Portfolio Active Return and IR

[turtle2]

David,

On page 11 of 2010.8.a.Investment.pdf,
r(PA) = r(P) - r(B), P=Portfolio, B= Benchmark, PA= Portfolio Active Return

r(P) =Theta(P) + beta(P)*r(B) as defined earlier,

Then
r(PA) = Theta(P) + beta(P)*r(B) - r(B).

The above is missing - r(B) term on page 11.

Can you please explain Std( r(PA) ) formula's derivation on page 11.

Thanks.

Turtle2

 

[David Harper CFA FRM]

Hi Turtle2,

I just noticed, due to your question that I do have a typo on page 11:
r(PA) = Theta(P) + beta(P)*r(B) - r(B).

That should be:
r(PA) = Theta(P) + beta(PA)*r(B).

… so it's not missing "-r(B)" but rather the beta should be beta(PA). Sorry!!

r(PA) = Theta(P) + beta(PA) *r(B)
Variance(PA) = variance(Theta(P)) + variance(beta(PA)*r(B)) + 2*COV(Theta(P),beta(PA) *r(B)),
... applies Gujarati's important variance property: VAR(A+B) = VAR(A)+VAR(B) +2*COV(A,B)
But since Theta is uncorrelated by definition, COV()=0,
Variance(PA) = variance(Theta(P)) + variance(beta(PA)*r(B))
StdDev(PA) = SQRT[variance(Theta(P)) + variance(beta(PA)*r(B))]
… note: variance(beta*r(b)) = beta(PA)^2*variance(b)

It is analogous to a CAPM (not the same, but analogous):
CAPM/market model where E(R) = Rf + b*ERP + alpha
Variance(R) = variance(b*MRP) + variance (alpha); i.e., Rf variance = 0 and correlation of Rf is 0
Variance(R) = variance(alpha) + b^2*variance(MRP); i.e., VaR(aX) = a^2*VaR(X)
StdDev(R) = SQRT[variance(alpha) + b^2*variance(MRP)]
… note structural similarity to final formula on page 11

David

 

[turtle2]

David,
So,
r(PA) = Theta(P) + beta(P)*r(B) - r(B) = Theta(P) + { beta(P)-1}* r(B)

then
Variance(PA) = variance(Theta(P)) + variance( {beta(P)-1}*r(B)) + 2*COV(Theta(P),{beta(P)-1} *r(B)),
so on as you proceed to derive.

Do you define beta(PA) as {beta(P) - 1} here ?

I am also not sure about your decomposition of total active systematic return later reading into only three components ?

Turtle2

 

[David Harper CFA FRM]

Turtle2,

No, it's not missing -r(B). The formula for active return should be:
r(PA) = Theta(P) + Beta(PA)*r(B)

(sorry to confuse, I had to edit after i posted, as i often need to do, so your email doesn't match the entry....)

David