In this paper, we study the optimal reinsurance contracts that minimize the convex combination of the Conditional Value-at-Risk (CVaR) of the insurer’s loss and the reinsurer’s loss over the class of ceded loss functions such that the retained loss function is increasing and the ceded loss function satisfies Vajda condition. Among a general class of reinsurance premium principles that satisfy the properties of risk loading and convex order preserving, the optimal solutions are obtained. Our results show that the optimal ceded loss functions are in the form of five interconnected segments for general reinsurance premium principles, and they can be further simplified to four interconnected segments if more properties are added to reinsurance premium principles. Finally, we derive optimal parameters for the expected value premium principle and give a numerical study to analyze the impact of the weighting factor on the optimal reinsurance.

E. Franckx [1] has established the distribution function of the largest individual claim of a portfolio. By assuming the number of claims to be Poisson distributed, H. Ammeter was able to develop the distribution function of the total loss excluding the largest individual claim [2] as well as the distribution function of the nth largest claim [3].

Of course, the nth largest claim is dependent on the largest claim, second largest claim and so on, down to the (nth — 1) largest claim. If we assume the number of claims to be Poisson distributed and the amount of the individual claim to be Pareto distributed, the correlation between the mth largest and the nth largest claim can be expressed by an analytical formula which is susceptible to numerical computation.

With this knowledge we shall be able to compute the variance of the sum of the n largest claims and moreover the correlation between the sum of the n largest claims and the total loss amount. Although an excess of loss reinsurance treaty and a treaty reinsuring the n largest claims are very different in their construction, this paper will show that from a practical point of view there exists a similarity between the two treaties. The correlation coefficient between the sum of the n largest claims and the sum of all claims exceeding a certain limit enables us to assess the degree of similarity.

The correlation coefficient and thus the degree of similarity will prove to be high even in case of the reinsurance of only a small number of largest claims.

Finally, the knowledge of the two first moments of the sum of the n largest claims allows us to compute the premium and the security or variance loading for the reinsurance of the n largest claims.

In the classical optimal dividends problem, dividend decisions are allowed to be made at any point in time — according to a continuous strategy. Depending on the surplus process that is considered and whether dividend payouts are bounded or not, optimal strategies are generally of a band, barrier or threshold type. In reality, while surpluses change continuously, dividends are generally paid on a periodic basis. Because of this, the actuarial literature has recently considered strategies where dividends are only allowed to be distributed at (random) discrete times — according to a periodic strategy.

In this paper, we focus on the Brownian risk model. In this context, the optimal continuous and periodic strategies have previously been shown (independently of one another) to be of barrier type. For the first time, we consider a model where both strategies are used. In such a hybrid strategy, decisions are allowed to be made either at any time (continuously), or periodically at a lower cost. This proves optimal in some cases. We also determine under which combination of parameters a pure continuous, pure periodic or hybrid (including both continuous and periodic dividend payments) barrier strategy is optimal. Interestingly, the hybrid strategy lies in-between periodic and continuous strategies, which provides some interesting insights. Results are illustrated.

Copula function has been widely used in insurance and finance for modeling inter-dependency between risks. Inspired by the Bernstein copula put forward by Sancetta and Satchell (2004, Econometric Theory, 20, 535–562), we introduce a new class of multivariate copulas, the composite Bernstein copula, generated from a composition of two copulas. This new class of copula functions is able to capture tail dependence, and it has a reproduction property for the three important dependency structures: comonotonicity, countermonotonicity and independence. We introduce an estimation procedure based on the empirical composite Bernstein copula which incorporates both prior information and data into the estimation. Simulation studies and an empirical study on financial data illustrate the advantages of the empirical composite Bernstein copula estimation method, especially in capturing tail dependence.

We investigate the application of natural hedging strategies for long-term care (LTC) insurers by diversifying both longevity and disability risks affecting LTC annuities. We propose two approaches to natural hedging: one built on a multivariate duration, the other on the Conditional Value-at-Risk minimization of the unexpected loss. Both the approaches are extended to the LTC insurance using a multiple state framework. In order to represent the future evolution of mortality and disability transition probabilities, we use the stochastic model of Cairns et al. (2009) with cohort effect under parameter uncertainty through a semi-parametric bootstrap procedure. We calculate the optimal level of a product mix and measure the effectiveness provided by the interaction of LTC stand alone, deferred annuity and whole-life insurance. We compare the results obtained by the two approaches and find that a natural hedging strategy for LTC insurers is attainable with a product mix of LTC and annuities, but including low proportion of LTC.

The Kolmogorov distance is used to transform arithmetic severities into equispaced arithmetic severities in order to reduce the number of calculations when using algorithms like Panjer's formulae for compound distributions. An upper bound is given for the Kolmogorov distance between the true compound distribution and the transformed one. Advantages of the Kolmogorov distance and disadvantages of the total variation distance are discussed. When the bounds are too big, a Berry-Esseen result can be used. Then almost every case can be handled by the techniques described in this paper. Numerical examples show the interest of the methods.

While extensive literature exists on the valuation and risk management of financial guarantees embedded in insurance contracts, both the corresponding longevity guarantees and interactions between financial and longevity guarantees are often ignored. The present paper provides a framework for a joint analysis of financial and longevity guarantees, and applies this framework to different annuity conversion options in deferred unit-linked annuities. In particular, we analyze and compare different versions of so-called guaranteed annuity options and guaranteed minimum income benefits with respect to the value of the option and the resulting risk for the insurer. Furthermore, we examine whether and to what extent an insurance company is able to reduce the risk by typical risk management strategies. The analysis is based on a combined stochastic model for both financial market and future survival probabilities. We show that different annuity conversion options have significantly different option values, and that different risk management strategies lead to a significantly different risk for the insurance company.

This paper considers the model suggested by Schnieper (1991), which separates the true IBNR claims from the IBNER. Stochastic models are defined, using both recursive and non-recursive procedures, within the framework of the models described in England and Verrall (2002). Approximations to the prediction error of the reserves are derived analytically.

Some extensions to the original Schnieper model are also disussed, together with other possible applications of this type of model.

Gerber (1974) has given a characterization of the exponential utility function by proving the fact that a premium calculation principle of zero utility is iterative iff the underlying utility function is linear or exponential. In the present note we prove the more general result that the premium calculation principle introduced by Bühlmann, Gagliardi, Gerber and Straub (1977) is iterative iff the underlying function v is linear or exponential or when the principle is a mean value principle.

In this paper we explore the bias in the estimation of the Value at Risk and Conditional Tail Expectation risk measures using Monte Carlo simulation. We assess the use of bootstrap techniques to correct the bias for a number of different examples. In the case of the Conditional Tail Expectation, we show that application of the exact bootstrap can improve estimates, and we develop a practical guideline for assessing when to use the exact bootstrap.

An interesting class of stochastic claims reserving methods is given by the models with conditionally independent loss increments (CILI), where the incremental losses are conditionally independent given a risk parameter Θi,j depending on both the accident year i and the development year j. The Bühlmann–Straub credibility reserving (BSCR) model is a particular case of a CILI model where the risk parameter is only depending on i. We consider CILI models with additive diagonal risk (ADR), where the risk parameter is given by the sum of two components, one depending on the accident year i and the other depending on the calendar year t = i + j. The model can be viewed as an extension of the BSCR model including random diagonal effects, which are often declared to be important in loss reserving but rarely are specifically modeled. We show that the ADR model is tractable in closed form, providing credibility formulae for the reserve and the mean square error of prediction (MSEP). We also derive unbiased estimators for the variance parameters which extend the classical Bühlmann–Straub estimators. The results are illustrated by a numerical example and the estimators are tested by simulation. We find that the inclusion of random diagonal effects can be significant for the reserve estimates and, especially, for the evaluation of the MSEP. The paper is written with the purpose of illustrating the role of stochastic diagonal effects. To isolate these effects, we assume that the development pattern is given. In particular, our MSEP values do not include the uncertainty due to the estimation of the development pattern.

Machine learning has recently entered the mortality literature in order to improve the forecasts of stochastic mortality models. This paper proposes to use two pure, tree-based machine learning models: random forests and gradient boosting, based on the differenced log-mortality rates to produce more accurate mortality forecasts. These forecasts are compared with forecasts from traditional, stochastic mortality models and with forecasts from random forests and gradient boosting variants of the stochastic models. The comparisons are based on the Model Confidence Set procedure. The results show that the pure, tree-based models significantly outperform all other models in the majority of cases considered. To address the lack of interpretability issue associated with machine learning models, we demonstrate how to extract information about the relationships uncovered by the tree-based models. For this purpose, we consider variable importance, partial dependence plots, and variable split conditions. Results from the in-sample fit suggest that tree-based models can be very useful tools for detecting patterns within and between variables that are not commonly identifiable with traditional methods.

Ohlin [7] showed optimality properties of Stop Loss reinsurance when the ceding insurer uses a continuous loss function to evaluate his risks. We generalize this property to the range of the distribution; we then show that Stop Loss reinsurance is no longer the best form when the company uses a percentile parameter. Finally we prove an optimality theorem concerning chance games which allows us to determine the retention as a function of the size of the portfolio.

While the traditional R 2 value is useful to evaluate the quality of a fit, it does not work when it comes to evaluating the predictive power of estimated financial models in finite samples. In this paper we introduce a validated value useful for prediction. Based on data from the Danish stock market, using this measure we find that the dividend-price ratio has predictive power. The best horizon for prediction seems to be four years. On a one year horizon, we find that while inflation and interest rate do not add to the predictive power of the dividend-price ratio then last years excess stock return does.

Let F(x) be the distribution function of the total amount ξ of claims of an insurance company. It is assumed that this company reduces its liability by means of one or several reinsurance arrangements. Let the remaining part of claims amount be equal to η, which is another random variable. Reinsurance arrangements are supposed to fulfil the following consistency condition: if ξ = x, then almost certainly o ≤ η ≤ x. Presumably all reinsurance arrangements occuring in practice can be supposed to fulfil this requirement.

The problem is to find an optimal reinsurance arrangement, i.e. an optimal random variable η, in the sense that, the net reinsurance premium being given, the variance of η reaches its minimum. In other words, a variable η is looked for, which gives

E{η} = P = const.; V{η} = minimum

In the sequel we use conditional expectations in the sense defined by DOOB ([I]).

Let η be an arbitrary random variable satisfying the consistency condition. If

R(x) = E{η|ξ =x},

then evidently ≤ R(x) ≤ x. Further we have

For the variance the inequality

holds true, since

Clearly the arrangement E {η|ξ} gives also to the reinsurer a smaller variance than the original arrangement.

In this paper, we propose an analytic analogue to the simulation procedure described in Taylor (1997). We apply the formulas to a Belgian data set and discuss the interaction between a priori and a posteriori ratemakings.

One of the basic problems in life is: Given information (from the past), make decisions (that will affect the future). One of the classical actuarial examples is the adaptive ratemaking (or credibility) procedures; here the premium of a given risk is sequentially adjusted, taking into account the claims experience available when the decisions are made.

In some cases, the rates are fixed and the premiums cannot be adjusted. Then the actuary faces the question: Should a given risk be underwritten in the first place, and if yes, what is the criterion (in terms of claims performance) for cancellation of the policy at a later time?

Recently, Cozzolino and Freifelder [6] developed a model in an attempt to answer these questions. They assumed a discrete time, finite horizon, Poisson model. While the results lend themselves to straightforward numerical evaluation, their analytical form is not too attractive. Here we shall present a continuous time, infinite horizon, diffusion model. At the expense of being somewhat less realistic, this model is very appealing from an analytical point of view.

Mathematically, the cancellation of policies amounts to an optimal stopping problem, see [8], [4], or chapter 13 in [7], and (more generally) should be viewed within the framework of discounted dynamic programming [1], [2].

La théorie du bonus n'en est qu'à ses débuts et, afin d'en étudier une première approximation, Monsieur le Professeur Fréchet a utilisé la théorie des chaînes de Markoff-Poincaré; la première synthèse de cette théorie a été donnée dans son livre „Recherches théoriques modernes sur le calcul des probabilités — 2ème livre — Gauthier Villars”.

Nous trouvons que l'idée de Monsieur Fréchet, précisée dans un cadre bien défini d'hypothèses, est particulièrement intéressante et nous voudrions, quant à nous, essayer de dégager pour les actuaires les conséquences qu'une telle hypothèse introduit. On pourra en déduire une marche à suivre pour vérifier dans quelle mesure cette hypothèse cadre avec la réalité.

L'hypothèse fondamentale de Monsieur Fréchet consiste à admettre:

1°) l'existence d'une probabilité pour qu'un véhicule ayant eu i sinistres au cours de l'année calendrier, subisse j sinistres au cours de l'exercice suivant;

2°) que cette probabilité est indépendante de la voiture et de l'année considérée.

Si nous considérons les possibilités réduites à r = 3 cas (de façon à simplifier l'exposé, ce qui ne limite en rien le raisonnement), les possibilités seront complètement définies par le tableau ou matrice de la chaîne de Markoff-Poincaré (voir page suivante).

L'existence d'une chaîne de Markoff-Poincaré entraîne un certain nombre de conséquences, développées et démontrées dans l'ouvrage cité de Monsieur Fréchet.

Modeling policyholders’ lapse behaviors is important to a life insurer, since lapses affect pricing, reserving, profitability, liquidity, risk management, and the solvency of the insurer. In this paper, we apply two machine learning methods to lapse modeling. Then, we evaluate the performance of these two methods along with two popular statistical methods by means of statistical accuracy and profitability measure. Moreover, we adopt an innovative point of view on the lapse prediction problem that comes from churn management. We transform the classification problem into a regression question and then perform optimization, which is new to lapse risk management. We apply the aforementioned four methods to a large real-world insurance dataset. The results show that Extreme Gradient Boosting (XGBoost) and support vector machine outperform logistic regression (LR) and classification and regression tree with respect to statistic accuracy, while LR performs as well as XGBoost in terms of retention gains. This highlights the importance of a proper validation metric when comparing different methods. The optimization after the transformation brings out significant and consistent increases in economic gains. Therefore, the insurer should conduct optimization on its economic objective to achieve optimal lapse management.

This study investigates a combined optimal financing, reinsurance and dividend distribution problem for a big insurance portfolio. A manager can control the surplus by buying proportional reinsurance, paying dividends and raising money dynamically. The transaction costs and liquidation values at bankruptcy are included in the risk model. Under the objective of maximising the insurance company's value, we identify the insurer's joint optimal strategies using stochastic control methods. The results reveal that managers should consider financing if and only if the terminal value and the transaction costs are not too high, less reinsurance is bought when the surplus increases or dividends are always distributed using the barrier strategy.