P( s )[r ( s ) - E (r )] s
2
2
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 = W1r1 + W2r2 W1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2 r1 = Expected return on Security 1 r2 = Expected return on Security 2
Portfolio Risk with Risk-Free Asset
Rule 4: When a risky asset is combined with a riskfree asset, the portfolio standard deviation equals the risky asset's standard deviation multip...