AHMADI, SEYED SAEED; LI, JOHNNY SIU-HANG. *Coherent mortality forecasting with generalized linear models: a modified time-transformation approach.* 194-221. In this paper, we propose an alternative approach for forecasting mortality for multiple populations jointly. Our contribution is developed upon the generalized linear models introduced by Renshaw *et al.*, (1996) [A.E. Renshaw, S. Haberman, P. Hatzoupoulos (1996), The modelling of recent mortality trends in United Kingdom male assured lives, British Actuarial Journal 2 (1996) 2: 449-477] and Sithole *et al.*, (2000) [T.Z. Sithole, S. Haberman, R.J. Verrall (2000), An investigation into parametric models for mortality projections, with applications to immediate annuitants’ and life office pensioners’ data, Insurance Mathematics and Economics (2000) 27: 285-312], in which mortality forecasts are generated within the model structure, without the need of additional stochastic processes. To ensure that the resulting forecasts are coherent, a modified time-transformation is developed to stipulate the expected mortality differential between two populations to remain constant when the long-run equilibrium is attained. The model is then further extended to incorporate a structural change, an important property that is observed in the historical mortality data of many national populations. The proposed modeling methods are illustrated with data from two different pairs of populations: (1) Swedish and Danish males; (2) English and Welsh males and U.K. male insured lives.

AVRAM, FLORIN; PISTORIUS, MARTIJN. *On matrix exponential approximations of ruin probabilities for the classic and Brownian perturbed Cramér-Lundberg processes.* 57-64. Padé rational approximations are a very convenient approximation tool, due to the easiness of obtaining them, as solutions of linear systems. Not surprisingly, many matrix exponential approximations used in applied probability are particular cases of the first and second order “admissible Padé approximations” of a Laplace transform, where admissible stands for nonnegative in the case of a density, and for nonincreasing in the case of a ccdf (survival function). Our first contribution below is the observation that for Cramér-Lundberg processes and Brownian perturbed Cramér-Lundberg processes there are three distinct rational approximations of the Pollaczek-Khinchine transform, corresponding to approximating (a) the claims transform, (b) the stationary excess transform, and (c) the aggregate loss transform. A second contribution is providing three new always admissible second order approximations for the ruin probabilities of the Cramér-Lundberg process with Brownian perturbation, one of which reduces in the absence of perturbation to De Vylder’s approximation. Our third contribution is a method for comparing the resulting approximations, based on the concept of largest weak-admissibility interval of the compounding/traffic intensity parameter p.

BENKHELIFA, LAZHAR. *Kernel-type estimator of the reinsurance premium for heavy-tailed loss distributions.* 65-70. In this paper, we generalize the classical estimator of the reinsurance premium for heavy-tailed loss distributions with a kernel-type estimator. Since this estimator exhibits a bias, we propose its bias-reduced version by using a least-squares method. The asymptotic normality of the proposed estimators is established under suitable assumptions. A small simulation study is carried out to prove the performance of our approach.

BOHNERT, ALEXANDER; BORN, PATRICIA; GATZERT, NADINE. *Dynamic hybrid products in life insurance: assessing the policyholders’ viewpoint.* 87-99. Dynamic hybrid life insurance products are intended to meet new consumer needs regarding stability in terms of guarantees as well as sufficient upside potential. In contrast to traditional participating or classical unit-linked life insurance products, the guarantee offered to the policyholders is achieved by a periodical rebalancing process between three funds: the policy reserves (i.e. the premium reserve stock, thus causing interaction effects with traditional participating life insurance contracts), a guarantee fund, and an equity fund. In this paper, we consider an insurer offering both, dynamic hybrid and traditional participating life insurance contracts and focus on the policyholders’ perspective. The results show that higher guarantees do not necessarily imply a higher willingness-to-pay, but that in case of dynamic hybrid contracts, a minimum guarantee level should be offered in order to ensure that the willingness-to-pay exceeds the minimum premium the insurer has to charge when selling the contract. In addition, strong interaction effects can be found between the two products, which particularly impact the willingness-to-pay of the dynamic hybrids.

BRANDTNER, MARIO; KÜRSTEN, WOLFGANG. *Solvency II, regulatory capital, and optimal reinsurance: How good are Conditional Value-at-Risk and spectral risk measures?* 156-167. We study the problem of optimal reinsurance as a means of risk management in the regulatory framework of Solvency II under Conditional Value-at-Risk and, as its natural extension, spectral risk measures. First, we show that stop-loss reinsurance is optimal under both Conditional Value-at-Risk and spectral risk measures. Spectral risk measures thus constitute a more general class of suitable regulatory risk measures than specific Conditional Value-at-Risk. At the same time, the established type of stop-loss reinsurance can be maintained as the optimal risk management strategy that minimizes regulatory capital. Second, we derive the optimal deductibles for stop-loss reinsurance. We show that under Conditional Value-at-Risk, the optimal deductible tends towards restrictive and counter-intuitive corner solutions or “plunging”, which is a serious objection against its use in regulatory risk management. By means of the broader class of spectral risk measures, we are able to overcome this shortcoming as optimal deductibles are now interior solutions. Especially, the recently discussed power spectral risk measures and the Wang risk measure are shown to avoid any plunging. They yield a one-to-one correspondence between the risk parameter and the optimal deductible and, thus, provide economically plausible risk management strategies.

CHIU, MEI CHOI; WONG, HOI YING. *Mean-variance asset-liability management with asset correlation risk and insurance liabilities.* 300-310. Consider an insurer who invests in the financial market where correlations among risky asset returns are randomly changing over time. The insurer who faces the risk of paying stochastic insurance claims needs to manage her asset and liability by taking into account of the correlation risk. This paper investigates the impact of correlation risk to the optimal asset-liability management (ALM) of an insurer. We employ the Wishart process to model the stochastic covariance matrix of risky asset returns. The insurer aims to minimize the variance of the terminal wealth given an expected terminal wealth subject to the risk of paying out random liabilities of compound Poisson process. This ALM problem then becomes a linear-quadratic stochastic optimal control problem with stochastic volatilities, stochastic correlations and jumps. The recognition of an affine form in the solution process enables us to derive the explicit closed-form solution to the optimal ALM portfolio policy, obtain the efficient frontier, and identify the condition that the solution is well behaved.

CHOI, MICHAEL C H; CHEUNG, ERIC C K. *On the expected discounted dividends in the Cramér-Lundberg risk model with more frequent ruin monitoring than dividend decisions.* 121-132. In this paper, we further extend the insurance risk model in Albrecher *et al.* (2011b), who proposed to only intervene in the compound Poisson risk process at the discrete time points {Lk}k=08 where the event of ruin is checked and dividend decisions are made. In practice, an insurance company typically balances its books (and monitors its solvency) more frequently than deciding on dividend payments. This motivates us to propose a generalization in which ruin is monitored at {Lk}k=08 whereas dividend decisions are only made at {Ljk}k=08 for some positive integer j. Assuming that the intervals between the time points {Lk}k=08 are Erlang(n) distributed, the Erlangization technique (e.g. Asmussen *et al.*, 2002) allows us to model the more realistic situation with the books balanced e.g. monthly and dividend decisions made e.g. quarterly or semi-annually. Under a dividend barrier strategy with the above randomized interventions, we derive the expected discounted dividends paid until ruin. Numerical examples about dividend maximization with respect to the barrier bb and/or the value of j are given.

DENUIT, MICHEL; LIU, LIQUN; MEYER, JACK. *A separation theorem for the weak s-convex orders.* 279-284. The present paper extends to higher degrees the well-known separation theorem decomposing a shift in the increasing convex order into a combination of a shift in the usual stochastic order followed by another shift in the convex order. An application in decision making under risk is provided to illustrate the interest of the result.

DJEHICHE, BOUALEM; LÖFDAHL, BJÖRN. *Risk aggregation and stochastic claims reserving in disability insurance.* 100-108. We consider a large, homogeneous portfolio of life or disability annuity policies. The policies are assumed to be independent conditional on an external stochastic process representing the economic-demographic environment. Using a conditional law of large numbers, we establish the connection between claims reserving and risk aggregation for large portfolios. Further, we derive a partial differential equation for moments of present values. Moreover, we show how statistical multi-factor intensity models can be approximated by one-factor models, which allows for solving the PDEs very efficiently. Finally, we give a numerical example where moments of present values of disability annuities are computed using finite-difference methods and Monte Carlo simulations.

ELING, MARTIN. *Fitting asset returns to skewed distributions: are the skew-normal and skew-student good models?* 45-56. Vernic (2006), Bolancé *et al.* (2008), and Eling (2012) identify the skew-normal and skew-student as promising models for describing actuarial loss data. In this paper, we change the focus from the liability to the asset side and ask whether these distributions are also useful for analyzing the investment returns of insurance companies. To answer this question, we fit various parametric distributions to capital market data which has been used to describe the investment set of insurance companies. Our results show that the skew-student is an especially promising distribution for modeling asset returns such as those of stocks, bonds, money market instruments, and hedge funds. Combining the results of Vernic (2006), Bolancé *et al.* (2008), Eling (2012), and this paper, it appears that the skew-student is a promising actuarial tool since it describes both sides of the insurer’s balance sheet reasonably well. References: R. Vernic (2006), Multivariate skew-normal distributions with applications in insurance, Insurance Mathematics and Economics (2006) 38: 413-426; C. Bolancé, M. Guillen, E. Pelican, R. Vernic (2008), Skewed bivariate models and nonparametric estimation for the CTE risk measure, Insurance Mathematics and Economics (2008) 43(3): 386-393; M. Eling (2012), Fitting insurance claims to skewed distributions: are the skew-normal and skew-student good models? Insurance Mathematics and Economics (2012) 51: 239-248.

FENG, RUNHUAN; SHIMIZU, YASUTAKA. *Potential measures for spectrally negative Markov additive processes with applications in ruin theory.* 11-26. The Markov additive process (MAP) has become an increasingly popular modeling tool in the applied probability literature. In many applications, quantities of interest are represented as functionals of MAPs and potential measures, also known as resolvent measures, have played a key role in the representations of explicit solutions to these functionals. In this paper, closed-form solutions to potential measures for spectrally negative MAPs are found using a novel approach based on algebraic operations of matrix operators. This approach also provides a connection between results from fluctuation theoretic techniques and those from classical differential equation techniques. In the end, the paper presents a number of applications to ruin-related quantities as well as verification of known results concerning exit problems.

GBARI, SAMUEL; DENUIT, MICHEL. *Efficient approximations for numbers of survivors in the Lee-Carter model.* 71-77. In portfolios of life annuity contracts, the payments made by an annuity provider (an insurance company or a pension fund) are driven by the random number of survivors. This paper aims to provide accurate approximations for the present value of the payments made by the annuity provider. These approximations account not only for systematic longevity risk but also for the diversifiable fluctuations around the unknown life table. They provide the practitioner with a useful tool avoiding the problem of simulations within simulations in, for instance, Solvency 2 calculations, valid whatever the size of the portfolio.

GIJBELS, IRÈNE; HERRMANN, KLAUS. *On the distribution of sums of random variables with copula-induced dependence.* 27-44. We investigate distributional properties of the sum of d possibly unbounded random variables. The joint distribution of the random vector is formulated by means of an absolutely continuous copula, allowing for a variety of different dependence structures between the summands. The obtained expression for the distribution of the sum features a separation property into marginal and dependence structure contributions typical for copula approaches. Along the same lines we obtain the formulation of a conditional expectation closely related to the expected shortfall common in actuarial and financial literature. We further exploit the separation to introduce new numerical algorithms to compute the distribution and quantile function, as well as this conditional expectation. A comparison with the most common competitors shows that the discussed Path Integration algorithm is the most suitable method for computing these quantities. In our example, we apply the theory to compute Value-at-Risk forecasts for a trivariate portfolio of index returns.

HEILPERN, STANISLAW. *Ruin measures for a compound Poisson risk model with dependence based on the Spearman copula and the exponential claim sizes.* 251-257. This paper is devoted to an extension to the classical compound risk model. We relax the independence assumption of claim amounts and interclaim times. The dependent structure between these random variables is described by the Spearman copula. We study the Laplace transform of the discounted penalty function and we give the explicit expression of it for the exponential claim size.

HUANG, XIAOXIA; ZHAO, TIANYI. *Mean-chance model for portfolio selection based on uncertain measure.* 243-250. This paper discusses a portfolio selection problem in which security returns are given by experts’ evaluations instead of historical data. A factor method for evaluating security returns based on experts’ judgment is proposed and a mean-chance model for optimal portfolio selection is developed taking transaction costs and investors’ preference on diversification and investment limitations on certain securities into account. The factor method of evaluation can make good use of experts’ knowledge on the effects of economic environment and the companies’ unique characteristics on security returns and incorporate the contemporary relationship of security returns in the portfolio. The use of chance of portfolio return failing to reach the threshold can help investors easily tell their tolerance toward risk and thus facilitate a decision making. To solve the proposed nonlinear programming problem, a genetic algorithm is provided. To illustrate the application of the proposed method, a numerical example is also presented.

LEE, WING YAN; WILLMOT, GORDON E. *On the moments of the time to ruin in dependent Sparre Andersen models with emphasis on Coxian interclaim times.* 1-10. The structural properties of the moments of the time to ruin are studied in dependent Sparre Andersen models. The moments of the time to ruin may be viewed as generalized versions of the Gerber-Shiu function. It is shown that structural properties of the Gerber-Shiu function hold also for the moments of the time to ruin. In particular, the moments continue to satisfy defective renewal equations. These properties are discussed in detail in the model of Willmot and Woo (2012) [G.E. Willmot, J.-K. Woo, On the analysis of a general class of dependent risk processes, Insurance Mathematics and Economics (2012) 51: 134-141], which has Coxian interclaim times and arbitrary time-dependent claim sizes. Structural quantities needed to determine the moments of the time to ruin are specified under this model. Numerical examples illustrating the methodology are presented.

MALINOVSKII, VSEVOLOD K; KOSOVA, KSENIA O. *Simulation analysis of ruin capital in Sparre Andersen’s model of risk.* 184-193. Ruin capital is a function of premium rate set to render the probability of ruin within finite time equal to a given value. The analytical studies of this function in the classical Lundberg model of risk with exponential claim sizes done in Malinovskii (2014) [V.K. Malinovskii (2014), Improved asymptotic upper bounds on ruin capital in Lundberg model of risk, Insurance Mathematics and Economics (2014) 55: 301-309] have shown that the ruin capital’s shape is surprisingly simple. This work presents the results of related simulation studies. They are focused on the question whether this shape remains similar in Sparre Andersen’s model of risk.

PANTELOUS, ATHANASIOS A; YANG, LIN. *Robust LMI stability, stabilization and [H8] control for premium pricing models with uncertainties into a stochastic discrete-time framework.* 133-143. The premium pricing process and the reserve stability under uncertainty are very challenging issues in the insurance industry. In practice, a premium which is sufficient enough to cover the expected claims and to keep stable the derived reserves is always required. This paper proposes a premium pricing model for General (Non-Life) Insurance products, which implements a negative feedback mechanism for the known reserves with time-varying, bounded delays. The model is developed into a stochastic, discrete-time framework and norm-bounded parameter uncertainties have been also incorporated. Thus, the stability, the stabilization and the robust [H8] control for the reserve process are investigated using Linear Matrix Inequality (LMI) criteria. For the robust [H8] control, attention will be focused on the design of a state feedback controller such that the resulting closed-loop system is robustly stochastically stable with disturbance attenuation level y>0. Numerical examples and figures illustrate the main findings of the paper.

PENG, XINGCHUN; WEI, LINXIAO; HU, YIJUN. *Optimal investment, consumption and proportional reinsurance for an insurer with option type payoff.* 78-86. This paper is devoted to the study of optimization of investment, consumption and proportional reinsurance for an insurer with option type payoff at the terminal time under the criterion of exponential utility maximization. The surplus process of the insurer and the financial risky asset process are assumed to be diffusion processes driven by Brownian motions which are non-Markovian in general. Very general constraints are imposed on the investment and the proportional reinsurance processes. Based on the martingale optimization principle, we use BSDE and BMO martingale techniques to derive the optimal strategy and the optimal value function. Some interesting particular cases are studied in which the explicit expressions for the optimal strategy are given by using the Malliavin calculus.

PENG, XINGCHUN; CHEN, FENG; HU, YIJUN. *Optimal investment, consumption and proportional reinsurance under model uncertainty.* 222-234. This paper considers the optimal investment, consumption and proportional reinsurance strategies for an insurer under model uncertainty. The surplus process of the insurer before investment and consumption is assumed to be a general jump-diffusion process. The financial market consists of one risk-free asset and one risky asset whose price process is also a general jump-diffusion process. We transform the problem equivalently into a two-person zero-sum forward-backward stochastic differential game driven by two-dimensional Lévy noises. The maximum principles for a general form of this game are established to solve our problem. Some special interesting cases are studied by using Malliavin calculus so as to give explicit expressions of the optimal strategies.

PETERS, GARETH W; DONG, ALICE X D; KOHN, ROBERT. *A copula based Bayesian approach for paid-incurred claims models for non-life insurance reserving.* 258-278. Our article considers the class of recently developed stochastic models that combine claims payments and incurred losses information into a coherent reserving methodology. In particular, we develop a family of hierarchical Bayesian paid-incurred claims models, combining the claims reserving models of Hertig (1985) [J. Hertig, A statistical approach to the IBNR-reserves in marine insurance, Astin Bulletin (1985) 15 (2): 171-183] and Gogol (1993) [D. Gogol, Using expected loss ratios in reserving, Insurance Mathematics and Economics (1993) 12(3): 297-299]. In the process we extend the independent log-normal model of Merz and Wüthrich (2010) by incorporating different dependence structures using a Data-Augmented mixture Copula paid-incurred claims model. In this way the paper makes two main contributions: firstly we develop an extended class of model structures for the paid-incurred chain ladder models where we develop precisely the Bayesian formulation of such models; secondly we explain how to develop advanced Markov chain Monte Carlo sampling algorithms to make inference under these copula dependence PIC models accurately and efficiently, making such models accessible to practitioners to explore their suitability in practice. In this regard the focus of the paper should be considered in two parts, firstly development of Bayesian PIC models for general dependence structures with specialised properties relating to conjugacy and consistency of tail dependence across the development years and accident years and between Payment and incurred loss data are developed. The second main contribution is the development of techniques that allow general audiences to efficiently work with such Bayesian models to make inference. The focus of the paper is not so much to illustrate that the PIC paper is a good class of models for a particular data set, the suitability of such PIC type models is discussed in Merz and Wüthrich (2010) [M. Merz, M.V. Wüthrich, Paid-incurred chain claims reserving method, Insurance Mathematics and Economics (2010) 46(3): 568-579] and Happ and Wüthrich (2013) [M. Merz, M.V. Wüthrich, Estimation of tail factors in the paid-incurred chain reserving method, Variance (2013) 7(1): 61-73]. Instead we develop generalised model classes for the PIC family of Bayesian models and in addition provide advanced Monte Carlo methods for inference that practitioners may utilise with confidence in their efficiency and validity.

SPREEUW, JAAP. *Archimedean copulas derived from utility functions.* 235-242. The inverse of the (additive) generator of an Archimedean copula is a strictly decreasing and convex function, while utility functions (applying to risk averse decision makers) are nondecreasing and concave. This provides a basis for deriving an inverse generator of an Archimedean copula from a utility function. If we derive the inverse of the generator from the utility function, there is a link between the magnitude of measures of risk attitude (like the very common Arrow-Pratt coefficient of absolute risk aversion) and the strength of dependence featured by the corresponding Archimedean copula. Some new copula families are derived, and their properties are discussed. A numerical example about modeling dependence of coupled lives is included.

SUN, YING; WEI, LI. *The finite-time ruin probability with heavy-tailed and dependent insurance and financial risks.* 178-183. Consider a discrete-time insurance risk model in which the insurer makes both risk-free and risky investments. Assume that the one-period insurance and financial risks form a sequence of independent and identically distributed copies of a random pair (X,Y) with dependent components. When the product XY is heavy tailed, under a mild restriction on the dependence structure of (X,Y), we establish for the finite-time ruin probability an asymptotic formula, which coincides with the long-standing one in the literature. Various important special cases are presented, showing that our work generalizes and unifies some of recent ones.

TAN, CHONG IT; LI, JACKIE; LI, JOHNNY SIU-HANG; BALASOORIYA, UDITHA. *Parametric mortality indexes: from index construction to hedging strategies.* 285-299. In this paper, we investigate the construction of mortality indexes using the time-varying parameters in common stochastic mortality models. We first study how existing models can be adapted to satisfy the new-data-invariant property, a property that is required to ensure the resulting mortality indexes are tractable by market participants. Among the collection of adapted models, we find that the adapted Model M7 (the Cairns-Blake-Dowd model with cohort and quadratic age effects) is the most suitable model for constructing mortality indexes. One basis of this conclusion is that the adapted model M7 gives the best fitting and forecasting performance when applied to data over the age range of 40-90 for various populations. Another basis is that the three time-varying parameters in it are highly interpretable and rich in information content. Based on the three indexes created from this model, one can write a standardized mortality derivative called K-forward, which can be used to hedge longevity risk exposures. Another contribution of this paper is a method called key K-duration that permits one to calibrate a longevity hedge formed by K-forward contracts. Our numerical illustrations indicate that a K-forward hedge has a potential to outperform a q-forward hedge in terms of the number of hedging instruments required.

TANG, QIHE; YANG, FAN. *Extreme value analysis of the Haezendonck-Goovaerts risk measure with a general Young function.* 311-320. For a risk variable XX and a normalized Young function f(·), the Haezendonck-Goovaerts risk measure for X at level q (0,1) is defined as Hq[X]=infx R(x+h), where h solves the equation E[f((X-x)+/h)]=1-q if Pr(X>x)>0 or is 0 otherwise. In a recent work, we implemented an asymptotic analysis for Hq[X] with a power Young function for the Fréchet, Weibull and Gumbel cases separately. A key point of the implementation was that hh can be explicitly solved for fixed x and q, which gave rise to the possibility to express Hq[X] in terms of x and q. For a general Young function, however, this does not work anymore and the problem becomes a lot harder. In the present paper, we extend the asymptotic analysis for Hq[X] to the case with a general Young function and we establish a unified approach for the three extreme value cases. In doing so, we overcome several technical difficulties mainly due to the intricate relationship between the working variables x, h and q.

YANG, JIANPING; ZHUANG, WEIWEI; HU, TAIZHONG. *Lp-metric under the location-independent risk ordering of random variables.* 321-324. The Lp-metric h,p(X) between the survival function F of a random variable X and its distortion F is a characteristic of the variability of X. In this paper, it is shown that if a random variable X is larger than another random variable Y in the location-independent risk order or in the excess wealth order, then h,p(X)= h,p(Y) whenever p (0,1] and the distortion function h is convex or concave. An alternative and simple proof of the corresponding known result in the literature for the dispersive order is given. Some applications are also presented.

ZHANG, HUIMING; LIU, YUNXIAO; LI, BO. *Notes on discrete compound Poisson model with applications to risk theory.* 325-336. Probability generating function (p.g.f.) is a powerful tool to study discrete compound Poisson (DCP) distribution. By applying inverse Fourier transform of p.g.f., it is convenient to numerically calculate probability density and do parameter estimation. As an application to finance and insurance, we firstly show that in the generalized CreditRisk+ model, the default loss of each debtor and the total default of all debtors are both approximately equal to a DCP distribution, and we give Le Cam’s error bound between the total default and a DCP distribution. Next, we consider geometric Brownian motion with DCP jumps and derive its rth moment. We establish the surplus process of the difference of two DCP distributions, and numerically compute the tail probability. Furthermore, we define the discrete pseudo compound Poisson (DPCP) distribution and give the characterizations and examples of DPCP distribution, including the strictly decreasing discrete distribution and the zero-inflated discrete distribution with P(X=0)>0.5.

ZHANG, ZHIMIN; YANG, HAILIANG. *Nonparametric estimation for the ruin probability in a Lévy risk model under low-frequency observation.* 168-177. In this paper, we propose a nonparametric estimator for the ruin probability in a spectrally negative Lévy risk model based on low-frequency observation. The estimator is constructed via the Fourier transform of the ruin probability. The convergence rates of the estimator are studied for large sample size. Some simulation results are also given to show the performance of the proposed method when the sample size is finite.

ZHENG, YANTING; CUI, WEI. *Optimal reinsurance with premium constraint under distortion risk measures.* 109-120. Recently distortion risk measure has been an interesting tool for the insurer to reflect its attitude toward risk when forming the optimal reinsurance strategy. Under the distortion risk measure, this paper discusses the reinsurance design with unbinding premium constraint and the ceded loss function in a general feasible region which requiring the retained loss function to be increasing and left-continuous. Explicit solution of the optimal reinsurance strategy is obtained by introducing a premium-adjustment function. Our result has the form of layer reinsurance with the mixture of normal reinsurance strategies in each layer. Finally, to illustrate the applicability of our results, we derive the optimal reinsurance solutions with premium constraint under two special distortion risk measures – VaR and TVaR.

ZHU, YUNZHOU; CHI, YICHUN; WENG, CHENGGUO. *Multivariate reinsurance designs for minimizing an insurer’s capital requirement.* 144-155. This paper investigates optimal reinsurance strategies for an insurer with multiple lines of business under the criterion of minimizing its total capital requirement calculated based on the multivariate lower-orthant Value-at-Risk. The reinsurance is purchased by the insurer for each line of business separately. The premium principles used to compute the reinsurance premiums are allowed to differ from one line of business to another, but they all satisfy three mild conditions: distribution invariance, risk loading and preserving the convex order, which are satisfied by many popular premium principles. Our results show that an optimal strategy for the insurer is to buy a two-layer reinsurance policy for each line of business, and it reduces to be a one-layer reinsurance contract for premium principles satisfying some additional mild conditions, which are met by the expected value principle, standard deviation principle and Wang’s principle among many others. In the end of this paper, some numerical examples are presented to illustrate the effects of marginal distributions, risk dependence structure and reinsurance premium principles on the optimal layer reinsurance.

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