H12 - Excess Returns and Beta: Deriving the Security Market...

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Excess Returns and Beta: Deriving the Security Market Line W.L. Silber I. We showed that market forces combined with a search by investors for efficient portfolios would produce the following relationships for each security (i, j = 1 , . .., n ) in a portfolio: (1) m f m i m f i R R X R R s s - = - / This expression can be rewritten as (2) i m m f m f i X R R R R - + = / s s II. Expression (2) says that the equilibrium expected return ( R i ) on security i should equal the risk-free rate ( R f ) plus the market price of risk ( 29 m f m R R s / - times i m X / s . What is i m X / s ? It is the increase in risk ( m s ) associated with a small increase in asset i , in other words, it is the risk contribution of security i to portfolio risk, m s . We can derive i m X / s by taking the derivative of the expression for the variance of a portfolio with respect to X i . The result, as shown in Garbade (p. 175, fn 12) is: (3) ( 29
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H12 - Excess Returns and Beta: Deriving the Security Market...

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