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LECTURE10 - Lecture 10 Lecture The Capital Asset Pricing...

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Lecture 10 Lecture 10 The Capital Asset Pricing Model The Capital Asset Pricing Model
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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
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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 π μ Expectation (mean) ( 29 ( 29 = - = = S 1 s 2 s s ER R R var π σ 2 Variance 0 2 = σ σ Standard error
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( 29 ( 29 ( 29 B Bs S 1 s A As s B A AB R R R R R , R Cov Ε Ε - - = = = π σ 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 σ ρ = 1 1 1 2 2 2 2 2 2 2 - = AB B A AB AB B A AB ρ σ σ σ ρ σ σ σ ( 29 B A AB B A AB R , R corr σ σ σ ρ = = Correlation is a normalized covariance Note !
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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 = + + = = + + = μ μ 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 = - + - + - = = - + - + - = σ σ
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Standard errors 0.03536 0.07071 B A = = = = 2 2 B A σ σ σ σ Covariance ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0.0025 0.1 0.15 0.1 0 0.25 0.1 0.1 0.1 0.1 0.5 0.1 0.05 0.1 0.2 0.25 2 AB - = - - + - - + - - = σ Correlation ( 29 ( 29 1.0 0.03536 0.07071 0.0025 B A AB AB - = - = = σ σ σ ρ 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
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Historical Data Based Approach From historical data, calculate the percentage returns R 1 , R 2 , …, R T 0 2 = σ σ Sample standard deviation (or standard deviation) Sample average percentage return = = + + = = T 1 t t T 1 R T 1 T R ... R R μ Sample Variance ( 29 = - - = T 1 t 2 t 2 R R 1 T 1 σ
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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 σ Sample correlation of R A and R B B A AB AB σ σ σ ρ = ( 29 = - - = = T 1 t 2 A At R R 1 T 1 2 2 ; A A A σ σ σ ( 29 = - - = = T 1 t 2 B Bt 2 B R R 1 T 1 σ σ σ ; 2 B B
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Expected Return and Variance of Returns on Portfolios A portfolio is an investment in stocks. Let be the proportion invested in stock n. Then 2 N 1 x N 1 n n = = [0,1] x n If the return on stock n is R n , then the return on the portfolio is = = + + = N 1 n n n N N 1 1 p R x R x ... R x R and the expected return on the portfolio is = = = = = = N 1 n n n N 1 n n n N 1 n n n p R x ΕR x R x Ε μ
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( 29 ( 29 ( 29 ( 29 ( 29 ∑∑ ∑∑ = - = = = - = = = = = + = - - + - = - = - = - = N 2 n 1 n 1 m nm m n N 1 n 2 n 2 N 2 n m m 1 n 1 m n n m n 2 N 1 n n n 2 2 N 1 n n n n 2 N 1 n
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