TVaR-based capital allocation with copulas

Because of regulation projects from control organisations such as the European solvency II reform and recent economic events, insurance companies need to consolidate their capital reserve with coherent amounts allocated to the whole company and to each line of business. The present study considers an insurance portfolio consisting of several lines of risk which are linked by a copula and aims to evaluate not only the capital allocation for the overall portfolio but also the contribution of each risk over their aggregation. We use the tail value at risk (TVaR) as risk measure. The handy form of the FGM copula permits an exact expression for the TVaR of the sum of the risks and for the TVaR-based allocations when claim amounts are exponentially distributed and distributed as a mixture of exponentials. We first examine the bivariate model and then the multivariate case. We also show how to approximate the TVaR of the aggregate risk and the contribution of each risk when using any copula.     

TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts

In this paper, we consider a portfolio of n dependent risks X₁,…,X_n and we study the stochastic behavior of the aggregate claim amount S=X₁+?+X_n. Our objective is to determine the amount of economic capital needed for the whole portfolio and to compute the amount of capital to be allocated to each risk X₁,…,X_n. To do so, we use a top-down approach. For (X₁,…,X_n), we consider risk models based on multivariate compound distributions defined with a multivariate counting distribution. We use the TVaR to evaluate the total capital requirement of the portfolio based on the distribution of S, and we use the TVaR-based capital allocation method to quantify the contribution of each risk. To simplify the presentation, the claim amounts are assumed to be continuously distributed. For multivariate compound distributions with continuous claim amounts, we provide general formulas for the cumulative distribution function of S, for the TVaR of S and the contribution to each risk. We obtain closed-form expressions for those quantities for multivariate compound distributions with gamma and mixed Erlang claim amounts. Finally, we treat in detail the multivariate compound Poisson distribution case. Numerical examples are provided in order to examine the impact of the dependence relation on the TVaR of S, the contribution to each risk of the portfolio, and the benefit of the aggregation of several risks.     

Dynamic capital allocation with irreversible investments

Capital allocation models generally assume that the risk portfolio is constructed at a single point in time, when the underwriter has full information about available underwriting opportunities. However, in practice, opportunities are not all known at the beginning but instead arrive over time. Moreover, a commitment to an opportunity is not easy to change as time passes. Thus, to optimize a portfolio, the underwriter must make decisions on opportunities as they arrive while making use of assumptions about what will arrive in the future. This paper studies capital allocation rules in this setting, finding important differences from the static setting. The pricing of an opportunity is based on an expected future marginal cost of risk associated with that opportunity—one that will be fully understood only after the risk portfolio is finalized. The risk charge for today’s opportunity is thus a probability-weighted average of the product of the marginal value of capital in future states of the world and the amount of capital consumed by the opportunity in those future states. Our numerical examples illustrate how the marginal cost of risk for an opportunity is shaped by when it arrives in time, as well as what has arrived before it.     

Excess based allocation of risk capital

In this paper we propose a new rule to allocate risk capital to portfolios or divisions within a firm. Specifically, we determine the capital allocation that minimizes the excesses of sets of portfolios in a lexicographical sense. The excess of a set of portfolios is defined as the expected loss of that set of portfolios in excess of the amount of risk capital allocated to them. The underlying idea is that large excesses are undesirable, and therefore the goal is to determine the allocation for which the largest excess is as small as possible. We show that this allocation rule yields a unique allocation, and that it satisfies some desirable properties. We also show that the allocation can be determined by solving a series of linear programming problems.     

GlueVaR risk measures in capital allocation applications

GlueVaR risk measures defined by Belles-Sampera et al. (2014) generalize the traditional quantile-based approach to risk measurement, while a subfamily of these risk measures has been shown to satisfy the tail-subadditivity property. In this paper we show how GlueVaR risk measures can be implemented to solve problems of proportional capital allocation. In addition, the classical capital allocation framework suggested by Dhaene et al. (2012) is generalized to allow the application of the Value-at-Risk (VaR) measure in combination with a stand-alone proportional allocation criterion (i.e., to accommodate the Haircut allocation principle). Two new proportional capital allocation principles based on GlueVaR risk measures are defined. An example based on insurance claims data is presented, in which allocation solutions with tail-subadditive risk measures are discussed.     

Bivariate option pricing with copulas

The adoption of copula functions is suggested in order to price bivariate contingent claims. Copulas enable the marginal distributions extracted from vertical spreads in the options markets to be imbedded in a multivariate pricing kernel. It is proved that such a kernel is a copula function, and that its super-replication strategy is represented by the Fréchet bounds. Applications provided include prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, no-arbitrage pricing bounds, as well as values consistent with the independence of the underlying assets are provided. As a final reference value, a copula function calibrated on historical data is used.     

Some results on the CTE-based capital allocation rule

Tasche [Tasche, D., 1999. Risk contributions and performance measurement. Working paper, Technische Universität München] introduces a capital allocation principle where the capital allocated to each risk unit can be expressed in terms of its contribution to the conditional tail expectation (CTE) of the aggregate risk. Panjer [Panjer, H.H., 2002. Measurement of risk, solvency requirements and allocation of capital within financial conglomerates. Institute of Insurance and Pension Research, University of Waterloo, Research Report 01-15] derives a closed-form expression for this allocation rule in the multivariate normal case. Landsman and Valdez [Landsman, Z., Valdez, E., 2002. Tail conditional expectations for elliptical distributions. North American Actuarial J. 7 (4)] generalize Panjer’s result to the class of multivariate elliptical distributions.In this paper we provide an alternative and simpler proof for the CTE-based allocation formula in the elliptical case. Furthermore, we derive accurate and easy computable closed-form approximations for this allocation formula for sums that involve normal and lognormal risks.     

Optimal capital allocation in a hierarchical corporate structure

We consider capital allocation in a hierarchical corporate structure where stakeholders at two organizational levels (e.g., board members vs line managers) may have conflicting objectives, preferences, and beliefs about risk. Capital allocation is considered as the solution to an optimization problem whereby a quadratic deviation measure between individual losses (at both levels) and allocated capital amounts is minimized. Thus, this paper generalizes the framework of Dhaene et al. (2012), by allowing potentially diverging risk preferences in a hierarchical structure. An explicit unique solution to this optimization problem is given. In several examples, it is shown how the optimal capital allocation achieves a compromise between conflicting views of risk within the organization.     

Robust portfolio optimization with copulas

Conditional Value at Risk (CVaR) is widely used in portfolio optimization as a measure of risk. CVaR is clearly dependent on the underlying probability distribution of the portfolio. We show how copulas can be introduced to any problem that involves distributions and how they can provide solutions for the modeling of the portfolio. We use this to provide the copula formulation of the CVaR of a portfolio. Given the critical dependence of CVaR on the underlying distribution, we use a robust framework to extend our approach to Worst Case CVaR (WCVaR). WCVaR is achieved through the use of rival copulas. These rival copulas have the advantage of exploiting a variety of dependence structures, symmetric and not. We compare our model against two other models, Gaussian CVaR and Worst Case Markowitz. Our empirical analysis shows that WCVaR can asses the risk more adequately than the two competitive models during periods of crisis.     

Randomly weighted sums of subexponential random variables with application to capital allocation

We are interested in the tail behavior of the randomly weighted sum \( \sum _{i=1}^{n}\theta _{i}X_{i}\) , in which the primary random variables X ₁, …, X _n are real valued, independent and subexponentially distributed, while the random weights ?? ₁, …, ?? _n are nonnegative and arbitrarily dependent, but independent of X ₁, …, X _n . For various important cases, we prove that the tail probability of \(\sum _{i=1}^{n}\theta _{i}X_{i}\) is asymptotically equivalent to the sum of the tail probabilities of ?? ₁ X ₁, …, ?? _n X _n , which complies with the principle of a single big jump. An application to capital allocation is proposed.     

A class of multivariate copulas with bivariate Fréchet marginal copulas

In this paper, we present a class of multivariate copulas whose two-dimensional marginals belong to the family of bivariate Fréchet copulas. The coordinates of a random vector distributed as one of these copulas are conditionally independent. We prove that these multivariate copulas are uniquely determined by their two-dimensional marginal copulas. Some other properties for these multivariate copulas are discussed as well. Two applications of these copulas in actuarial science are given.     

A capital allocation based on a solvency exchange option

In this paper we propose a new capital allocation method based on an idea of [Sherris, M., 2006. Solvency, capital allocation and fair rate of return in insurance. J. Risk Insurance 73 (1), 71-96]. The proposed method explicitly accommodates the notion of limited liability of the shareholders. We show how the allocated capital can be decomposed, so that each stakeholder can have a clearer understanding of their contribution. We also challenge the no undercut principle, one of the widely accepted allocation axioms, and assert that this axiom is merely a property that certain allocation methods may or may not meet.     

Fitting bivariate loss distributions with copulas

Various processes in casualty insurance involve correlated pairs of variables. A prominent example is the loss and allocated loss adjustment expenses on a single claim. In this paper the bivariate copula is introduced and an approach to conducting goodness-of-fit tests is suggested. A large example illustrates the concepts.     

Multiobjective optimization of credit capital allocation in financial institutions

The evolution of international regulation leads to new capital requirements imposed on globally active companies. Financial services firms allocate capital to business lines in order to withstand the materializing credit losses and to measure the performance of various business lines. In this study, we introduce a methodology for optimal credit capital allocation based on operations research approach. In particular, we focus on the efficient allocation of capital to business lines characterized by credit risk losses and cost of capital. We compare different allocation methods and provide a rationale behind using the OR approach. Finally, we formulate a multiobjective optimization model to capital allocation problem and apply it to a real-world case of two financial conglomerates.     

Modeling defaults with nested Archimedean copulas

In 2001, Schönbucher and Schubert extended Li’s well-known Gaussian copula model for modeling dependent defaults to allow for tail dependence. Instead of the Gaussian copula, Schönbucher and Schubert suggested to use Archimedean copulas. These copulas are able to capture tail dependence and therefore allow a standard intensity-based default model to have a positive probability of joint defaults within a short time period. As can be observed in the current financial crisis, this is an indispensable feature of any realistic default model. Another feature, motivated by empirical observations but rarely taken into account in default models, is that modeled portfolio components affected by defaults show significantly different levels of dependence depending on whether they belong to the same industry sector or not. The present work presents an extension of the model suggested by Schönbucher and Schubert to account for this fact. For this, nested Archimedean copulas are applied. As an application, the pricing of collateralized debt obligations is treated. Since the resulting loss distribution is not analytical tractable, fast sampling algorithms for nested Archimedean copulas are developed. Such algorithms boil down to sampling certain distributions given by their Laplace-Stieltjes transforms. For a large range of nested Archimedean copulas, efficient sampling techniques can be derived. Moreover, a general transformation of an Archimedean generator allows to construct and sample the corresponding nested Archimedean copulas.     

On two families of bivariate distributions with exponential marginals: Aggregation and capital allocation

In this paper, we consider two main families of bivariate distributions with exponential marginals for a couple of random variables $(X_1,X_2)$. More specifically, we derive closed-form expressions for the distribution of the sum $S=X_1+X_2$, the TVaR of $S$ and the contributions of each risk under the TVaR-based allocation rule. The first family considered is a subset of the class of bivariate combinations of exponentials, more precisely, bivariate combinations of exponentials with exponential marginals. We show that several well-known bivariate exponential distributions are special cases of this family. The second family we investigate is a subset of the class of bivariate mixed Erlang distributions, namely bivariate mixed Erlang distributions with exponential marginals. For this second class of distributions, we propose a method based on the compound geometric representation of the exponential distribution to construct bivariate mixed Erlang distributions with exponential marginals. Notably, we show that this method not only leads to Moran–Downton’s bivariate exponential distribution, but also to a generalization of this bivariate distribution. Moreover, we also propose a method to construct bivariate mixed Erlang distributions with exponential marginals from any absolutely continuous bivariate distributions with exponential marginals. Inspired from Lee and Lin (2012), we show that the resulting bivariate distribution approximates the initial bivariate distribution and we highlight the advantages of such an approximation.     

Shuffles of copulas

We show that every copula that is a shuffle of Min is a special push-forward of the doubly stochastic measure induced by the copula M. This fact allows to generalize the notion of shuffle by replacing the measure induced by M with an arbitrary doubly stochastic measure, and, hence, the copula M by any copula C.     

Spatially homogeneous copulas

We consider spatially homogeneous copulas, i.e. copulas whose corresponding measure is invariant under a special transformations of \([0,1]^2\), and we study their main properties with a view to possible use in stochastic models. Specifically, we express any spatially homogeneous copula in terms of a probability measure on [0, 1) via the Markov kernel representation. Moreover, we prove some symmetry properties and demonstrate how spatially homogeneous copulas can be used in order to construct copulas with surprisingly singular properties. Finally, a generalization of spatially homogeneous copulas to the so-called (m, n)-spatially homogeneous copulas is studied and a characterization of this new family of copulas in terms of the Markov \(*\)-product is established.