Portfolio VaR

[frm_daniel]

Hi David,

This question is difficult. I have a hard time on it.

A firm has a porfolio of traded assets worth $100 million with a VaR of $15 million. The standard derivation of the return on the portfolio is 0.5. The firm is considering the sale of a position worth $1 million in an asset that has an expected return o 6% and a covariance of return with the portfolio of 0.2. The position that would be added has an expected return of 10% and a covariance of return with portfolio of 0.5 The VaR is based on a 95% confidence level.

The impact of the trade on the voltility of the portfolio is an increase of:
A. 0.004
B. 0.002
C. 0.006
D. 0.008
Ans : C

Ans : The beta, relative to the overall portfolio, of the proposed position is Beta(ip) = 0.5/0/0.5^2 = 2.0 (** this is ok to understand**), and the beta of the replace position is Beta(jp) = 0.2/0/5^2 = 0.8. The volality impact of the trade is equal to (0.2-0.8) * (1/100)*0.5 = 0.006. (** cannot figure out this equation logic**)

The impact of the trade on the VaR of the portfolio is a(an):
A. decrease of 0.95 million
B. decrease of 1.03 million
C. increase of 0.95 million
D. increase of 1.03 million
Ans : Impact on VaR = -(E(Ri)-E(Rj)*change in weight * portfolio + (Beta(ip) - Beta(jp)*1.65*Vol(Rp)*change in weight *portfolio =
-0.1-0.06)*1/100*100+(2-0.8)*1.65*0.5*(1/100)*100 = 0.95

I don't understand the logic. and seems not cover in AIM.

Daniel

 

[David Harper CFA FRM]

Hi Daniel,

It refers to Stulz Chapter 4 (4.2.1), which has not been formally assigned since 2008
(very low testability per se, but still instructive....)

On the first, Stulz Chapter 4 includes the derivation of:
Change in Volatility (Portfolio) = Volatility (Portfolio) * Beta(ip)*change in weight(%), so
The volatility impact of trade x% = (Beta(ip) - Beta(jp)) * trade[i.e., change in weight] * Volatility (Portfolio)
Here:
[(.5/.5^2) - (.2/.5^2)] * 1% trade * 0.5 portfolio volatility = + 0.006
... .5/.5^2 is the always useful beta = COV(i,P)/variance(P)

The second is really just Absolute VaR [i.e., Dowd: absolute VaR = -return*T + volatility*deviate*SQRT(T)] except it nets the two trades.
First, this trade has positive impact on exp return, trading up from 6% to 10%:
(10% - 6%) * 1% * $100 = +0.04 million in additional expected return (aka, "drift")

Second, this trade increases volatility as above:
[(.5/.5^2) - (.2/.5^2)] * 1% trade * 0.5 portfolio volatility = + 0.006
... but we want VaR so multiply by the deviate, and multiply by the portfolio to go from (%) to ($):
[beta(ip) - beta(jp)] * 1% trade * 0.5 portfolio volatility * $100 portfolio * 1.645 deviate = $0.99 MM

so the "relative VaR" here is 0.99 million
... but the net impact includes/net the benefit of +.04 return
... so we have relative VaR impact = .99 - .04, or as the formula has: -.04 + .99 = 0.95
... again, this is essentially: relative VaR = -drift + volatility*deviate (but in trade/difference terms)

Hope that helps, David