For the sake of the argument, let's just take two risky assets in the portfolio.
Given that correlation between returns on those 2 risky assets decreased, (rho1,2), VaR should naturally decrease because the risk (volatility of portfolio) has just been decreased. However, an interview question asked me in what situation/portfolio can the VaR actually increase??? I'm all confused

Best Response

One of the assets could have gotten riskier while the other stays the same. Consider a portfolio of two assets. One returns 5, 10 or 15, and the other returns 10, 20 or 30. Currently correlation=1 and VaR=0. Now let the first asset still return 5, 10, or 15, but now the second asset returns 0,-19.3, or-12. Correlation is now -.62 and VaR has increased.

A simpler example of this is a perfectly hedged portfolio of two assets. Say correlation=1 and the gains from the first asset will always perfectly offset losses from the second and vice versa, so VaR=0. If the correlation decreases there will now be some probability of a loss so VaR>0.

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mrme355:

A simpler example of this is a perfectly hedged portfolio of two assets. Say correlation=1 and the gains from the first asset will always perfectly offset losses from the second and vice versa, so VaR=0. If the correlation decreases there will now be some probability of a loss so VaR>0.

Just curious... Wouldn't you say that this is an example of correlation=-1, rather than 1?

You are correct

Thanks for the input mrme335!
I get the idea that if correlation decreases, then the SD of portfolio decreases as well, and hence VaR increases given the VaR formula: VaR=-(mu+1.65*SDportfolio) for 95% CI.
However, I'm confused at why does VaR=0 if correlation=1?

VaR is 0 in the correlation = -1 or 1 sense because you can just buy/sell the proper ratio to offset all gains and losses.

For example if you know with certainty, which you do because correlation=1 or -1, that whenever asset A increases in price by \$1 asset B decreases in price by \$3, then you could buy 3 of A and 1 of B. In this portfolio no matter what happens you will make, and lose no money, hence VaR = 0. The downside is obviously that you can't make money either. 