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Unformatted text preview: Capital Asset Pricing In this chapter, we study the first of the three main theorems in financial economics, namely the Capital Asset Pricing Model. (The other two theorems are the Modigliani-Miller theorem of corporate finance and the Black-Scholes formula of derivatives.) This model gives us a surprisingly simple relationship between the expected return of a portfolio and its beta risk (to be defined below). This relationship turns out to be linear. Although this formula has been empirically tested with only mixed results, it gives us a good approximation of the market pricing of risky portfolios. We will see how this formula can be applied in a number of settings. 1. Capital Asset Pricing Model (CAPM) Recall our portfolio model. In a Markowitz equilibrium, all investors are holding a portfolio made up of the risk-free asset and the market portfolio. The market portfolio is made up of all the risky assets, the weight of each is equal to its share in the value of all assets. We now ask: how is the price of a single risky asset determined in equilibrium? The answer is provided by the CAPM. Note that the price of a risky asset is inversely related to its expected rate of return. If an asset gives an expected return of $10 in a year, then its expected rate of return is 10% if its price today is $100. A price lower than $100 would imply a higher expected rate of return. Refer to the following market equilibrium diagram. Return CML Std. Dev. E(x R f P X M M x Now consider an asset X which has a rate of return of E( X ) and a risk of x . Let the covariance between X and the market portfolio be COV( X , M ). It is clear from the diagram that there exists a β such that E( X ) = β E( M ) + (1- β ) R f (1) We can form a portfolio, called P , by mixing M and the risk-free asset at the proportion β . P = (1- β ) R f + β M It follows that the expected return and variance of this portfolio are E( P ) = (1- β ) R f + β E( M ) = E( X ) VAR( P ) = β 2 VAR( M ) Now form another portfolio by mixing γ of X and (1- γ ) of P . Since E( X )=E( P ), the expected return of this portfolio is also E( X ). The variance of this portfolio is γ 2 VAR( X ) + (1- γ ) 2 VAR( P ) + 2 γ (1- γ )COV( X , P ) = γ 2 VAR( X ) + (1- γ ) 2 β 2 VAR( M ) + 2 γ (1- γ )COV( X , β M +(1- β ) R f ) = γ 2 VAR( X ) + (1- γ ) 2 β 2 VAR( M ) + 2 γ (1- γ ) β COV( X , M ) From the diagram, this variance is smallest when γ =0. Therefore, the first derivative of the =0....
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This note was uploaded on 03/03/2012 for the course ECON 3420 taught by Professor Kwong during the Spring '11 term at CUHK.
- Spring '11