2010 FRM L1 sample paper Part1 - question 16

[alexjr]

while I fully understand how (in explanation) undiversified VAR was derived, I am struggling to understand diversified VAR formulae.. Anyone?

thanks

 

[David Harper CFA FRM]

Hi step4cfa,

The elaboration in this thread might help: http://forum.bionicturtle.com/viewthread/2504/
(it references this spreadsheet, which has more than you want, but just in case it helps https://www.dropbox.com/s/4vexzbu5pm4s5tt/2010_sample_L1_16.xls)

Diversified VaR is, here, just the portfolio volatility scaled (multiplied) by the 1.654 deviate.
So it's the variance that includes a benefit (reduction) for imperfect (< 1.0) correlation

In this way, it's using the two-asset portfolio variance:
variance(aX + bY) = a^2*variance(x) + b^2*variance(Y) + 2*a*b*covariance(X,Y)
i.e., a variance property http://en.wikipedia.org/wiki/Variance

and our constants are the portfolio weights:
a = 0.25,
b = 0.75

Thanks, David

 

[alexjr]

variance(aX + bY) = a^2*variance(x) + b^2*variance(Y) + 2*a*b*covariance(X,Y)

but we are given correlation between two returns not a covariance. should it be

covariance(X,Y)=correlation(X,Y)* SQRT(variance(X))*SQRT(variance(Y))

so the final formulae
variance(aX + bY) = a^2*variance(x) + b^2*variance(Y) + 2*a*b*correlation(X,Y)* SQRT(variance(X))*SQRT(variance(Y))

OR

variance(aX + bY) = a^2*variance(x) + b^2*variance(Y) + 2*a*b*correlation(X,Y)* Std(X)*Std(Y)

thanks

 

[David Harper CFA FRM]

Hi step4cfa,

Yours are both correct (there is no difference between yours?), with regard to portfolio variance it should be:
variance(aX + bY) = a^2*variance(x) + b^2*variance(Y) + 2*a*b*correlation(X,Y)* SQRT(variance(X))*SQRT(variance(Y)), which is the same as
variance(aX + bY) = a^2*variance(x) + b^2*variance(Y) + 2*a*b*covariance(X,Y)

But that is the answer given (see step 2 in explain).

It may confuse that with regard to portfolio Value at Risk, the formula is:
Diversified Portfolio VaR = SQRT[ Individual VaR(A)^2 + Individual VaR(B) ^ 2 + Individual VaR(A)*Individual VaR(B)*correlation

But this is still the same formula because the volatilities are embedded in the Individual VaR:
Individual VaR(A) = position (A) *Volatility (A)
... the volatilities have not been dropped from the final term in portfolio VaR

Maybe i missed your point? Thanks, David