Managing to a VaR Limit Can Make Your Losses Worse: Basak and Shapiro

Every risk framework is an incentive scheme, whether or not anyone intended it to be. Tell a trader "your Value-at-Risk must stay below 10 million" and you have not merely constrained them. You have handed them an objective function, and they will optimize against it.

Basak and Shapiro asked the obvious follow-up that shockingly few people had asked: what portfolio does a rational, self-interested investor choose when they are constrained by a VaR limit? They solved the problem properly, and the answer is genuinely alarming.

A VaR-constrained investor often takes on MORE exposure to risky assets than an unconstrained one, and suffers LARGER losses when losses occur.

The risk limit, applied to a rational optimizer, makes the tail worse.

The problem: VaR has a blind spot, and blind spots get exploited

Recall exactly what VaR promises. A 99 percent VaR of 10 million says: on 99 percent of days, you will lose less than 10 million.

Now read that sentence again as an optimizer would read it. It says absolutely nothing about the other 1 percent. Losing 11 million and losing 500 million are identical from the point of view of the constraint. Both are breaches. Neither is worse than the other. The constraint simply does not distinguish them.

That is not a subtlety. That is a gaping hole, and any optimizer worth its salt will find it.

The key idea via analogy: the exam where 59 and 0 are the same grade

Imagine an exam where anything below 60 is a fail and all failures are recorded identically as "fail" on your transcript. A student who is going to fail anyway has zero incentive to score 59 rather than 5. And more perversely: a student who could safely score 65 by studying might instead take a wild gamble that gives them a good chance of 95 and a small chance of 5, because the downside of the gamble, a fail, is no worse than any other fail.

The exam's grading scheme has, unintentionally, made gambling rational.

A VaR constraint does the same thing to a portfolio. Basak and Shapiro show the optimal policy has a distinctive and quite disturbing shape:

In the good and moderate states of the world, the investor manages their exposure to keep VaR within the limit. Everything looks fine. The risk report is green.

In the worst states, the investor abandons the tail entirely. Since the constraint is already violated in those states no matter what, and since the constraint imposes no additional penalty for making them worse, the rational thing to do is stop protecting them and use the freed-up resources to do better everywhere else.

The mechanics: rather than paying to hedge the very worst outcomes, the investor sells off the protection in the deep tail. That is cheap to do, because deep tail protection is expensive to hold. The money saved lets them take more risky exposure in the states that the VaR constraint actually measures. So the portfolio looks better in the 99 percent of the time that the risk system observes, and is catastrophically worse in the 1 percent it does not.

The constraint has not reduced risk. It has taken the risk and pushed it out into the region where the measurement is blind, and concentrated it there.

Why this is more than a theoretical curiosity

The strategy that emerges from this optimization is a recognizable real-world behavior: selling deep out-of-the-money options. Collect premium, look great almost all the time, report a low VaR, and hold a position that will destroy you in the rare bad state. That is not a hypothetical. It is what a very large number of blow-ups have actually looked like, and it is exactly the behavior that a VaR-based risk limit fails to penalize and can positively encourage.

The paper also looks at the general equilibrium consequences, asking what happens to asset prices when many investors manage to VaR limits. The answer is that this collective behavior feeds back into market volatility, which is another way of saying that a risk framework adopted by everyone becomes a source of risk in itself.

The comparison that makes the point

Basak and Shapiro contrast the VaR constraint with a constraint based on expected losses in the tail, meaning a limit on the average size of the loss when things go badly, which is essentially expected shortfall.

That constraint has no blind spot. Making the tail worse makes the measured quantity worse. So an investor optimizing under it cannot buy improvement in the middle by sacrificing the extremes: every dollar of additional tail loss shows up in the number. The optimal policy under a tail-loss-based limit behaves the way a risk manager would actually want, reducing exposure and limiting the damage in bad states.

The two constraints look similar and produce opposite behavior in exactly the region you built them for.

Why it mattered

• It reframed risk measures as incentive systems. A risk limit is not a passive measurement, it is a rule that agents optimize against. Once you see that, you have to ask of any risk framework: what behavior does this reward? That question is now standard, and this paper is a large part of why.
• It supplied the economic case against VaR, complementing the mathematical one. Artzner and coauthors showed VaR is incoherent as a mathematical object. Basak and Shapiro showed that even setting that aside, using it as a constraint produces perverse behavior by rational agents. Two independent lines of attack reaching the same conclusion is much more convincing than either alone.
• It predicted a failure mode we then watched happen. The pattern of institutions with clean risk reports carrying enormous hidden tail exposure, and blowing up spectacularly when the tail arrived, is a recurring story. Reading this paper after the fact is uncomfortable.
• It strengthened the case for expected shortfall in regulation. If your constraint must not be gameable by pushing risk into the unmeasured region, you need a measure that looks at the whole tail. That argument is central to why the regulatory framework moved.

The honest limitations

• It assumes a fully rational optimizer with rich instruments. The result requires the investor to be able to reshape their payoff freely, effectively trading a complete set of state-contingent claims. Real traders face limits on what they can do, and the extreme version of the pathology requires more freedom than most desks have.
• It assumes a single, static constraint. Real risk management is not one VaR limit. It is a system: stress tests, concentration limits, scenario analysis, position limits, and human oversight. Many of these exist precisely because everyone knows VaR has a blind spot. The paper is a critique of VaR used alone as the binding constraint, which is a strawman in a well-run firm and a documentary in a badly-run one.
• The result is about the deep tail, which is also the least estimable part of the distribution. Some of the pathology is a statement about an idealized model where the investor knows the true probabilities out in the extremes. In practice, nobody knows those probabilities, which cuts both ways.
• Expected shortfall constraints are not a panacea. They remove this specific blind spot. They do not remove model risk, they are harder to estimate, and they are harder to backtest. Swapping the measure fixes the incentive problem, not the epistemic one.

The one-line takeaway

Basak and Shapiro showed that because a Value-at-Risk limit is completely indifferent to how bad a breach is, a rational investor constrained by one will stop protecting the worst outcomes and reinvest the savings into more risk elsewhere, ending up with larger exposure and bigger losses precisely when losses happen, which means a VaR limit can make the disaster it was installed to prevent more severe.