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.     

“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.     

On The Decomposition Of The Ruin Probability For A Jump-Diffusion Surplus Process Compounded By A Geometric Brownian Motion

Abstract

If one assumes that the surplus of an insurer follows a jump-diffusion process and the insurer would invest its surplus in a risky asset, whose prices are modeled by a geometric Brownian motion, the resulting surplus for the insurer is called a jump-diffusion surplus process compounded by a geometric Brownian motion. In this resulting surplus process, ruin may be caused by a claim or oscillation. We decompose the ruin probability in the resulting surplus process into the sum of two ruin probabilities: the probability that ruin is caused by a claim, and the probability that ruin is caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When claim sizes are exponentially distributed, asymptotical formulas of the ruin probabilities are derived from the integro-differential equations, and it is shown that all three ruin probabilities are asymptotical power functions with the same orders and that the orders of the power functions are determined by the drift and volatility parameters of the geometric Brownian motion. It is known that the ruin probability for a jump-diffusion surplus process is an asymptotical exponential function when claim sizes are exponentially distributed. The results of this paper further confirm that risky investments for an insurer are dangerous in the sense that either ruin is certain or the ruin probabilities are asymptotical power functions, not asymptotical exponential functions, when claim sizes are exponentially distributed.     

Absolute Ruin Probabilities in a Jump Diffusion Risk Model with Investment

Abstract

This article considers the compound Poisson insurance risk model perturbed by diffusion with investment. We assume that the insurance company can invest its surplus in both a risky asset and the risk-free asset according to a fixed proportion. If the surplus is negative, a constant debit interest rate is applied. The absolute ruin probability function satisfies a certain integro-differential equation. In various special cases, closed-form solutions are obtained, and numerical illustrations are provided.     

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.     

The surplus prior to ruin and the deficit at ruin for a correlated risk process

This paper presents an explicit characterization for the joint probability density function of the surplus immediately prior to ruin and the deficit at ruin for a general risk process, which includes the Sparre-Andersen risk model with phase-type inter-claim times and claim sizes. The model can also accommodate a Markovian arrival process which enables claim sizes to be correlated with the inter-claim times. The marginal density function of the surplus immediately prior to ruin is specifically considered. Several numerical examples are presented to illustrate the application of this result.     

Optimal investment of an insurer with regime-switching and risk constraint

We investigate an optimal investment problem of an insurance company in the presence of risk constraint and regime-switching using a game theoretic approach. A dynamic risk constraint is considered where we constrain the uncertainty aversion to the ‘true’ model for financial risk at a given level. We describe the surplus of an insurance company using a general jump process, namely, a Markov-modulated random measure. The insurance company invests the surplus in a risky financial asset whose dynamics are modeled by a regime-switching geometric Brownian motion. To incorporate model uncertainty, we consider a robust approach, where a family of probability measures is cosidered and the insurance company maximizes the expected utility of terminal wealth in the ‘worst-case’ probability scenario. The optimal investment problem is then formulated as a constrained two-player, zero-sum, stochastic differential game between the insurance company and the market. Different from the other works in the literature, our technique is to transform the problem into a deterministic differential game first, in order to obtain the optimal strategy of the game problem explicitly.     

Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift

Abstract

We extend the work of Browne (1995) and Schmidli (2001), in which they minimize the probability of ruin of an insurer facing a claim process modeled by a Brownian motion with drift. We consider two controls to minimize the probability of ruin: (1) investing in a risky asset and (2) purchasing quota-share reinsurance. We obtain an analytic expression for the minimum probability of ruin and the corresponding optimal controls, and we demonstrate our results with numerical examples.     

Asymptotics of ruin probabilities for controlled risk processes in the small claims case

We consider a risk process with the possibility of investment into a risky asset. The aim of the paper is to obtain the asymptotic behaviour of the ruin probability under the optimal investment strategy in the small claims case. In addition we prove convergence of the optimal investment level as the initial capital tends to infinity.     

Minimizing the Probability of Lifetime Ruin under Random Consumption

Abstract

We determine the optimal investment strategy in a financial market for an individual whose random consumption is correlated with the price of a risky asset. Bayraktar and Young consider this problem and show that the minimum probability of lifetime ruin is the unique convex, smooth solution of its corresponding Hamilton-Jacobi-Bellman equation. In this paper we focus on determining the probability of lifetime ruin and the corresponding optimal investment strategy. We obtain approximations for the probability of lifetime ruin for small values of certain parameters and demonstrate numerically that they are reasonable ones. We also obtain numerical results in cases for which those parameters are not small.     

Authors’ Reply: Corporate Hedging in the Insurance Industry: The Use of Financial Derivatives by U.S. Insurers - Discussion by L. Lee Colquitt; Arlette C. Wilson; Gray G. Venter; Morton Lane; Joan Lamm-Tennant,January 1997

Abstract

This paper studies the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transforms, which can naturally be interpreted as discounting. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. The expected discounted penalty, considered as a function of the initial surplus, satisfies a certain renewal equation, which has a probabilistic interpretation. Explicit answers are obtained for zero initial surplus, very large initial surplus, and arbitrary initial surplus if the claim amount distribution is exponential or a mixture of exponentials. We generalize Dickson’s formula, which expresses the joint distribution of the surplus immediately prior to and at ruin in terms of the probability of ultimate ruin. Explicit results are obtained when dividends are paid out to the stockholders according to a constant barrier strategy.     

On a renewal risk process with dependence under a Farlie–Gumbel–Morgenstern copula

In this article, we consider an extension to the renewal or Sparre Andersen risk process by introducing a dependence structure between the claim sizes and the interclaim times through a Farlie–Gumbel–Morgenstern copula proposed by Cossette et al. (2010) for the classical compound Poisson risk model. We consider that the inter-arrival times follow the Erlang(n) distribution. By studying the roots of the generalised Lundberg equation, the Laplace transform (LT) of the expected discounted penalty function is derived and a detailed analysis of the Gerber–Shiu function is given when the initial surplus is zero. It is proved that this function satisfies a defective renewal equation and its solution is given through the compound geometric tail representation of the LT of the time to ruin. Explicit expressions for the discounted joint and marginal distribution functions of the surplus prior to the time of ruin and the deficit at the time of ruin are derived. Finally, for exponential claim sizes explicit expressions and numerical examples for the ruin probability and the LT of the time to ruin are given.     

Ruin Problems for Phase-Type(2) Risk Processes

In this paper we consider a risk process in which claim inter-arrival times have a phase-type(2) distribution, a distribution with a density satisfying a second order linear differential equation. We consider some ruin related problems. In particular, we consider the compound geometric representation of the infinite time survival probability, as well as the (defective) distributions of the surplus immediately prior to ruin and of the deficit at ruin. We also consider explicit solutions for the infinite time ruin probability in the case where the individual claim amount distribution is phase-type.     

Moments of Discounted Dividends for a Threshold Strategy in the Compound Poisson Risk Model

Abstract

We consider a compound Poisson risk model in which part of the premium is paid to the shareholders as dividends when the surplus exceeds a specified threshold level. In this model we are interested in computing the moments of the total discounted dividends paid until ruin occurs. However, instead of employing the traditional argument, which involves conditioning on the time and amount of the first claim, we provide an alternative probabilistic approach that makes use of the (defective) joint probability density function of the time of ruin and the deficit at ruin in a classical model without a threshold. We arrive at a general formula that allows us to evaluate the moments of the total discounted dividends recursively in terms of the lower-order moments. Assuming the claim size distribution is exponential or, more generally, a finite shape and scale mixture of Erlangs, we are able to solve for all necessary components in the general recursive formula. In addition to determining the optimal threshold level to maximize the expected value of discounted dividends, we also consider finding the optimal threshold level that minimizes the coefficient of variation of discounted dividends. We present several numerical examples that illustrate the effects of the choice of optimality criterion on quantities such as the ruin probability.     

Minimizing the Probability of Ruin When Consumption is Ratcheted

Abstract

We assume that an agent’s rate of consumption is ratcheted; that is, it forms a nondecreasing process. We assume that the agent invests in a financial market with one riskless and one risky asset, with the latter’s price following geometric Brownian motion as in the Black-Scholes model. Given the rate of consumption, we act as financial advisers and find the optimal investment strategy for the agent who wishes to minimize his probability of ruin. To solve this minimization problem, we use techniques from stochastic optimal control.     

Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin

Abstract

I study the problem of how individuals should invest their wealth in a risky financial market to minimize the probability that they outlive their wealth, also known as the probability of lifetime ruin. Specifically, I determine the optimal investment strategy of an individual who targets a given rate of consumption and seeks to minimize the probability of lifetime ruin. Two forms of the consumption function are considered: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of his or her wealth. The first is arguably more realistic, but the second has a close connection with optimal consumption in Merton’s model of optimal consumption and investment under power utility.

For constant force of mortality, I determine (a) the probability that individuals outlive their wealth if they follow the optimal investment strategy; (b) the corresponding optimal investment rule that tells individuals how much money to invest in the risky asset for a given wealth level; (c) comparative statics for the functions in (a) and (b); (d) the distribution of the time of lifetime ruin, given that ruin occurs; and (e) the distribution of bequest, given that ruin does not occur. I also include numerical examples to illustrate how the formulas developed in this paper might be applied.     

In the insurance business risky investments are dangerous: the case of negative risk sums

We investigate models with negative risk sums when the company invests its reserve into a risky asset whose price follows a geometric Brownian motion. Our main result is an exact asymptotic of the ruin probabilities for the case of exponentially distributed benefits. As in the case of non-life insurance with exponential claims, the ruin probabilities are either decreasing with a rate given by a power function (the case of small volatility) or equal to one identically (the case of large volatility). The result allows us to quantify the share of reserve to invest into such a risky asset to avoid a catastrophic outcome, namely the ruin with probability one. We address also the question of smoothness of the ruin probabilities as a function of the initial reserve for generally distributed jumps.     

On a conjecture related to the ruin probability for nonhomogeneous exponentially distributed claims

Recently, some recursive formulas have been obtained for the ruin probability evaluated at or before claim instants for a surplus process under the assumptions that the claim sizes are independent, nonhomogeneous Erlang distributed, and independent of the inter-claim revenues, which are assumed to be independent, identically distributed, following an arbitrary distribution. Based on numerical examples, a conjecture has also been stated relating the order in which the claims arrive to the magnitude of the corresponding ruin probability. In this paper, we prove this conjecture in the particular case when the claims are all exponentially distributed with different parameters.     

Ruin Probabilities And Capital Requirement for Open Automobile Portfolios With a Bonus-Malus System Based on Claim Counts

For a large motor insurance portfolio, on an open environment, we study the impact of experience rating in finite and continuous time ruin probabilities. We consider a model for calculating ruin probabilities applicable to large portfolios with a Markovian Bonus-Malus System (BMS), based on claim counts, for an automobile portfolio using the classical risk framework model. New challenges are brought when an open portfolio scenario is introduced. When compared with a classical BMS approach ruin probabilities may change significantly. By using a BMS of a Portuguese insurer, we illustrate and discuss the impact of the proposed formulation on the initial surplus required to target a given ruin probability. Under an open portfolio setup, we show that we may have a significant impact on capital requirements when compared with the classical BMS, by having a significant reduction on the initial surplus needed to maintain a fixed level of the ruin probability.