Stress Testing Value-at-Risk

by Romain Berry
J.P. Morgan Investment Analytics & Consulting
romain.p.berry@jpmorgan.com

Value-at-Risk answers the question of how much a portfolio may lose if it remains unchanged over a given time horizon under normal market conditions at a given level of confidence. But VaR does not say how bad this portfolio may be hit if a sharp adverse movement occurs in these so-called normal markets. We actually find this last thought easier to grasp and thus to measure than trying to estimate the absolute (or relative to a benchmark or to a set of liabilities) level of risk in a portfolio with one single number.

We review in this article various approaches to stress a portfolio and derive the pros and cons of each methodology in terms of time, complexity, cost, resources, level of reporting, frequency and specific needs (regulatory requirements, for instance).

Overview of Stress Testing

We have spent three articles describing the three main methodologies widely used to compute VaR, although we have not yet touched on the core activities that a risk manager should perform on an ongoing basis: Stress Testing. In our view, a risk manager should not rely too much on VaR calculations since they may be underestimated quite often. This may sound like a controversial and frustrating statement, but remember that VaR is only valid under normal market conditions and a series of theoretical assumptions. In a nutshell, VaR is only as good as the model it stems from and must always be interpreted within the set of assumptions. That is something that most of VaR critics have forgotten too easily. VaR is an ingenious concept which should not be eradicated from the risk manager toolbox simply because it has been misused.

VaR should not be taken for the panacea of risk measures. The main advantage of VaR is that it summarizes within a single number the level of risk embedded in a portfolio. It is a starting point that we should tweak, distort and stress in order to better understand its behavior as we amend the assumptions it relies on. It is a tool that requires further tuning to provide a richer view of the risk of a portfolio.

Stress testing is a tuning process by which we can explore how the portfolio would react to small (Sensitivity Analysis) or more drastic (Stress Tests) changing conditions in the markets. Table 1 employs this clustering according to the size of the shock to exhibit various forms of stress tests.

Sensitivity Analysis

Sensitivity Analysis consists of shocking various risk factors of the portfolio with small upward or downward increments. It is very simple to implement and can be quickly automated in a systematic way. Examples of shocks are imposing a fall of all equity prices in the portfolio or proceeding to a parallel shift of the yield curve to shock the bonds included in the portfolio. In the former example, we will impose all equities to have a price 10% lower than they are at the moment. We rerun our VaR calculation and compare it with the VaR without the 10% equity shock. That might give us a rough idea of the “sensitivity” of the equities compared to the other asset classes in the portfolio.

From there, it is possible to determine a range of larger increments to analyze how the sensitivity evolves as the risk factors are shocked more dramatically. Monte Carlo Simulations VaR is sometimes seen as a “black box” as it is difficult to predict the outcome of the simulations. Sensitivity Analysis can provide some information on how the VaR reacts to shocks of various amplitudes on various risk factors. Furthermore, it is feasible to shock various factors at the same time within the same scenario. But without running individual shocks first, it will be hard to interpret which factor explains most of the new VaR estimate as correlations might have changed. In this instance, the risk manager needs to think as an economist more than as a quantitative analyst or to work with her colleagues in the Research department to integrate their view on various asset classes, regions, currencies, etc.

It is also interesting to shock these risk factors over a longer period of time than only at the end of the analysis horizon (generally, one day or one month). This way, you can generate different shocks that will apply to some risk factors over a few time horizons. For instance, we can recalculate a VaR on a portfolio where we have amended the exchange rates amongst the two main currencies with the following shocks: -10%, +5%, and -15%. Studying the historical movements of these two currencies can provide some information to project various scenarios on how the market could evolve over the next few time intervals. This is where Sensitivity Analysis meets with Stress Tests as this last approach could be seen as a Customized Stress Test.

Despite its simplicity, this methodology also has some pitfalls. The risk manager needs to exercise some judgment in determining the optimal size of each shock which may differ from one asset class to another. These shocks must also be reassessed on a regular basis to avoid missing a change in the pattern of one specific asset (sudden and brief increase in volatility) or a correlation increasing between two assets. Also, since we only shock one factor at a time and with a very small change, the analysis is very local.

Stress Testing

Stress testing aims to identify extreme events that could trigger catastrophic losses in a given portfolio. Per definition, the shocks that are applied to the portfolio are of much greater amplitude than those used in a standard Sensitivity Analysis. Mainly, there are three types of stress testing: historical, customized and reverse stress testing.

Historical stress tests or scenarios intend to test the healthiness of a portfolio by analyzing what would happen to the portfolio if particularly adverse and unexpected movements which occurred in the past would hit the portfolio in the near future. Some well-known examples of historical scenarios are the Russian Crisis, the attacks of 9/11, and more recently the Sub-Prime Crisis. Some of these historical scenarios could last a few days only like the Black Monday (October 19th, 1987) scenario. Some others like the Dotcom Bubble spanned over several months. The main advantage of these types of scenarios is that they really did happen! But even if the temptation is great to use these historical scenarios off the shelf and to systematically apply them on any types of portfolio, the risk manager should choose his historical scenarios very carefully and review them on a regular basis as the composition of the portfolio changes, but also because of a few dangers.

First, we need to select the historical stress tests that are the most relevant to the portfolio. Recreating the shocks that occur during the Russian Crisis to a portfolio that does not contain any bonds will have limited interest. Second, one should determine the start and end dates of the historical scenario. This is not as easy as it seems as there may be different interpretations on what these two dates are. Third, what do we do with the instruments that will reach maturity during the re-enactment of these events? You can roll them over or not depending on your strategy or on the size of these positions. In either case, cash flows need to be taken into account appropriately. Fourth, do you apply an absolute or a relative shock to the risk factors? Generally, we perform relative shocks but that depends on the risk factor (for instance, it is better to shock volatility on a relative basis). Fifth, what do we do about missing instruments? What is the point of applying the Black Monday scenario to a portfolio of CDSs? Further, if historical data are not available on all risk factors, one should either proxy them or proceed to an interpolation (more relevant to fixed income instruments where the term structure may need to be filled in and out throughout). Finally, we should point out that historical stress tests produce a loss estimate and not a VaR. Therefore, the likelihood of seeing a historical stress test come true remains unknown.

In order to fix some of these drawbacks, one can design some specific stress tests based on historical stress tests or on areas of vulnerability in the portfolio. These stress tests are called customized since they respond to a particular purpose such as shocking correlations, stressing a liquidity squeeze, or creating a scenario which is more likely to impact a portfolio than historical stress tests would. These scenarios can be economic, political or financial. The complexity of the scenario depends on various factors such as number of risk factors taken into account, period of time the pre-defined scenario is expected to last, complexity of the portfolio, number of positions in the portfolio, running time, staff, cost, etc. In a nutshell, there must be a trade-off between the constraints of establishing a complete program of customized stress tests and the desired outcome.

Reverse Stress Tests try to identify the risks that would lead an institution to fail. This is an appealing idea in the sense that instead of starting from the existing standpoint and seeing how close we can go towards the ridge of the cliff without falling, reverse stress testing tells you what risks you could take to fall directly off the cliff. That makes so much sense that you may wonder why we have not carried out reverse stress tests for ages. Well, the main problem with reverse stress testing is “how” to do it? There are so many reasons why an institution would fail that it may take some time to determine meaningful stress tests. When we conduct other types of stress testing, we always start from the known: the portfolio itself and its VaR and try to progress more or less in the dark to gauge the risks ahead. With reverse stress test, we start from the unknown and try to figure out how we became lost on the way home. This intellectually challenging thought could soon become a tedious task where one tries to assess which events could have triggered the failure and how this event has contaminated the entire system. There is no easy answer to this problem, but since contagion is the result of increasing correlation, working with copula statistical analysis could be a starting point.

Conclusion

We do believe that stress testing is more important than calculating the VaR itself. Of course, a minimum of effort needs to be put in place to compute an accurate VaR estimate, but at least an equal amount of time and effort must be spent to analyze its sensitivity to various shocks and stress events. We like to think that we construct a view of the risk profile of a portfolio by estimating VaR, a single number which helps manage the portfolio at a high level. But we should deconstruct this estimate with stress testing to obtain a more granular understanding of its sensitivity and weaknesses.

Erratum:

A typo occurred in our previous article entitled “An Overview of Value-at-Risk Part III – Monte Carlo Simulations VaR” in equation (4) which should have read as follows: