The Basel Accords oblige banks to use two methods to measure and limit risk – value at risk and expected shortfall – but research shows these are insufficient to curtail the behaviour of rogue traders. Damiano Brigo and John Armstrong suggest a different approach. 

It is crucial for society to set limits on the risks that banks are allowed to take. Within any bank, it is equally important to set limits on the risks taken by individual traders. But how should risk be measured in order to decide whether or not a particular trading or investment position is acceptably risky?

The international standards set by the Basel Accords require banks to use two standard measures of risk: value at risk (VaR) and, more recently, expected shortfall (ES). So are the risk limits defined using these standards sufficient to curb the behaviour of rogue traders? We found that, disturbingly, the answer is no.

The utility curve

We think of a rogue trader as someone who acts to maximise their expected utility. Their utility takes as input the profit or loss of their financial position at the end of the trade. We assume the trader likes making money, so their utility goes up with profits and goes down with losses. However, importantly, we assume that a rogue trader is not risk-averse.

A risk-averse individual will be moderately happy if they make $1bn, but will be extremely sad if they lose $1bn

Simplifying, a risk-averse individual will be moderately happy if they make $1bn, but will be extremely sad if they lose $1bn. As a result, their utility will be a classic increasing concave utility: if we plot a graph of their utility against their final position it will slope upwards, but the curve becomes less and less steep as the amount of money made increases, and becomes steeper and steeper in the opposite direction as the amount of money lost increases.

A rogue trader, instead, has an S-shaped utility curve. They are not particularly concerned about losing larger and larger amounts of money beyond a point because they have some kind of limited liability. Indeed, after they have lost their job and their reputation, the size of the loss for the institution beyond that point will not really matter that much to them, and the utility will go down less and less steeply for larger and larger losses.

The Nobel-awarded experimental psychologists Daniel Kahneman and Amos Tversky showed that most individuals behave according to S-shaped utility curves. This does not apply just to individuals. As financial institutions are owned by shareholders who have limited liability the utility curve of many institutions will be S-shaped too.

Applying new measures

What strategies would a rogue trader adopt in order to maximise their S-shaped utility when under VaR or ES risk limits? The trader could use, for example, digital options to generate a position that made larger and larger profits in more and more likely good scenarios but which would lead to more and more catastrophic losses in less and less likely bad scenarios.

There would be no limit on the expected utility a rogue trader could achieve with this strategy, regardless of the risk limits. Also, all such strategies would respect the same budget constraint, namely a limit on the initial cost of setting up the trading strategy. So what type of limits should be imposed instead?

One simple option is to measure the risk of a trader using a different expected utility for wider society, or for the regulator or the institution’s risk manager. This second utility would be given by a risk-averse curve, with a classic concave utility function where larger and larger losses would be weighted increasingly faster.

A typical rogue trader would feel the effect of limits set in this way and the maximum utility they could achieve would be reduced, although the precise curve a regulator should impose on banks and the incentives a trader should receive for being compliant with this curve are to be investigated further.

A benefit of this approach is that it makes explicit the fundamental fact that different parties within the same institution – for example, traders and risk managers – will have different objectives and utilities.

Professor Damiano Brigo holds the chair in mathematical finance at Imperial College London, where he co-heads the mathematical finance research group and is part of the stochastic analysis research group. Dr John Armstrong is senior lecturer in financial mathematics, probability and statistics at King's College London. 

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