# LO 70.4: Determine the statistical significance of a performance measure using

LO 70.4: Determine the statistical significance of a performance measure using standard error and the t-statistic.
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. In order to assess a managers ability to generate alpha, we conduct a r-test under the following hypotheses:
Null (i70): True alpha is zero.
Alternative ( N f : True alpha is not zero.
_ a 0 ~~ ct/V n
where: i = alpha estimate O o = alpha estimate volatility N = sample number of observations standard error of alpha estimate = a / Vn
In order to compute the r-statistic, we will need to know the alpha estimate, the sample number of observations, and the alpha estimate of volatility. From the volatility and sample size estimates, we can compute the standard error of the alpha estimate, which is shown in the denominator of the r-statistic calculation.
At a 95% confidence level (5% significance level) we reject the null hypothesis if we estimate a r-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.
Professors Note: Using a t-value o f 2 is a general test o f statistical significance. >From the F R M Part I curriculum, we know that the actual t-value with a 95% confidence level given a large sample size is 1.96.
If we assume an excess (alpha) return of 0.09% and a standard error of the alpha of 0.093%, the r-statistic would be equal to 0.97 (t = 0.09% / 0.093%); therefore, we fail to reject H Q and conclude that there is no evidence of superior (or inferior) performance.
Professors Note: Using statistical inference when evaluating performance is extremely challenging in practice. By the tim e you are reasonably confident that a managers returns are in fa ct due to skill, the manager may have moved elsewhere.
2018 Kaplan, Inc.
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Topic 70 Cross Reference to GARP Assigned Reading – Bodie, Kane, and Marcus, Chapter 24
M e a s u r in g H e d g e Fu n d P e r f o r m a n c e