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.
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