Information Ratio and Alpha

[Liming]

Dear David,

In a practice question on information ratio from page http://forum.bionicturtle.com/viewthread/1237/ you s,aid that:

36d. Assume instead the benchmark is a style-based index with beta of 1.5. What is the revised Jensen’s alpha and information ratio?
Alpha = 14% pr. - 4% rf. - [3% excess market return * 1.5 beta] = 5.5%
IR = alpha/TE = 5.5%/7% = 0.786
as IR is basically same as the t-distributed variable alpha/standard error(alpha).

I can't understand the sentences highlighted in red color because I have the following thought:

1) I do agree that IR is a t-distributed variable alpha/standard error(alpha) but only when beta = 1.
My reasoning is that: Since Rp - Rf = alpha + beta*(RB - Rf) where Rp, Rf, RB refers to return of portfolio, riskless and benchmark respectively. It logically follows that when beta = 1, Rp - Rf = alpha + RB - Rf, therefore Rp - RB = alpha, which means excess return of portfolio over its benchmark and Volatility(Rp-RB) = Volatility(alpha) Since information ratio = (Rp - RB)/Volatility(Rp-RB), it logically equals to alpha/standard error(alpha) as well.

2) Based on my above reasoning, it's clear that the formula transformation can't be done when beta <> 1.

As a result, I don't understand your general statement that: IR = alpha/TE = 5.5%/7% = 0.786
as IR is basically same as the t-distributed variable alpha/standard error(alpha). because I think it doesn't apply in all scenarios.

Thank you for your clarification!


Cheers!
Liming

07/10/09

 

[David Harper CFA FRM]

Hi Liming,

Interesting but shouldn't instead you start with:
Rp - Rf = alpha + beta*(Rm - Rf)

such that Rb = beta*(Rm - Rf)
and tracking error = StdDev (Rp - Rb) = StdDev (Rp - beta*(Rm - Rf))

or, even more specifically, Amenc (assigned FRM) defines TE = StdDev (Rp - Rb)
... but I have come to think of this as potentially confusing if we are not careful about the definition of Rb

rather, Grinold is more precise with: TE = StdDev (alpha) or StdDev (residual)
...this ensure consistency, I think, because alpha and residual, by definition, are "what's left over" after multiplying beta by the common factors

or, let me put yet another way: the reason I am challenged by all of this, maybe you will agree/disagree, is in the issue of how to define the benchmark; e.g., you can define as either the index (S&P 500) or as beta*index ... in my view, Grinold avoids your valid objections by carefully defining the benchmark as beta*index

...this also related to Grinold's careful use of active return versus residual; active = Rp - Rb which leads to confusion but residual return = Rp - beta*Rb

The statement above (in red) mimics Grinold and comes from the perspective that alpha is the intercept in the regression. If you think of alpha as the intercept, then we have the significance stat for the intercept with t-stat = intercept coefficient / StdError (intercept coefficient) = alpha/StdError (alpha) = alpha / StdDev(alpha)
In my view, the important thing is to understand why alpha is the intercept

Then the IR is effectively an annualized t-stat (please see this thread: http://forum.bionicturtle.com/viewthread/1242/)
such that:
t-stat = alpha /StdDev(alpha) * SQRT[T], and as Grinold shows
t-stat = IR * SQRT(T)

...I can't refute your logic, I still think your approach has merit and may reconcile, I just don't see the direct link at the moment

Thanks, David