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• Multivariate Copulas for FinancialModeling

by Gary Venter, Jack Barnett, Rodney Kreps, and John Major

ABSTRACT

Although the copula literature has many instances of bi-variate copulas, once more than two variates are correl ated,the choice of copulas often comes down to selection of thedegrees-of-freedom parameter in the t-copula. In search fora wider selection of multivariate copulas we review a gen-eralization of the t-copula and some copulas defined byHarry Joe. Generalizing the t-copula gives more flexibilityin setting tail behavior. Possible applications include in-surance losses by line, credit risk by issuer, and exchangerates. The Joe copulas are somewhat restricted in the rangeof correlations and tail dependencies that can be produced.However, both right- and left-positive tail dependence ispossible, and the behavior is somewhat different from thet-copula.

KEYWORDS

Multivariate copulas, grouped t-copula, copula parameter estimation,copula fit testing, foreign exchange dependencies

• Variance Advancing the Science of Risk

1. Introduction

Copulas provide a convenient way to expressmultivariate distributions. Given the individualdistribution functions Fi(Xi) the multivariate dis-tribution can be expressed as a copula functionapplied to the probabilities, i.e., F(X1, : : : ,Xn) =C[F1(X1), : : : ,Fn(Xn)]. Venter [6] discusses manyof the basic issues of using copulas with theheavy-tailed distributions of property and liabil-ity (P&L) insurance. One of the key conceptsis that copulas can control where in the rangeof probabilities the dependence is strongest. Anycopula C is itself a multivariate distribution func-tion but one that applies only to uniform distribu-tions on the unit square, unit cube,: : : , dependingon dimension. The uniform [0,1] variables areinterpreted as probabilities from other distribu-tions and are represented by U, V, etc.One application is correlation in losses across

lines of business. Lines tend to be weakly cor-related in most cases but can be strongly cor-related in extreme cases, like earthquakes.Belguise and Levi [1] study copulas applied tocatastrophe losses across lines. With financialmodeling growing in importance, other poten-tial applications of copulas in insurance com-pany management include the modeling of de-pendence between loss and loss expense, depen-dence among asset classes, dependence amongcurrency exchange rates, and credit risk amongreinsurers.The upper and lower tail dependence coeffi-

cients of a copula provide quantification of tailstrength. These can be defined using the rightand left tail concentration functions R and L on(0,1):

R(z) = Pr(U > z j V > z) and

L(z) = Pr(U < z j V < z):The upper tail dependence coefficient is the limitof R as z! 1, and the lower tail dependence co-

efficient is the limit of L as z! 0. For the nor-mal copula and many others, these coefficientsare zero. This means that for extreme values thedistributions are uncorrelated, so large-large orsmall-small combinations are not likely. How-ever, this is somewhat misleading, as the slopesof the R and L functions for the normal andt-copulas can be very steep near the limits.Thus there can be a significant degree of de-pendence near the limits even when it is zeroat the limit. Thus looking at R(z) for z a bit lessthan 1 may be the best way to examine large lossdependencies.While a variety of bivariate copulas is avail-

able, when more than two variables are involvedthe practical choice comes down to normal vs.t-copula. The normal copula is essentially the t-copula with high degrees of freedom (df), so thechoice is basically what df to use in that cop-ula. Venter [5] discusses the use of the t-copulain insurance. The t takes a correlation param-eter for each pair of variates, and any correla-tion matrix can be used. The df parameter addsa common-shock effect. This can be seen in thesimulation methodology, where first a vector ofmultivariate normal deviates is simulated thenthe vector is multiplied by a draw from a sin-gle inverse gamma distribution. The df parameteris the shape parameter of the latter distribution,which is more heavy-tailed with lower df. Thefactor drawn represents a common shock thathits all the normal variates with the same fac-tor. This increases tail dependence in both theright and left tails but also increases the like-lihood of anti-correlated results, so the overallcorrelation stays the same as in the normalcopula.Two problems for the t-copula are the symme-

try between right and left tails and having onlya single df parameter. The stronger tail depen-dence in insurance tends to happen in the righttail. Putting the same dependence in the left tailwould be inaccurate, but it would probably have

104 CASUALTY ACTUARIAL SOCIETY VOLUME 01/ISSUE 01

• Multivariate Copulas for Financial Modeling

little effect on risk measurement overall, whichis largely affected by the right tail. The singledf parameter is more of a restriction. It resultsin giving more tail dependence to more stronglycorrelated pairs. This is reasonable but is not al-ways consistent with data.Thus more flexible multivariate copulas would

be useful. One alternative provided by Daul etal. [2] is the grouped t-copula, which uses differ-ent df parameters for different subgroups of vari-ables, such as corporate bonds grouped by coun-try. We introduce a special case of that calledthe individuated t-copula, or IT, which has a dfparameter for each variable. Another directionis that of Joe [3], who develops a method forcombining simple copulas to build up multivari-ate copulas. Three of thesecalled MM1, MM2,and MM3have closed-form expressions. Weinvestigate some of the properties of thesecopulas.The t-copula for n variates has (n2 n+2)=2

parameters. The MMC copulas have one moreparameter for each variate and so have (n2 + n+2)=2 all together, while the IT has (n2 + n)=2.This situation does not include enough parame-ters to have separate control of the strength ofthe tail dependency for every pair of variates,but it does add some flexibility. The IT copulageneralizes the t so it can be tried any time thet is too limiting in the possibilities for tail be-havior. The MMC copulas have a parameter foreach pair of variates, but this does not give themfull flexibility in matching a covariance matrix.When all the bivariate correlations are fairly low(below 50% in the trivariate case but decreasingwith more dimensions) they can usually all bematched, but the higher they get the more simi-lar they are forced to be. This would probably befine for insurance losses by line of business, asthey tend to have low correlations overall but canbe related when the losses are large. In modelingthem, some additional flexibility in tail behavior

vs. overall correlation could be helpful, so thiscould be an application where the MMC copulaswould have an advantage over the t-copula.

2. IT copula

This copula is most readily described by thesimulation procedure from its parameters, whichare a correlation matrix and a parameter n foreach of the N variables. The simulation startswith the generation of a multivariate normal vec-tor fzng with correlation matrix by the usualapproach (Cholesky decomposition, etc.). Thena uniform (0,1) variate u is drawn. The inversechi-squared distribution quantile with probabilityu and df n, denoted by wn = hn(u), is calculated.Then tn = zn[n=wn]

1=2 is t-distributed with n df.To get the copula value, which is a probability,that t-distribution is applied to tn. The only differ-ence between this and simulation of the t-copulais the t uses the same inverse chi-square draw foreach variate.The chi-squared distribution is a special case

of the gamma. The ratio wn=n is a scale trans-form of the chi-squared variate, so is a gammavariate. If the gamma density is parameterized tobe proportional to x1ex= , then wn=nhas parameters = 2=n and = n=2. This isa distribution with mean 1. It can be simulatedeasily if an inverse gamma function is available,as in some spreadsheets. It is not necessary forthe df to be an integer for this to work. If notan integer, a beta distribution can be used tocalculate the t-distribution probability, as inVenter [5].Although the simulation is straightforward,

the copula density and probability functions aresomewhat complicated. In terms of hn, the in-verse chi-squared function above, and denotingthe matrix inverse of as J and the inverset-distribution of un with n df as tn, the copula

VOLUME 01/ISSUE 01 CASUALTY ACTUARIAL SOCIETY 105

• Variance Advancing the Science of Risk

density at (u1, : : : ,un) can be shown to be:

c(~u) =Z 10

QNn=1

8 x) with n+1 df. If we denote the bivari-ate standard normal survival function as S(x,y)= Pr(X > x,Y > y), then the tail dependencecoefficients for Xm and Xn for the IT copulaare:Z 1

0S(cny

1=n ,cmy1=m)dy,

where cj =p2

"

1+ j2

!,p4

#1=j:

These tend to be between the t-copula depen-dence coefficients for the two dfs but closer tothat for the higher of the two dfs if these are verydifferent.Fitting the IT by maximum likelihood is possi-

ble but involves several numerical steps. Alterna-tively, the df parameter for the t-copula could beestimated for each pair of variables by matchingtail behavior, as in Venter [5], and then individ-ual dfs assigned to be consistent with this. Daulet al. [2] propose separate t-copulas for differentgroups of variables, which are combined into asingle copula with the overall correlation matrixand the separate dfs.

3. MMC copulas

Joes MM1, MM2, and MM3 copulas eachhave an overall strength parameter , a param-eter for each pair of variables ij (not the Kro-necker delta), and add an additional parameterpj , with 1=pj m 1, for eac

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