1. Risk and Risk Aversion
Chapter 6

2. Risk - Uncertain Outcomes
p = .6
W1 = 150 Profit = 50
W = 100
1-p = .4
W2 = 80 Profit = -20
E(W) = pW1 + (1-p)W2 = 6 (150) + .4(80) = 122
s2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2 =
.6 ( )2 + .4(80=122)2 = 1,176,000
s =

3. Risky Investments with Risk-Free
p = .6
W1 = 150 Profit = 50
Risky Inv.
1-p = .4
100
W2 = 80 Profit = -20
Risk Free T-bills
Profit = 5
Risk Premium = 17

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

5. Risk Aversion and Value:
U = E ( r ) A s 2
= A (34%) 2
Risk Aversion A Value
High
Low
T-bill = 5%

6. Dominance Principle • 2 dominates 1; has a higher return
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

7. Utility and Indifference Curves
Represent an investor’s willingness to trade-off return and risk.
Example
Exp Ret St Deviation U=E ( r ) As2

8. Indifference Curves Increasing Utility Expected Return
Standard Deviation

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

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

11. 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

12. 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.

13. Portfolio Risk
Rule 5: When two risky assets with variances s12 and s22, respectively, are combined into a portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is given by:
p2 = w1212 + w2222 + 2W1W2 Cov(r1r2)
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2