1. Risk Measurement Risk = Actual return deviated from Expected Return = ROIiE(R)

2. E(R) =  P i R i = R = Expected Return Ex-ante = Future Events Ex-post = Historical Data  = Standard Deviation = risk    = Variance =  P(R-R) 2

3. Standard Deviation, Sigma, Expected Return  R  - 100 0 +100

4. Returns on Alternative Investments I. Discrete Probability Distribution Estimated Rate of Return State of the T- High U.S. Market 2-Stock Economy Probability Bills Tech Collections Rubber Portfolio Portfolio Recession0.18.0% -22% 28.0% 10.0% -13.0% 3.0% Below average0.28.0 -2.0 14.7 -10.0 1.0 6.4 Average0.48.0 20.0 0.0 7.0 15.0 10.0 Above average0.28.0 35.0 -10.0 45.0 29.0 12.5 Boom0.18.0 50.0 -20.0 30.0 43.0 15.0 Expected Ret (k) 8.0 17.4% 1.7% 13.8% 15.0% 9.6% Std. Dev. (  0.0 20.0 13.4 18.8 15.3 3.3 Coef. of Var. (CV) 0 1.1 7.9 1.4 1.0 0.3 Risk (b) 0.0 1.29 -0.86 0.68 1.00

5. Returns on Alternative Investments II. Continuous Probability Distribution Probability of Occurrence 0.4 Market Portfolio -45 -30 -15 0 15 30 45 60 75 k Rate of Return (%)

6. Calculation of k n k =  P i k i i=1 k High Tech = 0.10(-22.0%) + 0.20(-2.0%) + 0.40(20.0%) + 0.20(35.0%) + 0.10(50.0%) = 17.4% k T-bills = 8.0% k Collections = 1.7% k U.S.Rubber = 13.8% k M = 15.0%

7. Calculation of  n  = VARIANCE =  2 =  (k i - k) 2 P i i=1  High Tech = [(-22.0 - 17.4) 2 0.10 + (-2.0 - 17.4) 2 0.20 + (20.0 - 17.4) 2 0.40 + (35.0 - 17.4) 2 0.20 + (50.0 - 17.4) 2 0.10] 1/2 = (401.1) 1/2 = 20.0%  T-bills = 0.0%  Collections = 13.4%  U.S.Rubber = 18.8%  M = 15.3%

8. Continuous Probability Distributions: High Tech, U.S. Rubber, & T-Bills Probability of Occurrence T-Bills High Tech U.S. Rubber -45 -30 -15 0 8 15 30 45 60 Rate of Return (%)

9. Calculation of CV CV =  k CV T-bills = 0.0% / 8.0% = 0.0 CV HighTech = 20.0% / 17.4% = 1.1 CV Collections = 13.4% / 1.7% = 7.9 CV U.S. Rubber = 18.8% / 13.8% = 1.4 CV Market = 15.3% / 15.0% = 1.0

10. Ranking of Investment Alternatives Expected Return Risk  CV Security k  Ranking CV Ranking High Tech 17.4% 20.0% 5 1.1 3 Market 15.0 15.3 3 1.0 2 U.S. Rubber 13.8 18.8 4 1.4 4 T-bills 8.0 0.0 1 0.0 1 Collections 1.7 13.4 2 7.9 5 1 = Least risky 5 = Most risky

11. Portfolio Return & Standard Deviation 2-Stock Portfolio Return: 50% High Tech and 50% Collections k p =  n w i k i i=1 k p = 0.5(17.4%) + 0.5(1.7%) = 9.6% Standard Deviation: State of the Expected Return Economy Prob. High Tech Collections 2-Stk Portfolio Recession 0.10 -22.0% 28.0% 3.0% Below average 0.20 -2.0 14.7 6.4 Average 0.40 20.0 0.0 10.0 Above average 0.20 35.0 -10.0 12.5 Boom 0.10 50.0 -20.0 15.0

12. Portfolio Return & Standard Deviation By considering the portfolio return in each state of the economy, we have another way of calculating k p : k p = 0.10(3.0%) + 0.20(6.4%) + 0.40(10.0%) + 0.20(12.5%) + 0.10(15.0%) = 9.6% Given the distribution of returns for the portfolio, we can calculate the portfolio’s  p and CV:  p = [(3.0 - 9.6) 2 0.10 + (6.4 - 9.6) 2 0.20 + (10.0 - 9.6) 2 0.40 + (12.5 - 9.6) 2 0.20 + (15.0 - 9.6) 2 0.10] 1/2 = 3.3% and CV p = 3.3% / 9.6% = 0.34

13. Portfolio Returns & Risk: High Tech & Collections optional question integrated case Rate of Return (%) 20 16 12k P 8 4 0 0 20 40 60 80 100 % in High Tech Standard Deviation  P (%) 20 16 12  P 8 4 0 0 20 40 60 80 100 % in High Tech

14. Portfolio Size & Risk Density Portfolio of Stocks with K p=16% One Stock 0 16 Percent 1.  gets smaller as more stocks are combined. 2. k p remains constant. 3. So, if you don’t like risk, hold a portfolio (or a mutual fund). Portfolio Risk,  p (%) 33 30Minimum attainable risk in a portfolio of average stocks 25Diversifiable, Risk s M = 20.6 15 Stand-alone Risk Market Risk 10 0 10 20 30 40 1,500+ # of stocks in portfolio

15. Chapter 6 The Concept of Beta Return on Stock i,k i (%) High Tech (slope = beta = 1.29) 40 Market (slope = beta = 1.0) U.S. Rubber (slope = beta = 0.68) 20 -20 20 40 Return on the Market, k M (%) -20 Year HighTech T-Bills Collections U.S.Rubber The Market 1990 -12.3% 8.0% 21.6% -1.9% -8.0% 1991 14.1 8.0 3.9 12.1 12.5 1992 17.4 8.0 1.8 13.8 15.0 1993 20.6 8.0 -0.6 15.5 17.5 1994 47.0 8.0 -18.1 29.4 38.0 mean 17.0% 8.0% 1.7% 13.8% 15.0% beta 1.29 0.00 -0.86 0.68 1.00

16. Security Market Line Equation k RF = T-Bill reate = 8% k M = k m = 15% k i = k RF +(k M - k RF )b i k High Tech = 8.0% + (15.0% - 8.0%)1.29 = 8.0% + (7.0%) 1.29 = 8.0% +9.0% = 17.0% k M = 8.0% + (7.0%) 1.00 = 15.0% k U.S.Rubber = 8.0% + (7.0%) 0.68 = 12.8% k T-bills = 8.0% + (7.0%) 0.00 = 8.0% k Collections = 8.0% + (7.0%) (-0.86) = 2.0%

17. Security Market Line Graph Required & Expected Rates of Return (%) SML: k i = k rf + (k M - k RF ) b i 22 = 8% + 7%(b i ) 20 18 High Tech 16 k M 14 U.S. Rubber 12 10 8 k RF 6 4 2 Collections 0 -2 -4 -6 -2 -1 0 1 2 Beta

18. Changes in the Security Market Line Required & Expected Rates of Return (%) K i 30Increased Risk Aversion 25 20 Increased Inflation 15 10 5 Original Situation 0 0.00 0.50 1.00 1.50 2.00 Beta

19. Portfolio size and risk Large company stock : 12.6% + 20% = 32.5% 12.6% - 20% = -7.5% Small company stock : 17.7% + 34.4% = 52.1% 17.7% - 34.4% = -16.7% Long term bonds : 6% + 8.7% 6% - 8.7% U.S bill : 3% + 3.3% 3% - 3.3%