Two-Sided Bounds for the Finite Time Probability of Ruin

Explicit, two-sided bounds are derived for the probability of ruin of an insurance company, whose premium income is represented by an arbitrary, increasing real function, the claims are dependent, integer valued r.v.s and their inter-occurrence times are exponentially, non-identically distributed. It is shown, that the two bounds coincide when the moments of the claims form a Poisson point process. An expression for the survival probability is further derived in this special case, assuming that the claims are integer valued, i.i.d. r.v.s. This expression is compared with a different formula, obtained recently by Picard & Lefevre (1997) in terms of generalized Appell polynomials. The particular case of constant rate premium income and non-zero initial capital is considered. A connection of the survival probability to multivariate B -splines is also established.     

Optimal Dynamic Premium Control in Non-life Insurance. Maximizing Dividend Pay-outs

In this paper we consider the problem of finding optimal dynamic premium policies in non-life insurance. The reserve of a company is modeled using the classical Cramér-Lundberg model with premium rates calculated via the expected value principle. The company controls dynamically the relative safety loading with the possibility of gaining or loosing customers. It distributes dividends according to a 'barrier strategy' and the objective of the company is to find an optimal premium policy and dividend barrier maximizing the expected total, discounted pay-out of dividends. In the case of exponential claim size distributions optimal controls are found on closed form, while for general claim size distributions a numerical scheme for approximations of the optimal control is derived. Based on the idea of De Vylder going back to the 1970s, the paper also investigates the possibilities of approximating the optimal control in the general case by using the closed form solution of an approximating problem with exponential claim size distributions.     

On the probability of ruin in the collective risk theory for insurance enterprises with oly negative risk sums

Abstract

1. The determination of the probability that an insurance company once in the future will be brought to ruin is a problem of great interest in insurance mathematics. If we know this probability, it does not only give us a possibility to estimate the stability of the insurance company, but we may also decide which precautions, in the form of f. ex. reinsurance and loading of the premiums, should be taken in order to make the probability of ruin so small that in practice no ruin is to be feared.     

Ruin under stochastic dependence between premium and claim arrivals

We investigate, focusing on the ruin probability, an adaptation of the Cramér–Lundberg model for the surplus process of an insurance company, in which, conditionally on their intensities, the two mixed Poisson processes governing the arrival times of the premiums and of the claims respectively, are independent. Such a model exhibits a stochastic dependence between the aggregate premium and claim amount processes. An explicit expression for the ruin probability is obtained when the claim and premium sizes are exponentially distributed.     

On the Probability of (Non-) Ruin in Infinite Time

In the context of the classical Poisson ruin model Gerber (1988a,b) and Shiu (1987, 1989) have obtained two formulae for the ruin and non ruin probabilities in infinite time. Here these two formulae are generalized to the case of an arbitrary premium process when all claims are integer-valued, as in Picard & Lefèvre (1997). Moreover, this generalization throws a new light on the two known formulae and it then leads very simply to a third new formula.     

“Skewness and Stock Option Prices”, Hans Gerber and Bruno Landry,July 1997

We determine the optimal amount of life insurance for a household of two wage earners. We consider the simple case of exponential utility, thereby removing wealth as a factor in buying life insurance, while retaining the relationship among life insurance, income, and the probability of dying and thus losing that income. For insurance purchased via a single premium or premium payable continuously, we explicitly determine the optimal death benefit. We show that if the premium is determined to target a specific probability of loss per policy, then the rates of consumption are identical under single premium or continuously payable premium. Thus, not only is equivalence of consumption achieved for the households under the two premium schemes, it is also obtained for the insurance company in the sense of equivalence of loss probabilities.     

A diffusion approximation for the ruin function of a risk process with compounding assets

Abstract

The traditional theory of collective risk is concerned with fluctuations in the capital reserve {Y(t): t ?O} of an insurance company. The classical model represents {Y(t)} as a positive constant x (initial capital) plus a deterministic linear function (cumulative income) minus a compound Poisson process (cumulative claims). The central problem is to determine the ruin probability ψ(x) that capital ever falls to zero. It is known that, under reasonable assumptions, one can approximate {Y(t)} by an appropriate Wiener process and hence ψ(.) by the corresponding exponential function of (Brownian) first passage probabilities. This paper considers the classical model modified by the assumption that interest is earned continuously on current capital at rate β > O. It is argued that Y(t) can in this case be approximated by a diffusion process Y*(t) which is closely related to the classical Ornstein-Uhlenbeck process. The diffusion {Y*(t)}, which we call compounding Brownian motion, reduces to the ordinary Wiener process when β = O. The first passage probabilities for Y*(t) are found to form a truncated normal distribution, which approximates the ruin function ψ(.) for the model with compounding assets. The approximate expression for ψ(.) is compared against the exact expression for a special case in which the latter is known. Assuming parameter values for which one would anticipate a good approximation, the two expressions are found to agree extremely well over a wide range of initial asset levels.     

A ruin problem for the insurance of expanding portfolios

Abstract

The purpose of this short note is to demonstrate the power of very straightforward branching process methods outside their traditional realm of application. We shall consider an insurance scheme where claims do not necessarily arise as a stationary process. Indeed, the number of policy-holders is changing so that each of them generates a random number of new insurants. Each one of these make claims of random size at random instants, independently but with the same distribution for different individuals. Premiums are supposed equal for all policy-holders. It is proved that there is, for an expanding portfolio, only one premium size which is fair in the sense that if the premium is larger than that, then the profit of the insurer grows infinite with time, whereas a smaller premium leads to his inevitable ruin. (Branching process models for the development of the portfolio may seem unrealistic. However, they do include the classical theory, where independent and identically distributed claims arise at the points of a renewal process.)     

Valuation of Equity-Linked Insurance and Annuity Products with Binomial Models

Abstract

In this paper we develop a valuation method for equity-linked insurance products. We assume that the premium information of term life insurances, pure endowment insurances, and endowment insurances at all maturities is obtainable within a company or from the insurance market. Using a method similar to that of Jarrow and Turnbull (1995), we derive three martingale probability measures associated with these basic insurance products. These measures are agedependent, include an adjustment for the mortality risk, and reproduce the premiums of the respective insurance products. We then extend the martingale measures to include the financial market information using copulas and use them to evaluate equity-linked insurance contracts and equity-indexed annuities in particular. This is different from the traditional approach under which diversification of mortality risk is assumed. A detailed numerical analysis is performed for various existing equity-indexed annuities in the North American market.     

Optimal constrained investment in the Cramer-Lundberg model

We consider an insurance company whose surplus is represented by the classical Cramer-Lundberg process. The company can invest its surplus in a risk-free asset and in a risky asset, governed by the Black-Scholes equation. There is a constraint that the insurance company can only invest in the risky asset at a limited leveraging level; more precisely, when purchasing, the ratio of the investment amount in the risky asset to the surplus level is no more than a; and when short-selling, the proportion of the proceeds from the short-selling to the surplus level is no more than b. The objective is to find an optimal investment policy that minimizes the probability of ruin. The minimal ruin probability as a function of the initial surplus is characterized by a classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We study the optimal control policy and its properties. The interrelation between the parameters of the model plays a crucial role in the qualitative behavior of the optimal policy. For example, for some ratios between a and b, quite unusual and at first ostensibly counterintuitive policies may appear, like short-selling a stock with a higher rate of return to earn lower interest, or borrowing at a higher rate to invest in a stock with lower rate of return. This is in sharp contrast with the unrestricted case, first studied in Hipp and Plum, or with the case of no short-selling and no borrowing studied in Azcue and Muler.     

Probability of ruin with variable premium rate

Abstract

The probability of ruin is investigated under the influence of a premium rate which varies with the level of free reserves. Section 4 develops a number of inequalities for the ruin probability, establishing upper and lower bounds for it in Theorem 4. Theorem 5 gives an expression for the ruin probability, and it is seen in Section 5 that this amounts to a generalization of the ruin probability given by Gerber for the special case of a negative exponential claim size distribution. In that same section it is shown the Lundberg's inequality is not derivable from the generalized theory of Section 4, and this is seen as a drawback of the methods used there. Sections 6 and 7 deal with some special cases, including claim size distributions with monotone failure rates. Section 8 shows that, in contrast with the result for a constant premium that the probability of ruin for zero initial reserve is independent of the claim size distribution, the same result does not hold when the premium rate is allowed to vary. Section 9 gives some comments on the possible effect of “dangerousness” of a claim size distribution on ruin probability.     

“Social Security-Adequacy,Equity, and Progressiveness: A Review of Criteria Based on Experience in Canada and the United States,” Robert L. Brown and Jeffrey Ip,January 2000

Abstract

We consider an optimal dynamic control problem for an insurance company with opportunities of proportional reinsurance and investment. The company can purchase proportional reinsurance to reduce its risk level and invest its surplus in a financial market that has a Black-Scholes risky asset and a risk-free asset. When investing in the risk-free asset, three practical borrowing constraints are studied individually: (B1) the borrowing rate is higher than lending (saving) rate, (B2) the dollar amount borrowed is no more than K > 0, and (B3) the proportion of the borrowed amount to the surplus level is no more than k > 0. Under each of the constraints, the objective is to minimize the probability of ruin. Classical stochastic control theory is applied to solve the problem. Specifically, the minimal ruin probability functions are obtained in closed form by solving Hamilton-Jacobi-Bellman (HJB) equations, and their associated optimal reinsurance-investment policies are found by verification techniques.     

Authors’ Reply: Credit Standing and the Fair Value of Liabilities: A Critique - Discussion by Marsha Wallace

Abstract

As investment plays an increasingly important role in the insurance business, ruin analysis in the presence of stochastic interest (or stochastic return on investments) has become a key issue in modern risk theory, and the related results should be of interest to actuaries. Although the study of insurance risk models with stochastic interest has attracted a fair amount of attention in recent years, many significant ruin problems associated with these models remain to be investigated. In this paper we consider a risk process with stochastic interest in which the basic risk process is the classical risk process and the stochastic interest process (or the stochastic return-on-investmentgenerating process) is a compound Poisson process with positive drift. Within this framework, we first derive an integro-differential equation for the Gerber-Shiu expected discounted penalty function, and then obtain an exact solution to the equation. We also obtain closed-form expressions for the expected discounted penalty function in some special cases. Finally, we examine a lower bound for the ruin probability of the risk process.     

An optimal stopping problem in risk theory

Abstract

In classical risk theory often stationary premium and claim processes are considered. In some cases it is more convenient to model non-stationary processes which describe a movement from environmental conditions, for which the premiums were calculated, to less favorable circumstances. This is done by a Markov-modulated Poisson claim process. Moreover the insurance company is allowed to stop the process at some random time, if the situation seems unfavorable, in order to calculate new premiums. This leads to an optimal stopping problem which is solved explicitly to some extent.     

Oversigt over de udenlandske Aktuartidsskrifter

Abstract

In the classical compound Poisson risk model, Lundberg's inequality provides both an upper bound for, and an approximation to, the probability of ultimate ruin. The result can be applied only when the moment generating function of the individual claim amount distribution exists. In this paper we derive an upper bound for the probability of ultimate ruin when the moment generating function of the individual claim amount distribution does not exist.     

An Actuarial Analysis of Participating Life Insurance

An actuarial model is developed to reveal the intrinsic nature of participating life insurance. The basic safe-side criterion is examined. It is established how the first-order prospective net premium reserve includes safety margins or bonus loadings, and it is demonstrated how the bonus loadings are currently released. It is demonstrated how surplus may be distributed and accumulated as a terminal bonus in an equitable way. The level premium is divided into a variable recurrent single premium and a variable natural premium, and an alternative to the prospective net premium reserve is examined. A capitalization of future safety margins or bonus loadings, which are related to past premiums and the paid-up benefit, may allow the insurance company a considerable increase in investment freedom. The theory is illustrated by numerical results.     

Self-Annuitization and Ruin in Retirement

Abstract

At retirement, most individuals face a choice between voluntary annuitization and discretionary management of assets with systematic withdrawals for consumption purposes. Annuitization–buying a life annuity from an insurance company–assures a lifelong consumption stream that cannot be outlived, but it is at the expense of a complete loss of liquidity. On the other hand, discretionary management and consumption from assets–self-annuitization–preserves flexibility but with the distinct risk that a constant standard of living will not be maintainable.

In this paper we compute the lifetime and eventual probability of ruin (PoR) for an individual who wishes to consume a fixed periodic amount–a self-constructed annuity–from an initial endowment invested in a portfolio earning a stochastic (lognormal) rate of return. The lifetime PoR is the probability that net wealth will hit zero prior to a stochastic date of death. The eventual PoR is the probability that net wealth will ever hit zero for an infinitely lived individual.

We demonstrate that the probability of ruin can be represented as the probability that the stochastic present value (SPV) of consumption is greater than the initial investable wealth. The lifetime and eventual probabilities of ruin are then obtained by evaluating one minus the cumulative density function of the SPV at the initial wealth level. In that eventual case, we offer a precise analytical solution because the SPV is known to be a reciprocal gamma distribution. For the lifetime case, using the Gompertz law of mortality, we provide two approximations. Both involve “moment matching” techniques that are motivated by results in Arithmetic Asian option pricing theory. We verify the accuracy of these approximations using Monte Carlo simulations. Finally, a numerical case study is provided using Canadian mortality and capital market parameters. It appears that the lifetime probability of ruin–for a consumption rate that is equal to the life annuity payout–is at its lowest with a well-diversified portfolio.     

Some properties of the ruin function in the collective theory of risk

Abstract

It is well known that the chief aim of all theory of risk is to attain a sort of objective and somehow confirmed opinion of how and to which extent an insurance company ought to reinsure its risks in order that the probability of ruin by random fluctuations of the risk process shall become so small that it can be overlooked in practice.     

Asymptotic Investment Behaviors under a Jump-Diffusion Risk Process

We study an optimal investment control problem for an insurance company. The surplus process follows the Cramer-Lundberg process with perturbation of a Brownian motion. The company can invest its surplus into a risk-free asset and a Black-Scholes risky asset. The optimization objective is to minimize the probability of ruin. We show by new operators that the minimal ruin probability function is a classical solution to the corresponding HJB equation. Asymptotic behaviors of the optimal investment control policy and the minimal ruin probability function are studied for low surplus levels with a general claim size distribution. Some new asymptotic results for large surplus levels in the case with exponential claim distributions are obtained. We consider two cases of investment control: unconstrained investment and investment with a limited amount.