Hi David, could you help me solve this question?

72/ garp 2016. An analyst regresses the returns of 100 stocks against the returns of a major market index. The resulting pool

of 100 alphas has a residual risk of 18% and an information coefficient of 9%. If the alphas are normally

distributed with a mean of 0%, roughly how many stocks have an alpha greater than 3.24% or less than -3.24%?

This question is a straight application of the formula giving the "structure" of alphas (in a forecast, based on the skills of a manager) from the Grinold and Kahn (G&K) chapter that is "thrown" in the curriculum without much context, unfortunately. If you accept the formula the question is easy to solve:

alpha =( residual risk) * (Information Coefficient) * N(0,1)

This describes alpha as a random variable: distribution of outcomes over many forecasts, or independent "picks" the manager would trade on. The Information Coefficient (IC) is the

*correlation* of the forecasted returns with the realized returns, and measures the

*skill* of a manager (quality of investment decisions, in a way, making good "picks" repeatedly). The standard normal distribution, called "score" in the formula, is an assumption (approximation).

In a very crude way (I know

@David Harper CFA FRM read the whole book so he can provide more detail and correct me

), this equation is the version for forecasting purposes of the relation IR = alpha/TE --> alpha = TE * IR, which is based on realized returns (as opposed to ex-ante, i.e. forecasted). The G&K book actually introduces the "fundamental law of asset management" that ties IR with Information Coefficient: IR^2 = IC^2 * N(picks). The number of "picks" (independent, hence the sum in quadrature) is referred to as the

*breadth* of the manager. It starts making sense when you read it a couple of times (although there's quite a bit of math behind it if you want to establish it formally).

With that said, on to the solution: we know E(alpha)=0, given. As per formula above:

Sigma(alpha)=(residual risk)*IC*Sigma(N(0,1)) = 18%*9% = 1.62%

You note that 2*Sigma(alpha)=3.24%, and the alphas being normally distributed, the probability mass outside of +/-3.24% will be 1-95.45%=4.55%, so about 100*4.55% = 5 stocks. That's it, really.

I'm not a fan of this question as there's

*so much* that goes into it...

. Hope it helps.

## Stay connected