Jensen Alpha question across multiple years of returns

[sneakyplacebo]

Hi David / team, I sat the exam in May and there was a question on calculating the Jensen's alpha across multiple years of returns. I forget the entire question and have trawled through the forums to find links to it to no avail really. I was wondering if you could remember such a question yourself and if you could if you could maybe put together a practice one with a worked answer as I still am none the wiser in how to approach a multiple year jensen alpha question 6 months on..

Thanks.

 

[David Harper CFA FRM]

Hi @sneakyplacebo

It sounds like the sort of question here at 404.3 https://forum.bionicturtle.com/thre...-significance-of-performance.7744/#post-29777

ie., 404.3. Over an historical measurement period, a long/short equity hedge fund produced an alpha of +100 basis points per month. The monthly standard deviation of the residual (non-systemic) risk was only 5.0%. If we want a two-tailed 95% significance level, approximately how many years (N) are required to determine that the fund demonstrated skill, that is, such that we can reject the null hypothesis that his true alpha is zero?

there is some additional discussion/derivation here @ https://forum.bionicturtle.com/threads/question-on-t-statistic.2588/#post-8122
At that time, I was thinking:

IR = t-stat = alpha / StdDev (alpha) = alpha / SQRT [ Variance(alpha) ]
i.e., on the right we have same critical t/critical Z as we have for, say a regresssion coefficient, only now I have made explicit the variance in order to treat the time dimension
So over T years:
t-stat = (T*alpha) / SQRT[ T * Variance(alpha) ]
t-stat = (T*alpha) / ( SQRT[T] * StdDev(alpha) )
t-stat = alpha /StdDev(alpha) * T/SQRT[T]
t-stat = alpha /StdDev(alpha) * SQRT[T]

... so you'll note the key here is that alpha is an intercept and a t-stat is just a "test of the intercept."

Although it appears to me now that maybe a more intuitive derivation is:

t-stat [coefficient/standard error (coefficient)]
= t-stat [alpha / annualized standard error (alpha)]
.... where annualized standard error (alpha) = annualized tracking error = TE * SQRT(1/T), such that:

t-stat = [alpha / TE * SQRT(1/Y) ] = (alpha / TE) * SQRT(T) = IR * SQRT(T)
... I am glad you asked (!) because i find this a more satisfying interpretation: the t-stat is here a typical "critical t" (i.e., coefficient/standard error of coefficient) except that it "scales" the tracking error into an annualized tracking error, to product annualized/annualized ratio