Performance Evaluation

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Example 5.2

Consider a stock with mean return of 20 percent and standard deviation of 40 percent. The mean market rate of return is 20 percent, with standard deviation equal to 12 percent. The covariance of these two risky asset returns is assumed to be equal to 4 percent. The risk-free rate is 8 percent. With this information, we compute . Thus, we have:

This suggests that percent. This stock is not a good buy because it is performing below its CAPM risk-adjusted rate of return. What we've computed here is Jensen's index, the abnormal return on the risky asset, which, in this case, is about –21.36 percent. The intuition is that if the CAPM holds, then and the actual excess return to the asset is proportional to the excess market return as indicated by (this beta estimate says that this asset is about 2.78 times as risky as the market portfolio). In this example, however, the asset's excess return is too low, that is, its return lies below the capital market line. It is what we might call an underperformer. (Jensen's index—sometimes called Jensen's alpha in the empirical literature—is really a test of the CAPM.) Significantly nonzero estimates of the intercept α in a regression of the asset's excess return on the market excess return invalidates the CAPM.

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The Sharpe ratio, on the other hand, examines the ratio of the excess return on an asset to its standard deviation (or risk). That is, is the Sharpe ratio.

For pricing, recall that the one period return (ignoring dividends) is , where P is the asset's market price and the subscripts index time. Thus, according to the CAPM, and substituting for r_i:

What's interesting here is that the asset's beta and the market excess return together price the asset. If the asset has a beta of zero (indicating that it doesn't covary with the market), then P₀ is simply the discounted (at the risk-free rate) value of the payoff P₁.

The payoff, P₁ is usually not known with certainty. As such, it is a random variable that is state dependent. We are interested in its expected value E(P₁). It is reasonable to think that the expected future price is a function of the stream of cash flows, or dividends, if we are thinking about a share of stock. Thus, we can write a simple dividend discount model as:

More importantly, we can show how the CAPM extends interpretation of the discount rate used in the DDM of Chapter 4. There, the pricing model discounted expected future cash flows at a constant rate, r:

The capital asset pricing model produces a logically consistent result but uses a more meaningful discounting function—we replace r with .