Analytical scores for stress scenarios
AAnalytical scores for stress scenarios
Pierre Cohort ∗ , Jacopo Corbetta † , and Ismail Laachir ‡ Zeliade Systems, 56 rue Jean-Jacques Rousseau, Paris, FranceJuly 7, 2020
Abstract
In this work, inspired by the Archer-Mouy-Selmi approach ([2]), we present two methodologies forscoring the stress test scenarios used by CCPs for sizing their Default Funds. These methodologies canbe used by risk managers to compare different sets of scenarios and could be particularly useful whenevaluating the relevance of adding new scenarios to a pre-existing set.
After the financial crisis of 2008, the topic of stress testing got more and more attention from the financialinfrastructures environment, specifically from Central Clearing Counterparties (CCPs). Both the PFMI-IOSCO [7] and EMIR regulations [6], [4] require CCPs to specify extreme but plausible scenarios for thesizing of their default funds. Moreover the EMIR regulation requires a review of the same stress scenarios(at least annually according to Article 31 of EMIR RTS [4] and according to the key explanations of Principles4, 5, and 7 of PFMI [7]). As such a risk manager may face the conundrum of assessing the benefits of addingor changing sets of scenarios. In this work we provide two methodologies giving a quantitative assessmentof the advantages or disadvantages of the new set of scenarios.
A hypothetical scenario is, as the name suggests, a stylized scenario designed in order to capture a tail risk.Hypothetical stress scenarios can be designed starting from risk manager views, as for example a parallelshift of all yield curves for fixed income products, or using as a base a scenario obtained via quantitativemethods, coming from the fit of a distribution, or a PCA.By construction, the hypothetical scenarios do not refer directly to the history. So, it is a good indicatorof the scenarios quality to estimate their plausibility. Moreover, the size of the hypothetical scenarios aregenerally calibrated separately for each risk factor, so that the plausibility of the joint scenarios does nothave the same confidence level as the one dimentional quantile level. The plausibility can be estimated, forinstance, by evaluating the log-likelihood of the hypothetical scenarios, using a joint-distribution calibratedon the risk factors. A joint Student distribution can be calibrated on the historical data of the risk factors,then used to estimate the likelihood of the scenario. ∗ [email protected] † [email protected] ‡ [email protected], Corresponding author a r X i v : . [ q -f i n . R M ] J u l he problem of the design of extreme but plausible scenarios has been tackled in the literature, forexample by Thomas Breuer and co-authors in [3]. They look for the scenario that gives the worst loss, undera plausibility constraint (likelihood less than a cap). While this approach is valid for portfolio management,it is not well suited for the Stress Testing context of CCPs. In their second paper [5], the authors consider adual problem whose solution does not depend on additional dimensionality of the problem and which closelyresembles the problems faced by a risk manager at a CCP. This approach is made even more explicit byQ.Archer, P.Mouy and M.Selmi (LCH), who proposed (cf. [2]) a framework for the design of extreme butplausible scenarios. Their methodology considers a linear portfolio P , so that the P&Ls from risk factorreturns s is simply P t s , and assumes a calibrated historical distribution of the risk factors, with density f .The approach can be formalized by the following maximum likelihood problem:max P t s ≤ q f θ (s) (1)where • s is the vector of the risk factors returns. • P is the portfolio positions. • f θ is the density function of the joint-distribution of all the risk factor returns s . • q is a cap constraint on the portfolio loss.For instance a heavy-tail distribution, like a T- Student law can be calibrated on the risk factors returns. q can be chosen as the α -quantile of the distribution of the portfolio loss P t s . For example, if s is a Gaussianvector, then P t s is Gaussian so that q can be chosen as a (one-dimensional) Gaussian quantile.This methodology has the advantage of yielding extreme scenarios, with a loss cap, so that we are ensuredto have meaningful scenarios. Moreover, if the distribution is chosen to be a standard elliptic (Student orGaussian, with correlation Σ, null average and marginal standard deviation 1), then the problem (1) admitsa simple closed formula solution: S ∗ ( P ) = q Σ P t P Σ P . (2)As pointed out in [2] and [5] another advantage of this methodology is the fact that, at least in anelliptical setting, the solution of (1) is not dependent on the presence of additional risk factors not appearingin the portfolio, while the primal problem considered in [3] gives losses dependent on the introduction of riskfactors unrelated to the portfolio.
We apply now the ideas from [2] to the comparison of sets of stress scenarios.We start by considering a single linear portfolio P (such that the P&L associated to the risk factor return s is t P s ) and a set of stress scenarios S := { S , . . . , S n } for the risk factor returns s .We can thus calculate the stress loss associated to the portfolio P as: l ( P ) := min ≤ i ≤ n t P S i , we then select the scenario ˆ S ( P ) among S that drive l ( P ). If several drivers are found, we select the driverthat maximize the density function (another criterion could be applied, such as minimizing the Mahalanobisdistance t ˆ S ( P )Σ − ˆ S ( P ) among the candidate drivers).We then compute the scenario S ∗ ( P ) solving sup S ; t P S ≤ t P ˆ S f θ ( S ) , (3)that is to say the most plausible scenario generating a loss equal (or greater) to the worst loss obtainedwith the stress scenarios. Equivalently, the two scenarios ˆ S ( P ) and S ∗ ( P ) generate the same loss l ( P ) but S ∗ ( P ) is the most plausible with respect to the distribution assumption.2e finally introduce the two score functions. The first score function measures the quality of the ratioloss to plausibility of ˆ S ( P ), and is given by φ S ( P ) := f θ (cid:16) ˆ S ( P ) (cid:17) f θ ( S ∗ ( P )) ∈ ]0 ,
1] (4)The higher the score, the better it is, as a high φ indicates that the stress scenarios contained in theset S are close, in a plausibility sense, to the most likely scenario inducing the same level of losses for theportfolio.The second score is a geometrical criterion, measuring to what extent a driver is in the same directionas the optimal scenario. ψ S ( P ) := (cid:104) ˆ S ( P ) , S ∗ ( P ) (cid:105) (cid:13)(cid:13)(cid:13) ˆ S ( P ) (cid:13)(cid:13)(cid:13) (cid:107) S ∗ ( P ) (cid:107) ∈ ] − ,
1] (5)Also in this case the higher the score, the better it is, as it indicates that the “risk direction” of theportfolio is captured by the stress scenario set S . Suppose now that we have two sets of stress scenarios: S = { S , · · · , S n } , and T = { T , · · · , T m } , possiblypartially overlapping, and we want to evaluate the advantages of one set with respect to the other. Thescore functions we introduced in the previous section allows us to do it in the following way: • Select a reference set of portfolios P , . . . , P M . • Calculate the values φ S , ψ S ( P i )’s and φ T , ψ T ( P i )’s. • Define a final score from those values.The choice of the final score depends on the risk manager view. In our numerical result parts we proposetwo different approaches. • Scenario approach , which is particularly meaningful when the set T is a modification of the set S .For each stress scenario we compute the average and standard deviation of the function ψ and φ onthe set of portfolios for which the scenario is the driver. This approach allows to have a view on whichscenarios could be eventually modified, or even eliminated as being very far from optimality, eitherfrom a plausibility or geometrical points of view. • Portfolio approach . We compare the score functions on each portfolio. This approach allows to betterunderstand for which portfolios the risk is not correctly sized, and it can be used for understandingwhich test portfolios are not sufficiently stressed by the current set of stress scenarios.We point out that the proposed scores should be used more as a non rejection indicator and not as anacceptance one, similarly to the Kupiec Test which says that an hypothesis can not be rejected, not that itshould be accepted.
For elliptical distributions, the solution of (3) can be found exactly as described in [2]. However, for themore generic meta-elliptical distributions (introduced in [1]) this is no more the case. As these are thedistributions we will fit the risk factors returns on, we provide two possible alternatives for finding the mostplausible scenario at a given loss.We recall that a meta-elliptical distribution f θ is a multidimensional distribution with elliptical copula.The setting we will consider consists of a T-Student copula with T-Student marginals, and it is consequentlycharacterized by a parameter θ ∗ containing: • the location vector µ and scale vector σ the correlation matrix Σ • the vector of marginal degrees of freedom ν • the degrees of freedom ¯ ν of the copula(denoting also in the sequel the d-dimensional constant vector(¯ ν, . . . , ¯ ν )). The first method was proposed by Mouy et al. [2] and it is based on approximating the meta-ellipticaldistribution by an elliptical distribution, i.e. using the same degree of freedom for the copula and themarginals.We start by normalizing the distribution, via the linear transformation ˜ s := ( s − µ ) /σ , and we get theequivalent problem sup ˜ s ; t ˜ P ˜ s ≤ ˜ q ˜ f θ ∗ (˜ s )where • ˜ f θ ∗ (˜ s ) := f θ ∗ ( s ) • ˜ P := σP • ˜ q := t P ( ˆ S − µ ).If the distribution ˜ f θ ∗ was elliptical, the optimal scenario for the problem above would be given by (2)˜ S ∗ ( P ) = ˜ q Σ ˜ P t ˜ P Σ ˜
P ,
Transporting it back to the original problem, one has: S ∗ ( P ) := µ + σ (cid:32) ˜ q Σ ˜ P t ˜ P Σ ˜ P (cid:33) . As stated above, the approximated solution is obtained by approximating the meta-elliptical distributionwith an elliptical distribution, obtaining the sub-optimal scenario¯ S ( P ) := µ + σT − ν ◦ T ¯ ν (cid:32) ˜ q Σ ˜ P t ˜ P Σ ˜ P (cid:33) where T ν ( x ) is the vector ( T ν i ( x i )) ≤ i ≤ d , T ν i being the CDF of a standard T-Student distribution with ν i degrees of freedom.We point out that the approximation quality is strongly linked to the “almost linearity” of the function T − ν ◦ T ¯ ν around 0. In the case where ¯ ν and ν are very different, the approximation could be poor, withsignificant discrepancies both in term of optimal density value and on loss constraint violation. An exact solution can also be recovered numerically using classical optimizers. In fact the target function iseasy and fast to calculate and the constraint is linear.Moreover, as the applications for which our methodologies are devised require the score calculation to bedone once for all or at low frequency so using a time consuming resolution method is not an issue.Finally, to calculate a score, it is possible to restrict the relevant portfolios to involve a low number ofrisk factors (e.g. spreads, involving each only 2 risk factors). The effective dimension of the exact resolutioncan then be lowered and the optimization made easier.4
Numerical experiments
We thus performed our experiments on the synthetic Yields curves provided by the European Central Bankand downloadable at http://sdw.ecb.europa.eu/: • AAA : synthetic curves aggregated from the
AAA issuers of the EURO zone (dynamic basket). • ALL : synthetic curves aggregated from all the issuers of the EURO zone (dynamic basket).We used the pillars 6 M, Y, Y, Y, Y, Y of those yield curves.We assume the following setting: • Reference set of portfolios : we consider spread portfolios of the form ( B i , − B j ) where: – B i , B j are some Bonds with semi-annual coupons, with a time-to-maturity equal to one of thepillars’ maturity – β = − β = − D i /D j . – D i , D j are the durations of the bonds B i and B j . – we obtain 2 × × (cid:0) × (cid:1) = 264 portfolios. • Distribution assumption : a meta-t distribution on the Yield rate returns. • Bond pricing : we approximate the P&L for a bond to be (∆ Y ) D where ∆ Y is the Yield rate moveand D the base bond duration.We will obtain the optimal scenarios via numerical optimization. Should we fit a single distribution on all the risk factor at the same time, or one on each single test portfolio?While for Gaussian distribution this does not have an impact, in the case of meta-elliptical distributionsthat we are considering, the situation is a little bit more complicated. This is because, while the marginaldistributions are fixed, the copula can vary.From a stability point of view, our decision makes the scores dependent on the number of risk factorschosen, as the copula is fitted on each group separately. However, we believe that these additional degreesof freedom allow to better measure the risk and give a better understanding of the differences between setsof stress scenarios.
We consider 2 sets of stress scenarios: a base and an enriched one. As the scenario generation methodologyis not the focus of this work we decided to use over-simplified and stylized sets. Moreover, this choice allowsus to better highlight the contribution of our scores, as the difference between the sets havealso a clearinterpretation.Both the base and the enriched set are obtained starting from the first three components of a PrincipalComponent Analysis performed separately on the returns of the
AAA and
ALL curves. The vectors arerescaled by a factor 3 × σ i where σ i is the explained standard deviation associated and combined as follows: • The base set S considers only combination of the same level, and with the same sign: ( ± n th component AAA , ± n th component ALL ), n = 1 , ,
3, for a total 6 possible stressed scenarios; • The enriched set S (cid:48) considers the scenarios in the base set, plus the combination given by ( ± n th component AAA , ∓ n th component ALL ), n = 2 ,
3, for a total 10 possible stressed scenarios.We plot here the three drivers of the risk scenarios for the two curves. Notice that the PCA analysisprovides three main directions that indeed qualitatively correspond to the shift, slope change and curvaturechange (displaying respectively the sign patterns +, − /+ and +/ − /+) often used in devising hypotheticalscenarios for rate curves. 5 .3 Result analysis In the table below we compare the average and standard deviation for the loss to plausibility φ · ( P ) and thegeometrical score ψ · ( P ) aggregated at the driving scenario level, and considered as a whole (last line). Thecolumns Quantity indicates for how many portfolios the selected scenario produces the largest losses.Base Scenarios Enriched ScenariosScenario Quantity φ mean φ std ψ mean ψ std Quantity φ mean φ std ψ mean ψ std(+1, +1) 80 0.334 0.258 0.858 0.193 64 0.929 0.085 0.397 0.248(-1, -1) 80 0.348 0.263 0.872 0.171 64 0.934 0.079 0.414 0.250(+2, +2) 28 0.442 0.244 0.512 0.331 17 0.537 0.324 0.381 0.256(-2, -2) 28 0.437 0.246 0.497 0.343 17 0.522 0.330 0.374 0.254(+3, +3) 24 0.666 0.259 0.746 0.274 21 0.733 0.285 0.640 0.261(-3, -3) 24 0.658 0.262 0.736 0.283 21 0.723 0.293 0.632 0.263(+2, -2) 0 0 0 0 0 23 0.868 0.218 0.797 0.236(-2, +2) 0 0 0 0 0 23 0.869 0.225 0.802 0.237(+3, -3) 0 0 0 0 0 7 0.879 0.131 0.839 0.142(-3, +3) 0 0 0 0 0 7 0.867 0.148 0.834 0.143Total 264 0.420 0.285 0.766 0.282 264 0.530 0.301 0.833 0.243 Table 1: Comparison of the scores on the different scores obtained with the different stress scenarios sets.
We can see that the introduction of the new scenarios does not deteriorate significantly the score of theexisting scenarios (it actually improves it in some cases), and that the new scenarios have average scoresquite elevated.It is up to the risk manager to decide if, according to his/her expertise, the new scenarios are acceptableor not. Particular attention should be paid in case the scores of the new scenarios are high, but the averagescore for some of the old scenarios has been lowered. Again, we would like to highlight that our scores arenot to be intended as an acceptance tool, but more as a non rejection one.We have compared above the score functions ψ and φ at a scenario level. In the figure below we comparethe two scores at a portfolio level. • Left graph : the functions P → φ S ( P ) (blue), and P → φ S (cid:48) ( P ) (green). The two functions are plottedwith the test portfolios in the ascending order for P → φ S ( P ). • Right graph : the functions P → ψ S ( P ) (blue), and P → ψ S (cid:48) ( P ) (green). The two functions areplotted with the test portfolios in the ascending order for P → ψ S ( P ).6e can see that, with the exception of very few portfolios, the scores obtained by the enriched scenariosare higher than the one obtained by the base set (when the score is the same the scenario driving the stressedvalue is the same in the two sets). One natural question is whether or not the new scenarios, even if obtaining better scores for the analyzedportfolios, are plausible enough.A priori some of the new scenarios may be very close or even coincide with their corresponding mostplausible scenarios, however, when compared with the scenarios previously available they may be way lessplausible. A typical example could be a rescaling of one of the existing scenarios by a factor >
1. Thiscould induce (for some test portfolios) bigger losses, but at the same time the stress scenarios would be lessplausible.A trade-off between scenario plausibility and losses may happen and it is up to the risk manager toanalyze it and decide if it is acceptable or not, but it can also happen that the new drivers not only generatebigger losses but are also more probable, simply because they explore new direction with respect to the oldones and result in a more significant position with respect to the reference portfolios.In the figure below we present the two cases: • Left Graph : Long one bond
AAA with maturity 3 Y , short one ALL with maturity 3 Y . In this casethe new driver not only provides a higher loss but also has a higher probability. • Right Graph : Long one bond
ALL with maturity 6 M , short one AAA with maturity 6 M . In thiscase the new driver there is a trade-off higher losses lower plausibility.The dashed lines represent the level line of the distribution on the scenarios (either driver or optimal).7he graph on the left shows the most interesting case from a risk manager point of view. The originalset of scenario did not tackle in a good way the risk associated with the specific test portfolio, as the driveris orientated almost perpedicularly with respect to the optimal scenario. The enlarged set introduces ascenario which is in a “good direction” respect to this specific test portfolio, providing a larger loss ( ≈ . In this work we have presented two methodologies which can help risk managers to compare sets of stressscenarios and in particular to assess the benefits of the introduction of new scenarios the existing ones. Thetwo methodologies allow the risk manager to analyze different aspects of the stress scenarios, notably theirposition and relevance for the reference sets of portfolios.The proposed methodologies have a clear and natural meaning which allows to better understand thebenefit of one set of scenarios with respect to the other.