## AP RPt [Rp 3PRM Rp

where:

ap = ex post alpha

Rpt = actual return on portfolio in period t [Rp + (3P(RM - Rp)] = CAPM (predicted return)

### Accounting for Performance

Alpha (a) plays a critical role in determining portfolio performance. A positive alpha produces an indication of superior performance; a negative alpha produces an indication of inferior performance; and zero alpha produces an indication of normal performance matching the benchmark. The performance indicated by alpha, however, could be a result of luck and not skill. We conduct a ¿-test under the following hypotheses:

Null (H0): True alpha is zero

Alternative (H^j: True alpha is not zero a —0 1 ~ S.E.(q)

where:

At a 95% confidence level (5% significance level) we reject the null hypothesis if we estimate a t value of 2 or larger. That is, the probability of observing such a large estimated alpha by chance is only 5%, assuming returns are normally distributed.

If we assume an excess return of 0.09% and a standard error of the alpha of 0.093%, the ¿-statistic would be equal to 0.97 (t = 0.09%/0.093%); therefore, we fail to reject HQ and conclude that there is no evidence of superior (or inferior) performance.

Professor's Note: As a rule of thumb, a t-statistic of two or larger is sufficient to reject the null hypothesis of true alpha equal to zero at the 95% confidence level. Higher confidence levels such as 97.5%, 99%, and99.5% require increasingly higher t-statistic values to reject the null hypothesis for a given level of observations.

### Accounting for Risk

According to the CAPM, market risk is the only risk in a well diversified portfolio that accounts for the variation in portfolio returns. That is, risk-adjusted excess returns should exactly match the realized excess returns in a portfolio. Thus, alpha should be zero. Portfolio managers should earn returns in accordance with the risk they take. Therefore, according to the Jensen measure, realized returns over and above risk-adjusted returns is evidence of superior performance for which managers should receive credit.

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