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

2. 6-2 W = 100 W 1 = 150 Profit = 50 W 2 = 80 Profit = -20 p =.6 1-p =.4 E(W) = pW 1 + (1-p)W 2 = 6 (150) +.4(80) = 122  2 = p[W 1 - E(W)] 2 + (1-p) [W 2 - E(W)] 2 = 0.6 (150-122) 2 + 0.4(80-122) 2 = 1,176,000   Risk - Uncertain Outcomes

3. 6-3 W 1 = 150 Profit = 50 W 2 = 80 Profit = -20 p =.6 1-p =.4 100 Risky Inv. Risk Free T-billsProfit = 5 Expected profit on risky is: 0.6*50 + 0.4*(-20) = 22 Risk Premium = 22 – 5 = 17 (on a 100 investment) Risky Investments with Risk-Free

4. 6-4 Investor’s view of risk Risk Averse – must be paid to take a fair bet Risk Neutral – is indifferent to a fair bet Risk Seeking – will pay you to take a fair bet Utility – a measure of gain from taking an investment (rather than measuring $) Utility Function (many possible) U = E ( r ) - 0.005 A  2 A = measure of the degree of risk Aversion 0.005 is a scaling factor to make the range of A more pleasing when expressing the inputs as percent rather than decimals Risk Aversion & Utility

5. 6-5 Certainty Equivalent A Certainty Equivalent is the amount you would take for sure, in place of a risky bet. It is simply the value of the utility, for the function noted above If the certainty equivalent is lower than the risk free rate, the investor should choose the risk free asset A risk lover would have a negative value for A in the Utility formula

6. 6-6 Risk Aversion and Value: U = E (r) ― 0.005 A  2 = 22 ―.005 A (34) 2 Risk AversionAValue High5-6.90 3 4.66 Low 116.22 T-bill = 5% For T-bill: U = E(r) = 5

7. 6-7 Mean Variance Dominance D dominates C if: E(r D ) ≥ E(r C ) and σ D ≤ σ C And at least one inequality is strict (i.e. two equal things cannot dominate one another)

8. 6-8 Dominance Principle 1 23 4 Expected Return 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

9. 6-9 Utility and Indifference Curves Represent an investor’s willingness to trade-off return and risk. Example (for A=4) Exp RetSt Deviation U=E ( r ) -.005A  2 1020.02 1525.52 2030.02 2533.92

10. 6-10 Indifference Curves Expected Return Standard Deviation Increasing Utility Note: Where the indifference curve hits the y axis is the certain equivalent

11. 6-11 Expected Return Rule 1 : The return for an asset is the probability weighted average return in all scenarios.

12. 6-12 Variance of Return Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return.

13. 6-13 Return on a Portfolio Rule 3: The expected rate of return on a portfolio is a weighted average of the expected rates of return of each asset comprising the portfolio, with the portfolio proportions as weights. Example: A two asset portfolio E(r p ) = w 1 E(r 1 ) + w 2 E(r 2 ) w 1 = Proportion of funds in Security 1 w 2 = Proportion of funds in Security 2 E(r 1 ) = Expected return on Security 1 E(r 2 ) = Expected return on Security 2 Note: w 1 + w 2 = 1

14. 6-14 Portfolio Risk with Risk-Free Asset Rule 4: When a risky asset is combined with a risk-free asset, Rule 5 shows that the portfolio standard deviation equals the risky asset’s standard deviation multiplied by the portfolio proportion invested in the risky asset.

15. 6-15 Covariance and Correlation Covariance measures the tendency for two random variable to move together For example, it makes sense that the returns on Ford and the returns on GM will usually move together (Why?) Calculated like a variance, but for two assets

16. 6-16 Covariance and Correlation Covariance can be positive or negative and from plus to minus infinity, so often one chooses to standardize it to the -1 though 1 interval using a measure called covariance

17. 6-17 Rule 5: When two risky assets with variances  1 2 and  2 2, respectively, are combined into a portfolio with portfolio weights w 1 and w 2, respectively, the portfolio variance is given by:   p 2 = w 1 2  1 2 + w 2 2  2 2 + 2w 1w w 2 Cov(r 1 r 2 ) Cov(r 1 r 2 ) = Covariance of returns for Security 1 and Security 2 Portfolio Risk