Limitations of VaR as a Risk Measure

OUTLINE OF THE BOOK

2.2 VALUE AT RISK

2.2.3 Limitations of VaR as a Risk Measure

VaR also has its drawbacks as a risk measure. Some of these are fairly obvious — that VaR estimates can be subject to error, that VaR systems can be subject to model risk (i.e., the risk of errors arising from inappropriate assumptions on which models are based) or implementation risk (i.e., the risk of errors arising from the way in which systems are implemented). However, such problems are common to all risk measurement systems, and are not unique to VaR.

2.2.3.1 VaR Uninformative of Tail Losses

Yet VaR does have its own distinctive limitations. One of these is that VaR only tells us the most we can lose if a tail event does not occur — it tells us the most we can lose 95% of the time, or whatever — but tells us nothing about what we can lose in the remaining 5% of occasions. If a tail event (i.e., a loss in excess of VaR) does occur, we can expect to lose more than the VaR, but the VaR figure itself gives us no indication of how much that might be.

This can lead to some awkward consequences. A trader or asset manager might ‘spike’ his firm by entering into deals that produce small gains under most circumstances and the occasional very large loss. If the probability of loss is low enough, then this position would have a low VaR and so appear to have little risk, and yet the firm would now be exposed to the danger of a very large loss.

A single VaR figure can also give a misleading impression of relative riskiness: we might have two positions with equal VaRs at some given confidence level and holding period, and yet one position might involve much heavier tail losses than the other. The VaR measure taken on its own would incorrectly suggest that both positions were equally risky.

Fortunately, we can sometimes ameliorate these problems by using more VaR information. For example, the trader who spikes his firm might be detected if the VaR of his position were also estimated at very high confidence levels. A solution to our earlier problems is, therefore, to look at the curve of VaR against confidence level, and not just to look at a single VaR figure — which is in effect to look at the VaR at only one point on its surface.

2.2.3.2 VaR Can Create Perverse Incentive Structures

But it is not always feasible to use information about VaRs at multiple confidence levels, and where it is not, the failure of VaR to take account of losses in excess of itself can create some perverse outcomes. For instance, a rational investor using a VaR risk measure can easily end up with perverse positions precisely because a VaR-based risk–return analysis fails to take account of the magnitude of losses in excess of VaR. If a particular investment has a higher expected return at the expense of the possibility of a higher loss, a VaR-based decision calculus will suggest that we should make that investment if the higher loss does not affect (i.e., and therefore exceeds) the VaR, regardless of the size of the higher expected return and regardless of the size of the higher possible loss. Such a categorical acceptance of any investment that increases expected return — regardless of the possible loss, provided only that it is insufficiently probable — makes a mockery of risk–return analysis, and the investor who makes decisions in this way is asking for trouble.⁹ Admittedly, this example is rather extreme, because the VaR itself will often rise with the expected return, but the key point is that we cannot expect a VaR-based rule to give us good risk–return decisions except in particular circumstances (i.e., to be precise, unless risks are elliptically distributed or are rankable by first- order stochastic dominance, which is a very demanding and empirically unusual condition;¹⁰see, e.g., Yoshiba and Yamai (2001, pp. 16–17)).

If an investor working on his/her own behalf can easily end up with perverse positions, there is even more scope for mischief where decision-making is decentralised and traders or asset managers are working to VaR-constrained targets or VaR-defined remuneration packages. After all, traders or asset managers will only ‘spike’ their firm if they work to an incentive structure that encourages them to do so. If traders face a VaR-defined risk target, they will often have an incentive to sell out-of-the-money options to increase ‘normal’ profits and hence their bonus; the downside is that the institution takes a bigger hit once in a while, but it is difficult to design systems that force traders to care about these bigger hits: the fact that VaR does not take account of what happens in ‘bad’

states can distort incentives and encourage traders or managers to ‘game’ a VaR target (and/or a VaR-defined remuneration package), and promote their own interests at the expense of the interests of the institutions they are supposed to be serving.

9There is also a related problem. The VaR is not (in general) a convex function of risk factors, and this makes it difficult to program portfolio optimisation problems involving VaR: the optimisation problem has multiple local equilibria, and so forth (see, e.g., Mausser and Rosen (1998) or Yamai and Yoshiba (2001b, p. 15)).

10See note 1 above.

2.2.3.3 VaR Can Discourage Diversification

Another drawback is that VaR can discourage diversification, and a nice example of this effect is provided by Eberet al.(1999). Suppose there are 100 possible future states of the world, each with the same probability. There are 100 different assets, each earning reasonable money in 99 states, but suffering a big loss in one state. Each of these assets loses in a different state, so we are certain that one of them will suffer a large loss. If we invest in one of these assets only, then our VaR will be negative at, say, the 95% confidence level, because the probability of incurring a loss is 1%. However, if we diversify our investments and invest in all assets, then we are certain to incur a big loss. The VaR of the diversified portfolio is therefore much larger than the VaR of the undiversified one. So, a VaR measure can discourage diversification of risks because it fails to take into account the magnitude of losses in excess of VaR.

2.2.3.4 VaR Not Sub-additive

But there is also a deeper problem with VaR. In order to appreciate this problem, we need first to introduce the notion of sub-additivity. A risk measureρ(·) is said to be sub-additive if the measured risk of the sum of positionsAandBis less than or equal to the sum of the measured risks of the individual positions considered on their own, i.e.:

ρ(A+B)≤ρ(A)+ρ(B) (2.9)

Sub-additivity means that aggregating individual risks does not increase overall risk. Sub-additivity matters for a number of reasons:

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If risks are sub-additive, then adding risks together would give us an overestimate of combined risk, and this means that we can use the sum of risks as a conservative estimate of combined risk.

This facilitates decentralised decision-making within a firm, because a supervisor can always use the sum of the risks of the units reporting to him as a conservative risk measure. But if risks are not sub-additive, adding them together gives us an underestimate of combined risks, and this makes the sum of risks effectively useless as a risk measure. In risk management, we want our risk estimates to be unbiased or biased conservatively.

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If regulators use non-sub-additive risk measures to set capital requirements, a financial firm might be tempted to break itself up to reduce its regulatory capital requirements, because the sum of the capital requirements of the smaller units would be less than the capital requirement of the firm as a whole.

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Non-sub-additive risk measures can also tempt agents trading on an organised exchange to break up their accounts, with separate accounts for separate risks, in order to reduce their margin require- ments. This could be a matter of serious concern for the exchange because the margin requirements on the separate accounts would no longer cover the combined risks.

Sub-additivity is thus a highly desirable property for any risk measure. Unfortunately, VaR is not generally sub-additive, and can only be made to be sub-additive if we impose the (usually) implausible assumption that P/L (or returns) are normally (or slightly more generally, elliptically) distributed (Artzner et al. (1999, p. 217)).

A good counter-example that demonstrates the non-sub-additivity of VaR is a portfolio consisting of two short positions in very-out-of-the-money binary options. Suppose each of our binary options has a 4% probability of a payout (to us) of−$100, and a 96% probability of a payout of zero. The underlying variables (on which the payouts depend) are independently distributed, so the payout on

Table 2.1 Non-sub-additive VaR

(a) Option Positions Considered Separately

PositionA PositionB

Payout Probability Payout Probability

−100 0.04 −100 0.04

0 0.96 0 0.96

VaR at 95%cl 0 VaR at 95%cl 0

(b) Option Positions Combined

Payout Probability

−200 0.04²=0.0016

−100 2(0.04)0.96=0.0768

0 0.96²=0.9216

VaR at 95%cl 100

one binary option is independent of the payout on the other. If we take the VaR confidence level to be 95% and the holding period to be equal to the period until the options expire, then each of our positions has a VaR of 0 at the 95% level. If we combine the two positions, however, the probability of a zero payout falls to less than 95% and the VaR is positive (and, in this case, equal to $100).

The VaR of the combined position is therefore greater than the sum of the VaRs of the individual positions; and the VaR is not sub-additive (Table 2.1).