Portfolio VaR

[wrongsaidfred]

Hi David,

I have run into the formula VaR(port)=sqrt(VaR1^2+VaR2^2+2*rho*VaR1*VaR2) a couple of times but it does not always seem to work. For instance, I when given the following problem there seem to be two ways to do it that give me different answers.

Asset A: $800,000, E(R)=9%, sigma=15%
Asset B: $200,000, E(R)=18%, sigma=15%

If I do this with the dollar VaR formula above I get $146,811.

If instead I find the portfolio E(R) and portfolio sigma and THEN apply
VaR=1,000,000(1.65*sigma-E(R)) I get $128,280.

I checked my numbers a couple of times and I do not seem to be making any arithmetic mistakes. I assume that the dollar VaR formula is inappropriate in the case for some reason. My guess is that it has something to do with the drift but I have no idea if that is correct or not.

Any hep you could provide would be greatly appreciated.

Thanks in advance for any help you could provide,
Mike

 

[David Harper CFA FRM]

Hi Mike,

The first formula only works for relative VaR (i.e., without drift term): as you suggest, absolute VaR includes the drift, and if you see this thread from last year, I illustrated its failure for absolute VaR (http://forum.bionicturtle.com/threads/var-converting-time-horizon.4013/#post-10554). I imagine there is an analytical equivalent for the absolute VaR, but it appears non-trivial and not worth the effort. I think it only make sense to apply the portfolio drift AFTER. I assume your numbers work if you break your second into two parts?:

VaR=1,000,000*1.65*sigma-1,000,000*E(R); and just apply the formula to get (1,000,000*1.65*sigma). This term is relative VaR and they should definitely match on this term.

Here is the "proof" in case it's interesting:
VaR(P$) = W(P$)*deviate*SQRT[w(a%)^2*sigma(a%)^2 + w(b%)^2*sigma(b%)^2 + 2*w(a%)*w(b%)*COV(a,b)],
VaR(P)^2 = W(P$)^2*deviate^2*[w(a%)^2*sigma(a)^2 + w(b%)^2*sigma(b)^2 + 2*w(a%)*w(b%)*COV(a,b),
VaR(P)^2 = [W(P$)^2*deviate^2*w(a%)^2*sigma(a)^2)] + (W(P$)^2*deviate^2*w(b%)^2*sigma(b)^2) + W(P$)^2*deviate^2*2*w(a%)*w(b%)*COV(a,b),

as W($P)^2*deviate^2*w(a%)^2*sigma(a)^2 = [W(P)*deviate*w(a%)*sigma(a)]^2, and W($P)*w(a%) = w($a):
VaR(P)^2 = VaR(a$)^2 + VaR(b$)^2 + W(P$)^2*deviate^2*2*w(a%)*w(b%)*COV(a,b),
VaR(P)^2 = VaR(a$)^2 + VaR(b$)^2 + [W(P$)*w(a%)*deviate] * [W(P$)*w(b%)*deviate]*2*COV(a,b); as COV = sigma(a)*sigma(b)*correlation(a,b):
VaR(P)^2 = VaR(a$)^2 + VaR(b$)^2 + [W(P$)*w(a%)*deviate*sigma(a)] * [W(P$)*w(b%)*deviate*sigma(b)]*2*correlation(a,b),
VaR(P)^2 = VaR(a$)^2 + VaR(b$)^2 + 2*VaR(a$) * VaR(B$) * correlation(a,b),
VaR(P$) = SQRT[VaR(a$)^2 + VaR(b$)^2 + 2*VaR(a$) * VaR(B$) * correlation(a,b)]

Thanks, David

 

[wrongsaidfred]

Awesome.

Much appreciated.

Thanks,
Mike