Virtually every financial institution with a risk management function uses the value-at-risk or expected shortfall measure to set limits for traders. So it is unsettling to find that, according to new research, these popular gauges are not a good way of curbing risk-taking behaviour.
In a recent Risk.net article, Rogue traders versus value-at-risk and expected shortfall, John Armstrong, a senior lecturer at King’s College London, and Damiano Brigo, the head of the mathematical finance group at Imperial College London, show risk limits set according to classical VAR and expected shortfall-based models are not breached when rogue traders take more tail risk – exposing their positions to extreme losses – while staying within a set budget constraint.
The quants do so by first defining a rogue trader as someone whose primary objective is to maximise his expected utility, or rather his satisfaction from the gains or losses on his positions. They argue such a trader should have an S-shaped utility curve, an idea originally proposed by behavioural scientists Daniel Kahneman and Amos Tversky as part of the popular prospect theory the pair developed in 1979.
The S-shaped utility curve is shaped such that, when gains increase, the increase in utility experienced by a trader for an additional unit of gain begins to decline. The same holds in reverse: as losses deepen, the trader’s utility declines at a slower rate. The behaviour at the lower end of the tail is especially important, because it shows the limited liability of a trader when faced with very large losses.
“Essentially, a trader or investor has limited liability. Once he has lost his job and reputation, then whichever bigger loss the bank takes doesn’t really matter that much to him. So it is wrong to characterise the utility as going down faster and faster, because roughly speaking, beyond a point the trader or investor has nothing else to lose,” says Brigo.
Using the example of losses and gains on a risky asset priced using the Black-Scholes model, the quants show that traditional risk limits set by banks using VAR or expected shortfall fail to pick up on the risk-taking behaviour of a trader when faced with low-probability, high-loss scenarios – that is, the limits do not get breached as the trader tries to maximise his expected utility by taking greater risk under less likely scenarios.
The payoff for the trader in this case is similar to that of a digital option, where larger losses are possible over a narrow range of less likely scenarios, whereas large gains are possible over a wider range of more likely scenarios. The result is a payoff with an expected utility that is quite high, because there is more likelihood of large gains.
Ever since the financial crisis and scandals such as Libor-fixing, a lot of academic research has gone into how to model and manage trading incentives
“If you weight bigger and bigger losses less and less, you can build a financial payoff that is basically a kind of digital option on the underlying asset that bypasses the VAR or expected shortfall constraint. It still gives you a trade that is within the limit, but can push your expected utility larger and larger,” says Brigo.
The quants show further that if, instead of VAR or expected shortfall, one were to set risk limits based on a concave utility function, rogue trading behaviour can be curbed. A concave utility function is one in which utility declines at a much faster rate as losses increase – essentially, giving a much greater weight to larger losses than VAR or expected shortfall would.
The concave utility function can be characterised as something that fights against the tendencies of traders, because its shape is the opposite of the S-shaped utility curve in the region of losses.
Ever since the financial crisis and scandals such as Libor-fixing, a lot of academic research has gone into how to model and manage trading incentives. Since the crisis was in part spurred by excessive risk-taking behaviour, regulators reacted by raising capital adequacy, limiting bonuses and increasing disclosure requirements.
Under the Fundamental Review of the Trading Book, for instance, exotic instruments – which are considered more risky – are penalised using a capital add-on.
All these changes were meant to incentivise businesses to take less risk and be more transparent about their activities. And while all of them indirectly address risk-taking behaviour in one way or another, Armstrong’s and Brigo’s paper is one of the first to look at the issue from the point of view of basing internal risk management on that behaviour. If risk-taking behaviour is what a risk manager is supposed to limit, then modelling that behaviour might be an essential step towards doing a good job.
Editing by Tom Osborn