OUTLINE OF THE BOOK
2.2 VALUE AT RISK
2.2.1 VaR Basics
A much better approach is to allow the P/L or return distribution to be less restricted, but focus on the tail of that distribution — the worstppercent of outcomes. This brings us back again to the notion of the VaR, and the reader will recall that the VaR on a portfolio is the maximum loss we might expect over a given holding or horizon period, at a given level of confidence.4Hence, the VaR is defined contingent on two arbitrarily chosen parameters — a holding or horizon period, which is the period of time over which we measure our portfolio profit or loss, and which might be daily, weekly, monthly, or whatever; and a confidence level, which indicates the likelihood that we will get an outcome no worse than our VaR, and which might be 50%, 90%, 95%, 99%, or indeed any fraction between 0 and 1.
The VaR is illustrated in Figure 2.5, which shows a common pdf of profit/loss (P/L) over a chosen holding period.5Positive P/L values correspond to profits, and negative observations to losses, and
4The roots of the VaR risk measure go back to Baumol (1963, p. 174), who suggested a risk measure equal toµ−kσ, where µandσare the mean and standard deviation of the distribution concerned, andkis a subjective confidence-level parameter that reflects the user’s attitude to risk. As we shall see, this risk measure is the same as the VaR under the assumption that P/L is elliptical, and the class of elliptical distributions includes the normal and the Studentt, among others. Of course, Baumol did not use the term ‘value at risk’, which only came into use later.
5The figure is constructed on the assumption that P/L is normally distributed with mean 0 and standard deviation 1 over a holding period of 1 day.
0.4 0.35 0.3
0.15 0.25 0.2
0.1 0.05 0
Probability
4 Profit () / loss ()
Probability density for P/L VaR at 95% cl
1.645
1 2
0 3
−2 −1
−4 −3
Figure 2.5 Value at risk.
Note: Produced using the ‘normalvarfigure’ function.
positive values will typically be more common than negative ones. To get the VaR, we must choose a confidence level (cl). If this is 95%, say, then the VaR is given by the negative of the point on the x-axis that cuts off the top 95% of P/L observations from the bottom 5% of tail observations. In this case, the relevantx-axis value is –1.645, so the VaR is 1.645. The negative P/L value corresponds to a positive VaR, indicating that the worst outcome at this level of confidence is a loss of 1.645.
In practice, the point on thex-axis corresponding to our VaR will usually be negative and, where it is, will correspond to a (positive) loss and a positive VaR. However, thisx-point can sometimes be positive, in which case it indicates a profit rather than a loss, and in this case the VaR will be negative. This also makes sense: if the worst outcome at this confidence level is a particular profit rather than a loss, then the VaR, the likely loss, must be negative.
As mentioned already, the VaR is contingent on the choice of confidence level, and will gen- erally change when the confidence level changes. This is illustrated in Figure 2.6, which shows the corresponding VaR at the 99% level of confidence. In this case, the VaR is determined by the cut-off between the top 99% and the bottom 1% of observations, so we are dealing with a 1% tail rather than the earlier 5% tail. In this case, the cut-off point is−2.326, so the VaR is 2.326. The higher confidence level means a smaller tail, a cut-off point further to the left and, therefore, a higher VaR.
This suggests the more general point that, other things being equal, the VaR tends to rise as the confidence level rises.6This point is further illustrated in the next figure (Figure 2.7), which shows
6Strictly speaking, the VaR is non-decreasing with the confidence level, which means that the VaR can remain the same asclrises. However, the VaR will never fall as the confidence level rises, and cases where the VaR remains flat are not too common.
0.4 0.35 0.3
0.15 0.25 0.2
0.1 0.05 0
Probability
4 Profit () / loss ()
Probability density for P/L VaR at 99% cl
= 2.326
1 2
0 3
−2 −1
−4 −3
Figure 2.6 VaR at the 99% confidence level.
Note: Produced using the ‘normalvarfigure’ function.
2.6 2.4
1.8 2.2
2
1.6
1.4
1.2
VaR
0.99 0.98 Confidence level
0.95 0.96
0.94 0.97
0.92 0.93 0.9 0.91
Figure 2.7 VaR and confidence level.
Note: Produced using the ‘normalvarplot2D cl’ function.
18
12 14 16
6 10 8
4 2 0
VaR
100 90 80 Holding period
50 60
40 70
20 30
0 10
Figure 2.8 VaR and holding period.
Note: Produced using the ‘normalvarplot2D hp’ function.
how the same VaR varies as we change the confidence level and keep other parameters constant. In this particular case, the VaR not only rises with the confidence level, but also rises at an increasing rate — a point that risk managers might care to note.
We should also remember that the VaR is contingent on the choice of holding period as well, and so we should consider how the VaR varies with the holding period. This behaviour is illustrated in Figure 2.8, which plots the VaR at the 95% confidence level against a holding period that varies from 1 day to 100 days. In this case, the VaR rises with the square root of the holding period, from a value of 1.645 at the start to 16.449 at the end. This ‘square root’ case is commonly cited in the literature, but we should recognise that VaR might rise in a different way, or even fall, as the holding period rises.
Of course, each of the last two figures only gives a partial view of the relationship between the VaR and the confidence level/holding period: the first takes the holding period as given and varies the confidence level, and the second varies the holding period whilst taking the confidence level as given. To form a more complete picture, we need to see how VaR changes as we al- low both parameters to change. The result is a VaR surface — illustrated in Figure 2.9 — that enables us to read off the value of the VaR for any given combination of these two parameters.
The shape of the VaR surface shows how VaR changes as the underlying parameters change, and conveys a great deal of risk information. In this particular case, which is also typical of many, the surface rises with both confidence level and holding period to culminate in a spike — indicating where our portfolio is most vulnerable — as both parameters approach their maximum values.
0
0 0.9 0.92 0.94
0.96 0.98 1 100
50 5
10 15 20 25
Confidence level Holding period
VaR
Figure 2.9 A VaR surface.
Note: Produced using the ‘normalvarplot3D’ function.
Box 2.2 Value at Risk as a Regulatory Risk Measure
Value at risk is also used by bank regulators to determine bank capital requirements against market risk.7 Under the 1996 Amendment to the Basle Accord, institutions judged to have sound risk management practices are allowed the option of having their capital requirements determined by their own VaR estimates. This is known as the ‘internal models’ approach to regulatory capital requirements. The effective daily capital requirement is the maximum of the previous day’s VaR andktimes the average of the daily VaR over the last 60 days, wherekis a multiplier in the range between 3 and 4. This multiplier is set by the bank’s supervisors, conditional on the results of a set of standardised backtests (see Box 9.1: Regulatory Backtesting Requirements), with better backtesting results leading to a lower value ofk. The application of this multiplier is sometimes justified as providing insurance against model risk, non-normal market moves, and similar factors.
The Amendment also requires that VaR be derived at the 99% confidence level using a 10-day holding period.8However, in the initial implementation of this approach, banks are allowed to proxy the 10-day VaR by multiplying the 1-day VaR by the square root of 10. Banks are allowed to calculate the VaR using their own preferred models, subject to certain minimum criteria (e.g., that the model covers non-linear Greek factors, and so forth). Finally, there are also certain additional
7For good accounts of the current regulatory capital requirements, see Crouhyet al.(1998; 2001, ch. 4).
8These parameters imply that the VaR will be exceeded in only about one 10-day period in every four years. This should lead to a very low probability of failure, because the capital requirement itself is at least three times the VaR. Our estimated failure probability will then depend on what we assume about the P/L distribution. If we assume that the P/L is normal — which, strictly speaking, we shouldn’t, because of extreme value theory — my calculations lead to a probability of failure indistinguishable from zero; but if we assume a Gumbel distribution, whichisconsistent with extreme value theory, then the probability of failure is no more than 0.25% per year, and less if we have a multiplier of greater than 3. This means that our institutions should be pretty safe — unless, like Barings or LTCM, they have a poor risk measurement model.
capital charges for ‘specific risk’ or credit-related risks on market instruments (e.g., counter- party default risk on OTC positions), and these too can be determined using an internal models approach.
Even if we grant that there is any need for regulatory capital requirements in the first place — and I would suggest there isn’t — then perhaps the best thing we can say about the internal models approach is that it does at least make some effort to tie capital requirements to a rea- sonably respectable measure of market risk. Unfortunately, it does so in a very arbitrary and indefensible way. The multiplier is essentially pulled out of thin air, give or take a certain amount of adjustment for the results of a primitive backtest. The confidence level enshrined in the regulation — 99% — is also of no real relevance to bank solvency, and the regulations give con- siderable scope for manipulation and regulatory avoidance (i.e., they encourage institutions to seek ways to minimise the impact of capital regulations whilst technically complying with them).
In some respects the regulations also tend to discourage the development of good market prac- tice and there are good reasons to believe that they might make the financial system less rather than more stable (see, e.g., Danielsson (2001) and Danielssonet al.(2001)). If regulators wished to determine market risk capital requirements in an intellectually coherent fashion, they would be better advised to work from some target probability of financial distress and use extreme value theory to work out the capital charges — but it would be much better if governments ab- stained from capital regulation and other forms of intervention altogether (e.g., such as deposit insurance) and allowed banks to determine their own capital requirements under free-market conditions.