FRM Fun 14. Ex ante information ratio

[David Harper CFA FRM]

FRM Fun 14 is P1 only: I would appreciate if P2 and quantitative experts would kindly recuse themselves.

Risk-adjusted performance measures in the FRM can often be calculated on an ex ante (expected) or ex post (i.e., after actual risk and performance is observed) basis. Even the Sortino, which you might think can only be employed ex post, can arguably be calculated ex ante (why is Sortino naturally ex post? how might we calculate it ex ante? ... this is not today's question, just musing here ....).

P1 level question:

• If a portfolio's benchmark is the S&P 500 and the portfolio's beta is 1.20, what do we need to calculate an ex ante information ratio for the portfolio?
  (note: in my opinion, there is more than one correct answer)

 

[Pedram]

Hi

Information ratio can be calculated as active return divided by tracking error.
from portfolio's beta we can obtain the active returns ( difference between portfolio's returns and benchmark's returns) and the tracking error is the standard deviation of active returns.
with the assumption that we already have the benchmark's returns, I think !! we don't need any further information to calculate the IR for the portfolio.

Try my best
cheers

 

[David Harper CFA FRM]

Hi Pedram - Thank you, I agree ... I was looking to implement your idea with the math, as it applies variance property(ies) and CAPM ... i was trying to see if only beta is needed but i think we need a bit more (?)... thanks!

 

[RiskNoob]

I agree with Pedram we need more assumptions ... here is my first try with more assumptions:



Let Rf be the interest free rate, Rp be the portofolio return and Rm be the S&P 500 benchmark return.

We are asked to find IR, it can be express as calculate:

IR = ( E[Rp] – E[Rm] ) / std (Rp – Rm)

For calculating numerator, from CAPM,

E[Rp] = Rf + 1.2( E[Rm] – Rf) (where 1.2 is beta)

Subtracting E[Rm] both sides and some arithmetic gives

E[Rp] - E[Rm] = 0.2( E[Rm] – Rf )

This tells the excess return is characterized by risk price multiplied by excess beta.

For calculating denominator,

Var(Rp – Rm) = Var(Rp) + Var (Rm) – Cov(Rp, Rm)
= Var(Rp) + Var (Rm) – 1.2Var(Rm) by definition of beta

Taking the square root to calculate std (Rp –Rm) and finally obtain IR

 

[David Harper CFA FRM]

Hi RiskNoob, yes, agreed, thank you ... I was thinking that, and of course these all have expected values, E(), Var(Rp – Rm) = Var([Rf + B*E] - [Rf + E]), where Rf = riskfree, B = beta, and E = ERP.

So that Var([Rf + B*E] - [Rf + E]) = Var(B*E - E) = Var[E*(B-1)] = (B-1)^2*Var[E].

In this way. E[active IR] = E*(B-1)/[(B-1)^2*Var(E)], such that the only additional needed information is variance(ERP). Yes, it assumes uncorrelated and E[error] = 0

... maybe I'm wrong, it feels like it "escapes" without the Covariance term that you correctly show? I hope it's interesting at least!

 

[RiskNoob]

I realized that my solution for calculating denominator has a hole - term still contains E[Rp]. I agree with you the latter approach is concrete and complete. Thanks for the correction!

 

[David Harper CFA FRM]

Thanks RiskNoob, good deal, you still win the star for the best answer given so far. Thanks!

 

[Aleksander Hansen]

Since it's been a good day or so and the question has been answered and discussed I'll throw in a little something.

The issue with the above derivation is that it skips the covariance. Since the portfolio return is based on and derived from the benchmark, the two are not independent.
Loosely speaking, (and this is by no means rigorous) the heuristics can be given by:

The fact, by defining the benchmark as the market portfolio, it is a subset x of the universe of asset.

That is, we have that the market portfolio and the benchmark are both finite [sets]. Moreover, for ever asset r ∈ Rm there is a unique x ∈ X in the benchmark; Rm and X have the same number of elements and hence there exist a bijection between the two sets [skipping some technicalities here for readability but they hold true, so it is with no loss of generality]. Furthermore, there is an injection (one-to-one) from Rp to X. However, then it must be true that Rp is a subset of Rm.

The derivation above is thus incorrect without additional (and different) assumptions as we have proved it so ad absurdium.

 

[ShaktiRathore]

David has escaped covariance term now I would like to show what IR depends when Covariance is taken into account,
Information Ratio:
IR= [E(Rp)-E(Rb)]/sqrt(Var(Rp-Rb))...(1)
Now,
E(Rp)=Rf+B*(ERP)
or E(Rp)-E(Rb)=Rf-E(Rb)+B*(ERP)
or E(Rp)-E(Rb)=-(-Rf+E(Rb))+B*(ERP)
or E(Rp)-E(Rb)=-ERP+B*(ERP)
or E(Rp)-E(Rb)=(B-1)*ERP...(2)
from (1) and (2),
IR=(B-1)*ERP/sqrt(Var(Rp-Rb))...(3)
Var(Rp-Rb)=VaR(Rp)+VaR(Rb)-2*Cov(Rp,Rb)...a
VaR(Rp)=VaR(Rf+B*ERP)=VaR(Rf)+VaR(B*ERP)=B^2*VaR(ERP)....b
VaR(Rb)=VaR(ERP-Rf)=VaR(ERP)...c
Cov(Rp,Rb)=B*sqrt(VaR(Rb)*VaR(Rp))=B*sqrt(B^2*VaR(ERP)*VaR(ERP))
or Cov(Rp,Rb)=B^2*VaR(ERP)....d
put b,c,d in a gives us,
Var(Rp-Rb)=B^2*VaR(ERP)+VaR(ERP)-2*B^2*VaR(ERP)
or Var(Rp-Rb)=VaR(ERP)-B^2*VaR(ERP)
or Var(Rp-Rb)=(1-B^2)*VaR(ERP)....(4)
or VaR(Rp-Rb)=modulus of ((1-B^2)*VaR(ERP)) depending on whether beta is less than or greater than 1.
from (3) and (4) above,
IR=(B-1)*ERP/sqrt(modulus of (1-B^2)*VaR(ERP)).
if beta B is given =1.2 then IR=.2*ERP/sqrt(.44*VaR(ERP))
or IR is proportional to ERP/VaR(ERP)
So, Apart from VaR(ERP) which david has suggessted that IR depends upon when we actually takes into account covariance then the IR terms also depends upon the ERP. So basically we require 2 inputs the ERP, VaR(ERP).I hope its clear from above derivation. Hope someone does not find mistake in the above proof.i fthen kindle tell .

 

[David Harper CFA FRM]

@Alex:
I agree covariance (Rp, benchmark) is required but it's in the beta(Rp, benchmark) already, as demonstrated by ShaktiRathore's derivation.
Isn't it more accurate to say that my omission (simplification) is the covariance between portfolio return and the error, which has ex ante expected value of zero but we cannot really say that about Var (e); i.e., while valid is E[e] = 0, it is not true that Var(E[e]) = 0.
What i mean is, Var (E[Rp], E[benchmark]) = Var (Rf + B*ERP + e, B*ERP), such that we can treat Rf and B as constants, but not really (e), such that the omitted term is Cov(e, ERP) and I am assuming that is equal to zero but I'm not sure that's justified (i.e., an ex post IR based on realized performance obviously includes the noise/error).


@ShaktiRathore,
Brilliant, thank you! I basically agree ...

But is this line correct? Cov(Rp,Rb)=B*sqrt(VaR(Rb)*VaR(Rp))=B*sqrt(B^2*VaR(ERP)*VaR(ERP))

I get:
Cov(Rp,Rb)=Cov(B*ERP + Rf, ERP + Rf) = B*Cov(ERP,ERP) = B*Var(ERP), such that:
Var(Rp-Rb)=B^2*Var(ERP)+Var(ERP)-2*B*Var(ERP) = Var(ERP)*[B^2 + 1 - 2*B] = (B-1)^2*Var(ERP). If you agree with that mod, then we agree (!!):
IR=(B-1)*ERP/ ([1-B^2)*VaR(ERP)].

 

[ShaktiRathore]

Yes David I totally Agree with you infact I misused the formula for beta which instead of Cov(Rp,Rb)=B*sqrt(VaR(Rb)*VaR(Rp)) is
Cov(Rp,Rb)=B*VaR(Rb) =B*VaR(ERP+Rf)=B*VaR(ERP) so in this respect we tally,
Var(Rp-Rb)=B^2*VaR(ERP)+VaR(ERP)-2*B*VaR(ERP)=(B-1)^2*VaR(ERP)
and thus finally, IR comes as (B-1)*ERP/(B-1)*sqrt(VaR(ERP)) or as ERP/sqrt(VaR(ERP)) I hope u agree with this final result David

 

[David Harper CFA FRM]

Hi ShaktiRathore - Yes!! I completely forgot to SQRT() the denominator, as TE is a volatility not a variance. My original above E*(B-1)/[(B-1)^2*Var(E)], should read E*(B-1)/SQRT[(B-1)^2*Var(E)] = E*(B-1)/[(B-1)*Volatility(E)] = ERP/volatility(ERP). We agree, very exciting ... thank you

 

[Aleksander Hansen]

@Alex:
I agree covariance (Rp, benchmark) is required but it's in the beta(Rp, benchmark) already, as demonstrated by ShaktiRathore's derivation.
Isn't it more accurate to say that my omission (simplification) is the covariance between portfolio return and the error, which has ex ante expected value of zero but we cannot really say that about Var (e); i.e., while valid is E[e] = 0, it is not true that Var(E[e]) = 0.
What i mean is, Var (E[Rp], E[benchmark]) = Var (Rf + B*ERP + e, B*ERP), such that we can treat Rf and B as constants, but not really (e), such that the omitted term is Cov(e, ERP) and I am assuming that is equal to zero but I'm not sure that's justified (i.e., an ex post IR based on realized performance obviously includes the noise/error).

I agree with most of what you wrote, however, it does not change what I wrote above (so I must for the first time I think, check the disagree button on your post ).

Notice that my argument, says nothing about (rp, benchmark) not having included, it simply demonstrates that it must be included.

However, the main point of the argument is that "it skips the covariance", and I proceed to show that, following the logic of the argument to its conclusion, by definition the CAPM implies that any covariance term in this model cannot, by construction, be independent, due to its dependence on the universe of assets spanned by the benchmark.
That is a much more general proof.