T5 MPT and CAPM T Statistic

[yisuho97]

Hi, here's a college student studying statistics who just started his FRM courses.

In Chapter 5, p24, there's a t statistic of 0.101 at the bottom of spreadsheet screen. I feel like this may not be asked during the exam but my statistics background made me wonder where 4.49% * sqrt 5.0 came from. Trying my conventional statistical formula for t statistic did not fit in with the information in spreadsheet. Thanks!

 

[David Harper CFA FRM]

Hi @yisuho97 That's our fault: I see that we explained the other ratios but we didn't add a bullet for the final t-stat (I have tagged it for an edit, at the next batch edit since it's not an error per se). You can see that sheet performs a lot of work (many concepts) and, admittedly, the alpha-based t-stat is an FRM P2.T10 measure (Chapter 8, Portfolio Performance Evaluation). See image below (source: 2020 Financial Risk Management Part II: Risk Management and Investment Management, 10th Edition. Pearson Learning Solutions, 2019. VitalBook file.)

Here is derivation at https://forum.bionicturtle.com/threads/question-on-t-statistic.2588/post-8122 i.e.,

Hi Agin,

That's from Amenc Chapter 4 where the derivation isn't given but please see this thread with Paul from last year:
http://forum.bionicturtle.com/viewreply/3096/

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.

Thanks, David

I hope that's helpful! GARP reference: