Showing papers in "Insurance Mathematics & Economics in 1998"

On the discounted penalty at ruin in a jump-diffusion and the perpetual put option

Abstract: We consider the jump-diffusion that is obtained if an independent Wiener process is added to the surplus process of classical ruin theory. In this model, we examine the expected discounted value of a penalty at ruin; we show that it satisfies a defective renewal equation which has a probabilistic interpretation. For this purpose, results for the jump-diffusion process are derived concerning the first record low caused by a jump and downcrossings before the first record low caused by a jump. As an application, we determine the optimal exercise boundary for a perpetual put option.

Ordering risks: Expected utility theory versus Yaari's dual theory of risk

Abstract: We introduce a class of partial orderings of risks that are dual to stochastic dominance orderings. These arise as “distortion-free” orderings in Yaari's dual theory of risk (1987). We show that these dual orderings are equivalent to inverse stochastic dominance orderings (Muliere and Scarsini, 1989). We motivate third dual stochastic dominance via insurance economics, while providing an alternative interpretation for second (dual) stochastic dominance. We apply dual stochastic dominance to actuarial science and show how the dual ordering of risks is related to ordering income distributions in the economics of income inequality.

Comonotonicity, correlation order and premium principles

Abstract: In this paper, we investigate the notion of dependency between risks and its effect on the related stop-loss premiums. The concept of comonotonicity, being an extreme case of dependency, is discussed in detail. For the bivariate case, it is shown that, given the distributions of the individual risks, comonotonicity leads to maximal stop-loss premiums. Some properties of stop-loss preserving premium principles are considered. A simple proof is given for the sub-additivity property of Wang's premium principle.

Optimal risk and dividend control for a company with a debt liability

Abstract: We consider a model of a corporation, which can choose a production/business policy among an available set of control policies with different expected profits and associated risks. In addition, there is a choice of the amount of dividends to be paid out to the shareholders. Notwithstanding any policy decision there is a constant payment of a corporate debt, such as bond liability or loan amortization. The objective is to find the policy which maximizes the expected total discounted dividend pay-outs until the time of bankruptcy. We model the dynamics of the corporate assets as a diffusion process with the diffusion and drift coefficients being affine functions of the risk control variable. In other words, the potential profit in our model is proportional to risk. The cumulative dividends are modeled by an increasing process. The resulting problem becomes a mixed regular-singular control problem for diffusion processes. We show that there exists an optimal level b 1 such that the optimal dividend policy is to keep the company's wealth below b 1 and pay out as dividends all the amounts in excess of this level. On the other hand, the profit/risk control policy depends on the ratio of the maximal possible expected profit to the liability payments. When this ratio is small, the optimal policy is always to undertake the maximal risk. When this ratio is large, the risk depends on the current amount of the company's wealth. The risk as a function of the wealth is monotone and increasing, reaching its maximum at some b 0 b 1 .

Pension schemes as options on pension fund assets: implications for pension fund management

Abstract: This paper shows that the three principal types of funded pension scheme (defined benefit, defined contribution and targeted money purchase) are related through a set of options on the underlying financial assets held in the fund. The value of these options depends on both contribution inflows and the financial asset allocation chosen by the fund manager. The option values can therefore be used to assess both the appropriateness of the funding level and the effectiveness of the asset allocation in achieving the objectives of asset-liability management. In particular, they can be used to determine the probability of scheme insolvency.

Ruin probabilities for Erlang(2) risk processes

Abstract: In this paper we consider a risk process in which claim inter-arrival times have an Erlang(2) distribution. We consider the infinite time survival probability as a compound geometric random variable and give expressions from which both the survival probability from initial surplus zero and the ladder height distribution can be calculated. We consider explicit solutions for the survival/ruin probability in the case where the individual claim amount distribution is phase-type, and show how the survival/ruin probability can be calculated for other individual claim amount distributions.

Optimal proportional reinsurance policies for diffusion models with transaction costs

Abstract: This paper extends the results of Hojgaard and Taksar (1997a) to the case of posititve transactions costs. The setting here and in Hojgaard and Taksar (1997a) is the following: When applying a proportional reinsurance policy π the reserve of the insurance company Rtπ is governed by a SDE dRtπ = (μ − (1 − aπ (t))λ dt + aπ (t)σ dWt, where Wt is a standard Brownian motion, μ, σ > 0 are constants and λ ≥ μ. The stochastic process aπ (t) satisfying 0 X≤ aπ (t) ≤ 1 is the control process, where 1 − aπ (t) denotes the fraction of all incoming claims, that is reinsured at time t. The aim of this paper is to find a policy that maximizes the return function Vπ (x) = E ∫τπ0 e−ct Rπt dt, where c > 0, τπ is the time of ruin and x refers to the initial reserve. In Hojgaard and Taksar (1997a) a closed form solution is found in case of λ = μ by means of Stochastic Control Theory. In this paper we generalize this method to the more general case where we find that if λ ≥ 2μ, the optimal policy is not to reinsure, and if μ > λ > 2μ, the optimal fraction of reinsurance as a function of the current reserve monotonically increases from 2(λ − μ)/λ to 1 on (0, x1) for some constant x1 determined by exogenous parameters.

Ruin theory with compounding assets — a survey

Abstract: The paper discusses existing results in ruin theory when assets earn interest. Both analytical and numerical approaches are considered, with main emphasis on recent publications.

Optimal reinsurance and stop-loss order

Abstract: During the last two decades, the interest of the actuarial literature in the stochastic orderings has been growing to such a point that they become one of the most important tools to compare the riskiness of different random situations. Our purpose in this note is to derive new results about the optimal reinsurance coverages for the ceding company, when the optimality criterion consists in minimizing the retained risk with respect to the stop-loss order. We so slightly complete the study initiated in Van Heerwaarden, Kaas and Goovaerts [Insurance: Mathematics and Economics 8 (1989) 11–17].

An actuarial approach to option pricing under the physical measure and without market assumptions

Abstract: As the title may indicate, this paper uses merely probabilistic and actuarial considerations for pricing options There are no economical considerations involved, and our approach is valid even when an equilibrium price measure does not exist (arbitrage, non-equilibrium) or is not unique (incompleteness) We only make use of the physical measure that generates the pay-out distributions The approach does not in general carry over to general derivative securities, since we use an interpretation of the securities under consideration as being potential losses or claims from the issuers point of view Under this interpretation we calculate the price of the security as the fair premium needed to insure the potential loss As a special case of our formula we derive the Black and Scholes formula

Stochastic cooperative games in insurance

Abstract: This paper shows how problems in ‘non-life’-(re)insurance can be modeled as cooperative games with stochastic payoffs. Pareto optimal allocations of the risks faced by the insurers and the insureds are determined. It is shown that the core of the corresponding insurance games is nonempty. Moreover, it is shown that specific core allocations are obtained when the zero-utility principle is used for calculating premiums. Finally, game theory is used to give a justification for subadditive premiums.

Double barrier hitting time distributions with applications to exotic options

Abstract: In this paper we derive two defective density functions related to double barrier hitting probabilities of a geometric Brownian motion. A technique developed by Gerber and Shiu (1994, 1996) and Laplace transforms are used. Our approach is simple and straightforward, and purely analytical. We then apply the formulas to value some exotic options whose payoffs are contingent on barrier hitting time. This work is motivated by a recent development of equity indexed annuities in the United States.

On the distribution of a sum of correlated aggregate claims

Abstract: Assume that a book of business is the union of disjoint classes of business and each of which has an aggregate distribution. The classes of business are correlated. We present necessary formulas to compute the aggregate distribution for the whole book when claim counts distribution takes a certain form. As a special case when claim counts distribution is multivariate Poisson, we show that the aggregate distribution for the whole book is compound Poisson.

Burr regression and portfolio segmentation

Abstract: Two Burr regression models are proposed. These models extend existing log-logistic regression models. An algorithm for computing the maximum likelihood estimators is proposed. Graphical techniques for model validation are incorporated. An actuarial application to portfolio segmentation for fire insurance is included.

Perception of ecological risk to water environments

Abstract: This paper examines lay and expert perceptions of the ecological risks associated with a range of human activities that could adversely affect water resource environments It employs the psychometric paradigm pioneered in characterizing perceptions of human health risks, which involves surveys to obtain judgments from subjects about risk items in terms of several important characteristics of the risks The paper builds on a previous study that introduced ecological risk perception This second study employs a larger, more diverse sample, a more focused topic area, and comparisons between lay and expert judgments The results confirm that a small set of underlying factors explain a great deal of variability in lay judgments about ecological risks These have been termed Ecological Impact, Human Benefits, Controllability, and Knowledge The results are useful in explaining subjects' judgments of the general riskiness of, and need for regulation of, various risk items The results also indicate several differences and areas of agreement among the lay and expert samples that point to potential key issues in future ecological risk management efforts for water resources

On some filtering problems arising in mathematical finance

Abstract: Three situations in which filtering theory is used in mathematical finance are illustrated at different levels of detail. The three problems originate from the following different works: 1. (1) On estimating the stochastic volatility model from observed bilateral exchange rate news, by Mahieu and Schotman (1997). 2. (2) A state space approach to estimate multi-factors CIR models of the term structure of interest rates, by Geyer and Pichler (1996). 3. (3) Risk-minimizing hedging strategies under partial observation in pricing financial derivatives, by Fischer et al. (1996). In the first problem we propose to use a recent nonlinear filtering technique based on geometry to estimate the volatility time series from observed bilateral exchange rates. The model used here is the stochastic volatility model. The filters that we propose are known as projection filters, and a brief derivation of such filters is given. The second problem is introduced in detail, and a possible use of different filtering techniques is hinted at. In fact the filters used for this problem in (2) and part of the literature can be interpreted as projection filters and we will make some remarks on how more general and possibly more suitable projection filters can be constructed. The third problem is only presented briefly.

Ruin probabilities in perturbed risk models

Abstract: We consider the asymptotical behaviour of the ruin function in perturbed and unperturbed non-standard risk models when the initial risk reserve tends to infinity. We give a characterization of this behaviour in terms of the unperturbed ruin function and the perturbation law provided that at least one of both is subexponential. By a number of examples for the claim arrival process as well as the perturbation process we show that our result is a generalization of previous work on this subject.

Individuals' estimates of the risks of death: Part I — A reassessment of the previous evidence

Abstract: It is widely argued that individuals have biased perceptions of health and safety risks. A reconsideration of the best-known evidence suggests that this view is the erroneous result of a failure to consider the implications of scarce information. Our findings imply that the hypothesis that people make unbiased estimates of hazard rates fails to be rejected by the very data that were initially used to reject it. Thus, we are able to reconcile the alleged existence of widespread bias in risk perception with other findings that such bias is less apparent in the case of job-related hazards. The seeming bias in estimating population-average death rates and the lack of such bias in assessing job risks are two manifestations of the same behavior, which is the optimal acquisition of costly information.

Availability crises in insurance markets: Optimal contracts with asymmetric information and capacity constraint

Abstract: Insurance markets sometimes exhibit "crises" in which prices rise dramatically and coverage is unavailable or is rationed at the new prices. A recent explanation for such crises is the "capacity constraint" model of Gron and Winter. Crises usually follow sudden and large depletions in insurers' equity or surplus. The capacity constraint model argues that frictional costs in replacing surplus, and limited liability, give rise to a kinked insurance supply function and that crises arise from discontinuous short term adjustments around the kink. While this model explains much about liability insurance crises, it still leaves unexplained their most prominent feature; that insurance is rationed or unavailable. We follow their insight in looking to equity shocks and capital market frictions to explain crises and combine this with a model of optimal risk sharing contracts under the information conditions characteristic of this market. We use implicit long term contracts with truth telling constraints to address information asymmetries and this allows us to model crises that exhibit rationing. Our model is tested in the market most dramatically affected by such crises in the 1980‘s, the general liability insurance market.

Pricing insurance contracts — an economic viewpoint

Abstract: This paper presents a new approach for pricing insurance contracts, based both on economic and probabilistic arguments. The novel property of this approach is that it uses the demand for insurance to find the optimal premium an insurer should charge. Our approach stands in contrast to the standard loading factor methods used in actuarial science, where the number of insureds is constant regardless of the charged premium. The insurer maximizes its expected profit, defined as the difference between the expected net revenue from selling insurance contracts and the expected loss due to insolvency. We show how to find the expected-profit maximizing premium, π ∗ , and its corresponding optimal number of insureds, n ∗ . The first proposition presented in our paper identifies the premium (and number of insureds that minimize the expected loss due to insolvency). The second proposition gives, for a broad class of demand curves, sufficient conditions for the existence and uniqueness of an internal optimal solution. The third proposition asserts that, due to the suggested expected loss function, the insurer's objective function demonstrates economies to scale. Lastly, we provide a numerical solution for the case of a linear demand curve, giving the optimal premium and number of insureds.

On piecewise deterministic Markov control processes: Control of jumps and of risk processes in insurance

Abstract: Dynamic programming for piecewise deterministic Markov processes is studied where only the jumps but not the deterministic flow can be controlled. Then one can dispense with relaxed controls. There exists an optimal stationary policy of feedback form. Further, a piecewise deterministic Markov model for the control of dividend pay-out and reinsurance is introduced. This model can be transformed to a model with uncontrolled flow. It is shown that a classical solution to the Bellman equation exists and that a non-relaxed optimal policy of feedback form can be obtained via the Bellman equation. Lipschitz continuity of the one-dimensional vector field defining the controlled flow will be replaced by strict positivity.

On the computation of aggregate claims distributions: some new approximations

Abstract: This paper proposes a new approximation to the aggregate claims distribution based on the inverse Gaussian (IG) distribution. It is compared to several other approximations in the literature. The IG approximation compares favorably to the well-known gamma approximation. We also propose an IG-gamma mixture that approximates the true distribution extremely accurately, in a large variety of situations.

Exact and approximate properties of the distribution of surplus before and after ruin

Abstract: We obtain explicit expressions for the distribution of surplus immediately before and after ruin which allow for simple derivation of bounds as well as simple evaluation for certain choices of the claim size distribution. We then use these expressions to construct Tijms-type approximations which are often exact.

A realistic non-homogeneous stochastic pension fund model on scenario basis

Abstract: As far as we know, this paper presents for the first time a general, rigorous and tractable stochastic evolution time model for pension fund called the discrete time non-homogeneous semi-Markov model, or in short, the DTNHSM pension fund model taking into account both economic, financial and demographic evolution factors so that it becomes a real-life model. The most important factors are: seniority, general age dependence, rate of inflation and salary lines. The model starts from a set of m states and each member of the fund is necessarily in one and only one of these states at each time epoch, for example each year. The main probabilistic assumption is that the successive state transitions together with transition time epochs constitute a two-dimensional non-homogeneous Markov additive process on which the state at any time epoch t is defined by the imbedded non-homogeneous semi-Markov process. Let us say that we introduce as other fundamental tool the concept of scenario both with strategic ch...

The moments of ruin time in the classical risk model with discrete claim size distribution

Abstract: Firstly exact simple expressions are given for the moments Mr = E0(Tr 1{T λm and c ≤ λm are treated.

Concepts and methods for discrete and continuous time control under uncertainty

Abstract: We present some concepts and solution methods for finite horizon control problems under uncertainty in discrete as well as continuous time. We discuss exact and approximate solution methods and mention possible applications. The uncertainty is mainly of a stochastic nature, but also other forms of uncertainty are considered.

A note on interest rate term structure estimation using tension splines

Abstract: Tension splines are proposed as a flexible tool for interest rate term structure estimation to circumvent some difficulties arising with the ordinary cubic spline estimators. A few computational experiments on test problems support the merits of the proposed approach.

Non-Poissonian claims' arrivals and calculation of the probability of ruin

Abstract: Collective risk model with a special emphasize on non-Poissonian claims' arrival processes is considered. Exact and approximate techniques for the calculation of the probabilities of ruin are examined. Simulation going back to the importance sampling is applied to two particular cases of non-Poissonian claims arrival processes to illustrate strong dependence of the probabilities of ruin on the interclaims distribution.

Conjoint prediction of paid and incurred losses

Abstract: Actuaries use paid and incurred methods to predict losses. Then one method is selected, and usually the information contained in the other(s) is jettisoned. Sometimes methods are weighted together, but without statistical justification. And even when an incurred method is deemed sufficient, present valuing will require predictions of loss payments. This paper will present a statistical model whereby paid and incurred losses are predicted together, or interdependently. As a result, the predictions of both paid and incurred losses will attain to the same ultimate amounts. The model will be developed from statistical theory, using a simple example. Then a realistic example will be treated, and results therefrom will be present valued. One of the appendices will treat the financial theory of valuing stochastic cash flows, in which accepted theory regarding risk-adjusted rates of return will be challenged.

On the tradeoff between the law of large numbers and oligopoly in insurance

Abstract: For a fixed number of customers in an insurance market, changing the number of insurance firms creates both scale effects on the individual firms (as the number of customers per firm is altered), and an oligopolistic effect on market supply. The most prominent effect of scale is the impact on solvency associated with the law of large numbers (LLN). In this paper, we focus on the relationship between the LLN and the oligopolistic effect of the number of firms in the market, and use a game-theoretic model of insurance market equilibrium to study this problem. For constant absolute risk averse buyers and risk neutral sellers, we find that there is a natural tradeoff between the effects of the LLN and oligopoly that causes both equilibrium quantity and the equilibrium payoff to customers to possess unique interior maxima over the number of insurance firms.