Regarding Value at Risk and Tail Value at Risk, which statement is true?

• A. VaR satisfies subadditivity, TVaR does not.
• B. VaR fails subadditivity, TVaR is fine.
• C. Both VaR and TVaR fail subadditivity.
• D. TVaR fails subadditivity, VaR is fine.

Answer: (D)

Subadditivity is the idea that diversifying a portfolio should not increase risk: the risk of X plus Y should not exceed the sum of the risks of X and of Y.

Value at Risk can fail this property because it only looks at a cutoff point in the loss distribution. It ignores how large losses can be beyond that threshold, so the tail interaction of two separate risks can produce a bigger combined tail than the sum of their individual VaRs. A simple way to see this is with two binary-loss risks: each one has a small chance of a big loss and a large chance of no loss. Individually, their 95% VaR is zero, but the VaR of their sum at 95% can jump to a substantial amount, violating subadditivity.

Tail Value at Risk, or expected shortfall, addresses this by averaging losses that occur beyond the VaR threshold. This tail average captures more of the tail behavior and, in common practice, is subadditive. Therefore, the true relationship is that VaR can fail subadditivity while TVaR remains subadditive.

The statement that TVaR fails subadditivity while VaR is fine does not hold in general, so it’s not the correct description of how these risk measures behave.