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Unformatted text preview: Lecture 10 Lecture 10 The Capital Asset Pricing Model The Capital Asset Pricing Model Expectation, variance, standard error (deviation), covariance, and correlation of returns may be based on (i) fundamental analysis (ii) historical data Preliminaries Fundamental or Theoretical Analysis S possible states s probability of state s = 1,2,,S R s likely return is state s Notation 4 business cycle states (boom, normal, recession, depression) 3 industry demand states 2 firm demand share states 3 firm cost states Then, there are 4*3*2*3 = 72 possible states (or situations) Example: Suppose there are ( 29 = = = S 1 s s s R R E Expectation (mean) ( 29 ( 29 = = = S 1 s 2 s s ER R R var 2 Variance 2 = Standard error ( 29 ( 29 ( 29 B Bs S 1 s A As s B A AB R R R R R , R Cov  = = = returns on stock A R As s = 1,,S returns on stock B R Bs s = 1,,S Covariance measures how two random variables are related ( 29 ( 29 AB AB sign sign = 1 1 1 2 2 2 2 2 2 2  = A B B A A B A B B A A B ( 29 B A A B B A A B R , R corr = = Correlation is a normalized covariance Note ! Example: Suppose we have a theoretical model that predicts the following returns on stocks A and B in 3 states. States s R A R B Boom 0.25 20% 5% Normal 0.50 10% 10% Recession 0.25 0% 15% Expected returns ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0.10 0.15 0.25 0.10 0.50 0.05 0.25 0.10 0.00 0.25 0.10 0.50 0.20 0.25 B A = + + = = + + = Variances ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0.00125 0.10 0.15 0.25 0.10 0.10 0.50 0.10 0.05 0.25 0.005 0.10 0.00 0.25 0.10 0.10 0.50 0.10 0.2 0.25 2 2 2 2 B 2 2 2 2 A = + + = = + + = Standard errors 0.03536 0.07071 B A = = = = 2 2 B A Covariance ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0.0025 0.1 0.15 0.1 0.25 0.1 0.1 0.1 0.1 0.5 0.1 0.05 0.1 0.2 0.25 2 AB = + + = Correlation ( 29 ( 29 1.0 0.03536 0.07071 0.0025 B A AB AB = = = Returns on stocks A and B are perfectly negatively correlated. Stocks A can be used as a hedge against the risk in holding stock B Historical Data Based Approach From historical data, calculate the percentage returns R 1 , R 2 , , R T 2 = Sample standard deviation (or standard deviation) Sample average percentage return = = + + = = T 1 t t T 1 R T 1 T R ... R R Sample Variance ( 29 = = T 1 t 2 t 2 R R 1 T 1 Historical Data Based Approach (continued) Sample covariance of returns on stocks A and B, calculated from the historical samples of R A and R B R A = (R A1 , , R AT ) ; R B = (R B1 , , R BT ) ( 29 ( 29 = = = = = = T 1 t Bt B T 1 t At A B Bt T 1 t A At AB R T 1 R ; R T 1 R R R R R 1 T 1 Sample correlation of R A and R B B A AB AB = ( 29 = = = T 1 t 2 A At R R 1 T 1 2 2 ; A A A ( 29 = = = T 1 t 2 B Bt 2 B R R 1 T 1 ; 2 B B Expected Return and Variance of Returns...
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 Spring '06
 Bejan

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