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.     

Capital Allocation for Sarmanov’s Class of Distributions

This paper is a follow-up of the study realized by Vernic (2014) on the aggregation of dependent random variables joined by Sarmanov’s multivariate distribution, with accent on the particular case of exponentially distributed marginals. More precisely, in this paper we present capital allocation formulas for a portfolio of risks following the just mentioned Sarmanov’s distribution. The overall capital and its allocation to the risk sources are evaluated using the TVaR rule. The resulting formulas are illustrated in some particular cases.     

Factor risk quantification in annuity models

Calculation of risk contributions of sub-portfolios to total portfolio risk is essential for risk management in insurance companies. Thanks to risk capital allocation methods and linearity of the loss model, sub-portfolio (or position) contributions can be calculated efficiently. However, factor risk contribution theory in non-linear loss models has received little interest. Our concern is the determination of factor risk contributions to total portfolio risk where portfolio risk is a non-linear function of factor risks. We employ different approximations in order to convert the non-linear loss model into a linear one. We illustrate the theory on an annuity portfolio where the main factor risks are interest-rate risk and mortality risk.     

Modeling of claim exceedances over random thresholds for related insurance portfolios

Large claims in an actuarial risk process are of special importance for the actuarial decision making about several issues like pricing of risks, determination of retention treaties and capital requirements for solvency. This paper presents a model about claim occurrences in an insurance portfolio that exceed the largest claim of another portfolio providing the same sort of insurance coverages. Two cases are taken into consideration: independent and identically distributed claims and exchangeable dependent claims in each of the portfolios. Copulas are used to model the dependence situations. Several theorems and examples are presented for the distributional properties and expected values of the critical quantities under concern.     

Modeling of claim exceedances over random thresholds for related insurance portfolios

Large claims in an actuarial risk process are of special importance for the actuarial decision making about several issues like pricing of risks, determination of retention treaties and capital requirements for solvency. This paper presents a model about claim occurrences in an insurance portfolio that exceed the largest claim of another portfolio providing the same sort of insurance coverages. Two cases are taken into consideration: independent and identically distributed claims and exchangeable dependent claims in each of the portfolios. Copulas are used to model the dependence situations. Several theorems and examples are presented for the distributional properties and expected values of the critical quantities under concern.     

On a compound Poisson risk model with dependence and in the presence of a constant dividend barrier

In this paper, we consider a classical risk process with dependence and in the presence of a constant dividend barrier. The dependence structure between the claim amounts and the interclaim times is introduced through a Farlie–Gumbel–Morgenstern copula. We analyze the expectation of the discounted penalty function and the expectation of the present value of the distributed dividends. For each function, an integro‐differential equation with boundary conditions is derived, and the solution is provided. Finally, we find an explicit solution for each function when the claim amounts are exponentially distributed. We illustrate the impact of the dependence on these two quantities. Copyright © 2012 John Wiley & Sons, Ltd.     

基于Copula函数TVaR和EVaR的风险评估和资金配置

In this paper,the expressions of tail value of risk(TVaR)and exponential tail value of risk(EVaR)for the total risk portfolio are given,which are splitted into two cases: the bivariate case and the multivariate case according to the number of the insurances.Then the risk contributions of the insurances portfolio and the credit portfolio are also obtained. Further more,for clarifying the above results,a numerical example is given.     

Equal Risk Bounding is better than Risk Parity for portfolio selection

Risk Parity (RP), also called equally weighted risk contribution, is a recent approach to risk diversification for portfolio selection. RP is based on the principle that the fractions of the capital invested in each asset should be chosen so as to make the total risk contributions of all assets equal among them. We show here that the Risk Parity approach is theoretically dominated by an alternative similar approach that does not actually require equally weighted risk contribution of all assets but only an equal upper bound on all such risks. This alternative approach, called Equal Risk Bounding (ERB), requires the solution of a nonconvex quadratically constrained optimization problem. The ERB approach, while starting from different requirements, turns out to be strictly linked to the RP approach. Indeed, when short selling is allowed, we prove that an ERB portfolio is actually an RP portfolio with minimum variance. When short selling is not allowed, there is a unique RP portfolio and it contains all assets in the market. In this case, the ERB approach might lead to the RP portfolio or it might lead to portfolios with smaller variance that do not contain all assets, and where the risk contributions of each asset included in the portfolio is strictly smaller than in the RP portfolio. We define a new riskiness index for assets that allows to identify those assets that are more likely to be excluded from the ERB portfolio. With these tools we then provide an exact method for small size nonconvex ERB models and a very efficient and accurate heuristic for larger problems of this type. In the case of a common constant pairwise correlation among all assets, a closed form solution to the ERB model is obtained and used to perform a parametric analysis when varying the level of correlation. The practical advantages of the ERB approach over the RP strategy are illustrated with some numerical examples. Computational experience on real-world and on simulated data confirms accuracy and efficiency of our heuristic approach to the ERB model also in comparison with some state-of-the-art local and global optimization codes.     

Extremes for coherent risk measures

Various concepts appeared in the existing literature to evaluate the risk exposure of a financial or insurance firm/subsidiary/line of business due to the occurrence of some extreme scenarios. Many of those concepts, such as Marginal Expected Shortfall or Tail Conditional Expectation, are simply some conditional expectations that evaluate the risk in adverse scenarios and are useful for signaling to a decision-maker the poor performance of its risk portfolio or to identify which sub-portfolio is likely to exhibit a massive downside risk. We investigate the latter risk under the assumption that it is measured via a coherent risk measure, which obviously generalizes the idea of only taking the expectation of the downside risk. Multiple examples are given and our numerical illustrations show how the asymptotic approximations can be used in the capital allocation exercise. We have concluded that the expectation of the downside risk does not fairly take into account the individual risk contribution when allocating the VaR-based regulatory capital, and thus, more conservative risk measurements are recommended. Finally, we have found that more conservative risk measurements do not improve the fairness of the cost of capital allocation when the uncertainty with parameter estimation is present, even at a very high level.     

Ruin-based risk measures in discrete-time risk models

For an insurance company, effective risk management requires an appropriate measurement of the risk associated with an insurance portfolio. The objective of the present paper is to study properties of ruin-based risk measures defined within discrete-time risk models under a different perspective at the frontier of the theory of risk measures and ruin theory. Ruin theory is a convenient framework to assess the riskiness of an insurance business. We present and examine desirable properties of ruin-based risk measures. Applications within the classical discrete-time risk model and extensions allowing temporal dependence are investigated. The impact of the temporal dependence on ruin-based risk measures within those different risk models is also studied. We discuss capital allocation based on Euler’s principle for homogeneous and subadditive ruin-based risk measures.     

资本配置视角下财产保险公司承保决策分析

探讨了产险公司在资本和收益双重约束条件下的承保决策问题．首先,从保险理论和实践的角度选择了TVaR资本配置方法,然后构造综合资本、收益双重因素的承保决策模型并进行了实证分析,结论显示从资本的角度进行承保决策是可行的．     

An algorithm for sequential tail value at risk for path-independent payoffs in a binomial tree

We present an algorithm that determines Sequential Tail Value at Risk (STVaR) for path-independent payoffs in a binomial tree. STVaR is a dynamic version of Tail-Value-at-Risk (TVaR) characterized by the property that risk levels at any moment must be in the range of risk levels later on. The algorithm consists of a finite sequence of backward recursions that is guaranteed to arrive at the solution of the corresponding dynamic optimization problem. The algorithm makes concrete how STVaR differs from TVaR over the remaining horizon, and from recursive TVaR, which amounts to Dynamic Programming. Algorithmic aspects are compared with the cutting-plane method. Time consistency and comonotonicity properties are illustrated by applying the algorithm on elementary examples.     

时变Copula下我国寿险公司投资市场风险经济资本测度

随着保险资金投资渠道的放宽,保险公司对于自身资金运用方面的管理显得日益重要,基于此,选取了国债和政府机构债券、企业债、证券投资基金以及股票这四种资产作为研究对象,将收益率低于同期银行存款利率的情形视为损失,结合样本数据进行了经济资本的测度分析.通过对比以往学者的研究,选定了用GARCH-偏正态分布进行收益率的拟合,并运用时变Copula函数进行风险相关性的测量,计算出了不同置信度下,寿险公司投资市场风险的经济资本.结果显示,时变Copula比常数Copula在风险相关性度量方面表现更好.     

Regulatory capital decisions in the Context of consumer loan portfolios

A topic of interest in recent literature is regulatory capital requirements for consumer loan portfolios. Banks are required to hold regulatory capital for unexpected losses, while expected losses are to be covered by either provisions or future income. In this paper, we show the set of efficient operating points in the market share and profit space for a portfolio manager operating under Basel II capital requirement and under capital constraints are a union of single-cutoff-score and double-cutoff-score operating points. For a portfolio manager to increase market-share beyond the maximum allowable under a single-cutoff score policy (eg, with binding capital constraints) requires granting loans to higher than optimal risk applicants. We show this result in greater portfolio risk but without an increase in regulatory capital requirement amount. The increase in forecasted losses is assumed to be absorbed by provisions or future margin income. Given portfolio managers take on higher risk under the same regulatory capital amount, our findings call for greater focus on provision amounts and future margin income under the supervisory review pillar of Basel II. This research raises the issue of whether the design of the regulatory formula for consumer loan portfolios is flawed.     

Inference and risk measurement with the pari-mutuel model

We explore generalizations of the pari-mutuel model (PMM), a formalization of an intuitive way of assessing an upper probability from a precise one. We discuss a naive extension of the PMM considered in insurance, compare the PMM with a related model, the Total Variation Model, and generalize the natural extension of the PMM introduced by P. Walley and other pertained formulae. The results are subsequently given a risk measurement interpretation: in particular it is shown that a known risk measure, Tail Value at Risk (TVaR), is derived from the PMM, and a coherent risk measure more general than TVaR from its imprecise version. We analyze further the conditions for coherence of a related risk measure, Conditional Tail Expectation. Conditioning with the PMM is investigated too, computing its natural extension, characterising its dilation and studying the weaker concept of imprecision increase.     

Solvency II reporting: How to interpret funds’ aggregate solvency capital requirement figures

Depending on the current risk exposure of an insurance company, the impact of buying an additional unit of a fund on an insurer’s overall Solvency II capital charges, i.e., the Solvency Capital Requirement (SCR), will differ. We call this impact the fund’s SCR contribution and show in which boundaries it lies if only the fund’s aggregate sub-SCR figures are known but not the risk exposures of the insurance company buying the fund. The upper bound of this range, the worst-case SCR contribution, can be used as a conservative measure to assess funds’ Solvency II risk contributions or to assign them to different Solvency II risk categories. We believe that providing funds’ worst-case SCR contributions can be useful information to insurance companies when screening from a broad investment universe.     

Das Vorhersagerisiko der Sterblichkeitsentwicklung – Kann es durch eine geeignete Portfoliozusammensetzung minimiert werden?

Annuities as well as term insurance create risks for the insurance companies due to changes in mortality/longevity – especially in low-interest phases. For the past decades an increase in life expectancy was observed. In this article, we examine whether an insurance company can minimise the longevity risk by means of an appropriate composition of its portfolio. We use stochastic interest rates and mortality trends. For annuities and term insurance different mortality trends are used. Based on an example we show the impact of the portfolio composition on the longevity risk. The results prove that a deliberate portfolio composition can significantly reduce the longevity risk for the insurance company.     

Insurance valuation: A computable multi-period cost-of-capital approach

We present an approach to market-consistent multi-period valuation of insurance liability cash flows based on a two-stage valuation procedure. First, a portfolio of traded financial instrument aimed at replicating the liability cash flow is fixed. Then the residual cash flow is managed by repeated one-period replication using only cash funds. The latter part takes capital requirements and costs into account, as well as limited liability and risk averseness of capital providers. The cost-of-capital margin is the value of the residual cash flow. We set up a general framework for the cost-of-capital margin and relate it to dynamic risk measurement. Moreover, we present explicit formulas and properties of the cost-of-capital margin under further assumptions on the model for the liability cash flow and on the conditional risk measures and utility functions. Finally, we highlight computational aspects of the cost-of-capital margin, and related quantities, in terms of an example from life insurance.     

Portfolio adjusting optimization with added assets and transaction costs based on credibility measures

In response to changeful financial markets and investor’s capital, we discuss a portfolio adjusting problem with additional risk assets and a riskless asset based on credibility theory. We propose two credibilistic mean–variance portfolio adjusting models with general fuzzy returns, which take lending, borrowing, transaction cost, additional risk assets and capital into consideration in portfolio adjusting process. We present crisp forms of the models when the returns of risk assets are some deterministic fuzzy variables such as trapezoidal, triangular and interval types. We also employ a quadratic programming solution algorithm for obtaining optimal adjusting strategy. The comparisons of numeral results from different models illustrate the efficiency of the proposed models and the algorithm.