EXPECTED TAIL LOSS

OUTLINE OF THE BOOK

2.3 EXPECTED TAIL LOSS

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).

Artzneret al.go on to prove that any coherent risk measure can be regarded as the maximum expected loss on a set of ‘generalised scenarios’, where a generalised scenario is a set of loss values and their associated probabilities (Artzneret al. (1999, p. 219)). This is a very powerful result with far-reaching implications. Suppose that we assume the P/L follows a particular distribution function (e.g., a normal distribution). Given this distribution, we can produce a set of possible loss values, each of whose probability obeys the assumed distribution. This set of loss values constitutes a distinct scenario, and we can define a risk measure — the expected tail loss (ETL) — given by the expected value of these losses. This ETL can also be regarded as the coherent risk measure associated with a single distribution function. Now suppose that we do the same again with another distribution function, leading to an alternative ETL. The maximum of the two ETLs is itself a coherent risk measure. And so forth: if we have a set of comparable ETLs, each of which corresponds to a different distribution function for P/L, then the maximum of these ETLs is also a coherent risk measure.

Another coherent risk measure is the highest loss, or the highest loss among a subset of considered scenarios. Moreover, because coherent risk measures involve scenarios, we can also regard the outcomes of stress tests as coherent risk measures as well. The theory of coherent risk measures therefore provides a theoretical justification for stress testing!

In short, the highest loss from a set of possible outcomes, the expected tail loss, the highest ETL from a set of comparable ETLs based on different distribution functions, and loss estimates from scenario analyses are all coherent risk measures — and the VaR, of course, is not coherent.

2.3.2 The Expected Tail Loss

The ETL is perhaps the most attractive coherent risk measure.¹¹This measure often goes by different names in the literature — including expected shortfall, conditional VaR,¹²tail VaR, tail conditional expectation, and worst conditional expectation, all of which are much the same¹³— but the basic concept itself is very simple.¹⁴ The ETL is the expected value of our losses,L, if we get a loss in excess of VaR:

ETL=E[L|L >VaR] (2.11)

11A brief heuristic proof of the coherence of ETL is suggested by Eberet al. (1999). Imagine a very simple world with 100 equiprobable states tomorrow, a listXof 100 possible numbers, and a confidence level of 95%. The ETL is minus the average of the five smallest numbers. Now consider choosing five states from the 100 available, and for each such choice assign 0.20 to each state, and 0 to every other state: this defines a probabilitypor generalised scenario on the state space. We now consider all such choices of five states from the 100 available, and this gives us a setPof probabilities. Next, note that searching for the average of the five biggest numbers is the same as searching for the biggest number of all averages of five possible numbers. This establishes that the ETL can be represented as the biggest value from a given set of scenarios, and application of Artzneret al’s representation theorem (proposition 4.1 in Artzneret al.(1999, p. 219)) then establishes that the ETL is coherent.

12As if the ETL (or whatever we call it) didn’t have enough names, one of its names sometimes means something quite different. The term ‘conditional VaR’ can also mean VaR itself conditional on something else, such as a set of exogenous variables (e.g., as in the conditional autoregressive VaR or CAViaR of Engle and Manganelli (1999)). When meeting this term, we must make certain from the context what it actually refers to. Naturally, this ambiguity disappears if we have conditional conditional VaR — or conditional ETL, for those who have no sense of humour.

13There are some subtle variations in precise definitions, but I prefer to avoid these complications here. These different definitions and their implications are discussed further in Acerbi and Tasche (2001a).

14The ETL risk measure has also been familiar to insurance practitioners for a long time: it is very similar to the measures of conditional average claim size that have long been used by casualty insurers. Insurers are also very familiar with the notion of the conditional coverage of a loss in excess of a threshold (e.g., in the context of reinsurance treaties). For more on ETL and its precursors, see Artzneret al.(1999, pp. 223–224).

0.4 0.35 0.3

0.15 0.25 0.2

0.1 0.05 0

Probability

4 Loss () / profit ()

VaR at 95% cl 1.645

ETL 2.061

1 2

0 3

−2 −1

−4 −3

Figure 2.10 Expected tail loss.

Note: Produced using the ‘normaletlfigure’ function.

The VaR tells us the most we can expect to lose if a bad (i.e., tail) event doesnotoccur, and the ETL tell us what we can expect to lose if a tail eventdoesoccur.¹⁵

An illustrative ETL is shown in Figure 2.10. If we express our data in loss/profit terms (i.e., we multiply P/L by minus 1, to make the loss terms positive), the VaR and ETL are shown on the right-hand side of the figure: the VaR is 1.645 and the ETL is (about) 2.061. Both VaR and ETL obviously depend on the underlying parameters and distributional assumptions, and these particular figures are based on a 95% confidence level and 1-day holding period, and on the assumption that daily P/L is distributed as standard normal (i.e., with mean 0 and standard deviation 1).

Since the ETL is conditional on the same parameters as the VaR itself, it is immediately obvious that any given ETL figure is only a point on an ETL curve or ETL surface. The ETL/confidence level curve is shown in Figure 2.11. This curve is similar to the earlier VaR curve shown in Figure 2.7 and, like it, tends to rise with the confidence level. There is also an ETL/holding period curve corresponding to the VaR/holding period curve shown in Figure 2.8.

There is also an ETL surface, illustrated in Figure 2.12, which shows how ETL changes as both confidence level and holding period change. In this case, as with its VaR equivalent in Figure 2.9, the surface rises with both confidence level and holding period, and spikes as both parameters approach their maximum values.

In short, the ETL has many of the same attractions as the VaR: it provides a common consistent risk measure across different positions, it takes account of correlations in a correct way, and it has many of the same uses as VaR. However, the ETL is also a better risk measure than the VaR for at least five different reasons:

15For those who want one, a thorough comparison of VaR and ETL is given in Pflug (2000).

2.8

2.4 2.6

1.8 2.2 2

1.6 1.4 1.2

VaR/ETL

0.99 0.98 Confidence level

ETL

VaR

0.95 0.96

0.94 0.97

0.92 0.93 0.9 0.91

Figure 2.11 ETL and the confidence level.

Note: Produced using the ‘normalvaretlplot2D cl’ function.

0

0 0.9 0.92 0.94

0.96 0.98 1 100

50 5

10 15 20 25 30

Confidence level Holding period

ETL

Figure 2.12 The ETL surface.

Note: Produced using the ‘normaletlplot3D’ function.

r

The ETL tells us what to expect in bad (i.e., tail) states — it gives an idea of how bad bad might be — whilst the VaR tells us nothing other than to expect a loss higher than the VaR itself.

r

An ETL-based risk–expected return decision rule is reliable under more general conditions than a VaR-based risk–expected return decision rule: in particular, the ETL-based rule is consistent with expected utility maximisation if risks are rankable by a second-order stochastic dominance rule, whilst a VaR-based rule is only consistent with expected utility maximisation if risks are rankable by a (much) more stringent first-order stochastic dominance rule (see Yoshiba and Yamai (2001, pp. 21–22)).¹⁶

r

Because it is coherent, the ETL always satisfies sub-additivity, whilst the VaR does not. The ETL therefore has the various attractions of sub-additivity, and the VaR does not.

r

The ETL does not discourage risk diversification, and the VaR sometimes does.

r

Finally, the sub-additivity of ETL implies that the portfolio risk surface will be convex, and convexity ensures that portfolio optimisation problems using ETL measures, unlike ones that use VaR measures, will always have a unique well-behaved optimum (see, e.g., Uryasev (2000, p. 1), Pflug (2000), Acerbi and Tasche (2001b, p. 3)). In addition, as Rockafeller and Uryasev (2000) and Uryasev (2000) demonstrate, this convexity ensures that portfolio optimisation problems with ETL risk measures can be handled very efficiently using linear programming techniques.¹⁷

The ETL thus dominates the VaR as a risk measure, and users of VaR would be well advised, where practicable, to use ETL measures instead.

Box 2.4 Other Coherent Risk Measures

There are other coherent risk measures besides the ETL. One of these is the outcome of a worst- case scenario analysis (Boudoukhet al.(1995), Baharet al.(1997)). We would normally carry out this analysis using a simulation method: if we take a confidence level of, say, 99%, we would run a simulation trial of 100 drawings from our chosen P/L distribution, and pick the minimum value; we then run M such trials to obtain a set ofM comparable minimum values, change the sign on these to make losses positive, and so obtain a simulated distribution of worst-case losses.

Our risk measure would then typically be a prespecified high quantile of this distribution (e.g., the quantile associated with the 95% confidence level, which cuts off the top 5% of worst-case losses from the bottom 95% of such losses). Alternatively, our risk measure might be the mean of this distribution, in which case our risk measure is the expected worst-case scenario, which is equivalent to the ETL. Leaving this special case aside, a worst-case scenario analysis based on a high quantile will produce a risk measure that exceeds the ETL, which in turn of course exceeds the VaR — and this risk measure is coherent.

Another coherent risk measure is provided by the Chicago Mercantile Exchange’s Standard Portfolio Analysis of Risk (SPAN) methodology, which is used to derive the margin requirements for positions in the interest-rate futures market (see, e.g., Artzner et al. (1999, p. 212)). The system considers 14 scenarios where volatility can go up or down, and where the futures price can remain the same or take one of three possible upward or downward movements. In addition,

16See also note 2 above.

17One might also add that there is some evidence that ETL might be less prone to sampling error than VaR, so estimates of ETL might be more accurate than estimates of VaR (Mausser and Rosen (2000, p. 218)).

it also considers two extreme upwards or downwards movements of the futures price. The prices of securities in each scenario are then generated by an appropriate model (e.g., the Black model) and the measure of risk is the maximum loss incurred, using the full loss for the first 14 scenarios, and 35% of the loss for the last two extreme scenarios. This risk measure can be interpreted as the maximum of the expected loss under each of 16 different probability measures, and is therefore a coherent risk measure.