Foundations: Amenc, Chapter 4

[HC78]

Hello David,
In pdf document you propose, for Q:32-4, I try to find a reasoning to reach at formula used (i.e. T=(t-stat/IR)^2).
1/If we set Alpha as random variable with mean 1% and Standard Deviation=10% (IR ?); and if we scale mean by nb of period (n) we obtain Mean(n)=1%*n and if we scale SD we obtain SD(n)=SD*sqrt(n). That's the scaling rule used to extrapolate SD from one to n period.
2/ if we want to calculate nb of period to wait until we are sure that Alpha is significantly different from 0, is it equivalent to construct a two-tailed test for which Ho (null hypothesis) is Alpha=0%, at 95% confidence level. We want to test when alpha is different from 0 (zero) with a level of confidence.
So I have these computations:
Alpha SD Alpha
Period
1 1,00% 10,00%
383 383% 195,82%

t calc=1,96 (=(1%*n-0%)/(SD*SQrt(n))
P(Alpha>0%)=0,974897286 (N(x))
And I solve for n to obtain P(Alpha>0%)=97.5% b/c it is a two tail test.

I not sure if I am correct or not. Could you help me?

Thank you very much
Hervé

 

[David Harper CFA FRM]

Hi Herve,

Apologies, I have a bit of trouble following your /2.
Can I refer you to some prior discussion on this here @ http://forum.bionicturtle.com/viewthread/1242/

I think the key idea is that alpha is a (linear) regression intercept; i.e., a coefficient estimate such that we are testing to reject the null that the true intercept = 0 (as you indeed assume).
The t-stat is then like any test of regression coefficient: coefficient / Standard Error (coefficient), except it happens to be the alpha-intercept: t-stat = alpha / Std Dev (alpha).

Now copying my math from the other thread:

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 regression coefficient, only now I have made explicit the variance in order to treat the time dimension

Now scale 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]
… such that SQRT[T] = t-stat/IR, and T = (t-stat/IR)^2

I hope that helps, David

 

[HC78]

Hello David,

sure, that help. I do agree with you : my explanations are far from clear. Thread you mentionned is very helpful (difference in treatment for “alpha” and “TE” is very instructive).

Thannks a lot
Hervé

 

[Yossarian]

Hi David below is my question

Market portfolio's Sharp ratio is 40%, the correlation between the market portfolio and the stock is 0.7, the stock's sharp ratio is

a) 12%
b) 28%
c) 32%
d) 30%

can you please explain the solution in detail..