Copula-based top-down approaches in financial risk aggregation

Transcription

1 Number 3 Working Paper Series by the University of Applied Sciences of bfi Vienna Copula-based top-down approaches in financial risk aggregation December 6 Christian Cech University of Applied Sciences of bfi Vienna

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3 Abstract This article presents the concept of a copula-based top-down approach in the field of financial risk aggregation. Selected copulas and their properties are presented. Copula parameter estimation and goodness-of-fit tests are explained and algorithms for the simulation of copulas and meta-distributions are provided. Further, the dependence structure between interest rate and credit risk factor changes that are computed from sovereign and corporate bond indices is examined. No clear pattern of the dependence structure can be observed as it varies substantially with the duration and the rating of the obligors. This could indicate that top-down approaches are too simplistic to be implemented in practice. However, the results also suggest that copula-based approaches for the data sample at hand seem preferable to the assumption of a multivariate Gaussian distribution as none of the marginal distributions examined are normally distributed and as the Gaussian copula s fit in terms of the AIC is worse than that of other copulas. Further, the Gaussian copula seems to underestimate the probability of joint strong risk factor changes for the data sample at hand. 3

4 Contents Introduction 5 Bottom-up and top-down approaches 7 3 Copula-based approaches 3. Introduction to copulas Modelling the marginal distributions Presentation of selected copulas Gaussian copula Student t copula BB copula Clayton copula Gumbel copula Frank copula Copula-parameter estimation Maximum likelihood estimation Parameter estimation using correlation measures Empirical copulas Goodness-of-fit tests Simulation of selected meta-distributions Simulation of meta-gaussian distributions Simulation of meta-student t distributions Simulation of bivariate meta-bb distributions Simulation of bivariate meta-clayton distributions Simulation of bivariate meta-gumbel distributions Simulation of bivariate meta-frank distributions Implementation of a top-down approach and empirical evidence 46 5 Conclusion 6 Appendix 6 Appendix A: Rank-based correlation measures Appendix B: GoF test probability integral transform Appendix C: Empirical results for non-autocorrelation-adjusted data 69 List of tables and figures 7 References 73 4

5 Introduction According to the Basel Committee on Banking Supervision [54], risk aggregation refers to the development of quantitative risk measures that incorporate multiple types or sources of risk. Amongst these types of risks are e.g. credit risk, market risk (interest rate risk, stock price risk, etc.), insurance risk (life and property and casualty insurance), operational risk, liquidity risk, asset liability management (ALM) risk, business risk, etc. These quantitative risk measures, defined over a specific time horizon, may then be used to estimate the economic capital that is needed to absorb unexpectedly high potential losses. Apart from the properties of the types of risks and the time horizon, the amount of economic capital depends on the rating that a financial institution aspires, as the probability of default (i.e. the probability that the economic capital cannot absorb the realised losses) is related to the confidence level of the risk measure. Such reasoning also forms the basis of the new Basel II regulatory framework (Basel Committee on Banking Supervision [55] and European Parliament and Council [56]), where banks are required to hold at least the minimum regulatory capital as a buffer against credit risk (regarded as the main source of banking risks), whose magnitude does not only depend on the size of the exposure but also on the riskiness of the credit portfolio. The one-year survival probability (of the financial institution) is targeted at 99.9% per year (i.e. the expected probability of default is no more than.%). Additionally, banks have to hold minimum regulatory capital for market risk in the trading book and for operational risk. The Basel II regulatory framework, however, does not account for diversification effects between risk types (credit, market and operational risk), as the minimum regulatory capital requirements for each risk-class are simply added to obtain the total minimum regulatory The economic capital to be held in this context is then defined as the value-at-risk with confidence level α which implies that a default probability of ( α) is conjectured. For critical remarks see e.g. Pézier [57]. While the value-at-risk has been criticised as it does not display the desirable feature of sub-additivity (see e.g. Arztner et al. [7]), it is widely used in practice. Alternative risk measure like e.g. the expected shortfall (also termed conditional value-at-risk, CVaR) are sub-additive and could easily be obtained by the methodology presented in this paper. However, there is no direct linkage between the expected shortfall and a financial institution s probability of default. The specific formulae to compute the regulatory minimum capital requirement for credit risk in the Basel II accord are derived on the basis of a structural model (Merton [5] model), where it is assumed that the credit portfolio is asymptotically fine-grained, i.e. it is assumed that idiosyncratic risk is diversified away completely (see e.g. Finger [5] and Gordy [33]). 5

6 capital. This conservative approach implicitly assumes perfect positive correlation between the risk types. The prudential rules of the Basel II accord should however not be mistaken as a guideline on how to allocate economic capital efficiently. Rather, institution-internal models that go beyond the minimum regulatory requirements of Basel II ( pillar ) are used in practice. 3 Risk aggregation models that quantify the diversification effects seem to be one necessary foundation for the efficient allocation of economic capital. Several approaches to risk aggregation have been proposed in the literature. Section gives an introduction to top-down approaches as opposed to bottom-up approaches in the context of risk aggregation and reviews existing literature. Copula-based approaches, presented in section 3, seem adequate and preferable to the widely employed assumption of multivariate Gaussian distributions 4 of risk factor changes, if the risk factor changes are not normally distributed. 5 Section 3. gives an introduction to copula-based approaches in the context of top-down risk aggregation. Various parametric distribution functions that are used to model marginal distributions in the context of risk aggregation are shortly mentioned in section 3.. Section 3.3 presents some selected bivariate and two multi-dimensional copulas in detail and compares their properties. Specific equations for copula functions and copula densities are also provided in this section. Different approaches to copula parameter estimation are presented in section 3.4, and goodness-of-fit tests are presented in section 3.5. Finally, section 3.6 provides algorithms for the simulation of the presented copulas. Section 4 first shortly addresses the results of two recent studies on the implementation of a top-down risk aggregation model. These studies, using institution-internal data, find that the risk-factor changes seem to be only slightly correlated. In the remainder of section 4, daily market data (bond index returns) are used to examine the dependence structure between interest rate risk and credit risk. The empirical results provide a very heterogeneous picture of the dependence structure between these two risk factor 3 Efforts on risk management system that go beyond the minimum capital requirements of pillar are also regulatorily required in pillar of the Basel II accord. 4 or other widely used assumptions on joint distributions that jointly model the marginal distributions and their dependence structure like e.g. the multivariate Student t or Weibull distributions and the highly flexible multivariate generalized hyperbolic distribution (see e.g. McNeil et al. [48], section 3..3). 5 Not everybody agrees on this statement, see e.g. Mikosch [5]. 6

7 changes, depending on the maturity bands examined and the credit quality. The goodness-of-fit of six copulas and empirical evidence of positive tail dependence is examined for 5 data pairs with a sample size of N =, 77 each. Section 5 concludes. Bottom-up and top-down approaches In general one can employ different approaches to aggregate different risk types (for a review article see e.g. Saita [6] or Alexander [4]). These approaches may broadly be classified into bottom-up and top-down approaches. 6 Bottom-up approaches try to model the distribution of various risk factors and their impact on risk types, such as credit risk, market risk, etc. One prominent example of a bottom-up approach is Credit Metrics, a credit risk model which derives the profit and loss distribution of a credit portfolio from the asset values of the obligors, which are modelled as linear combinations of correlated industry index returns (see e.g. Crouhy et al. [4]). A bottom-up approach in the context of risk aggregation would estimate the impact of these risk factors (industry index returns) and, if necessary, additional risk factors (such as the interest rate term structure, credit spreads, etc.) on the profits and losses of other lines of business (e.g. the market portfolio) and model the joint profit and loss distribution on that basis. Hence, the dependence between risk types (profits and losses of different lines of business) is modelled indirectly: a joint distribution of risk factor changes is estimated and the impact of these risk factor changes on the diverse financial portfolios profits and losses, defined as (generally non-linear) functions of the risk factor changes, is modelled. Top-down approaches, on the contrary, do not try to identify common single risk factors that influence different types of risk, but rather start from aggregated data, e.g. the profits and losses of different lines of business, such as the returns of the credit portfolio or the market portfolio. Operational risk in this context would be modelled as a portfolio of risk exposures with nonpositive profits and losses or returns. Empirical panel-data of (or assumptions on) the profits and losses or returns of these portfolios allow to estimate a joint distribution of the total returns, or the total risk. The single components that constitute the financial portfolios (or, alternatively expressed, the single risk factors that influence the portfolio profits and losses) are not 6 See e.g. Cech and Jeckle []. The approaches are also referred to as base-level and top-level aggregation, see e.g. Aas et al. []. 7

8 Bottom-up approach Distribution of economic risk factors, e.g. interest rate term structure credit spread term structure equity returns GDP growth etc. and dependence structure (copula) Top-down approach Distribution of portfolio returns, e.g. market portfolio returns credit portfolio returns insurance portfolio returns losses due to operational risk etc. and dependence structure (copula) Profit and loss functions (domain: economic risk factors), e.g. market portfolio returns credit portfolio returns insurance portfolio returns losses due to operational risk etc. Joint profits and losses / returns. Resulting in joint profits and losses / returns. Figure : Bottom-up and top-down approaches. addressed in this approach. Figure schematically depicts the bottom-up and the top-down approach. In both approaches a common time horizon for the estimation of risk factor changes has to be found. Ideally, the time horizon would correspond to the internal capital allocation cycle which conventionally is one year. Generally, the profit and losses or returns of credit and insurance risk are also measured at least at this frequency and risk measures are estimated for a one-year horizon. Market portfolio profits and losses and associated risk measures are often measured and estimated on a daily basis, as the average holding period of instruments in the market portfolio is generally short-term and also because of regulatory directives. If one assumes that the market portfolio profits and losses are normally distributed and i.i.d. 7, one can easily compute one-year risk measures for the market portfolio by using the squareroot-time formula and estimate the one-year unexpected loss 8 as a multiple of the one-day unexpected loss. The one-year unexpected loss is computed by multiplying the one-day unexpected loss by 6 6 (assuming 6 trading days per annum) and the one-year value at risk is the one-year unexpected loss minus the one-year expected profit. This approach however ignores the usual pre-defined market risk management intervention policies like stop-loss limits, etc. as the risk measures are computed on the basis of 7 Independent and identically distributed, i.e. the returns are not autocorrelated. 8 I.e. the negative value of the one-sided confidence interval lower bound for a deviation from the expected profit. 8

9 the current portfolio composition. I.e. the risk measures are computed for a buy-and-hold portfolio, which leads to an overestimation (upward-bias 9 ). On the other hand side, non-normality of daily market risk factor changes has been widely documented. The univariate risk factor changes are often leptokurtic and left-skewed; furthermore the probability of joint extremely negative returns is higher than implied by a multivariate normal distribution (see e.g. Fortin and Kuzmics [7]). This again leads to an underestimation of the risk measures (downward-bias) if normality of the market risk factor changes is falsely assumed. Aas et al. [] in their model incorporate risk management intervention policies by simulating daily market portfolio returns under predefined stop-loss policies and limits, using a constant conditional correlation (CCC) GARCH(,) model to account for volatility clustering and leptokurtic return distributions. The distribution of -year market portfolio returns that are obtained from the simulations of successive one-day returns are then used to model the one-year market risk and its correlation with other risk types. This very promising approach is, however, only useful if there exists a sufficiently large data-set of e.g. credit or insurance portfolio profits and losses on an annual basis so that a model for aggregated risk can be calibrated. Rosenberg and Schuermann [59] overcome the problem of short time-series by estimating linear regression functions to explain market and credit risk as functions of macroeconomic risk factors, using panel-data for quarterly returns of the market and credit portfolio returns of a set of large banks. The regression functions are calibrated using 9 years of historical data. Assuming constant regression parameters, market and credit portfolio returns and their dependence structure are simulated by 9 years of historical quarterly macroeconomic risk factor data as regressors (operational risk is modelled separately). If one wants to avoid the model risk associated with both models presented above, institution-internal time series of profits and losses or returns for the different lines of business may be used to estimate a risk-aggregation model. This again results in a very small data sample for the calibration of the model if annual data is used as generally there are no long time series of e.g. credit portfolio profits and losses available. Using monthly institutioninternal data seems to be a promising compromise. 9 Hickman et al. [38] show that risk management intervention policies can substantially reduce the risk. 9

10 In both top-down and bottom-up approaches, the task of estimating joint distributions (joint economic risk factors changes in the context of bottom-up approaches and joint portfolio profits and losses in the context of top-down approaches), may be decomposed into (a) the estimation of the marginal distributions (univariate risk factor changes or portfolio profits and losses) and (b) the estimation of the dependence structure, if a copula-based approach is used. Copulas may be thought of as a more flexible version of correlation matrices that are widely used in risk management models that assume joint normality. Copula-based approaches are discussed in detail in section 3. Work on top-down approaches has been done by Kuritzkes et al. [47] (insurance, market, credit, ALM, operational and business risk), Ward and Lee [69] (insurance, market, credit, ALM and operational risk), Dimakos and Aas [] (market, credit and operational risk), Rosenberg und Schuermann [59] (market, credit and operational risk) and Tang and Valdez [67] (different types of insurance risk). While Kuritzkes et al. [47] in their simplifying approach assume a joint normal distribution of the risk factor changes, the latter four articles describe a copula-based approach to aggregate the risk of financial portfolios. Ward and Lee [69] and Dimakos and Aas [] use a Gaussian copula to combine the marginal distributions. The latter study only models pairwise dependence between credit and market risk and credit and operational risk without specifically modelling the dependence between market and operational risk. Rosenberg and Schuermann [59] estimate the marginal distributions parameters and their correlation measures using market data and values that were reported in other studies and regulatory reports. The marginal distributions are combined by a Gaussian and a Student t copula to point out the effects of positive tail dependence (see section 3). They report that the choice of the copula (Gaussian or Student t) has a more modest effect on risk than has the business mix (the weights assigned to a bank s financial portfolios). Tang and Valdez [67] use semi-annual data for loss ratios for the aggregate Australian insurance industry from 99 to to calibrate a model that aggregates risks of different lines of insurance business (motor, household, fire and industrial special risks, liability, and compulsary third party insurance). The marginal distributions are modelled Hall [35] points out that the economic capital may be severely underestimated if a joint Gaussian distribution is assumed while indeed the marginal distributions are non-normal. The calibration of the marginal market and credit portfolio return distributions is done for data that was obtained by a simulation in a bottom-up manner. The aggregation of these risks is done in a top-down manner, where the correlation matrix reported in Kuritzkes et al. [47] is used.

11 as gamma, log-normal and pareto distributions and the consequences of assuming Gaussian, Student t and Cauchy copulas are addressed. Work on bottom-up approaches has been done by Medova and Smith [49] (market and credit risk), Alexander and Pézier [5] and Aas et al. []. Medova and Smith [49] use Monte Carlo simulations to allow for a varying exposure of the credit portfolio (employing a structural credit risk model). Alexander and Pézier [5] estimate multiple linear regression models, regressing the profits and losses of 8 business units on 6 risk factors 3. Pearson s correlation coefficient is used as dependence measure. To account for tail dependence (a higher probability of joint extreme events as compared to the Gaussian distribution/copula; see section 3) the authors suggest to use the tail correlations rather than the usual overall correlations. Aas et al. [] use a bottom-up approach to aggregate market, credit and ownership risk. For aggregating (additionally) operational and business risk, they use a top-down approach (employing a Gaussian copula). 3 Copula-based approaches 3. Introduction to copulas Copula-based approaches are a rather new methodology in risk management. The term copula was introduced by Sklar [66] in 959 (a similar concept for modelling dependence structures of joint distributions was independently proposed by Höffding [39] some twenty years earlier). Recent textbooks on copulas are e.g. Joe [4] and Nelsen [5], [53]. Copulas are functions that combine or couple (univariate) marginal distributions to a multivariate joint distribution. Sklar s theorem (using a slightly different notation in the original article) states that a n-dimensional joint distribution function F (x) evaluated at x = (x, x,..., x n ) may be expressed in terms of the joint distribution s copula C and its marginal distributions F, F,..., F n as The business units are: Corporate finance, Trading and sales, Retail banking, Commercial banking, Payment and settlement, Agency and custody, Asset management, and Retail brokerage. 3 Risk factors: Y treasury rate, Y - Y treasury rate (slope), implied interest rate volatility, S&P 5 index, S&P 5 implied volatility, Y credit spread.

12 F (x) = C (F (x ), F (x ),..., F n (x n )), x R n. () The copula function C is itself a multivariate distribution with uniform marginal distributions on the interval U = [, ], C : U n U. Reformulating formula yields C(u) = F ( F (u ), F (u ),..., F n (u n ) ), u U n, () where u = (u, u,..., u n ) = (F (x ), F (x ),..., F n (x n )) are the respective univariate marginal distributions. Thus, a copula-based approach allows a decomposition of a joint distribution into its marginal distributions and its copula. On the other hand marginal distributions may be combined to a joint distribution assuming a specific copula. The crucial point in using a copula-based approach is that it allows for a separate modelling of the marginal distributions (i.e. the univariate profit and loss or return distributions) and the dependence structure (the copula). Figure displays an example for the combination of two marginal distributions to a joint bivariate distribution. Assume that a financial institution holds two portfolios, a market portfolio and a credit portfolio. The market portfolio s annual return distribution is modelled as random variable r M.+.5 t 5, where t 5 is a Student s t distributed random variable with ν = 5 degrees of freedom. The credit portfolio s annual return distribution is modelled as r C ln(.5 B(3,.)), where B(3,.) is a beta distributed random variable. Both portfolios exhibit fat tails, assigning a higher probability to extreme events than a normal distribution. The returns of the credit portfolio are heavily left-skewed, assigning a higher probability to extreme losses than to extreme gains. Table displays mean and median values, the standard deviation, skewness, and excess-kurtosis of the two return distributions. 4 The data shows that both return distributions are non-normal. In a 4 The moments of the two distributions displayed in table are the sample estimates of,, simulated returns, using Monte Carlo simulation. Using this simulated values for the depiction of the credit portfolio returns density in figure as kernel smoothed densities with Gaussian kernels, this explains the small right tail above the.5-threshold displayed in figure. The simulated credit portfolio returns will not take on a value of greater than.5, as the beta distribution is defined on the [,]-interval. (For a primer on kernel smoothed densities see e.g. Scott and Sain [64]; on beta distributions, see e.g. Johnson et al. [44], Chapter 5.)

13 density r M = F M - (um ) density r C = F C - (uc ) density.5 density.5.5 u M = F M (r M ).5 u C = F C (r C ) marginal distributions and copula u C u M joint distribution Figure : Example of the combination of a market and a credit portfolio marginal return distributions to a joint returns distribution using a copulabased approach. 3

14 r M r C mean.. median..5 standard deviation skewness -.97 excess kurtosis Table : Sample moments for r M.+.5 t 5 and r C ln(.5 B(3,.)). copula-based approach these marginal distributions may easily be combined to a joint distribution, as shown in figure. Apart from the ability to combine arbitrary marginal distributions to a joint distribution, copula-based approaches allow for a specific modelling of the dependence structure, i.e. the copula. One frequently observed empirical evidence is that extreme joint market movements are more frequently observed than implied by a multivariate Gaussian distribution that is often used in market risk models. 5 This empirical evidence is sometimes referred to as correlation-breakdown. Copula-based approaches allow for a flexible modelling of the probability of joint extreme observations (unconditional on the marginal distributions). For example, a Student t copula assigns a higher probability to joint extreme observations than does a Gaussian copula. This higher probability of joint extreme observations as compared to the Gaussian copula is referred to as positive tail dependence. As an example, figure 3 shows scatter plots of two jointly distributed standard normal random variables. These standard normal marginal distributions are combined by a Gaussian copula, a Student t copula, a Clayton copula, and a Gumbel copula, respectively. The resulting joint distributions (for arbitrary marginal distributions) are referred to as meta-gaussian, meta- Student t, meta-clayton and meta-gumbel distributions. The top row shows scatter plots of simulated joint distributions that have a correlation (in terms of Spearman s rho 6 ) of.4. The bottom row shows corresponding scatter 5 In passing, note that the multivariate Gaussian distribution in copula-based approaches terms is a set of univariate Gaussian marginal distributions that are combined by a Gaussian copula. 6 In copula-based approaches, rank-based correlation measures such as Spearman s rho and Kendall s tau are preferable to the widely known Pearson correlation measure that is 4

15 ρ S =.4: (meta )Gaussian meta Student t meta Clayton meta Gumbel ρ S =.8: (meta )Gaussian meta Student t meta Clayton meta Gumbel Figure 3: Simulation scatter plots of bivariate meta-gaussian, meta-student t, meta-clayton and meta-gumbel distributions. The top row shows scatter plots of joint distributions with a Spearman s rho correlation measure of approximately.4, the bottom row shows scatter plots of joint distributions with a Spearman s rho of approximately.8. Both marginal are standard normally distributed. plots for joint distributions with a correlation of.8. It can be seen that for identical marginal distributions and Spearman s rho the Student t copula assigns a higher probability to joint extreme events than does the Gaussian copula. Assigning an equal probability to joint extreme positive deviations and to joint extreme negative deviations from the median value, the Student t copula displays symmetric tail dependence. Asymmetric tail dependence is prevalent if the probability of joint extreme negative realisations differs from that of joint extreme positive realisations. In figure 3 it can be seen that the Clayton copula assigns a higher probability to joint extreme negative events than to joint extreme positive events. The Clayton copula is said to display lower tail dependence, while it displays zero upper tail dependence. The converse can be said about the Gumbel copula (displaying upper but zero lower tail dependence). Table used in the context of multivariate normal distributions. A short note on Spearman s rho and Kendall s tau is given in Appendix A. 5

16 copula: tail dep. Gaussian Student t BB Clayton Gumbel Frank lower no yes yes yes no no upper no yes yes no yes no symmetric yes yes no no no yes Table : Summary of which bivariate copulas display lower and upper tail dependence and whether the positive tail dependence is symmetric. gives an overview of which of the copulas presented in this article display upper or lower tail dependence. 7 In section 3.3 we will give a formal definition of upper and lower tail dependence and provide explicit formulas for the magnitude of the tail dependence. Some copulas allow to model both positive and negative dependence in their standard versions by assigning appropriate copula-parameters. Amongst these copulas are e.g. the Gaussian, Student t and Frank copula. Figure 4 displays the bivariate densities of these 3 copulas for a Spearman s row of.4 (top row) and for a Spearman s rho of -.4 (bottom row). Other (bivariate) copulas like e.g. the BB copula and its two special cases, the Clayton and Gumbel copula in their standard version allow to model positive dependence only. 8 Copula rotation allows to transform copulas such that they may be used to model negative dependence also. Further, copula rotation allows to transform (bivariate) copulas depending on whether and/or where the empirical data at hand requires the copula to display lower, upper or zero tail dependence. Denoting a bivariate copula density as c(u, u ), the so-called survival copula s density is c (u, u ) = c( u, u ). 9 In the case of e.g. the Gumbel copula, the survival copula is used to model lower tail dependence and no upper tail dependence. In order to model discordance with e.g. a BB copula, 7 The flexible BB copula may also display either zero upper or zero lower tail dependence or symmetric tail dependence, depending on the parameterisation. In specific cases the Gaussian and Student t copula may display also positive and no tail dependence, respectively. See e.g. table 8 in section In fact, the Clayton copula may also be used in its standard version to model negative dependence if the copula parameter θ [, ). Such a parameterisation is not further considered in the present article. 9 While the bivariate copula C(u, u ) returns the probability that both uniformly distributed marginal distributions take on values less than or equal to u and u, the survival copula C (u, u ) returns the probability that both marginal distributions take on values greater than u and u, respectively. 6

17 Gaussian copula, ρ S =.4 Student t copula, ν = 3, ρ S =.4 Frank copula, ρ S = density 4 density 5 density.5 u u.5.5 u u.5.5 u u.5 Gaussian copula, ρ S =.4 Student t copula, ν = 3, ρ S =.4 Frank copula, ρ S = density 4 density 5 density.5 u u.5.5 u u.5.5 u u.5 Figure 4: Densities of bivariate Gaussian, Student t and Frank copulas. These copulas allow to model both concordance and discordance. The copulas in the top row display a Spearman s rho of approximately.4 (copula parameters: Gaussian: ρ =.4, Student t: ν = 3 and ρ =.43, Frank: θ =.6). The copulas in the bottom row display a Spearman s rho of approximately -.4 (copula parameters: Gaussian: ρ =.4, Student t: ν = 3 and ρ =.43, Frank: θ =.6). The densities are computed on the interval [.,.99]. 7

18 3. u u Clayton C ++ Clayton C u u u u Gumbel C ++ Gumbel C + Gumbel C u u u Clayton C + Clayton C u u u Gumbel C u u u u Figure 5: Contour plots of the densities of a Clayton (top row) and a Gumbel (bottom row) copula C, and of their rotated versions C +, C + and C. Spearman s rho for both copulas in their standard version is approximately.4 (copula parameters: Clayton: θ =.76, Gumbel: θ =.38). The densities are computed on the interval [.,.99]. the rotated versions C + or C + with densities c + (u, u ) = c( u, u ) and c + (u, u ) = c(u, u ) are used. Figure 5 displays contour plots of Clayton and Gumbel copulas densities and of the rotated versions densities. To present the consequences of assumptions on the copula in the context of economic capital estimation, let us return to our simplified example, where we assumed that a bank holds only two portfolios, a market portfolio with annual returns r M. +.5 t 5 and a credit portfolio with annual returns r C ln(.5 B(3,.)). The correlation in terms of Spearman s rho is ρ S =.4. Let us further assume that equal weights are assigned to these portfolios, such that the bank s total return in year t is r t =.5r M,t +.5r C,t, where r M,t and r C,t are the realised returns of the market and the credit portfolio in year t, respectively. Table 3 shows several quantiles of total return distributions. The quantiles correspond to average one-year default probabilities of Moody s ratings from 9 to 4 reported in Hamilton et al. [36], p.35. A bank that aspires a rating of e.g. Ba has to hold enough economic capital such that the total losses exceed the economic capital with a probability of no more than.3%. The quantiles were computed under The quantiles are obtained by a Monte Carlo simulation with,, simulations using antithetic sampling. 8

19 quantile.43 (B).3 (Ba).3 (Baa).6 (Aa) meta-gaussian meta-student t meta-clayton correlation= Table 3: Quantiles of the total return distribution, corresponding to average Moody s rating -year default probabilities, if meta-gaussian, meta-student t and meta-clayton distributions with a Spearman s rho of.4 are assumed. the assumption of a Gaussian copula, a Student t copula with ν = 3 degrees of freedom and a Clayton copula. Additionally, in order to demonstrate the diversification effect, the quantiles were computed under the assumption of perfect positive correlation. As can be seen in table 3, the effect of positive tail dependence (Student t copula and Clayton copula) increases, as the quantile decreases. In our simplistic example the economic capital to be held under the assumption of a Student t (Clayton) copula exceeds the economic capital under the assumption of a Gaussian copula by.94% (4.9%), if a B -rating is aspired, and by 7.% (8.34%), if a Aa -rating is aspired. Before presenting some selected copulas in detail in subsection 3.3, we shall shortly address how the marginal distributions may be modelled in the following subsection. 3. Modelling the marginal distributions In the context of top-down risk aggregation models, the following parametric distribution functions are widely used to model the marginal distributions: Market portfolio returns Generalized hyperbolic (GH) distribution, or one if its special cases such as the Normal inverse Gaussian (NIG) distribution or Student t and Gaussian distributions. Credit portfolio returns Beta distribution See e.g. Aas and Haff [] 9

20 Weibull distribution Insurance portfolio returns and operational risk Pareto distribution log-normal distribution Gamma distribution Alternatively nonparametric approaches like e.g. the use of kernel-smoothed empirical distribution functions are widely employed. 3.3 Presentation of selected copulas This section presents some selected copulas from the family of elliptical and Archimedean copulas. These are Elliptical copulas Gaussian copula Student t copula Archimedean copulas BB copula and its two special cases, the Clayton copula and the Gumbel copula. Frank copula Bivariate copula functions C(u, u ), i.e. the probability that both uniformly distributed marginal distributions jointly take on value less than or equal to u and u, respectively, are presented in table 4. Bivariate copula densities that are needed in the context of parameter estimation and for the depiction of data are presented in table 5. The Gaussian and Student t copulas in their standard versions allow for a higher flexibility than the Archimedean copulas by enabling a modelling of pairwise correlations that form the elements of the copula parameter matrix P ( capital Greek letter rho ). For a primer on kernel smoothing, see e.g. Scott and Sain [64]. More detailed information can be found in Wand and Jones [68] and Silverman [65]

21 copula parameters θ copula function C(u, u ; θ) Gaussian ρ [, ] Φ ρ (Φ (u ), Φ (u )) = Φ (u ) Φ (u ) π or equivalently (see Roncalli [58]) exp ( ) ρst s t ρ ( ρ ) dsdt ( ) u Φ Φ (u ) ρφ (s) ds ρ where Φ ρ is the bivariate standard normal distribution function with parameter ρ, and Φ is the functional inverse of the univariate standard normal c.d.f. Φ. Student t ν (, ) t ν,ρ (t ν (u ), t ν (u )) = ρ [, ] t ν (u ) t ν (u ) ( + s π ρ +t ρst ν( ρ ) ) ν+ dsdt or equivalently (see Roncalli [58]) ( ) u t ν+ t ν+ ν+t ν (u ) ρt ν (s) ds ν (s) ρ where t ν,ρ is the bivariate Student t distribution and t ν is the functional inverse of the univariate Student t c.d.f with ν degrees of freedom t ν (.). BB δ [, ), θ (, ) ( + [ (u θ ) δ ( + u θ ) δ ] ) θ δ Clayton θ (, ) ( u θ + u θ ) θ ( Gumbel θ [, ) exp [ ) ( ln u ) θ + ( ln u ) θ] θ ( ) Frank θ (, )\ ln + (e θu )(e θu ) θ e θ Table 4: Selected bivariate copula functions.

22 copula probability density function c(u, u ; θ) = C(u,u ;θ) u u ( ) Gaussian exp ρyy y y ρ ( ρ ) + y +y, where y = Φ (u ), y = Φ (u ) and Φ (.) is the functional inverse of the standard normal c.d.f. Φ(.). Student t BB Clayton Gumbel ( f ν,ρ t ν (u ), t ν (u ) ), f ν(t ν (u )) f ν(t ν (u )) where f ν,ρ is the p.d.f of the standard Student t distribution function with ν degrees of freedom and correlation matrix ρ, f ν is the p.d.f of the univariate standard Student t distribution and t ν is the functional inverse of the univariate Student t c.d.f with ν degrees of freedom t ν (.). ( u θ ) δ ( u θ ) δ u θ u θ ] [( + θ)a θ b δ + (δθ θ)a θ b δ, ( ) [ (u where a = + b θ δ and b = ) δ ( + u θ ) ] δ. ( + θ)u θ u θ ( u θ + u θ ) θ exp(a) ( ln u)θ ( ln u ) θ u u [ b θ + (θ )b θ ], where a = b θ, and b = [ ( ln u ) θ + ( ln u ) θ]. Frank θηe θ(u +u ) [η ( e θu )( e θu )], where η = e θ. Table 5: Probability density functions of selected bivariate copulas.

23 copula copula function C(u; θ) Gaussian parameters θ: P Φ (u )... Φ (u n) (π) exp ( n P x P x ) dx where Φ (.) is the functional inverse of the univariate standard normal c.d.f. Φ(.). Student t t ν (u )... t ν (u n) Γ( ν+n ) Γ( ν ) (πν) n P ( ) + x P ν+n x dx, ν parameters θ: where t ν is the functional inverse of the univariate Student t c.d.f. ν, P with ν degrees of freedom t ν (.) and Γ(.) is the Gamma function. Table 6: n-dimensional Gaussian and Student t copula functions. ρ, ρ,3 ρ,d ρ, ρ,3 ρ,d P = ρ,3 ρ, ρ,d ρ,d (3) Besides the copula parameter P, the Student t copula has an additional scalar parameter ν, the degrees of freedom. These can, however, not be used to explicitly model pairwise dependencies. Rather, the copula parameter ν, being a scalar, affects all pairwise dependencies in the same manner. Table 6 and 7 provide the n-dimensional copula functions of Gaussian and Student t copulas and their densities, respectively. If the dependence of more than two dependent variables is to be modelled, the Archimedean copulas flexibility seems very restricted as either only one (Clayton, Gumbel, Frank copulas) or only two (BB copula) scalar parameters are used to parameterise the joint multidimensional dependence structure. 3. This lack of flexibility can however be overcome by using hierarchical Archimedean copulas that are e.g. presented in Savu and Trede [6]. A hierarchical copula joins two (or more) bivariate (or higher dimensional) Archimedean copulas by another Archimedean copula. The structure of this approach is depicted in figure 6. If in the context of risk aggregation we want to combine the returns of, say, four financial portfolios we first 3 Formulas for n-dimensional Archimedean copulas can be found e.g. in Cherubini et al. [3], pp.47ff. 3

24 copula probability density function c(u; θ) = n C(u;θ) u... u n Gaussian φ P (Φ (u ),..., Φ (u n )) n i= φ(φ (u i )), where φ P (.) is the p.d.f. of the multivariate standard normal distribution with correlation matrix P, φ(.) is the p.d.f. of the univariate standard normal distribution, and Φ (.) is the functional inverse of the univariate standard normal c.d.f. Φ(.). Student t f ν,p ( t ν (u ),..., tν (u n ) ) n i= f ν(t ν (u i)), where f ν,p is the p.d.f of the standard Student t distribution function with ν degrees of freedom and correlation matrix P, f ν is the p.d.f of the univariate standard Student t distribution and t ν is the functional inverse of the univariate Student t c.d.f with ν degrees of freedom t ν (.). Table 7: n-dimensional Gaussian and Student t copula density functions. calibrate two copulas that combine the returns of portfolio and, and portfolio 3 and 4, respectively. These two copulas are then combined by a third copula. Parameter estimation is done in the same manner as for the other copulas (see subsection 3.4). For the simulation of hierarchical copulas, the conditional inversion method has to be used (see Savu and Trede [6], p.f). The concept of positive upper and lower tail dependence of bivariate copulas has already been introduced in section 3.. Loosely speaking, lower tail dependence λ L describes the conditional probability that one of the two random variables takes values below a very small value, given that also the other random variable takes very small values. Upper tail dependence λ U can be described analogously. Formally, λ L = lim C(α, α) α + α u α) = lim and α + α (4) λ U = lim α + C(α, α) α > α u > α) = lim, α α (5) provided the limit exists with λ L, λ U [, ]. For symmetric copulas λ L = λ U. Formulas for the magnitude of lower and upper tail dependence for the selected copulas are presented in table 8. The concept of copula rotation has also been introduced already (see figure 5 on p.8). Copulas may be rotated, depending on whether and/or where the empirical data at hand requires the copula to display positive, 4

25 copula 3: C 3 (C (U, U ), C (U 3, U 4 ) ) [,] copula : C (U, U ) [,] copula : C (U 3, U 4 ) [,] marginal distribution : U = F (X ) [,] marginal distribution : U = F (X ) [,] marginal distribution 3: U 3 = F 3 (X 3 ) [,] marginal distribution 4: U 4 = F 4 (X 4 ) [,] Figure 6: Structure of a four-dimensional hierarchical Archimedean copula. copula lower tail dependence λ L upper tail dependence λ U Gaussian λ L = λ U = (iff ρ < ; λ L = λ U = iff ρ = ) Student t ( ) λ L = λ U = t ν+ ν + ρ +ρ where t ν+ is the univariate Student t c.d.f with ν + degrees of freedom BB λ L = δθ λ U = δ Clayton λ L = θ λ U = Gumbel λ L = λ U = θ Frank λ L = λ U = Table 8: Lower and upper tail dependence, λ L and λ U, of selected bivariate copulas. 5

26 negative or zero tail dependence. Let us define the vector ū = (ū, ū ), where ū i = u i. 4 Then the following observations are true ū and ū have copula C (u, u ) = u +u +C( u, u ) with density c (u, u ) = c( u, u ). C is referred to as survival copula. ū and u have copula C + (u, u ) = u C( u, u ) with density c + (u, u ) = c( u, u ). u and ū have copula C + (u, u ) = u C(u, u ) with density c + (u, u ) = c(u, u ). If C(u, u ) is symmetric, then c(u, u ) = c (u, u ) and c + (u, u ) = c + (u, u ). If we want to use a copula C which is suited to describe upper tail dependence to model lower tail dependence, the corresponding C copula has to be employed. If we want to use a copula C which is only suited to describe positive dependence to model negative dependence, C + or C + have to be employed. In the sub-sections below, the densities of selected bivariate copulas are more closely regarded Gaussian copula The Gaussian copula is the most widely used copula. It is the copula that is implied by a multivariate Gaussian distribution (normal distribution). A multivariate Gaussian distribution is a set of normally distributed marginal distributions that are combined by a Gaussian copula. If other than normal marginal distributions are combined by a Gaussian copula, the resulting joint distribution is referred to as meta-gaussian distribution. Figure on p.3 contains an example of a meta-gaussian distribution. Figure 7 displays surface plots of Gaussian copula densities with a Spearman s rho of.4 (top left) and.8 (bottom left). The bivariate copula density goes to infinity at u = (, ), and u = (, ) for ρ > and at u = (, ) and u = (, ) for ρ <. On the right hand side, corresponding (meta-) Gaussian distribution densities with standard normal marginal distributions are displayed. 4 Note that Ū = U is uniformly distributed on the unit interval if U is uniformly distributed on the unit interval. 6

27 We shall use the Gaussian copula as benchmark to which we compare the other copulas Student t copula The Student t copula is the copula that is implied by a multivariate Student t distribution (Student t marginal distributions combined by a Student t copula). Like the Gaussian copula, the Student t copula has the parameter ρ in the bivariate case (table 4) or P in higher dimensions (table 6). Additionally it has the (scalar) parameter ν, the degrees of freedom. The higher ν, the higher the positive tail dependence (see table 8). Figure 8 displays surface plots of Student t copula densities with a Spearman s rho of.4 (top left) and.8 (bottom left). The bivariate copula density goes to infinity at u = (, ), u = (, ), u = (, ), and u = (, ). On the right hand side, corresponding contour plots of meta-student t distribution densities with standard normal marginal distributions are displayed. Additionally, contours of a Gaussian distribution with identical marginal distributions and Spearman s rho are plotted in light grey for comparison. It can be seen that the Student t copula assigns a higher density to events near all four corners than the Gaussian copula does. Differences between the Student t copula and meta-distribution s densities to those of the Gaussian copula with identical Spearman s rho are summarised in the contour plots at the bottom of figure 8, where grey shaded areas indicate that the densities of the Student t copula or meta-distribution exceed that of the Gaussian copula. As the degrees of freedom of a Student t copula increase, the copula approaches a Gaussian copula. The Gaussian copula can be regarded as a limiting case of the Student t copula, where ν. More in-depth information on Student t copulas can e.g. be found in Demarta and McNeil [8] BB copula The two-parametric BB copula allows for a high flexibility in modelling positively correlated bivariate dependence structures (copula parameters δ and θ). Figure 9 displays contour plots of BB copula densities with an identical Spearman s rho of.4. The plot on the very left and on the very right hand side are limiting cases of the BB copula. The very left BB copula has the copula parameter δ =. This special case of the BB copula is called a Clayton copula, and the BB copula parameter θ corresponds to the Clayton copula parameter θ. The very right BB copula has the parameter θ tends 7

28 Gaussian copula, ρ S =.4 (meta ) Gaussian distribution, ρ S =.4 6. density 4 density u..4 u m 3 N(,) 3 3 m N(,) Gaussian copula, ρ S =.8 (meta ) Gaussian distribution, ρ S =.8 6. density 4 density u..4 u m 3 N(,) 3 3 m N(,) Figure 7: Densities of bivariate Gaussian copulas (left hand side) with a Spearman s rho of.4 (copula-parameter ρ Gaussian =.4) and.8 (copulaparameter ρ Gaussian =.8) evaluated on the interval [.,.999] and corresponding meta-distributions with standard normal marginal distributions (right hand side). 8

29 Student t copula, ν=3, ρ S =.4 meta Student t dist., ν=3, ρ S = density u u m N(,) m N(,) Student t copula, ν=3, ρ S =.8 meta Student t dist., ρ S =.8 density u u m N(,) m N(,) ρ S =.4 ρ S =.8 3 ρ S =.4 3 ρ S =.8 u.5 u.5 m N(,) m N(,).5 u.5 u m N(,) m N(,) Figure 8: Densities of bivariate Student t copulas with ν = 3 degrees of freedom (left hand side) with a Spearman s rho of.4 (copula-parameter ρ Student t =.43) and.8 (copula-parameter ρ Student t =.83), evaluated on the interval [.,.9999]. Corresponding contour plots (contours at the.,.5,.,. and.3 level) of meta-student t distributions with standard normal marginal distributions are plotted on the right hand side. Additionally, contours of a Gaussian meta-distribution with identical Spearman s rho and marginal distributions are plotted in light grey. The graphs in the bottom row indicate in which areas the densities of the Student t copula or meta distribution exceed that of a Gaussian copula or meta distribution with identical Spearman s rho (grey-shaded areas). 9

30 3 3 u BB (δ =, θ =.76) u BB (δ =., θ =.48) u BB (δ =., θ =.) u BB (δ =.38, θ = + ) u u u u Figure 9: Densities of bivariate BB copulas with different parameterisation. All copulas have a a Spearman s rho of approximately.4. The densities are evaluated on the interval [.,.99]. towards zero. This special case of the BB copula is called a Gumbel copula, and the BB copula parameter δ corresponds to the Gumbel copula parameter θ. The next two sub-sections take a closer look on these two special cases of the BB copula and compare their densities to that of a Gaussian copula Clayton copula The Clayton copula displays lower tail dependence and zero upper tail dependence. These properties can be verified regarding the Clayton copula density plots displayed in figure on the left hand side. The top copula has a Spearman s rho of.4, the bottom copula has a Spearman s rho of.8. The triangle-shaped corresponding contour plots of meta-clayton distributions with standard normal marginal distributions are displayed on the right hand side. The contour plots on the bottom of figure show that the Clayton copula assigns a higher probability to joint extremely negative realisations as compared to the Gaussian copula, while it assigns a lower probability to joint extremely positive realisations Gumbel copula Figure displays the densities of a survival Gumbel copula with a Spearman s rho of.4 (top) and.8 (bottom). Like the Clayton copula, the survival Gumbel copula displays lower tails dependence and no upper tail dependence. The tear shaped corresponding contour plots of meta-survival Gumbel distributions with standard normal marginal distributions are displayed on the right hand side. The contour plots on the bottom of figure show that the survival Gumbel copula assigns a higher probability to joint extremely negative realisations as compared to the Gaussian copula, while it assigns a 3