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.     

“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.     

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.     

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.     

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.     

“Cash Flow Matching: A Risk Management Approach”, Garud Iyengar and Alfred Ka Chun Ma,July, 2009

Abstract

Consider a discrete-time risk model in which the insurer is allowed to invest a proportion of its wealth in a risky stock and keep the rest in a risk-free bond. Assume that the claim amounts within individual periods follow an autoregressive process with heavy-tailed innovations and that the log-returns of the stock follow another auto regressive process, independent of the former one. We derive an asymptotic formula for the finite-time ruin probability and propose a hybrid method, combining simulation with asymptotics, to compute this ruin probability more efficiently. As an application, we consider a portfolio optimization problem in which we determine the proportion invested in the risky stock that maximizes the expected terminal wealth subject to a constraint on the ruin probability.     

Optimal and Simple,Nearly Optimal Rules for Minimizing the Probability Of Financial Ruin in Retirement

Abstract

The increasing risk of poverty in retirement has been well documented; it is projected that current and future retirees’ living expenses will significantly exceed their savings and income. In this paper, we consider a retiree who does not have sufficient wealth and income to fund her future expenses, and we seek the asset allocation that minimizes the probability of financial ruin during her lifetime. Building on the work of Young (2004) and Milevsky, Moore, and Young (2006), under general mortality assumptions, we derive a variational inequality that governs the ruin probability and optimal asset allocation. We explore the qualitative properties of the ruin robability and optimal strategy, present a numerical method for their estimation, and examine their sensitivity to changes in model parameters for specific examples. We then present an easy-to-implement allocation rule and demonstrate via simulation that it yields nearly optimal ruin probability, even under discrete portfolio rebalancing.     

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.     

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.     

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.     

Continuity Estimates for Ruin Probabilities

A method of continuity analysis of ruin probabilities with respect to variation of parameters governing risk processes is proposed. It is based on the representation of the ruin probability as the stationary probability of a reversed process. We apply Kartashov's technique designed for continuity analysis of stationary distributions of general Markov chains in order to obtain desired continuity estimates. The method is illustrated by the Sparre Andersen and Markov modulated risk models.     

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.     

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.     

The Time of Recovery and the Maximum Severity of Ruin in a Sparre Andersen Model

Abstract

Phase-type distributions are one of the most general classes of distributions permitting a Markovian interpretation. Sparre Andersen risk models with phase-type claim interarrival times or phase-type claims can be analyzed using Markovian techniques, and results can be expressed in compact matrix forms. Computations involved are readily programmable in practice.

This paper studies some quantities associated with the first passage time and the time of ruin in a Sparre Andersen risk model with phase-type interclaim times. In an earlier discussion the present author obtained a matrix expression for the Laplace transform of the first time that the surplus process reaches a given target from the initial surplus. Using this result, we analyze (1) the Laplace transform of the recovery time after ruin, (2) the probability that the surplus attains a certain level before ruin, and (3) the distribution of the maximum severity of ruin. We also give a matrix expression for the expected discounted dividend payments prior to ruin for the Sparre Andersen model in the presence of a constant dividend barrier.     

Exact solutions for ruin probability in the presence of an absorbing upper barrier

Abstract

An explicit solution for the probability of ruin in the presence of an absorbing upper barrier was developed by Segerdahl (1970) for the particular case in which both the interoccurrence times between successive claims and the single claim amounts follow an exponential distribution with unit mean. In this paper we show that his method of solution may be extended to produce explicit solutions for two more general types of single claim amount distribution. These are the gamma distribution, denoted γ(a), where a is an integer, and the mixed exponential distribution. Comparisons are drawn between this approach when the upper barrier tends to infinity, and the classical solution for ruin probability in these particular cases given in Cramér (1955).     

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.     

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.     

Optimal Proportional Reinsurance Policies in a Dynamic Setting

We consider dynamic proportional reinsurance strategies and derive the optimal strategies in a diffusion setup and a classical risk model. Optimal is meant in the sense of minimizing the ruin probability. Two basic examples are discussed.     

Insurance premiums and default risk in mutual insurance

Abstract

Two types of default risk are discussed in the article: The traditional “probability of ruin” (insurer being unable to meet his obligations) and a “perceived probability of ruin” (the probability of the insured being affected by ruin). The explicit relationship between these probabilities on the actuarial loading factors of a mutual insurer were developed. The explicit mathematical formulae obtained for these complex relationships were followed also by numerical results. A second concept presented in the paper is related to the idea of actuarially fair premiums. It is shown that the premium must also be a function of the payments of the other insured as well as their claim distributions, reflecting thereby the simultaneity and mutual dependence of the insured.     

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.