Esscher Principle Research Articles

Enhancing an insurer's expected value by reinsurance and external financing

In this paper, we analyze a decision-making problem for an insurer with limited liability, who is subject to a solvency constraint and wants to maximize the expected value through reinsurance purchase and external financing. We impose mild conditions on the reinsurance premium principle, which are the axioms of law invariance, risk loading and preserving the convex order. These three axioms are satisfied by all the widely used premium principles, except the Esscher principle, listed in Young (2004). If value at risk (VaR) or conditional value at risk (CVaR) is adopted to calculate the insurer's regulatory capital, we show that the optimal reinsurance can be in the form of two layers. The optimal reinsurance form is reduced to one layer once the premium principle further satisfies a weak condition. In particular, under Wang's, expected value, variance and Dutch premium principles, we derive the insurer's optimal strategies of reinsurance and financing, which are obtained explicitly under the VaR risk measure but have to be solved numerically for the CVaR risk measure. Results indicate that the insurer has three types of optimal strategies: the reinsurance only strategy, external financing only strategy, and mixed strategy. Moreover, the insurer's attitude toward ceding the risk through reinsurance and increasing capital through external financing is often greatly affected by the regulatory regime, the financing cost and the reinsurance price.

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.

Reinsurance Arrangements Minimizing the Risk-Adjusted Value of an Insurer's Liability

In this paper, we investigate the problem of purchasing a reinsurance policy that minimizes the risk-adjusted value of an insurer's liability, where the valuation is carried out using a cost-of-capital approach. In order to exclude the moral hazard, we assume that both the insurer and reinsurer are obligated to pay more for larger loss in a typical reinsurance treaty. Moreover, the reinsurance premium principle is assumed to satisfy three axioms: law invariance, risk loading and preserving convex order. The proposed class of premium principles is quite general in the sense that it contains all the widely used premium principles except Esscher principle listed in Young(2004). When capital at risk is measured by the value at risk(VaR) or conditional value at risk(CVaR), we find it is optimal for the insurer to cede two separate layers over the prescribed premium principles. By imposing an additional weak constraint on the premium principle, we further get that the reinsurance in the form of a layer is optimal. Finally, to illustrate the applicability of our results, we derive explicitly the optimal one-layer reinsurance for expected value principle and Wang's premium principle, and show that two-layer reinsurance may be optimal for Dutch premium principle.

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.

Truncated Stop Loss as Optimal Reinsurance Agreement in One-period Models

We consider several one-period reinsurance models and derive a rule which minimizes the ruin probability of the cedent for a fixed reinsurance risk premium. The premium is calculated according to the economic principle, generalized zero-utility principle, Esscher principle or mean-variance principles. It turns out that a truncated stop loss is an optimal treaty in the class ofallreinsurance contracts. The result is also valid for models not involving ruin probability. An example is the Arrow model with the Kahneman-Tversky value function.

Truncated Stop Loss as Optimal Reinsurance Agreement in One-period Models

We consider several one-period reinsurance models and derive a rule which minimizes the ruin probability of the cedent for a fixed reinsurance risk premium. The premium is calculated according to the economic principle, generalized zero-utility principle, Esscher principle or mean-variance principles. It turns out that a truncated stop loss is an optimal treaty in the class of all reinsurance contracts. The result is also valid for models not involving ruin probability. An example is the Arrow model with the Kahneman-Tversky value function.

NONLIFE INSURANCE PRICING: STATISTICAL MECHANICS VIEWPOINT

We consider the insurance company as a physical system which is immersed in its environment (the financial market). The insurer company interacts with the market by exchanging the money through the payments for loss claims and receiving the premium. Here, in the equilibrium state, we obtain the premium by using the canonical ensemble theory, and compare it with the Esscher principle, the well-known formula in actuary for premium calculation. We simulate the case of car insurance for quantitative comparison.

A Unified Approach to Generate Risk Measures

The paper derives many existing risk measures and premium principles by minimizing a Markov bound for the tail probability. Our approach involves two exogenous functions v(S) and φ(S, π) and another exogenous parameter α ≤ 1. Minimizing a general Markov bound leads to the following unifying equation: E [φ (S, π)] = αE [v (S)].For any random variable, the risk measure π is the solution to the unifying equation. By varying the functions φ and v, the paper derives the mean value principle, the zero-utility premium principle, the Swiss premium principle, Tail VaR, Yaari's dual theory of risk, mixture of Esscher principles and more. The paper also discusses combining two risks with super-additive properties and sub-additive properties. In addition, we recall some of the important characterization theorems of these risk measures.

A Unified Approach to Generate Risk Measures

The paper derives many existing risk measures and premium principles by minimizing a Markov bound for the tail probability. Our approach involves two exogenous functions v(S) and φ(S, π) and another exogenous parameter α ≤ 1. Minimizing a general Markov bound leads to the following unifying equation: E [φ (S, π)] = αE [v (S)].For any random variable, the risk measure π is the solution to the unifying equation. By varying the functions φ and v, the paper derives the mean value principle, the zero-utility premium principle, the Swiss premium principle, Tail VaR, Yaari's dual theory of risk, mixture of Esscher principles and more. The paper also discusses combining two risks with super-additive properties and sub-additive properties. In addition, we recall some of the important characterization theorems of these risk measures.

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.

232024 (M30, M10) On a class of premium principles including the Esscher principle: Kamps U., Scandinavian Actuarial Journal 1998, No. 1

232024 (M30, M10) On a class of premium principles including the Esscher principle: Kamps U., Scandinavian Actuarial Journal 1998, No. 1

On a class of premium principles including the Esscher principle

A class of premium calculation principles is considered with the premiums obtained as expected values of suitably transformed distribution functions. The Esscher principle is a particular example. It is found that the likelihood ratio ordering of risks is preserved for any of these principles. A renewal theoretic interpretation of a special principle is given, and useful properties as well as a related characterization of the exponential distribution are shown.

On a class of premium principles including the Esscher principle : 074031 (M30) Kamps U., University of Dortmund, Forschungsbericht 95/4, 1995, pp. 1–8

On a class of premium principles including the Esscher principle : 074031 (M30) Kamps U., University of Dortmund, Forschungsbericht 95/4, 1995, pp. 1–8

On a class of premium principles including the Esscher principle.

On a class of premium principles including the Esscher principle.

Decision theoretic foundations of credibility theory

Most credibility formulas that have been developed till today refer to the net premium, and the problem of an adequate loading has not yet been solved satisfactorily. In the present paper an approach to premium calculation based on loss functions is introduced, and credibility formulas for the resulting premium principles (net premium principle, exponential principle, Esscher principle, e.g.) are derived. These formulas turn out to be Bayes rules in the sense of statistical decision theory. This gives rise to considering minimax rules as well.

Properties of premium calculation principles

All theoretical premium principles, which use a utility function (such as exponential principle, mean value principle, zero utility principle, Swiss premium calculation principle, Orlicz principle, Esscher principle) are analyzed in the light of practical properties such as homogeneity (as usual for quota shares) and sub-additivity. It is proved that a theoretical premium principle, which fulfills only very weakened forms of both practical properties, reduces necessarily to the net premium principle. Therefore it is impossible that the principles and the properties above are reasonable simultaneously.

Homogeneous Premium Calculation Principles

AbstractA premium calculation principle π is called positively homogeneous if π(cX) = cπ(X) for all c > 0 and all random variables X. For all known principles it is shown that this condition is fulfilled if it is satisfied for two specific values of c only, say c = 2 and c = 3, and for only all two point random variables X. In the case of the Esscher principle one value of c suffices. In short this means that local homogeneity implies global homogeneity. From this it follows that in the case of the zero utility principle or Swiss premium calculation principle, the underlying utility function is of a very specific type.A very general theorem on premium calculation principles which satisfy a weak continuity condition, is added. Among others the proof uses Kroneckers Theorem on Diophantine Approximations.

On the representation of additive principles of premium calculation

Abstract Two characterizations of mixtures of Esscher principles in terms of additivity combined with other conditions are given.