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# Information Ratio v t-stat on alpha

#### Kangaroo79

##### New Member
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I show IR = alpha/volatility(alpha)
and..... t-statistic = alpha coefficient/alpha s.d.

How are these two different?

#### Deepak Chitnis

##### Active Member
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Hi @Kangaroo79, Your IR formula is correct. But second formula which is t-statistic formula, it is the formula for hypothesis tests of whether coefficient alpha equals a prespecified value, hypothesis tests involving two statistically independent sample alphas as may arise when testing the equality of coefficient alpha across groups, and hypothesis tests involving two statistically dependent sample alphas as may arise when testing the equality of alpha across time or when testing the equality of alpha for two test scores within the same sample, here in foundation you don't need to worry about this, but remember that this two formulas are different they are not the same. You can see it more in Quantitative analysis. I hope that helps.
Thank you,

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#### Tania Pereira

##### Member
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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%?

#### berrymucho

##### Member
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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#### David Harper CFA FRM

##### David Harper CFA FRM
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Aweseome @berrymucho thank you! I am neither a fan of this question, it has been discussed previously, and I think the first version contained a basic error (which we submitted to GARP). @Tania Pereira We are trying to tag questions, so this can be found by searching on the tag = 'garp16-p2-72'