# Capm

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Asset Pricing and Risk Management

Copyright 1995 by Campbell R. Harvey. All rights reserved. No part of this lecture may be reproduced without the permission of the author.
Latest Revision: November 27, 1995.

1. The Derivation of the CAPM

In a world with all the assumptions made so far, all individuals should hold the market portfolio levered up or down according to risktolerance. A person with low risk tolerance (high risk aversion) will have most of her money in the riskfree security while a person with high risk tolerance (low risk aversion) will be borrowing to finance the purchase of the market portfolio.
For an individual at her optimum portfolio, consider a small additional borrowing to finance a purchase of asset i.

Portfolio Market Asseti Riskless
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Optimum w_m = 1 w_i = 0 0
Candidate w_m = 1 w_i = 0 + D -D
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Let's consider the mean and variance of the optimal portfolio.
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Next consider the derivative of the portfolio variance with respect to the weight in asset i:
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At the optimum, we know that w_m=1 and w_i=0. So let's evaluate this derivative at thesepoints:
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Next consider the derivative of the expected portfolio return with respect to the weight in asset i:
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At the optimum, the marginal change in portfolio expected return per unit of change in the variance must be equal for all securities (and the market portfolio m). This implies:
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Cross multiplying we get:
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This is the Capital Asset Pricing Model (CAPM).Substituting beta for the ratio of the covariance to the variance, we have the familiar form:
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This holds for all i.

2. Implementing the CAPM

In the previous section, we derived a relation between expected excess returns on an individual security and the beta of the security. We can write this as a regression equation. This is a special regression where the intercept is equal to zero.
[pic]This holds for all i. The beta is the covariance between the security i's return and the market return divided by the variance of the market return.
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So the CAPM delivers an expected value for security i's excess return that is linear in the beta which is security specific. We will interpret the beta as the individual security's contribution to the variance of the entire portfolio. Whenwe talk about the security's risk, we will be referring to its contribution to the variance of the portfolio's return -- not to the individual security's variance.
This relation holds for all securities and portfolios. If we are given a portfolio's beta and the expected excess return on the market, we can calculate its expected return. Finally, we have a tool which we can help us evaluate theadvertisement presented in Optional Portfolio Control.
The ad that appeared in the Wall Street Journal provided data on Franklin Income Fund and some other popular portfolios. The returns over the past 15 years were:
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The Franklin Income Fund 516%
Dow Jones Industrial Average 384%
Salomon's High Grade Bond Index 273%

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First, let's convert these returns intoaverage annual returns:
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The Franklin Income Fund 12.9%
Dow Jones Industrial Average 11.1%
Salomon's High Grade Bond Index 9.2%

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Note that the average annual returns are not nearly as impressive as the total return over 15 years. This is due to the compounding of the returns.
In order to use the CAPM, we need some extra data. We need the expected return on themarket portfolio, the security or portfolio betas and the riskfree rate. Suppose that the average return on the market portfolio is 13% and the riskfree return is 7%. Furthermore, suppose the betas of the portfolios are:
[pic]
The Franklin Income Fund 1.000
Dow Jones Industrial Average 0.683
Salomon's High Grade Bond Index 0.367

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These are reasonable beta...

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