Stress testing and systemic risk measures using multivariate conditional probability
aa r X i v : . [ q -f i n . R M ] A p r Stress testing and systemic risk measures using multivariateconditional probability
Tomaso Aste
Department of Computer Science, University College London,Gower Street, WC1E 6EA, London, United Kingdom
April 15, 2020
Abstract
The multivariate conditional probability distribution quantifies the effects of a set of vari-ables onto the statistical properties of another set of variables. In the study of systemicrisk in financial system, the multivariate conditional probability distribution can be usedfor stress-testing by quantifying the propagation of losses from a set of ‘stressing’ variablesto another set of ‘stressed’ variables. Here it is described how to compute such conditionalprobability distributions for the vast family of multivariate elliptical distributions, whichincludes the multivariate Student-t and the multivariate Normal distributions. Simple mea-sures of stress impact and systemic risk are also proposed. An application to the US equitymarket illustrates the potentials of this approach.
Keywords:
Stress testing, Systemic risk, Elliptical conditional probability
1. Introduction
The most general approach for stress testing consists in estimating the effects of thestress applied from a part of a system, say the ‘stressing’ variables X , onto the statisticalproperties of another part of the system, say the ‘stressed’ variables Y . This requires thecomputation of the conditional probability P r ( Y | X ). The computation of such a conditionalprobability in a multivariate system can be challenging, however it turns out to be quitesimple, and well known, for a vast set of probability distributions that belong to the ellipticalfamily [1]. In this paper I report how the expressions of these conditional probabilities can becalculated and I discuss the implications from the perspectives of stress testing and systemicrisk measures [2].There is a vast literature on systemic risk (see for instance [3] for an overview). Throughdifferent approaches, researchers study the propagation and amplification of losses in mar-kets caused by externalities that can trigger shortfall when the system is undercapitalized.Stress on a industry sector can cause fire-sales and trigger externalities on other institu-tions and different industry sectors. Spillovers effects can propagate and being amplified1hrough the system causing distorsions and systemic effects. To ensure the stability of thesystem it is important to be able to quantify the propagation of stress through the systemfrom institution to institution across sectors. Several methodologies and approaches havebeen proposed to quantify systemic risk with approaches ranging form a general economicperspective [4] to network theoretical approaches [5, 6, 7], specific econometric tools [8, 9],machine learning techniques [10] and game-theoretic reasoning [11]. Overall, all these ap-proaches are aiming to capture the propagation of stress and consequently losses throughthe system [12, 13, 14, 15, 16, 17] and assess the criticality of the system identifying spe-cific fragilities. This task requires the modeling of the complex set of variables, parametersand mechanisms characterizing the highly interconnected system of financial institutions,industries and banks at the basis of our economy.Mathematically this is a system comprised of many dependent variables that can bestatistically described by a joint probability distribution. The risk in the system is measuredby the probability of the losses. The systemic risk is associated with the interdependencyand therefore the probability of losses across different institutions. Propagation of stress isinstead associated with the conditional probability of a loss given that some other variablesare stressed at a certain amount of losses. This is the approach adopted in [18] where aconditional definition of value at risk was introduced. Such a conditional VaR is the valueat risk of the ‘stressed’ variable, Y , computed conditioned to the ‘stressing’ variable X being at its VaR value for a certain level of risk. This is an instance of the use of conditionalprobability to compute a measure of risk where two variables are involved.In this paper I argue that in general, for a system with many variables, the most im-portant contribution to risk in a system under stress is caused by the shift of the centroidof the conditional probability distribution with respect to the centroid of the unconditionedprobability distribution. Such a shift can be quantified in money value and interpreted aspropagation of stress.This paper is organized as follows: in section 2, to set the tone and introduce someimportant concepts, I briefly report about the CoVaR reasoning adopting however a slightlydifferent perspective with respect to the original paper, Then, in section 3 the concepts areextended to the multivariate domain providing a general multivariate computation of theexpressions for the conditional probability for elliptical distributions. Systemic risk measuresand stress testing tools from these conditional probabilities are introduced in section 4. Insection 5, I discuss an example concerning the study of systemic impacts between industrysectors under stress from the analysis of equity returns in the US market. Conclusions and2erspectives are given in section 6.
2. Univariate measure of risk in terms of Value at Risk and Conditional Valueat Risk
For continuous variables
V aR ( q ) is the q-quantile of the loss distribution, meaning thatwith probability q, losses will not exceed the value V aR ( q ). Specifically, for a randomvariable X representing the log-return losses, V aR X ( q ) is defined by P r ( X ≤ V aR X ( q )) = q. (1)When the random variable X belongs to the location scale family, then it has the propertythat the distribution function of the scaled variable a + bX also belongs to the same fam-ily. Therefore, I can arbitrary scale the variable while preserving the distribution and inparticular I can always write the distribution with respect to the standard form: P r (cid:20) X − µ X σ X ≤ V aR X ( q ) − µ X σ X (cid:21) = Φ (cid:20) V aR X ( q ) − µ X σ X (cid:21) = q. (2)Where µ X is the location parameter, σ X is the scale parameter and Φ( . ) is the standard formof the cumulative distribution function associated with the statistics of the standardizedvariable. If the inverse of such cumulative distribution exists, then V aR X ( q ) − µ X σ X = Φ − ( q ) . (3)and therefore I have a general expression for the V aR ( q ): V aR X ( q ) = µ X + Φ − ( q ) σ X . (4) In [19] a conditional version of the
V aR X ( q ) was proposed as a measure of impact ofa variable on the risk of another. In analogy with the definition of VaR form Eq.1 theconditional VaR is P r ( Y ≤ V aR Y | X ( q ) | X ) = q. (5)Which is identical to the definition of V aR X in Eq.1 except that the probability is now aconditional probability. For the whole location scale family the conditional probability as3he same form as the unconditional probability but with shifted location parameter µ Y | X = µ Y + ρ X,Y σ Y σ X ( X − µ X ) . (6)and a reduced scale parameter σ Y | X = (1 − ρ X,Y ) σ Y , (7)where ρ X,Y is the Pearson’s correlation coefficient between variable X and variable Y . From,Eq. 4 the conditional VaR is therefore V aR Y | X ( q ) = µ Y | X + Φ − ( q ) σ Y | X . (8)In [19] the value of X is stressed at its VaR value X = V aR Y ( q ).Note that the shift µ Y | X − µ Y of the location parameter in Eq.6 is the least square linearregression of Y given X . Essentially, the probability distribution shift its location by thisfactor and changes the scale by a factor q (1 − ρ X,Y ). This also implies that any increasein the value at risk of the conditioned variable is only driven by the linear shift because thevariability is reduced by the conditioning.Linear regression is a simple and intuitive and extremely useful formula that was alreadywell understood by Pearson and Galton over a century ago [20, 21, 22], and yet it still keepspopping out, as a novelty, in the current days’ literature, often called with different names.
3. Multivariate measure of systemic risk
Let now extend the reasoning illustrated in the previous section to a multivariate, casewith X ∈ R p X × and Y ∈ R p Y × . I consider the case when the multivariate probabilitydistribution of the random variables Z = ( X ⊤ , Y ⊤ ) ⊤ ∈ R p Z × (with p Z = p X + p Y ) belongsto the elliptical family, which implies that the multivariate probability density function canbe written as f Z ( Z ) = φ p z (( Z − µ Z ) ⊤ Ω − ( Z − µ Z )) (9)where φ p z ( . ) is a scalar function which is the standardized form of the distribution, indepen-dent form the location and scale parameters but dependent on dimension p z and eventuallyon other parameters (such as the degrees of freedom ν in the Student-t case). The matrix4 ∈ R p Z × p Z is a positive defined matrix, which is either the the covariance matrix or pro-portional to it, when it is defined. Here we call it generically shape matrix which is theterm used for the multivariate Student-t. µ Z ∈ R p Z × is the vector of means or locationparameters. The quantity ( Z − µ Z ) ⊤ Ω − ( Z − µ Z ) is the so-called Mahalanobis distance [23].For the elliptical family the conditional probability density function for the set of variables Y conditioned to the other set X can be always expressed as f Y | X ( Y ) = φ p Y (cid:2) ( Y − µ Y | X ) ⊤ J Y | X ( Y − µ Y | X ) (cid:3) (10)where µ Y | X is the vector of conditional location parameters, and J Y | X is the conditionalinverse shape matrix. They are: µ Y | X = µ Y + Ω Y X Ω − XX ( X − µ X ) , (11)and J Y | X = (cid:0) Ω Y Y − Ω Y X Ω − XX Ω XY (cid:1) − . (12)Where Ω XX , Ω XY , Ω Y Y and Ω
Y X are the block elements of the shape matrix ΩΩ = Ω XX Ω XY Ω Y X Ω Y Y ! . (13) The conditioning of a set of variables Y to another set of variables X produces as effecta shift in the position of the centroids of the probability distribution of the Y variables bythe vector Λ Y ( X ) = µ Y | X − µ Y = Ω Y X Ω − XX ( X − µ X ) . (14)It must be noted that, analogously to the uni-dimensional case also in this case the shift inthe locations coincide with the multivariate linear regression of Y with respect to X . Froma risk perspective this shift is the important factor because conditioning always reducesuncertainty and therefore it is this shift that increases any risk measure when stress isoperated. 5 .3. Reduction of uncertainty Conditioning always reduces variability and uncertainty. In this paper uncertainty isquantified in terms of the Shannon entropy defined as: H ( Z ) = H ( X, Y ) = log( | Ω | ) + H Z H ( X ) = log( | Ω XX | ) + H X H ( Y ) = log( | Ω Y Y | ) + H Y (15)Where | . | indicated the determinant of the matrix and H Z , H X , H Y are the entropiesassociated with the standardized random vector, which are therefore independent from thelocation vectors and the shape matrices. The effect on uncertainty of Y caused by theconditioning to X can be quantified from the difference between the entropy of the un-conditioned set of variables, H ( Y ) and the conditioned one H ( Y | X ). It should be notedthat H ( Y | X ) = H ( X, Y ) − H ( X ) and therefore the difference is H ( Y ) − H ( Y | X ) = H ( X ) + H ( Y ) − H ( X, Y ) = I ( X ; Y ) which is called mutual information. The mutual infor-mation I ( X ; Y ) is a non-negative quantity which is equal to zero when the two variables areindependent. This tells us that conditioning always reduces uncertainty on the conditionedvariable except in the case when the two sets are independent. From the previous espressosEq.15 the mutual information is I ( X ; Y ) = 12 log( | Ω XX || Ω Y Y || Ω | ) + const. , (16)where the constant might depend on the dimensions of the variable sets and on other pa-rameters of the distributions but not on the location vectors or the shape matrices.
4. Stress testing and systemic risk measures
The conditional probability density function f Y ( Y | X ) contains the full information aboutstress testing from which any other quantity, such as the CoVar can be derived. In theprevious sections an expression for the conditional probability density function was providedfor the rather large family of elliptical distributions that includes the Student-t distributionwhich is of great relevance for risk quantification in financial systems. It has also beenshown that the contribution to risk is driven by the shift in the distribution’s centroid. HereI propose to use the average centroid shift is as a simple scalar quantity directly associatedwith risk increase under stress. 6et consider a system of p Z variables Z = ( X ⊤ , Y ⊤ ) ⊤ representing losses (negativereturns) where I want to test the losses on the subset Y as consequence of losses on thesubset X . Let for instance think that X is the technology industry sector and I want thesee how much extreme losses in such a sector influences the financial industry sector Y .In analogy with the reasoning beyond the CoVaR approach [19], the idea is to stress theset of variables X to their q quantile values and measure the losses caused in the Y set.Differently from [19], instead of computing the value at risk of Y , which is problematicbecause the multivariate nature of this problem, I estimate the mean losses associated withthe shift of the Y centroid 1 /p Y P i ∈ Y Λ Y i . This simple scalar quantity is an estimate howmuch the sector Y can lose in average as a consequence of extreme variations of stress tothe sector X . This is just one possible meaningful and intuitive quantity among variouspossibilities. For instance, another possible quantity that has significant importance for riskwould be the maximum loss: max { Λ Y i ( V aR X ( q )) } .The systemic effect of conditional dependency can instead be estimated from the entropicquantities. The mutual information I ( X ; Y ) measures the dependency between the subsets X and Y and it is in itself a measure of risk with larger values associated with largerdependencies and therefore larger risks of stress propagation.
5. Stress testing experiment with US equity market
To provide an example of stress testing I collected data for 623 equities continuouslytraded on the US market between 01/02/1999 and 20/03/2020. I computed the log-returnsand performed the analysis as described in the previous sessions. In particular, I computethe centroid shift Λ Y ( X ) caused on the losses in an industry sector by stressing the losses at95% VaR on another industry sector X . The VaR is computed empirically from the historicdata. Results are reported in Fig.1 where the rows report in color map the average amount oflosses in the relative sector caused by stressing the sectors reported on columns. I observe forinstance that the technology industry sector is strongly impacted by other sector’s stress.I measure 4.0% average losses on technology industry sector caused by the stressing ofconsumer services industry sector. Whereas, on the other direction, the technology industrysector is impacting the consumer services industry sector for 2.9% in average. Note thatthis is not the CoVaR, it is the mean that is moved by this quantity. A shift of 2.9% of themean indicates that without any endogenous stress, the exogenous stress propagating fromthe technology sector moves already considerably the losses in the consumer services sectorfrom close to zero to 2.9%. To this shift one must then add the the variability and risk that7 a s i c M a t e r i a l s C on s u m e r G ood s C on s u m e r S e r v i ce s F i n a n c i a l s H ea lt h C a r e I ndu s t r i a l s O il & G a s T ec hno l ogy T e l ec o mm un i ca ti on s U tiliti e s Basic MaterialsConsumer GoodsConsumer ServicesFinancialsHealth CareIndustrialsOil & GasTechnologyTelecommunicationsUtilities
Figure 1: Intensity of influence of the stress in a industry sector over another sector. The color mapcorresponds to the average amount of losses (negative log-returns) on a the sector reported on the columnscaused by stressing at 95% VaR the sector reported on the rows. are endogenous to the sector, for instance the consumer services sector has an average VaRof 3.7%.I computed also the effect of each industry sector on the rest of the market and vice-versa. The results are reported in Fig.2 where one can see that most of the sectors aremore impacted than impacting with exception for the consumer goods that is slightly moreimpacting than impacted.Let us note that, in some analogy with the CoVaR approach, I stressed the variablesset X at their 95% VaR. However, this is relatively arbitrary way to set stressed values. Iverified that other level of VaR give similar results, for instance stressing the more extreme99% VaR produces very similar outcomes though with roughly doubled values of averagelosses. Another possibility is to stress with a uniform stress of, for instance, all equal to 1(the number itself is irrelevant because the results scale linearly with it). Results in this caseare consistent with the previous but quantitatively quite different. It is beyond the purposeof the present work to investigate which kind of stressing is the most adequate and, most8 .02 0.025 0.03 0.035 0.040.020.0250.030.035 Basic MaterialsConsumer GoodsConsumer ServicesFinancialsHealth CareIndustrialsOil & Gas TechnologyTelecommunicationsUtilities
Figure 2: Average losses caused on the entire system by stressing a given sector to 95% of its VaR vs. thelosses on the sector when the rest of the system is stressed to 95% VaR. The line is the diagonal separationwhere Impacting = Impacted and is reported for visual reference. likely, this depends on the purpose of the stress testing exercise.The effect of uncertainty reduction and consequent systemic coupling of the system isillustrated in Fig.3 where the values of the mutual informations are reported for the variousindustry sectors. Note that, in this case, the matrix is symmetric and the results do notdepends on the values of X and its level of stress. Interestingly, the map is quite differentform the one for the stress propagation in Fig.1. Note for instance the strong impactsbetween Technology and Consumer Services industry sectors in Fig.1 which is instead not apredominant interdependency as revealed by the mutual information in Fig.3.
6. Conclusions
The effects of stress testing a multivariate system of interacting variables are describedby the conditional probability distribution where all statistical properties of the stressedvariables can be expressed as function of the values of the stressing variables. In this workit has been shown that for the vast class of multivariate models described by ellipticaldistributions, the conditional probability has the same expression of the original multivariate9 a s i c M a t e r i a l s C on s u m e r G ood s C on s u m e r S e r v i ce s F i n a n c i a l s H ea lt h C a r e I ndu s t r i a l s O il & G a s T ec hno l ogy T e l ec o mm un i ca ti on s U tiliti e s Basic MaterialsConsumer GoodsConsumer ServicesFinancialsHealth CareIndustrialsOil & GasTechnologyTelecommunicationsUtilities
Figure 3: Intensity map for the mutual information between industry sectors. probability but with shifted centroid and a reduced variability. I showed that the shift inthe centroid coincides with the multivariate linear regression factor whereas the change inthe shape matrix is the same as for the change in the covariance in the multivariate normalcase. Also changes in entropy and mutual information are similar to the normal case besidea constant factor.It is argued that, for risk purposes, the linear shift of the centroid is the most importantfactor. Applications to an example of equity returns shows meaningful outcomes for whatconcerns the mutual effects on the average losses in industry sectors under stress conditions.This approach is very simple and easy to implement and it applies to multivariateStudent-t distributions that are quite realistic models for risk in financial systems. Thepractical limitation of this approach is the estimate of the parameters of the multivariatedistribution and in particular the shape matrix Ω which comprises p ( p − / L topological regularization using network filtering techniques [24] which overcome the courseof dimensionality problem. Stress testing using specific market states can be performed by10sing the clustering method presented in [25] which also overcome non-stationarity issues. Acknowledgments
The author acknowledges discussions with many of the members of the Financial Computingand Analytics group at UCL. In particular a special thank to Fabio Caccioli, Guido Mas-sara, Carolyn Phelan and Pier Francesco Procacci. Also, thanks for support from ESRC(ES/K002309/1), EPSRC (EP/P031730/1) and EC (H2020-ICT-2018-2 825215).
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