portfolio variance - Quick last minute question

[razu]

Hi David

I need to clarify on the portfolio variance calculation which is getting me a little worked up as I have seen a couple of different approaches and want to ensure I dont waste time in the exam.

If in a question it is given the you have 10 million of asset A with 20% volatility and 40 million of asset B with 25% volatility and correlation of .2 , do you calculate portfolio variance as

a) 10^2*.20^2+40^2*.25^2+2*10*40*.20*.25*.20 ; or
b).25^2*.20^2+.75^2*.25^2+2*.25*.75*.20*.25*.20 - I.e. using weights

I thought it would be b) but I saw a question (cant find the exact source right now!!!!) which used a) and I think both will give different answers (correct me if I am wrong)
Thanks

 

[David Harper CFA FRM]

Hi razu,

You can use either.

Typically we compute with weights on the way to a portfolio volatility in percentage (%) terms, as:

1. variance = 20%^2*20%^2 + 80%^2*25%^2 + 2*20%*80%*20%*25%*0.20 = 0.04480 ( think your weights should be 80/20 rather than 75/25 because portfolio = 40 + 10)
2. SQRT[20%^2*20%^2 + 80%^2*25%^2 + 2*20%*80%*20%*25%*0.20] = 21.1660% volatility
3. 21.1660% volatility * $50 portfolio = $10.58 dollar volatility

But we can instead compute a dollar variance by squaring the positions, as

1. dollar variance = $10^2*20%^2 + $40^2*25%^2 + 2*$10*$40*20%*25%*0.20 = $112.0 dollar^2
2. SQRT($112) = $10.583

and VaR is just a multiplier, so same with VaR:

• 99% VaR = 21.1660% portfolio volatiltiy * $50 portfolio = $24.62, or we can get to the same place with:
• PVaR = VaR(a)^2 + VaR(b)^2 + 2*VaR(a)*VaR(b)*correlation, where VaR(a) = individual VaR of position A, in this case:
  PVaR = ($10*20%*2.33)^2 + ($40*25%*2.33)^2 + 2*$4.65*$23.26*0.2 = $24.62

I hope that helps, good luck on the exam! Let us know how it goes please .... thanks,

 

[razu]

Thanks for the quick response David and clarrification. Should have got my weights right !! I checked it both ways and was getting different anwers in two questions. The stress is probably getting to me!

Will let you know how it goes! Thanks for all your help