F4160-5a - An Atheoretical Origin for a Linear Relation...

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Unformatted text preview: An Atheoretical Origin for a Linear Relation Between Risk and Return It is often thought, incorrectly, that a linear relationship between risk and return is a consequence of the CAPM. Consider an investor with a portfolio p of N assets with expected return E ( e r p ) and variance σ 2 ( e r p ) . Now assume that the investor has access to a risk-free asset with return r f and variance of zero. The investor may construct a new portfolio consisting partly of the risk-free asset and partly of portfolio p . Call this new portfolio q . Then the expected return of portfolio q is: E ( e r q ) = ( 1 & w p ) r f + w p E ( e r p ) . (1) Y. F. Chow (CUHK) Financial Economics 2009&10 First Term 1 / 77 By de&nition the variance of the risk-free asset is zero and it has zero covariance with the return on portfolio p , thus the variance of portfolio q is: σ 2 ( e r q ) = w 2 p & σ 2 ( e r p ) ) σ ( e r q ) = j w p j & σ ( e r p ) . (2) Re-arranging (2) and substituting back into (1) yields: E ( e r q ) = r f + & E ( e r p ) ¡ r f σ ( e r p ) ¡ σ ( e r q ) . Y. F. Chow (CUHK) Financial Economics 2009¡10 First Term 2 / 77 The Market Portfolio Let W i , > 0 be individual i &s initial wealth, and let w i , j be the proportion of W i , invested in the security j by individual i , where i = 1 , . . . , I and j = 1 , . . . , N . The total wealth in the economy is I ∑ i = 1 W i , & W m , . In equilibrium, W m , is equal to the total value of all securities. Let w m , j be the proportion of total wealth W m , contributed by the total value of the security j . That is, w m , j is the portfolio weights of the market portfolio m , which has a rate of return e r m = N ∑ j = 1 w m , j e r j . Y. F. Chow (CUHK) Financial Economics 2009¡10 First Term 3 / 77 For markets to clear, we must have I ∑ i = 1 w i , j W i , = w m , j W m , ) w m , j = I ∑ i = 1 w i , j W i , W m , . Therefore, it is obvious that the market portfolio weights are a convex combination of the portfolio weights for individuals. Y. F. Chow (CUHK) Financial Economics 2009&10 First Term 4 / 77 Assumptions of the Sharpe&Lintner&Mossin CAPM 1 All investors choose portfolios on the basis of their mean and variance of return; 2 All investors have identical subjective estimates of the joint probability distribution on the single period returns of all assets; 3 All investors can borrow and lend at a given risk-free rate of interest and there are no restrictions on short sales of any asset; 4 The quantities of all assets are given and all investors are price takers; 5 All assets are perfectly liquid and divisible; and 6 There are no taxes. Y. F. Chow (CUHK) Financial Economics 2009&10 First Term 5 / 77 Implications The &rst assumption is true if either of two conditions hold: Probability distributions for portfolio returns are all normally distributed. The assumption then holds because a normal distribution is completely described by its mean and its variance. As a symmetric distribution, risk corresponds to variance (or standard deviation) for a...
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F4160-5a - An Atheoretical Origin for a Linear Relation...

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