Copyright Incisive Media

Transcription

1 Extremely (un)likely: a plausibility approach to stress testing Pierre Mouy, Quentin Archer and Mohamed Selmi present a framework for generating extreme but plausible stress scenarios. The framework provides a closed-form solution for elliptical distributions and an extension for adjusted marginals. The stress scenarios generated are equally likely, equally harmful and not affected by the addition of risk factors orthogonal to the portfolio The question of financial market infrastructure stress testing, and central clearing counterparties (CCPs) specifically, has been a particularly live topic over the past few years. Regulation is not yet prescriptive on how to set and define a sound stress testing framework, and building both severe and plausible scenarios is a challenge. Broadly speaking, there are three main ways to come up with relevant stress scenarios. One can () replicate historical events, () extrapolate on macroeconomic or political assumptions (expert judgement) or () rely on distribution assumptions and quantitative methods. The first two options are usually easy to define and implement. However, relying on historical events limits the scenario universe to past observations, while expert judgement can be driven by risk aversion and induce bias and procyclical behaviours. Quantitative measures are usually a straightforward way to tackle unidimensional problems. It is fairly easy to set a quantile level or a time of return, and set the stress scenario accordingly at a once in x years or 99.x% level. Well-known techniques such as extreme value theory and Student s t distributions can be used to overcome pure historical distribution limits. In a multivariate environment, however, finding plausible stress scenarios respecting CCP operational constraints is more challenging. This article presents the practical problem the authors face as CCP risk managers and identifies the four core properties expected for a set of stress tests: exhaustivity, homogeneity, stability and plausibility. The authors first describe the concept of finding stress scenarios via an optimisation method introduced in Breuer & Krenn (). They illustrate using a simple example of two and three risk factors how the framework partially meets their upfront criteria. They then present a proposed variation with a closed-form solution for elliptical copulas. Finally, the method is extended to a proxy approach for meta-elliptical copulas, and potential improvements are suggested. Plausibility domain and hypothetical events Problem description and expected properties. To comply with regulatory standards under European Market Infrastructure Regulation (Emir) Article 4., a clearing house needs to define a set of historical and theoretical stress scenarios: A CCP shall develop scenarios of extreme but plausible market conditions. The scenarios shall include the most volatile periods that have been experienced by the markets for which the CCP provides its services and a range of potential future scenarios. Once defined, the set is used to assess the potential losses of all clearing members portfolios. Exposures not covered by the resources collected by the CCP (so-called stress test loss over initial margin, or STLOIM) are covered by the second layer of protection: the default fund. The default fund is a mutualised resource and is therefore meant to be resilient, stable and treat members bringing equivalent risks equally. Building a stress scenario reverses the usual risk management issue of estimating tail losses by finding a plausible set of joint risk factor moves leading to a specific loss. Defining a consistent set of stress scenarios is therefore a challenge. CCP risk managers look for the following operational properties. Exhaustivity: the stress scenario framework should be complete, that is, sufficiently diversified to cover the main risk factors and potential exposures to which the CCP could be exposed. Homogeneity: the framework should lead to equally likely, equally harmful scenarios. In other words, clearing members with identical risk profiles should be treated equally. Stability: the framework needs to be stable to the addition of risk factors orthogonal to a member portfolio. For instance, clearing new products should not affect the stress scenario of members not interested in trading them. Plausibility: the framework must lead to plausible losses on any risk factor or portfolio. In particular, if a return is not plausible at the individual risk factor level, it cannot be deemed plausible in a higher dimension. Plausibility domain under the Mahalanobis distance. Besides the size of the loss, it is just as important to attach a frequency of occurrence to a stress scenario to meet the homogeneity property. Furthermore, highly implausible scenarios undermine the credibility of stress tests. Let us consider a portfolio of n securities, with profit and loss P&L.S/ D P T S D i P i S i, where S i is the return of security i and P i is the value of the sub-portfolio composed of security i. The problem of finding the maximum loss over all possible returns is, in general, trivial. A more interesting problem is to find this maximum loss given additional constraints on the returns allowed (or considered). This can, for example, be achieved by specifying a bound on the plausibility of an indicator.s/ 6 p, where p is the -quantile of the.s/ distribution. When a covariance matrix of risk factor returns exists, a popular choice for.s/ is the Mahalanobis distance. The Mahalanobis distance of a vector of risk factor returns S to its expected value under a correlation matrix is defined by: q Maha.S/ D.S / T.S / 7 risk.net August 7

2 Intuitively, Maha.S/ is the size of the multivariate move from to S as measured in standard deviations. A high Maha.S/ means the scenario is far away from its expected value and, intuitively, less likely to happen. Points with the same Mahalanobis distance are located on an ellipse centered on, which therefore makes this indicator well suited for elliptical distributions. For simplicity, we assume the distribution is centered ( D ) throughout the rest of the paper. Academic approach: the plausible scenario with maximum loss. Optimisation problem. The question of finding a worst-loss scenario for a given portfolio has been tackled in the literature (see, for example, Breuer 8; Breuer & Krenn ). This approach looks for the scenario with the worst loss under a plausibility constraint on the Mahalanobis distance. It can be formalised by the minimisation problem: min P T S D min i P i S i () Maha.S/6p S T S6p where p is a cap constraint on the returns. It can be chosen, for example, as the -quantile of the distribution of the Mahalanobis distance p S T S. Problem solution. Suppose the matrix is symmetric and positivedefinite (if not, one can use a projection algorithm (Qi & Sun 6)). Hence, there exists a symmetric matrix M such that D M T M. We make the change of variable Y D MS and arrive at the equivalent optimisation problem: min Q T Y ky k6p where Q D M T P This problem can be solved explicitly. Indeed, for every vector Y such that ky k 6 p, the Cauchy-Schwarz inequality implies Q T Y > p kqk. Hence: min Q T Y > p kqk DQ T Y ky k6p where Y D p.q=kqk/, and Y obviously verifies the constraint ky k 6 p. This inequality constraint is binding, ie, ky kdp. Consequently, min ky k6p Q T Y D Q T Y. The minimiser of this problem is unique, thanks to the fact that () equality in the Cauchy- Schwarz inequality holds if and only if Y and Q are co-linear and () Y has a fixed norm. Then, this transformed problem has the unique solution Y D p.q=kqk/. Now, kqk D p, so the problem has the unique solution: S D M Y P D p p and the worst loss is given by P T S D p p. The dimensional dependence issue. The Mahalanobis distance gives iso-plausibility hypersurfaces as well as the proportion of observations inside the ellipse. As such, it is similar to a quantile. The papers we refer to propose a comprehensive description of the framework, its limitations and a proposed remediation. An equally interesting branch of the literature is dedicated to risk-reducing actions. For clarity, we limit our description to the concept of defining a stress scenario as the solution of an optimisation problem within an admissibility domain. () Having explicitly solved problem (), it remains now to fix the cap parameter p, or equivalently the plausibility level. Suppose S follows a multivariate normal distribution with covariance matrix. Then, S T S follows a.n/ law. p is defined as the -quantile of this distribution: p D F. / () where F n is the quantile function of a distribution with n degrees of freedom. The problem with dimensional dependence is caused by the occurrence of n in this equation. The optimal scenarios S and the losses P T S, which are functions of p, are therefore sensitive to the addition of new risk factors, even if they are irrelevant to the portfolio. This leads to two undesirable issues in practice: adding a new risk factor, even unrelated to the portfolio, affects the optimal scenario; and increasing the number of risk factors affects the maximum potential loss at the individual risk factor level. The number of risk factors is to some degree arbitrary. For example, one is free to include or exclude risk factors that are irrelevant to the portfolio value. This should not affect the measured risk. We now present a simple working example of how it affects the desired properties of stability and plausibility discussed above. Working example. Assume the CCP has a member portfolio comprised of CAC 4 and AEX indexes: long million of CAC 4 contracts and short million of AEX contracts (positions values), P D.; / T. The CCP needs to define an extreme but plausible stress test scenario over a five-day holding period with a confidence level set to D 99:9%. We assume the two risk factors are normally distributed with the following calibrated historical correlation and volatilities:! D :8 :8 where is the correlation matrix of the five-day returns between CAC 4 andaex. We also calibrate the historical standard deviation of the five-day returns CAC 4 D AEX D %. Then, one can easily compute the closed-form formula () and use () to obtain:!! S D :6 CAC 4 D :% : AEX 7:% The P&L obtained in this scenario is P T S D :67 million. Let us now assume the CCP wants to clear new products on a new commodity underlying, eg, wheat. Commodity-specialised members will trade this product; however, our first member, an index trader, will not be interested in trading on wheat and will not have any position on it in their portfolio. Now, we have three risk factors in the clearing universe with the following correlation matrix: n :8 : B C :8 :A : : risk.net 7

3 The first member has the same long million of CAC 4 contracts, short million of AEX contracts and no position on wheat. Their portfolio is represented as follows in a three-dimensional universe: P D.; ; / T. Using the closed-form formula () in three-dimensional space, one obtains: :% S B C 9% A :% with a corresponding P&L equal to :7 million. This is quite puzzling, since the first member did not change their portfolio positions; the simple addition of a risk factor to the clearing universe changes the optimal stress test scenario and the simulated stress test loss on this portfolio. Note also that the moves onaex in both two-dimensional space (7.%) and three-dimensional space (9%) are above the one-dimensional 99.9% of a Gaussian distribution with % standard deviation (.4%), and are thus using non-plausible returns and violating the plausibility criterion. From a CCP s point of view, such an approach is therefore not fully satisfactory for the purpose of defining stress test scenarios, as, if the homogeneity and exhaustivity properties are well met, the stability and plausibility criteria fail. The framework could potentially lead to unstable scenarios and increased volatility in default fund size when the risk factor universe evolves, even if members portfolio positions do not change. The next section describes a variation of the framework that meets all four criteria. Most plausible scenario among worst-loss scenarios The max likelihood optimisation. Here, we propose another perspective. We begin by defining a set of scenarios leading to a given loss level and then look for the most plausible one. This will be defined as the one with the largest likelihood value, as quantified by the density function f of the scenario. We can formalise this by using the maximum likelihood problem max P T S>q f.s/, where q is a cap constraint on the portfolio loss. It can be chosen to equal the -quantile of the portfolio loss distribution P T S. For example, if S is a Gaussian vector, then P T S is Gaussian, so q can be chosen as a (one-dimensional) Gaussian quantile. Discussion: cap on the portfolio loss. In general, the theoretical maximum loss of unbounded distributions can be infinite if all scenarios are considered. It will be impossible to find a market state in which the portfolio has its smallest value, since the loss potential is usually unlimited. Let us take the example of Breuer & Krenn () of a portfolio consisting only of a written call option: its value will fall without limit as long as the value of the underlying instrument rises. For this reason, it is useful to define a stress event as an event attached to a certain time of return, such as a once in every years event. Such plausibility statements are closely related to risk measures such as value-at-risk and expected shortfall. The confidence level is generally set to the financial institution s risk appetite and must respect regulatory requirements. This also applies to CCPs where margins are computed under a certain confidence level and stress losses are computed with a higher quantile level >. The cap used in the optimisation problem above is the -quantile of the portfolio loss distribution; this can be determined analytically if the loss distribution is known, and by relying on numerical techniques otherwise. Closed-form solution in the elliptical case. One interesting case is where the density f of the vector S is elliptical. This means f.s/ D g.s T S/ for some numerical function g. In the usual case, where g is a decreasing function (multivariate Gaussian or Student s t) and is a symmetric positive definite matrix, the problem above is equivalent to: min S T S (4) P T S>q Once again, we transform the problem, using the same transformation as in (). Using again the Cauchy-Scharwz inequality, we get a closed-form formula for the unique solution: S D q P and the worst loss is given by P T S D q. Contrary to problem (), the approach described in (4) does not suffer from the drawback of dimensionality dependency, since the -quantile q depends only on the assets in the portfolio. The dependency in dimension does not appear in the constraint any longer. Therefore, using the same will yield the same loss q. Moreover, if we add a new asset to the objective function without changing the portfolio, the optimal scenario () will not change, ie, the restriction of the worst-case scenario SN C to the first N components will correspond to the worst-case scenario SN. Now it remains to specify the loss cap parameter q. The natural choice is to choose q as a quantile of the distribution of P T S at some level. Suppose S follows a multivariate normal distribution with covariance matrix. Then, P T S follows a Gaussian distribution N.; / and the loss quantile q is equal to q D N. / p. Finally, the optimal scenario can be written as: S D N P. / p where N is the quantile function of a normal distribution. Note: one can easily compute the Mahalanobis distance of this optimal scenario p S T S, which is equal to N. /. The optimal scenario will therefore be on the ellipsoid of radius N. /. Working example. Let us recall the member portfolio comprised of CAC 4 and AEX indexes, long million of CAC 4 contracts and short million of AEX contracts, to check if the CCP constraints are fulfilled by the new approach, P D.; / T. Computing the solution () for the same portfolio P, correlation matrix and D 99:9% ina two-dimensional universe, we obtain:!! S D :7 CAC 4 D 8:% :9 AEX 4:% with a corresponding P&L equal to :6 million. Again, adding wheat as a new product in the CCP risk factors universe, and computing the optimal scenario () for the member in the three-dimensional universe, P D.; ; / T, we obtain: 8:% S B C 4:% A :% and the corresponding P&L is still :6 million. () 74 risk.net August 7

4 A two-dimensional illustration of the problem, showing a historical 99.9% loss quantile Portfolio direction Iso-loss line P&L < threshold P&L > threshold It is clear from this numerical example that adding new risk factors that do not intervene in the portfolio has no effect on the second approach (4). The loss implied by the scenario is the same, and the projection of the worst scenario on the risk factors intervening in the portfolio does not change. Besides, risk factor moves remain below the one-dimensional 99.9% Gaussian percentile :9. This is exactly what we want to achieve, since all four properties outlined previously are met. Problem (4) in this respect seems to be well suited to the purpose of generating plausible scenarios for the whole universe of risk factors. Graphical representation. Two-dimensional heuristic solution. Here, we provide a two-dimensional graphical representation of the numerical example above to help the reader intuitively understand problem (4) heuristically. All charts represent, observations of the two Gaussian risk factors, correlated at 8%. The P&L yielded by each point is the scalar product P T S and is therefore the projection of the point on the red line driven by the vector with the same co-ordinates as the member portfolio: P D.; / T. Any point on the blue line will have the same scalar product (same P&L) with the portfolio P. Indeed, the iso losses are orthogonal lines (figure ). The red observations above the orthogonal line yield a higher loss than the threshold, while those under it yield a lower loss. To find our optimal stress scenario empirically, we will first adjust the loss to the target level and then retain the best scenario. In our example, with, observations and x D 99:9%, the objective is to have red points (.%) above the blue line. The iso-loss line will therefore be gradually adjusted until the right number of red points is reached (figure ). Any point on the blue line would be a relevant, but not necessarily plausible, extreme scenario for this member portfolio in loss terms. In fact, the criterion about which scenario needs to be selected is not obvious. This is where the Mahalanobis distance is used: the Mahalanobis isoplausibility ellipse that is tangential to the iso-loss is drawn on the same chart, and the intersection gives the final optimal scenario (figure ). Whatever the portfolio, the iso-loss line for the final scenario will cut the iso-plausibility level; therefore, the loss on any portfolio is at least as plausible as for the original portfolio Final scenario loss on P Iso-loss line P optimal scenario iso loss Iso-plausibility ellipse New portfolio 4 P&L < threshold P&L > threshold Final scenario P optimal scenario Portfolio direction Therefore, the orange point given by the closed-form solution is the most plausible scenario (among all the blue line iso-loss curves), since it has the lowest Mahalanobis distance. This point is retained as the optimal stress test scenario for the portfolio. Note: this point (.7,.9) corresponds to the optimal scenario shifts obtained with the closed-form formula above and rescaled by the indexes standard deviations. Let us now consider a new portfolio direction P (dotted red line in figure ). The iso-loss line for the final scenario S (orange point) orthogonal to the new portfolio direction will cut the iso-plausibility ellipse, as it is no longer tangential (dotted blue line). Recall the optimal scenario for P (purple point) is also on the same ellipse with radius N.99:9%/ and on the tangent to the ellipse orthogonal to the P direction (black dotted line), thus giving a higher loss. The loss yielded by S on P is more plausible than on the original portfolio. This is true for any other portfolio direction and can be seen in higher dimensions by taking the projection onto the two-dimensional plane. Difference with the initial framework. The transformed problem can be interpreted as the most likely scenario giving a predefined quantile of the loss instead of the maximum loss over a plausibility ellipse. As an illustration, figure provides a comparison between the two optimisation problems. The larger ellipse is the 99.9% plausibility domain, and consequently only observations are outside the domain (red squares). Similarly, the lower ellipse leaves by construction only points above the tangent orthogonal to the portfolio, as explained in figure. It is clear in the example that the maximum loss over plausibility domain approach yields a higher loss (intersection of the dotted line with the red one) than the quantile loss method. Not one of the, simulated points yields a loss more punitive than that implied by the max loss over the risk.net 7

5 A comparison of the two frameworks Tangent to quantile ellipse 'Quantile' ellipse 4 Portfolio direction Iso-loss line Iso-plausibility ellipse P&L < threshold P&L > threshold Observations outside quantile ellipse Mahalanobis ellipse scenario (there is no point above the dotted line); the corresponding loss quantile is 99.99% instead of 99.9%. Extensions and limitations of the model Extensions. Case of multivariate Student s t distribution. The Student s t distribution has become the standard choice among heavy-tailed distributions. The benefits of moving from Gaussian to Student s t modelling are enormous in terms of plausibility of rare events. Breuer & Krenn () give an example of the plausibility of Black Monday, computed using both Gaussian and Student s t calibrated models on one-year daily samples, which is once every,74 years in the Gaussian case and once every years in the Student s t case (with four degrees of freedom). Therefore, the case of a multivariate Student s t distribution is interesting, as we are still in the context of an elliptical distribution and the closed-form formula still applies. The optimal scenario can be written as follows: S D T P. / p and the worst loss is equal to: p q D T. / where T is the distribution function of a multivariate Student s t with degrees of freedom. Meta-elliptical distributions. The main issue regarding multivariate Student s t distributions is the degree of freedom being the same for every marginal. In the real world, some risk factors may have fatter tails than others. Meta-elliptical distributions are multivariate statistical models in which the dependence structure is governed by an elliptical copula and the marginal distributions are arbitrary. For a Student s t copula, we obtain the so-called meta-t distribution. 4 An example showing the approximation for meta-elliptical distributions using actual market data Real scenarios Tangent to actual distribution Approx. scen projection Scenarios Student multivariate Modified portfolio Portfolio direction Tangent to Student multivariate 7 Calculation step Approx. final scen Best final scen Observations D.99, D.96, g D.9 and correlation D 8%. The chosen quantile was x D 99.9% We now solve (4), the maximum plausibility at a level of loss q, inthe setting of the models above. There are two issues: we are not in an elliptical setting, so the closed-form solution at a fixed level of loss q is not valid anymore; and the plausibility of the scenarios corresponding to a level of loss q is not available in closed form, since, unlike in the Gaussian or Student s t cases, the law of a linear combination of the risk factors is not known. The computation in this setting can be performed by solving the problem numerically or by computing an approximate solution. Proxy solution. A practical workaround (in order to avoid the numerical procedures above) is to transform the vector of returns S i into a vector X i by the classical copula-like transformation X D.F g.f i.s i ///. This distribution is constructed using a Student s t copula and adjusted marginals for each risk factor. Then, by definition, X follows a classical multivariate Student s t distribution with g degrees of freedom. One can solve (4) in terms of X instead of S and then return to the original space by the inverse transform. The approximate method to get the stress scenario can be described using the following steps: Define for each risk factor a Student s t distribution, with individual degree of freedom i. Map these to a single Student s t copula X i D F g.f i.s i //. Solve exactly the formal problem of worst loss in terms of X and using P i in the context of the (elliptical) multivariate Student s t distribution to obtain a proxy worst-loss scenario X. Get back the worst scenario from the proxy scenario by the inverse transformation: S final i D F i.f g.x i // 76 risk.net August 7

6 Note: one can also adjust the portfolio weights after step and solve step for the modified portfolio (figure ): Pi mod D Xi max P i, where Xi max is the maximum plausible move over unidirectional distribution X i. This is so the portfolio weight on each risk factor is adjusted to the factor s riskiness. This solution is an approximation but has been shown to be numerically very close to actual quantile losses for confidence levels under 99:9%. From our experience, calibration errors for the i parameters become more important than the approximations due to the method itself. The approximate solution can also be used as a starting point for a gradient method. Note: for calibration techniques of t copulas and meta-t copulas, refer to Demarta & McNeil (4). Limitations. Model risk. The framework relies on elliptical distributions.although the closed-form formula is useful, it still raises the concern of quality of fit and stability of correlation assumptions. The model is therefore only as good as the actual distribution assumption is at modelling the tail risk of the joint distribution of risk factors. The Student s t copula provides a degree of uncertainty in the correlation, and the meta-elliptical proxy adjustment above vastly improves the quality of the fit (blue ellipse versus red ellipse in figure 4). In our experience, the model is very handy and gives a reasonable idea of what could be a severe hypothetical event for the portfolios specified as input. This being said, any distribution assumption bears its model risk aspects, and one should not blindly follow a model output. Comparison with actual historical events is certainly a best-practice exercise. On a more quantitative note, the interested reader may refer to the relative entropy approach of Breuer & Csiszar (6). Exhaustive framework yet finite. The framework meets the exhaustivity criterion. Any equally plausible scenario (any point in the iso-plausibility ellipse) can theoretically be reached. However, in practice, only a finite number of scenarios are retained to form a CCP stress scenario set, leading to the following observation from Murphy & Nahai- Williamson (4): There are therefore two principal drivers of the stress loss over initial margin: the scenarios used; and the portfolios that they are applied to. The former should be revised when new REFERENCES Breuer T, 8 Overcoming dimensional dependence of worst case scenarios and maximum loss Journal of Risk (), pages 79 9 Breuer T and I Csiszar, 6 Measuring distribution model risk Mathematical Finance 6(), pages 9 4 Breuer T and G Krenn, What Is a Plausible Stress Scenario Computational Intelligence: Methods and Applications ICSC Academic Press, pages Breuer T, M Jandačka, K Rheinberger and M Summer, 9 How to find plausible, severe, and useful stress scenarios International Journal of Central Banking (), pages 4 vulnerabilities become apparent, but they often do not change from day to day. The latter, however, are highly variable, and hence CCPs must ensure that their stress tests are genuinely stressful for the portfolios they clear, and then perform these stressful stress tests daily on each of these portfolios. The proposed framework can easily be used to monitor and regularly challenge the stress scenario set, and it is an efficient tool to mitigate concerns. However, an interesting extension would be to start from a fixed number of scenarios and optimise their relative positioning so the size of the gaps (portfolios leading to plausible losses beyond those captured by the finite set) is minimised. Conclusion Our framework has been built from a CCP s point of view as a possible way to generate a resilient and coherent set of stress scenarios. It allows a risk manager to define equally plausible stress scenarios for either existing or hypothetical input portfolios at a given plausibility level. This article provides a closed-form formula for elliptical distributions and a proxy solution for adjusted marginals. The proposed approach follows existing footprints. However, by switching the optimisation problem considered, we overcome the dimensional dependence feature and can meet the upfront expected properties of homogeneity, exhaustivity, stability and plausibility. The framework can also be used to monitor the ongoing validity of a stress-testing suite. An interesting extension would be to optimise the number of stress scenarios within a set and their relative positioning. This would minimise the risk of significant gaps in the finite stress scenario set. Pierre Mouy is a senior risk analyst and Mohamed Selmi is head of risk methodology at LCH in Paris, and Quentin Archer is head of equities business risk at LCH in London. The views and opinions expressed in this article are those of the authors and do not reflect those of LCH. The authors thank their LCH colleagues, in particular Jean-Marie Boudet and Julien Dosseur Dutouquet. They are equally grateful to Claude Martini, Ismail Laachir and Thomas Breuer. pierre.mouy@lch.com, quentin.archer@lch.com, mohamed.selmi@lch.com. Demarta S and AJ McNeil, 4 The t copula and related copulas International Statistical Review 7(), pages 9 Murphy D and P Nahai-Williamson, 4 Dear Prudence, won t you come out to play? Approaches to the analysis of central counterparty default fund adequacy Financial Stability Paper, Bank of England Qi H and D Sun, 6 A quadratically convergent Newton method for computing the nearest correlation matrix SIAM Journal on Matrix Analysis and Applications 8(), pages 68 risk.net 77