Using the index model we saw earlier with ex post returns r i r f \u03b1 i \u03b2 i r m r

# Using the index model we saw earlier with ex post

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Using the index model we saw earlier with ex post returns, r i – r f = α i + β i (r m – r f ) + e i Taking expectations of both sides: E(r i ) – r f = α i + β i (r m – r f ) because E(e i ) = 0 Add r f to each side: E(r i ) = α i + r f + β i (r m – r f ) The CAPM says: E(r i ) = r f + β i (r m – r f ) Therefore, the CAPM assumes that α i = 0. That is, the expected value of α i for all securities is zero. The CAPM has every security’s characteristic line crossing the Y axis at zero (the origin). The index model says that realized alphas should (and they generally do) average out to zero, but they certainly are not all equal to zero. What do Practitioners do with the CAPM? Remember that the CAPM is a one-period model – but the length of the period can be anything. It can be one day, one month, one quarter, one year, or one ten-year period. To use the CAPM, we need to come up with a risk-free rate of return. Generally, practitioners use U.S. Treasuries as the proxy for the risk-free asset. R f = S.T. T-bill rate; or, the average expected S.T. T-bill rate over the time period in question. For this, we often assume that the L.T. Treasury rate is the average expected S.T. T-bill rate. β i = the expected beta going forward We usually start with the ex-post beta over the past 60 months by regressing the firm’s returns on the market returns and then, if desired, make adjustments to that value. Most practitioners usually use total returns rather than excess returns
Many practitioners will average the historical beta with an industry beta or with 1.0, since betas have been shown to migrate towards 1.0 over time (mean reverting). These are
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