Capital Asset Pricing Model, Part Four – The Equations

In the first three installments, we looked closely at the assumptions that underlie the Capital Asset Pricing Model (CAPM), as part of our overall project of investigating the role of alpha in hedge fund performance.  Today we will review the basic CAPM equations.  I promise you won’t need a PhD in Mathematics to get the gist.

Recall the purpose of CAPM: to calculate a theoretical price for an asset or portfolio of assets.  In the single asset case, we use the Security Market Line (SML) that projects the risk/reward characteristics of the asset compared to those of the overall market. The x-axis is beta and the y-axis is return. The slope of the SML is the market risk premium (i.e. the return in excess of the risk-free rate, which is the y-intercept). The slope is given by E(Rm) – Rf.  The entire SML equation is:

(E(Ri) – Rf ) / βi = E(Rm) – Rf

Where

E(Ri) is the expected return of asset i

Rf is the risk-free rate

βi , the beta of asset i, the sensitivity of  asset return to market return

E(Rm) is the expected return of the overall market

In other words, the excess return on an asset, deflated by the asset’s beta, equals the expected return of the market in excess of the risk-free rate.

Now, solve for E(Ri) to arrive at the CAPM equation:

E(Ri) = Rf + βi (E(Rm) – Rf)

So the expected return on asset i equals the excess return of the market multiplied by the assets beta, added to the risk-free rate.  This shows that beta is the sole explanation for an asset’s expected return.  As you know from previous blogs, this is a very shaky assumption.

One more rearrangement gives us:

E(Ri) – Rf  = βi (E(Rm) – Rf)

which shows that the asset’s risk premium is equal to the market premium multiplied by beta. In other words, the SML is a single-factor model, and beta, which is the asset’s relative covariance (i.e. covariance / variance) with the market, is that factor.

Now you see why we invested three blogs in gauging the real-world verisimilitude of CAPM.  In the next blog, we’ll explore how CAPM fares when applied to different hedge fund trading strategies.

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Copyright 2011 Eric Bank, Freelance Writer

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