Analytical Evaluation of Economic Risk Capital for Portfolios of Gamma Risks

Based on the notions of value-at-risk and expected shortfall, we consider two functionals, abbreviated VaR and RaC, which represent the economic risk capital of a risky business over some time period required to cover losses with a high probability. These functionals are consistent with the risk preferences of profit-seeking (and risk averse) decision makers and preserve the stochastic dominance order (and the stop-loss order). Quantitatively, RaC is equal to VaR plus an additional stop-loss dependent term, which takes into account the average amount at loss. Furthermore, RaC is additive for comonotonic risks, which is an important extremal situation encountered in the modeling of dependencies in multivariate risk portfolios. Numerical illustrations for portfolios of gamma distributed risks follow. As a result of independent interest, new analytical expressions for the exact probability density of sums of independent gamma random variables are included, which are similar but different to previous expressions by Provost (1989) and Sim (1992

Analytical Pricing of the Unit-Linked Endowment with Guarantees and Periodic Premiums

We consider the unit-linked endowment with guarantee and periodic premiums, where at each premium payment date the insurance company invests a certain fraction of the premium into a risky reference portfolio. In the dual random environment of stochastic interest rates with deterministic volatilities and mortality risk, and for a fixed guarantee, simple analytical lower and upper bounds for the fair periodic premium are explicitly derived. We also consider contracts with guaranteed minimum benefits that vary over time and we obtain tight lower and upper bounds for both fair periodic premiums and guaranteed minimum benefits that increase over time. The numerical illustrations of our results reveal that the analytical bounds are very tight. Moreover, the simple, fast and very reliable analytical numerical calculations with controlled accuracy avoid time consuming Monte Carlo calculations and are almost always preferred by practitioners. Some analytical closed-form solutions for one- and two-year maturity dates are also state

Optimal Reinsurance Revisited - Point of View of Cedent and Reinsurer

It is known that the partial stop-loss contract is an optimal reinsurance form under the VaR risk measure. Assuming that market premiums are set according to the expected value principle with varying loading factors, the optimal reinsurance parameters of this contract are obtained under three alternative single and joint party reinsurance criteria: (i) strong minimum of the total retained loss VaR measure; (ii) weak minimum of the total retained loss VaR measure and maximum of the reinsurer's expected profit; (iii) weak minimum of the total retained loss VaR measure and minimum of the total variance risk measure. New conditions for financing in the mean simultaneously the cedent's and the reinsurer's required VaR economic capital are revealed for situations of pure risk transfer (classical reinsurance) or risk and profit transfer (design of internal reinsurance or reinsurance captive owned by the captive of a corporate firm

Analytical Bounds for two Value-at-Risk Functionals

Based on the notions of value-at-risk and conditional value-at-risk, we consider two functionals, abbreviated VaR and CVaR, which represent the economic risk capital required to operate a risky business over some time period when only a small probability of loss is tolerated. These functionals are consistent with the risk preferences of profit-seeking (and risk averse) decision makers and preserve the stochastic dominance order (and the stop-loss order). This result is used to bound the VaR and CVaR functionals by determining their maximal values over the set of all loss and profit functions with fixed first few moments. The evaluation of CVaR for the aggregate loss of portfolios is also discussed. The results of VaR and CVaR calculations are illustrated and compared at some typical situations of general interes

Pseudo Compound Poisson Distributions in Risk Theory

Using Laplace transforms and the notion of a pseudo compound Poisson distribution, some risk theoretical results are revisited. A well-known theorem by Feller (1968) and Van Harn (1978) on infinitely divisible distributions is generalized. The result may be used for the efficient evaluation of convolutions for some distributions. In the particular arithmetic case, alternate formulae to those recently proposed by De Pril (1985) are derived and shown more adequate in some cases. The individual model of risk theory is shown to be pseudo compound Poisson. It is thus computable using numerical tools from the theory of integral equations in the continuous case, a formula of Panjer type or the Fast Fourier transform in the arithmetic case. In particular our results contain some of De Pril's (1986/89) recursive formulae for the individual life model with one and multiple causes of decrement. As practical illustration of the continuous case we construct a new two-parametric family of claim size density functions whose corresponding compound Poisson distributions are analytical finite sum expressions. Analytical expressions for the finite and infinite time ruin probabilities are also derive

A Gaussian Exponential Approximation to Some Compound Poisson Distributions

A three parameter Gaussian exponential approximation to some compound Poisson distributions is considered. It is constructed by specifying the reciprocal of the mean excess function as a linear affine function below some threshold and a positive constant above this threshold. As an analytical approximation to compound Poisson distributions, it is only feasible either for a limited range of the Poisson parameter or for higher coefficients of variation. A semiparametric determination of the unknown threshold parameter is proposed. The analysis of a real-life example from pension fund mathematics displays an improved quality of fit of the new model when compared with other simple good alternative approximations based on the zero gamma, translated gamma and zero translated gamm

Credible Loss Ratio Claims Reserves: the Benktander, Neuhaus and Mack Methods Revisited

The Benktander (1976) and Neuhaus (1992) credibility claims reserving methods are reconsidered in the framework of a credible loss ratio reserving method. As a main contribution we provide a simple and practical optimal credibility weight for combining the chain-ladder or individual loss ratio reserve (grossed up latest claims experience of an origin period) with the Bornhuetter-Ferguson or collective loss ratio reserve (experience based burning cost estimate of the total ultimate claims of an origin period). The obtained simple optimal credibility weights minimize simultaneously the mean squared error and the variance of the claims reserve. We note also that the standard Chain-Ladder, Cape Cod and Bornhuetter-Ferguson methods can be reinterpreted in the credible context and extended to optimal credible standard methods. The new approach is inspired from Mack (2000). Two advantages over the Mack method are worthwhile to be mentioned. First, a pragmatic estimation of the required parameters leads to a straightforward calculation of the optimal credibility weights and mean squared errors of the credible reserves. An advantage of the collective loss ratio claims reserve over the Bornhuetter-Ferguson reserve in Mack (2000) is that different actuaries come always to the same results provided they use the same actuarial premium