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

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

Standard Deviation, Sigma, Expected Return  R 

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 average Average Above average Boom Expected Ret (k) % 1.7% 13.8% 15.0% 9.6% Std. Dev. (  Coef. of Var. (CV) Risk (b)

Returns on Alternative Investments II. Continuous Probability Distribution Probability of Occurrence 0.4 Market Portfolio k Rate of Return (%)

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

Calculation of  n  = VARIANCE =  2 =  (k i - k) 2 P i i=1  High Tech = [( ) ( ) ( ) ( ) ( ) ] 1/2 = (401.1) 1/2 = 20.0%  T-bills = 0.0%  Collections = 13.4%  U.S.Rubber = 18.8%  M = 15.3%

Continuous Probability Distributions: High Tech, U.S. Rubber, & T-Bills Probability of Occurrence T-Bills High Tech U.S. Rubber Rate of Return (%)

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

Ranking of Investment Alternatives Expected Return Risk  CV Security k  Ranking CV Ranking High Tech 17.4% 20.0% Market U.S. Rubber T-bills Collections = Least risky 5 = Most risky

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 % 28.0% 3.0% Below average Average Above average Boom

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%) (6.4%) (10.0%) (12.5%) (15.0%) = 9.6% Given the distribution of returns for the portfolio, we can calculate the portfolio’s  p and CV:  p = [( ) ( ) ( ) ( ) ( ) ] 1/2 = 3.3% and CV p = 3.3% / 9.6% = 0.34

Portfolio Returns & Risk: High Tech & Collections optional question integrated case Rate of Return (%) k P % in High Tech Standard Deviation  P (%)  P % in High Tech

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 = Stand-alone Risk Market Risk ,500+ # of stocks in portfolio

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) Return on the Market, k M (%) -20 Year HighTech T-Bills Collections U.S.Rubber The Market % 8.0% 21.6% -1.9% -8.0% mean 17.0% 8.0% 1.7% 13.8% 15.0% beta

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%

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 ) High Tech 16 k M 14 U.S. Rubber k RF Collections Beta

Changes in the Security Market Line Required & Expected Rates of Return (%) K i 30Increased Risk Aversion Increased Inflation Original Situation Beta

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