Portfolio VaR

[Kitty]

Hi David,

There is a question that confuses me a lot...

Consider a portfolio of N assets worth $100 million with normally distributed returns. The standard deviations of the assets differ, but the correlation between any pair of assets is zero. As N becomes very large, the VaR of the portfolio will:
A. approach the value-weighted average of the standard deviations
B. approach the arithmetic average of the standard deviations
C. approach the standard deviation of the market portfolio
D. approach zero

The correct answer is D. While I just don't understand why C is not correct. In my understanding, a portfolio contains N assets when N is very large is very similar as a market portfolio, so the unsystematic risk should approach zero when N is large. However, I think the systematic risk still exists, and answer D doesn't account in systematic risk...

Could you correct me where I was wrong? Thanks in advance!!

 

[ShaktiRathore]

For N asset portfolio assuming they are equally weighted each assets weight is 1/N
Now std. deviation of portfolio = sum of variances of all assets (as correlation is 0 b/w them)
std. deviation of portfolio=(w1*sigma1)^2+(w2*sigma2)^2+........(wN*sigmaN)^2
std. deviation of portfolio=(1/N*sigma1)^2+(1/N*sigma2)^2+........(1/N*sigmaN)^2
std. deviation of portfolio=(1/N)^2[sigma1^2+(sigma2)^2+........(sigmaN)^2]
So as N approaches large value std. deviation of portfolio of portfolio approaches 0 =>Var=std. deviation of portfolio*z*Portfolio value
Var approaches 0 as std. deviation of portfolio approaches 0 as N becomes large.
C is not correct because the market std Deviation depends on relative weights of the constituents assets while the portfolio may not have the exact market constituents and would have different weightings of assets as compared to the market so in general its not necessarily true that portfolio risk will approach market standard deviation. Also the markets has a limited number of assets and portfolio could have assets which are not even in the market or a different variety of assets.
Yes unsystematic risk would diversify away to 0 and what is remaininng will be systematic risk but that systematic risk will not necessarily represent the market systmeatic risk due to different assets weighings or constituents.
thanks

 

[David Harper CFA FRM]

Hi Kitty,

I think (D) is basically correct, although I think I would fault the question for not specifying "equally weighted assets" (I agree with ShaktiRathore ... I prefer this assumption, too ... I am not certain the statement applies without it). The introduction of "market portfolio" makes the question more difficult (if a bit sneaky!) but it is instructive. Please note that as N becomes very large (tends to infinite), this does not imply the portfolio is tending toward the market portfolio (it actually implies it is moving away from the market portfolio: the market portfolio is the combination of RISKY assets with the highest Sharpe ratio ... put another way, the market portfolio has the highest return per risk).

In this question, as correlations are zero, the portfolio variance = w(1)^2*variance[x(1)] + w(2)^2*variance[x(2)] ... w(n)^2*variance[x(n)], where w(.) = 1/n. If the assets are equally weighted, then each weight is (1/n) such that each terms has w(.)^2 = (1/n)^2 which tends to zero as (n) increases. I hope that helps (@ShaktiRathore thanks for your response, I got distracted mid-post and ended up cross-posting with yours!),

 

[Kitty]

ShaktiRathore and @David Harper, thank you two very much for your explanation! I agree...