# Total portfolio VaR greater than aggregated individual VaRs

I am facing something weird in a simulation.

I have calculated a portfolio VaR: 100\$. Then I aggregated the VaR for individual position (loans) and obtain: 98\$.

I thought it was not possible for the diversification.

Is it because of the loss distribution shape?

Gratefully!

• Aug 20 '20 at 15:00
• @fesman I have asked there but I will ask you the same: I am using a Monte Carlo approach sampling from a Gaussian. Is it possible that I obtain "sometimes" not-subadditivity? Aug 20 '20 at 16:21

Since this question does not seem to be a duplicate, I will make up a simple (but not entirely unrealistic) numeric example.

Suppose some asset is now trading at some observable price, and suppose further that you have written two options: a put and a call that are slightly out of the money, i.e. whose strikes are, for concreteness, within 1 historical standard deviation below and above the current asset price. To over-simplify, assume that the implied volatility does not change, and that the P&L on the option depends only on the price of the underlying, as would be the case if both options expire tomorrow.

You use Monte Carlo to generate lots of possible scenarios in the changes in the asset price (as I said, we ignore the implied vol, which is an over-simplification) and take the 99% percentile loss to be the VaR of each option and also of the portfolio consisting of the two options.

What MC scenarios cause you to lose money on the put? If the asset price goes up, or goes down less than the put strike, you have zero P&L. But if the asset price goes down more than the put strike, then you have negative P&L linear in the asset price change below the strike. The exact scenario used for 99% VaR of the put is going to be close to the asset going down normsinv(99%) = 2.32635 standard deviations.

Similarly, the P&L on the call is going to be zero unless the asset price goes above the call strike, and then linear in the asset price change above the call strike. The exact scenario used for 99% VaR of the call is going to be close to the asset going up normsinv(99%) = 2.32635 standard deviations.

Now consider the 99% VaR of the portfolio. The portfolio loses money under more MC scenarios than either option alone: either if the asset is below the put strike or if the asset is above the call strike. The exact scenario used for 99% VaR of the portfolio is going to be either the put or the call losing money, because of the asset moving more than normsinv(99%) either up or down.

This counterintuitive behavior of VaR was already known back when regulators mandated the wide use of VaR in Basel II in the mid-1990s. Most people thought it is mostly a theoretical weakness. But it happens often enough in parctice and causes other related inconveniences, so later regulations (FRTB q.v.) use "espected shortfall" (ES) instead of VaR, which basically means that instead of the single scenario causing the loss at 99%, you look at many scenarios in the tail end of your losses, which generally remediates the behavior that we outlined.