Global senstvty analyss of credt rsk portfolos D. Baur, J. Carbon & F. Campolongo European Commsson, Jont Research Centre, Italy Abstract Ths paper proposes the use of global senstvty analyss to evaluate latent factor credt rsk models. Our clam s that ths type of senstvty analyss s superor to a local approach n provdng the rsk modeler wth a broader pcture of the rsk contrbutons of the key elements to a credt rsk model. The man fndng s that default probabltes and the correlaton of the latent varables are consderably more mportant than the multvarate dstrbuton and hence the copula of the latent varables. Keywords: credt rsk model, latent factor model, uncertanty analyss, global senstvty analyss. 1 Introducton Modellng credt rsk portfolos s one of the most challengng tasks n fnance of current days. In ths work we model credt portfolo losses followng a latent varable approach. In a latent varable credt rsk model, default of an oblgor occurs f a latent varable falls below a certan threshold. Dependence between ndvdual default events s caused by dependence between latent varable. These credt rsk models can result n very dfferent loss dstrbutons dependng on several factors. Obectve of our study s to assess the rskness of credt rsk portfolos and to study the senstvty of commonly used rsk measures wth respect to three key nput factors of the model. Prevous studes of the senstvty of latent factor credt rsk models to selected model nputs are avalable n the lterature (Frey et al. [1], and Kesel and Klenow [2]). However n these works the senstvty analyss has been performed by followng a local approach,.e. by evaluatng the mpact of changng one nput factor at a tme, whle our am s to evaluate the senstvty of
78 Computatonal Fnance and ts Applcatons a latent factor model more thoroughly by performng a global analyss. Results show that the global approach provdes the rsk manager wth more nformaton than the commonly appled local senstvty analyss and s therefore the proper approach to use n ths knd of problem settngs. The followng secton, 2, descrbes the credt rsk model under nvestgaton. ecton 3 ntroduces the concept of global senstvty analyss. ecton 4 presents the senstvty analyss experment wth ts results and secton 5 concludes. 2 The model Ths secton descrbes a credt portfolo model based on the latent varables approach. The dependence of m ndvdual oblgors s modelled va the dependence of m underlyng latent varables 1, 2..., m. The underlyng varables 1, 2..., m are assumed to be drven by one common factor Z and dosyncratc shocks ε accordng to the equaton: = a Z + ( 1 a ) ε = 1,..., m (1) where a, varyng n (0,1), s the factor loadng, representng the exposure of oblgor to factor Z. The random varables Z and ε are assumed to be ndependently and dentcally dstrbuted wth mean zero and varance one. The terms a and ( 1 a ) are used to ensure a constant varance of ndependent of the factor loadng. For = 1,.., m let the random varable Y be the default ndcator for oblgor takng values n {0,1}. We nterpret 1 as default and 0 as non-default. Let D be the cut off pont for default of oblgor so that Y = 1 D The probablty of default π s gven by π = P( Y = 1) = P( D ) whle the ont default probablty for oblgors and can then be wrtten as π = P Y = 1, Y = 1) = P( D, D ). (
Computatonal Fnance and ts Applcatons 79 In order to focus on the number of ont defaults we assume all losses to be equal to one. The number of ont defaults s obtaned by Monte Carlo smulaton accordng to the followng steps: () for each oblgor the dstrbuton s smulated accordng to equaton (1); () from the dstrbutons obtaned for the s and the nput default probabltes the cutoff ponts of the oblgors D, = 1,... m, are derved; () from the obtaned multvarate dstrbuton a number I of draws s randomly selected and compared wth the cutoff vector D so that the dstrbuton of ont defaults s obtaned. As we are nterested n extreme events, we concentrate on the upper tal of the dstrbuton and focus on ts 95 and the 99.5 quantles. In partcular we study the uncertanty n these varables (uncertanty analyss) and then ther senstvty to the followng model nput factors: (1) the dstrbuton of the factor Z, whch determnes the (margnal) dstrbutons of the latent varables and ther dependence structure; (2) the degree of correlaton among the latent varables, represented by the factor loadngs; (3) the probabltes of default, representng the ratng of oblgors n the portfolo. Note that, although the copula s not an explct element of ths model, changng the dstrbuton of or changng the copula are two alternatve ways to act on the multvarate dstrbuton. 3 On uncertanty and senstvty analyss Each mathematcal model ncorporates several nput factors whch characterze the real process beng modelled. These nputs are usually subect to many sources of uncertanty that consequently produces an uncertanty n the model output. Thus, an understandng of the senstvty of the model outputs to the uncertanty n the nput values s necessary n order to develop confdence n the model and ts predctons. Uncertanty analyss quantfes the uncertanty that arses n the model output due to the uncertanty n the model nput factors. Results of uncertanty analyss can be used to establsh whether or not, gven all the sources of uncertanty, the modeller can place confdence n the model outcomes. Uncertanty analyss s performed va a Monte Carlo approach. Each of the nput factors nvolved n the analyss s assgned a probablty densty functon (pdf), reflectng ts uncertanty,.e. our mprecse knowledge of the nput value.
80 Computatonal Fnance and ts Applcatons Then, a k-dmensonal sample of sze N, k beng the number of nput factors, s generated from the pre-selected pdf's through an approprate samplng desgn. Fnally, the model s evaluated at each sample pont to obtan N realzatons of the output used to estmate ts emprcal pdf. The estmaton of the output pdf allows to quantfyng the uncertanty assocated wth the model output whch s due to the uncertanty n the nputs. Note that results of the uncertanty analyss strongly depend on the pdf s assgned to the nputs. Furthermore, for an effectve analyss t s crucal that the entre range of varaton of each factor s explored. enstvty analyss (A) s the study of how the uncertanty n the output can be apportoned, qualtatvely or quanttatvely, to ts dfferent sources. In general senstvty analyss s used to order by mportance the nput factors accordng to the percentage of the output varaton (uncertanty) that they are accountng for. Results of senstvty analyss can be used to mprovng relablty of these outcomes. To ths end further research s addressed to mprove the estmates of the crucal nputs and to reduce the degree of uncertanty n the estmates of ther values. A large number of senstvty analyss methodologes are avalable n the lterature. The choce of the method to adopt to perform a senstvty experment on a model depends on a number of aspects: the propertes of the model under study (lnearty, addtvty, monotoncty), the number of nput factors nvolved n the analyss, the computatonal tme needed to evaluate the model, and, last but not least, the obectve of the analyss. Very often n the lterature, A s conceved as a local measure of the effect of a gven nput on the output, estmated by computng a dervatve. Ths type of approach, called local senstvty analyss, s based on samplng desgns that vary one nput factor at a tme, whle holdng all the others fxed at determned values. Although the local senstvty approach has some advantages, t has to be underlned that ths approach s practcable only when the nput factors are allowed small varatons around a base value, or when the nput-output relatonshp s assumed to be lnear (see altell et al. [3], [4]). In contrast n problem settngs such as rsk analyss, decson support, envronmental apprasal, where the degree of varaton of the nput factors s materal and/or the model s non-lnear, the lnear senstvtes alone are not lkely to provde a relable measure of senstvty. In these cases, the use of a global approach allowng all nput factors to vary smultaneously and that permts materal varatons, s mandatory. A revew of global senstvty analyss methodologes and a more detaled descrpton of ther features can be found n altell et al. [3], [4]. In ths work we nvestgate the senstvty of the portfolo model descrbed n secton 2 through a global senstvty approach, based on Monte Carlo smulaton. In our exercse ths approach s made necessary as we cannot rely on the assumptons needed to apply a local analyss,.e. we have no reasons to beleve a pror that the model s lnear. Furthermore we am to explore wder ranges of varaton for the nput factors wth respect to those examned n prevous works by Frey et al. [1] and Kesel and Klenow [2].
Computatonal Fnance and ts Applcatons 81 Our goal s to determne what nput factors n the credt rsk models are more responsble for varaton n the model outcome and therefore need a better determnaton. In other words, we are facng the followng problem, labelled as Factor Prortsaton, "what factor, once fxed to ts true albet unknown value, would gve the greatest reducton n the varance of the output?". The followng global senstvty measures are estmated for each nput factor :, whch s a measure of the man effect, and T, whch s a measure of the total effect,.e. a measure that takes nto account also the effect of factor due to ts nteractons wth other factors (two or more factors are sad to nteract when ther effect on the output cannot be expressed as the sum of ther sngle effects). tells us what percentage of the total output varance s due to the th factor, whle T s used to determne what factors are rrelevant and can be fxed at any gven value wthn ther range wthout sgnfcantly reduce the output varance ( T =0 f a suffcent condton to state that a factor s rrelevant). If a model s lnear the sum of the man effects s suffcent to explan the total output varance,.e. = 1. When a model s nonlnear, whch s nteractons among nputs are presents, than < 1, as the porton of output varance due to nteractons effects s not captured by the. A full descrpton of these measures, ncludng techncaltes on how to compute them can be found n altell et al. [3], [4]. It can be demonstrated that the ndces are the proper measure to rank factor n order of mportance when the problem s that of Factor Prortsaton also n the presence of nteractons. 4 The senstvty analyss experment In the work of Frey et al. [1] and Kesel and Klenow [2] senstvty analyss experments were performed on latent varable credt rsk models by followng a local approach,.e. by evaluatng the effect of changng one nput factor at a tme. The man fndng of ther work can be summarsed n the deducton that the copula, determnng the dstrbuton of, consderably contrbutes to determnng the number of ont defaults for a gven quantle of the loss dstrbuton. Here we am to perform a senstvty experment followng a global approach, whch allows for materal varaton of the nput factors values and whch does not requre any prelmnary assumptons on the model such as lnearty. Our experment starts wth the choce of the output varables of nterest, as dfferent choces can lead to very dfferent results. In ths work the output varables of nterest are the hgher quantles of the dstrbuton of ont defaults: the 95 and the 99.5 quantles.
82 Computatonal Fnance and ts Applcatons As second step we choose the nput factors and assgn the correspondent pdf. These are: () The dependence structure among the latent factors, defned through the choce of the dstrbuton for the common factor Z. Ths s modelled by a trgger factor whch may assume three possble values correspondng to the followng dstrbutons for Z: a Gaussan dstrbuton, a t-dstrbuton wth 10 or 4 degrees of freedom. () The degree of correlaton among the oblgors, represented by a m- dmensonal vector of factor loadngs a. A trgger factor s defned to choose among 5 possble determnatons of the vector of factor loadngs. These 5 determnatons are generated a pror to represent 5 dfferent correlaton structure among the oblgors: - very low correlaton, the a are sampled from unform dstrbutons on [0-0,15]; - low correlaton, the a are sampled from unform dstrbutons on [0,15-0,30]; - medum-low correlaton, the a are sampled from unform dstrbutons on [0,30-0,45]; - medum correlaton, the a are sampled from unform dstrbutons on [0,45-0,60]; - medum-hgh correlaton, the a are sampled from unform dstrbutons on [0,60-0,75]. () The ratng portfolo composton, represented by a m-dmensonal vector of default probabltes π. A trgger factor s used to sample among nne possble determnatons of the vector of default probabltes. These nne determnatons are randomly generated a pror under the assumpton that the π are unformly dstrbuted wthn a fxed range. The lower and upper bounds of the range depend on the oblgors ratng: - hgh rated oblgors (AAA class), the π are sampled from unform dstrbutons on [0-0,05]; - medum rated oblgors (BBB class), the π are sampled from unform dstrbutons on [0,05-0,10]; - low rated oblgors (CCC class), the π are sampled from unform dstrbutons on [0,10-0,15]. Then uncertanty and senstvty analyss are performed n a Monte Carlo fashon on a portfolo of 1000 oblgors (m=1000). An nput sample of sze N=
Computatonal Fnance and ts Applcatons 83 16.384 s generated through an approprate samplng strategy. Then the model s evaluated at each nput sample pont to produce N output realzatons that are analysed to produce uncertanty and senstvty analyss results. Results of the uncertanty analyss are n Table 1, whch shows the man descrptve statstcs of the output emprcal pdf. Table 1: Basc statstcs of the outputs emprcal pdf. 95 quantle 99.5 quantle Mean 303 569 t. Dev. 172 243 Mnmum 52 73 Maxmum 795 999 The obtaned statstcs pont out that the average number of ont defaults s rather dfferent for the two selected quantles. Ths underles the need for an analyst to set the obectve at the begnnng of the exercse. hftng the nterest from one quantle to another may n fact lead to dfferent results. The standard devaton values ndcate that results are rather volatle, especally those referrng to the hgher quantle. Ths stresses the need of a global senstvty analyss to assess the relatve contrbuton of the varous nput factors to such a consderable varablty. Results of senstvty analyss are n Table 2. The two senstvty measures, and T, are shown for each of the selected nput factors for the two quantles. Table 2: The senstvty measures for the 3 nput factors and the 2 outputs. 95 quantle 99.5 quantle T T Degree of correlaton. 0.237 0.318 0.572 0.632 Dependence structure 0.021 0.035 0.049 0.107 Ratng portfolo composton 0.672 0.774 0.329 0.396 The obtaned ndces show the relatve mportance of the three factors at dfferent quantles and allows for the concluson that more than the 80% of the total varance can be explaned by the degree of correlaton among oblgors and the portfolo ratng composton n both cases (the sum of ther frst order ndces s greater than 0.8). The relatve mportance of the portfolo ratng composton and the degree of correlaton among the oblgors depends on the quantle: at the 95 quantle the composton of the portfolo s accountng for most of the varablty n the output and therefore needs to be accurately determned; at the 99.5 quantle t s more relevant to focus on the determnaton of the degree of correlaton among oblgors.
84 Computatonal Fnance and ts Applcatons The effects of the dependence structure s almost neglgble for the 95 quantle ( = 0.035) and rather low wth respect to the others also at the 99.5 T quantle. The latter result mples that the choce of the dstrbuton of the common factor Z, whch determnes the dependence structure among the oblgors, s not dramatcally affectng the quantles of nterest. 5 Conclusons Ths paper has ntroduced the concept of global senstvty analyss to evaluate credt rsk models. One of our man fndngs s that n order to draw relable conclusons t s essental to establsh a pror the obectve functon of the analyss, as focusng on dfferent quantles of the dstrbuton of the number of ont defaults may lead to dfferent conclusons. In our framework we have shown that when the focus s on the 99.5 quantle of the dstrbuton of number of ont defaults, modellng the degree of correlaton s more effectve n reducng the output uncertanty than focusng on the portfolo composton. In contrast, when the nterest s on the 95 quantle, more careful must be placed n the credt portfolo constructon, snce the portfolo composton explans more than the 65% of the uncondtonal varance. Moreover results makes evdent that the dependence structure of the latent varables (.e. ther multvarate dstrbuton) s much less nfluent than other factors when the am s that of reducng the output varance. References [1] Frey, R., McNel, A.J., & Nyfeler, M., Copulas and credt models. RIK, October, pp. 111-114, 2001. [2] Kesel, R. & Klenow, T, enstvty analyss of credt portfolo models n Appled Quanttatve Fnance, W. Härdle, T. Klenow, G. tahl, pp.140-152, 2002. [3] altell, A., Chan, K., cott, E. M., enstvty Analyss, John Wley & ons, Probablty and tatstcs seres, 2000. [4] altell, A., Tarantola,., Campolongo, F., & M. Ratto, enstvty Analyss n Practce. A Gude to Assessng centfc Models. John Wley & ons publshers, Probablty and tatstcs seres, to appear March 2004.