1. Fall-02 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html Investments Zvi Wiener tel: 02-588-3049 mswiener@mscc.huji.ac.il Risk and Risk Aversion BKM Ch 6

2. Zvi WienerBKM Ch 6 slide 2 W = 100 W 1 = 150 Profit = 50 W 2 = 80 Profit = -20 p = 60% 1-p = 40% 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 =.6 (150-122)2 +.4(80=122)2 = 1,176,000   Risk - Uncertain Outcomes

3. Zvi WienerBKM Ch 6 slide 3 W 1 = 150 Profit = 50 W 2 = 80 Profit = -20 p =.6 1-p =.4 100 Risky Inv. Risk Free T-billsProfit = 5 Risk Premium = 17 Risky Investments with Risk-Free Investment

4. Zvi WienerBKM Ch 6 slide 4 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 & Utility

5. Zvi WienerBKM Ch 6 slide 5 Risk Aversion and Value: Using the Sample Investment U = E ( r ) -.005 A  2 =.22 -.005 A (34%) 2 Risk AversionAValue High5-6.90 3 4.66 Low 116.22 T-bill = 5%

6. Zvi WienerBKM Ch 6 slide 6 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

7. Zvi WienerBKM Ch 6 slide 7 Utility and Indifference Curves Represent an investor’s willingness to trade- off return and risk. Example Exp RetSt Deviation U=E ( r ) -.005A  2 1020.02 1525.52 2030.02 2533.92

8. Zvi WienerBKM Ch 6 slide 8 Indifference Curves Expected Return Standard Deviation Increasing Utility

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

10. Zvi WienerBKM Ch 6 slide 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. Zvi WienerBKM Ch 6 slide 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. r p = 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

12. Zvi WienerBKM Ch 6 slide 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. Zvi WienerBKM Ch 6 slide 13 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 1 W 2 Cov(r 1 r 2 ) Cov(r 1 r 2 ) = Covariance of returns for Security 1 and Security 2 Portfolio Risk

14. Zvi WienerBKM Ch 6 slide 14 Home Assignment Required: problems 1, 2, 3, 4, (3 rd ed). problems 1, 2, 3, 4, (5 th ed). Appendix b:1 – submit ! closely follow financial news!