Stop-loss Contracts Research Articles

An expansion formula for Hawkes processes and application to cyber-insurance derivatives

In this paper we provide an expansion formula for Hawkes processes which involves the addition of jumps at deterministic times to the Hawkes process in the spirit of the well-known integration by parts formula (or more precisely the Mecke formula) for Poisson functional. Our approach allows us to provide an expansion of the premium of a class of cyber insurance derivatives (such as reinsurance contracts including generalized Stop-Loss contracts) or risk management instruments (like Expected Shortfall) in terms of so-called shifted Hawkes processes. From the actuarial point of view, these processes can be seen as “stressed” scenarios. Our expansion formula for Hawkes processes enables us to provide lower and upper bounds on the premium (or the risk evaluation) of such cyber contracts and to quantify the surplus of premium compared to the standard modeling with a homogeneous Poisson process.

Pricing cumulative loss derivatives under additive models via Malliavin calculus

We show that integration by parts formulas based on Malliavin-Skorohod calculus techniques for additive processes help us to compute quantities like ${\E}(L_T h(L_T))$ for different suitable functions $h$ and different models for the cumulative loss process $L_T$. These quantities are important in Insurance and Finance. For example they appear in computing expected shortfall risk measures or stop-loss contracts. The formulas given in the present paper, obtained by simple proofs, generalize the formulas given in a recent paper by Hillairet, Jiao and Réveillac using Malliavin calculus techniques for the standard Poisson process, a particular case of additive process.

Optimal premium allocation under stop-loss insurance using exposure curves

Determining the retention level in the stop-loss insurance risk premium for both insurer and reinsurer is an important factor in pricing. This paper aims to set optimal reinsurance with respect to the joint behavior of the insurer and the reinsurer under stop-loss contracts. The dependence between the costs of insurer and reinsurer is expressed as a function of retention ($d$) and maximum-cap ($m$) levels. Based on the maximum degree of correlation, the optimal levels for $d$ and $m$ are derived under certain claim distributions (Pareto, Gamma and Inverse Gamma). Accordingly, the risk premium and exposure curves for both parties are based on the selected distributions. Quantification of the premium share over derived exposure curves based on the optimized retention and maximum levels and the maximum loss risk is obtained using VaR and CVaR as risk measures.

Time dependent stop-loss reinsurance and exposure curves

Stop-loss contracts are the most commonly used reinsurance agreements in insurance whose important factors are the retention and the maximum (cap) values attained on the random loss, which may occur within the policy period. Therefore, determining and forecasting the loss amounts is an important issue for both the insurer and the reinsurer. Along with many approaches in actuarial literature, we propose a geometric Brownian motion (BM) with the time-varying parameters to capture the time-dependent loss amounts. We implement the time-influence on stop-loss contract in the frame of the stochastic model and find the analytical derivations of costs associated with reinsurance contract for reinsurer and insurer with constraints on time, loss amount, retention, and both retention and cap levels. Additionally, the analytical forms of exposure curves are depicted to determine the premium share between reinsurer and insurer under time, loss, retention, and both retention and cap constraints. An application of the proposed methodology on real-life data and the calibration of time-varying parameters using dynamic maximum likelihood estimator and simulations on the proposed model are performed. Finally, we forecast the claim amounts, expected costs, and exposure curves on time-varying parameters using the cubic spline extrapolation and the dynamic ARIMA with trend search. It is shown that the time-varying approach using the stochastic model copes with the behavior of the claims and assures fair share between insurer and reinsurer.

New stochastic comparisons based on tail value at risk measures

In this article we provide a new criterion for the comparison of claims, when we have conditional claims arising in stop loss contracts or contracts with franchise deductible. These stochastic comparisons are made on the basis of the Tail Value at Risk (also known as conditional tail expectation), just for a fixed level and beyond. In particular, we explain the interest of comparing these quantities, study some preservation properties and, in addition, we provide sufficient conditions for its study. Finally we illustrate its usefulness with some examples.

Transform approach for discounted aggregate claims in a risk model with descendant claims

We consider a risk model with three types of claims: ordinary, leading, and descendant claims. We derive an expression for the Laplace–Stieltjes transform of the distribution of the discounted aggregate claims. By using this expression, we can then obtain the mean and variance of the discounted aggregate claims. For actuarial applications, the VaR and CTE are computed by numerical inversion of the Laplace transforms for the tail probability and the conditional tail expectation of the discounted aggregate claims. The net premium for stop-loss reinsurance contract is also computed.

Pricing formulae for derivatives in insurance using Malliavin calculus

In this paper, we provide a valuation formula for different classes of actuarial and financial contracts which depend on a general loss process by using Malliavin calculus. Similar to the celebrated Black–Scholes formula, we aim to express the expected cash flow in terms of a building block. The former is related to the loss process which is a cumulated sum indexed by a doubly stochastic Poisson process of claims allowed to be dependent on the intensity and the jump times of the counting process. For example, in the context of stop-loss contracts, the building block is given by the distribution function of the terminal cumulated loss taken at the Value at Risk when computing the expected shortfall risk measure.

Optimal stop-loss reinsurance: a dependence analysis

The stop-loss reinsurance is one of the most important reinsurance contracts in the insurance market. From the insurer point of view, it presents an interesting property: it is optimal if the criterion of minimizing the variance of the cost of the insurer is used. The aim of the paper is to contribute to the analysis of the stop-loss contract in one period from the point of view of the insurer and the reinsurer. Firstly, the influence of the parameters of the reinsurance contract on the correlation coefficient between the cost of the insurer and the cost of the reinsurer is studied. Secondly, the optimal stop-loss contract is obtained if the criterion used is the maximization of the joint survival probability of the insurer and the reinsurer in one period.

Optimal Stop-Loss Reinsurance: A Dependence Analysis

The stop-loss reinsurance is one of the most important reinsurance contracts in the insurance market. From the insurer point of view, it presents an interesting property: it is optimal if the criterion of minimizing the variance of the cost of the insurer is used. The aim of the paper is to contribute to the analysis of the stop-loss contract in one period from the point of view of the insurer and the reinsurer. Firstly, the influence of the parameters of the reinsurance contract on the correlation coefficient between the cost of the insurer and the cost of the reinsurer is studied. Secondly, the optimal stop-loss contract is obtained if the criterion used is the maximization of the joint survival probability of the insurer and the reinsurer in one period.

Stable solutions for optimal reinsurance problems involving risk measures

The optimal reinsurance problem is a classic topic in actuarial mathematics. Recent approaches consider a coherent or expectation bounded risk measure and minimize the global risk of the ceding company under adequate constraints. However, there is no consensus about the risk measure that the insurer must use, since every risk measure presents advantages and shortcomings when compared with others. This paper deals with a discrete probability space and analyzes the stability of the optimal reinsurance with respect to the risk measure that the insurer uses. We will demonstrate that there is a “stable optimal retention” that will show no sensitivity, insofar as it will solve the optimal reinsurance problem for many risk measures, thus providing a very robust reinsurance plan. This stable optimal retention is a stop-loss contract, and it is easy to compute in practice. A fast linear time algorithm will be given and a numerical example presented.

The financial hedging of contributions by oil firms to the IOPC Funds

The liability and compensation for the damage caused by maritime oil spills have been regulated by the 1992 International Convention on Civil Liability for Oil Pollution Damage (1992 Civil Liability Convention) and the 1992 International Oil Pollution Compensation Fund Convention (1992 IOPC Fund). In this compensation regime, the contributions of oil firms to the IOPC Funds are based on the aggregate risk of the Funds and are assessed each time an oil spill is registered. In this paper, we present the main characteristics of such a compensation regime and explain why oil firms would benefit from hedging their contributions to the Funds through standard insurance combined with investment on capital markets. A simulation of a combined hedging strategy using stop-loss contracts is also proposed.

Optimal reinsurance policy: The adjustment coefficient and the expected utility criteria

This paper is concerned with the optimal form of reinsurance from the ceding company point of view, when the cedent seeks to maximize the adjustment coefficient of the retained risk. We deal with the problem by exploring the relationship between maximizing the adjustment coefficient and maximizing the expected utility of wealth for the exponential utility function, both with respect to the retained risk of the insurer. Assuming that the premium calculation principle is a convex functional and that some other quite general conditions are fulfilled, we prove the existence and uniqueness of solutions and provide a necessary optimal condition. These results are used to find the optimal reinsurance policy when the reinsurance premium calculation principle is the expected value principle or the reinsurance loading is an increasing function of the variance. In the expected value case the optimal form of reinsurance is a stop-loss contract. In the other cases, it is described by a nonlinear function.

Optimal reinsurance under convex principles of premium calculation

Suppose an insurer wants to have a reinsurance contract minimizing a convex measure of his retained risk or maximizing a utility function. Suppose the reinsurer’s premium is fixed. The premium calculation principle of the reinsurer is a convex functional of his cover. Explicit forms of optimal reinsurance contracts are derived for some classes of convex principles including, among others, the exponential, p-mean value, semi-deviation, semi-variance, Dutch and Wang’s principles. The paper is a continuation of the work of Kaluszka [Kaluszka, M., 2004. An extension of Arrow’s result on optimality of a stop loss contract. Insur.: Math. Econ. 35, 527–536] which deals with mean-variance premium calculation principles.

An extension of Arrow’s result on optimality of a stop loss contract

The optimal, from the cedent’s point of view, reinsurance treaties for a fixed reinsurer’s premium are derived. We assume that the cedent wants to minimize a convex risk measure, e.g. the variance, semi-variance, upper partial moment or mean absolute deviation. The reinsurer’s premium is calculated on the basis of the expectation and variance of his/her part of the total risk. The obtained results provide an extension of Arrow’s result on the optimality of a stop loss treaty.

Indifference pricing of insurance contracts in a product space model: applications

This paper determines fair premiums and optimal strategies under the financial variance and standard deviation principles of Schweizer [Insur.: Math. Econ. 28 (2001b) 31] for some insurance contracts with financial risk. Examples considered include unit-linked pure endowment contracts and integrated risk management solutions such as stop-loss contracts with barrier and financial stop-loss contracts. In each example we illustrate how the amount of information available to the reinsurer affects premiums and investment strategies. In addition, we determine the simple upper and lower bounds for the fair premiums that were obtained in [Finan. Stochast., in press].

Non-optimality of a linear combination of proportional and non-proportional reinsurance

For the subclass of reinsurance contracts with maximum deductible contained in the class of all bivariate comonotonic risk-exchange structures associated to a given risk, we consider optimality with respect to a long-term actuarial mean self-financing property and competitiveness of the insurance premium. For arbitrary varying risks, the linear combination of proportional and stop-loss reinsurance is not optimal unless it is a pure stop-loss contract, at least if the variance premium principle is used to set insurance prices. By known distribution of the risk, it is shown how an optimal deductible of a stop-loss contract can be determined. Some applications to insurance and finance are briefly mentioned.

On the Loading of a Stop-Loss Contract: A Correction on Extrapolation and two Stable Price Methods

On the Loading of a Stop-Loss Contract: A Correction on Extrapolation and two Stable Price Methods

Best bounds for expected financial payoffs II: Applications

Based on a general algorithm to determine best bounds for expected piecewise linear payoffs, several important examples are treated in a unified manner. Tables of best bounds are given for the stop-loss, limited stop-loss, franchise and disappearing deductible, and two-layers stop-loss contracts. In the last example the maximal bound can only be obtained numerically

Insurance pricing and increased limits ratemaking by proportional hazards transforms

This paper proposes a new premium principle, where risk loadings are imposed by a proportional decrease in the hazard rates. This premium principle is scale invariant and additive for layers. It is shown that this principle will generate stop-loss contracts as optimal reinsurance arrangements in a competitive market when the reinsurer is less risk-averse than the direct insurer. Finally, increased limits factors are calculated based on this principle.

A dynamic reinsurance theory

In this paper we present a technique that can be used by the insurer, who reinsured part of his risk by means of a proportional stop-loss contract, to evaluate his residual risk position. Part of this technique consists of the calculation of the optimal reinsurance strategy. We also show how this same technique can be used by the reinsurer to evaluate his risk position.