Esscher Premium Research Articles

Calculating premium principles from the mode of a unimodal weighted distribution

Abstract The theory of utility is a well-known method of constructing insurance premiums (see e.g., Newton et al. (1986) Actuarial Mathematics. Itasca, Illinois: The Society of Actuaries.). Furman and Zitikis ((2008) Insurance: Mathematics and Economics, 42, 459–465.) proposed an alternative method using the mean value of a weighted random variable. According to this approach, for various choices of weighting, popular premiums such as net premium, modified variance premium, Esscher premium, and Kamps premium are obtained. On the other hand, some premiums cannot be obtained with this method, such as the premium of the exponential principle. In this paper, we provide a complementary theory by introducing a family of unimodal weighted distributions for which the mode is a premium principle.

Experience rating of risk premium for Esscher premium principle

In this article, a new method is introduced under the Bayesian framework to derive the credibility estimator of risk premiums based on the Esscher premium principle. This new estimator offers desirable statistical properties, making it more useful and practical compared to existing estimators. Additionally, Bayesian models for policy portfolios are established, and empirical Bayes methods are employed to estimate the structural parameters. The empirical Bayesian estimation of risk premiums is also discussed in detail. The convergence rate and goodness of the proposed estimators are verified through simulations. The results demonstrate the effectiveness and accuracy of the new estimator and its superior performance compared to other existing methods. Finally, an empirical analysis is conducted using real insurance data, which further confirms the applicability and reliability of the proposed credibility estimator and its superiority in practical insurance applications.

Endogenous Inverse Demand Functions

In this work we present an equilibrium formulation for price impacts. This is motivated by the Buhlmann equilibrium in which assets are sold into a system of market participants and can be viewed as a generalization of the Esscher premium. Existence and uniqueness of clearing prices for the liquidation of a portfolio are studied. We also investigate other desired portfolio properties including monotonicity and concavity. Price per portfolio unit sold is also calculated. In special cases, we study price impacts generated by market participants who follow the exponential utility and power utility.

Budget-constrained optimal retention with an upper limit on the retained loss

ABSTRACTUnlike sophisticated institutional insurance buyers, individual insurance seekers often use simple heuristic tools for risk management purposes, such as requiring that an insurance arrangement will not result in a retained loss that exceeds a certain predetermined and fixed level. In this paper, we re-examine the problem of budget-constrained demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss; but we further impose an additional upper-limit constraint on the retained loss and assume that the insurer distorts his subjective probability measure. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. We characterize the optimal retention function and show that it has a simple two-part structure: zero retention (full insurance) on an event to which the insurer assigns zero probability, and a retention that could be described as a limited variable deductible on the complement of this event. As an illustration, we examine the case of a distorted Esscher premium principle.

Budget-Constrained Optimal Insurance with an Upper Limit on the Retained Loss

Unlike sophisticated institutional insurance buyers, individual insurance seekers often use simple heuristic tools for risk management purposes, such as requiring that an insurance arrangement will not result in a retained loss that exceeds a certain predetermined and fixed level. In this paper, we re-examine the problem of budget-constrained demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss; but we further impose an additional upper-limit constraint on the retained loss and assume that the insurer distorts his subjective probability measure. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. We characterize the optimal retention function and show that it has a simple two-part structure: zero retention (full insurance) on an event to which the insurer assigns zero probability, and a retention that could be described as a limited variable deductible on the complement of this event. As an illustration, we examine the case of a distorted Esscher premium principle.

Skew-elliptical distributions with applications in risk theory

In this paper we derive important properties of the well-known skew-elliptical (SE) distributions which was introduced in Azzalini and Capitanio (J R Stat Soc Ser B (Stat Methodol) 65:367–389, 2003), and includes the more familiar skew-normal, skew-Student-t and skew-logistic distributions. We then derive the tail value at risk (TVaR) for a portfolio of SE risks. We provide the portfolio risk decomposition with TVaR. Furthermore, we obtain the Esscher premium principle, the weighted-premium principle, and the entropic risk measure with the underlying SE distributions. We also provide an explicit closed-form solution to the optimal portfolio selection with the SE distributions, and provide a numerical simulation of the results.

Limit behavior of the Esscher premium

Abstract Convergence from below of the Esscher premium for an arbitrary insurance risk to the essential supremum of the priced risk when the premium parameter tends to plus infinity is presented in the article. As a corollary of the above mentioned statement, we present convergence of the expected/averaged Esscher transform of an arbitrary random variable to its essential infimum from above when the transformation parameter tends to minus infinity.

On weighted balanced loss function under the Esscher principle and credibility premiums

This paper focuses on weighted balanced loss function under the Esscher principle (WBLF) of which we explore the modern practice of credibility theory and we generalize credibility premiums by using the WBLF. We obtain a distribution-free approach under the WBLF and the Esscher premium by using a minimization technique. Also, we discuss the consistency of the credibility premium generated by this distribution-free approach.

The credibility premiums based on estimated moment-generating function

ABSTRACTTraditionally, experience ratemaking is in principle based on the idea of Bühlmann’s credibility theory that, except for net premiums, was rarely applied to other premium calculation principles. This article uses Bühlmann’s credibility procedure to estimate moment-generating functions (MGFs) of risks and then deduces estimates of moments of those risks. For the premium calculation principles that can be expressed as functions of certain moments or more directly of the MGFs, this article develops a new type of experience ratemaking methods by means of the estimated MGFs and discusses their consistency and asymptotic normality. Numerical simulation shows that, under the Esscher and exponential premium principles, the new credibility estimates are better than existing credibilityestimates in the literature.

A note on weighted premium calculation principles

A prominent problem in actuarial science is to determine premium calculation principles that satisfy certain criteria. Goovaerts et al. [Goovaerts, M. J., De Vylder, F., Haezendonck, J., 1984. Insurance Premiums: Theory and Applications. North-Holland, Amsterdam, p. 84] establish an optimality-type characterization of the Esscher premium principle, but unfortunately their result is not true. In this note we propose a modified statement of this result.

Weighted premium calculation principles

A prominent problem in actuarial science is to define, or describe, premium calculation principles (pcp’s) that satisfy certain properties. A frequently used resolution of the problem is achieved via distorting (e.g., lifting) the decumulative distribution function, and then calculating the expectation with respect to it. This leads to coherent pcp’s. Not every pcp can be arrived at in this way. Hence, in this paper we suggest and investigate a broad class of pcp’s, which we call weighted premiums, that are based on weighted loss distributions. Different weight functions lead to different pcp’s: any constant weight function leads to the net premium, an exponential weight function leads to the Esscher premium, and an indicator function leads to the conditional tail expectation. We investigate properties of weighted premiums such as ordering (and in particular loading), invariance. In addition, we derive explicit formulas for weighted premiums for several important classes of loss distributions, thus facilitating parametric statistical inference. We also provide hints and references on non-parametric statistical inferential tools in the area.

Weighted Premium Calculation Principles

A prominent problem in actuarial science is to define, or describe, premium calculation principles (pcp's) that satisfy certain properties. A frequently used resolution of the problem is achieved via distorting (e.g., lifting) the de-cumulative distribution function, and then calculating the expectation with respect to it. This leads to coherent pcp's. Not every pcp can be arrived at in this way. Hence, in this paper we suggest and investigate a broad class of pcp's, which we call weighted premiums, that are based on loss distributions. Different weight function lead to different pcp's: any constant weight function leads to the net premium, an exponential weight function leads to the Esscher premium, and an indicator function leads to the conditional tail expectation. We investigate properties of weighted premiums such as ordering (and in particular loading), invariance. In addition, we derive explicit formulas for weighted premiums for several important classes of loss distributions, thus facilitating parametric statistical inference. We also provide hints and references on non-parametric statistical inferential tools in the area.

On the consistency of credibility premiums regarding Esscher principle

In this paper, we investigate the problems of convergence of experience-based ratemakings regarding the Esscher principle. In addition to the Bayes and the classical credibility premiums, we suggest a new credibility formula for the Esscher premium. Then we show the convergence of the Bayes and the newly defined credibility premiums towards the individual premium and point out that the classical credibility premium does not generally converge to the individual premium by presenting a sufficient and necessary condition under which the classical credibility Esscher premium converges to the individual premium. A simulation study is carried out to illustrate the theoretical conclusions.

Physical Premium Principle: A New Way for Insurance Pricing

In our previous work we suggested a way for computing the non-life insurance premium. The probable surplus of the insurer company assumed to be distributed according to the canonical ensemble theory. The Esscher premium principle appeared as its special case. The difference between our method and traditional principles for premium calculation was shown by simulation. Here we construct a theoretical foundation for the main assumption in our method, in this respect we present a new (physical) definition for the economic equilibrium. This approach let us to apply the maximum entropy principle in the economic systems. We also extend our method to deal with the problem of premium calculation for correlated risk categories. Like the Buhlman economic premium principle our method considers the effect of the market on the premium but in a different way.

Bounds for Ratios of Posterior Expectations: Applications in the Collective Risk Model

This paper considers the collective risk model for the insurance claims process. We will adopt a Bayesian point of view, where uncertainty concerning the specification of the prior distribution is a common question. The robust Bayesian approach uses a class of prior distributions which model uncertainty about the prior, instead of a single distribution. Relatively little research has dealt with robustness with respect to ratios of posterior expectations as occurs with the Esscher and Variance premium principles. Appropriate techniques are developed in this paper to solve this problem using the k -contamination class in the collective risk model.

The Esscher premium principle in risk theory: a Bayesian sensitivity study

In this paper the Esscher premium calculation principle is applied to the non-compound collective model in a robust Bayesian context. We consider that uncertainty with regard to the prior distribution can be represented by the assumption that the unknown prior distribution belongs to a class of distributions Γ and examine the ranges of the Bayesian premium when the priors belong to such a class. The assessment of the influence of the prior is termed sensitivity analysis or robustness analysis.

The Esscher Premium Principle: A Criticism. Comment.

Zehnwirth (1981) contains some flaws. Ifis the Esscher premium for a risk X, the loading is H(X) — E(X) and not h as Zehnwirth states. The first and third formulas on page 78 are wrong, since o(h) is a quantity such thatA correct statement would have been thator simply that H(X) is a continuous function of the parameter h. However, this continuity is not uniform in all risks, which is illustrated by (3). No matter how small h is, there is always an X such that the difference between H(X) and E(X) is substantial. In view of this what is the meaning of a statement like “… the Esscher premium is a small perturbation of the linearized credibility premium”?

The Esscher Premium Principle: A Criticism

The Esscher premium principle has recently had some exposure, namely, with the works of Bühlmann (1980) and Gerber (1980).Bühlmann (1980) devised the principle and coined the name for it within the framework of utility theory and risk exchange. Geruber (1980), on the other hand, gives further insight into the principle by studying it within the realm of forecasting in much the same spirit as credibility theorists forecast premiums. However, there is an important distinction: the choice of loss function.The present note sets out to criticize this relatively embryonic principle using decision theoretic arguments and indicates that the Esscher premium is essentially a small perturbation of the well established linearized credibility premium Bühlmann (1970).Let H denote the Esscher premium principle with loading h > o. That is, if X is an observable random variable and Y is a parameter (a risk or a random quantity) to be forecasted then the Esscher premium is given byThat is, H(Y ∣ X) is the Bayes decision rule for estimating Y given the data X relative to the loss functionwhere a is the estimate of Y, and of course the loading h is greater than zero.Now, for the clincher. This loss function is nonsensical from the point of view of estimation. It indicates a loss (or error) to the forecaster that is essentially the antithesis of relative loss.