Portfolio optimization under convex incentive schemes

We consider the terminal wealth utility maximization problem from the point of view of a portfolio manager who is paid by an incentive scheme, which is given as a convex function g of the terminal wealth. The manager’s own utility function U is assumed to be smooth and strictly concave; however, the resulting utility function U°g fails to be concave. As a consequence, the problem considered here does not fit into the classical portfolio optimization theory. Using duality theory, we prove wealth-independent existence and uniqueness of the optimal portfolio in general (incomplete) semimartingale markets as long as the unique optimizer of the dual problem has a continuous law. In many cases, this existence and uniqueness result is independent of the incentive scheme and depends only on the structure of the set of equivalent local martingale measures. As examples, we discuss (complete) one-dimensional models as well as (incomplete) lognormal mixture and popular stochastic volatility models. We also provide a detailed analysis of the case where the unique optimizer of the dual problem does not have a continuous law, leading to optimization problems whose solvability by duality methods depends on the initial wealth of the investor.     

Arbitrage and deflators in illiquid markets

This paper presents a stochastic model for discrete-time trading in financial markets where trading costs are given by convex cost functions and portfolios are constrained by convex sets. The model does not assume the existence of a cash account/numeraire. In addition to classical frictionless markets and markets with transaction costs or bid–ask spreads, our framework covers markets with nonlinear illiquidity effects for large instantaneous trades. In the presence of nonlinearities, the classical notion of arbitrage turns out to have two equally meaningful generalizations, a marginal and a scalable one. We study their relations to state price deflators by analyzing two auxiliary market models describing the local and global behavior of the cost functions and constraints.     

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.     

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.     

The supermartingale property of the optimal wealth process for general semimartingales

We consider an incomplete stochastic financial market where the price processes are described by a vector valued semimartingale that is possibly non locally bounded. We face the classical problem of utility maximization from terminal wealth, under the assumption that the utility function is finite-valued and smooth on the entire real line and satisfies reasonable asymptotic elasticity. In this general setting, it was shown in Biagini and Frittelli (Financ. Stoch. 9, 493–517, 2005) that the optimal claim admits an integral representation as soon as the minimax σ-martingale measure is equivalent to the reference probability measure. We show that the optimal wealth process is in fact a supermartingale with respect to every σ-martingale measure with finite generalized entropy, thus extending the analogous result proved by Schachermayer (Financ. Stoch. 4, 433–457, 2003) for the locally bounded case.      

Optimal investments for risk- and ambiguity-averse preferences: a duality approach

Ambiguity, also called Knightian or model uncertainty, is a key feature in financial modeling. A recent paper by Maccheroni et al. (preprint, 2004) characterizes investor preferences under aversion against both risk and ambiguity. Their result shows that these preferences can be numerically represented in terms of convex risk measures. In this paper we study the corresponding problem of optimal investment over a given time horizon, using a duality approach and building upon the results by Kramkov and Schachermayer (Ann. Appl. Probab. 9, 904–950, 1999; Ann. Appl. Probab. 13, 1504–1516, 2003). Supported by Deutsche Forschungsgemeinschaft through the SFB 649 “Economic Risk”.     

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.     

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.     

Minimizing the lifetime ruin under borrowing and short-selling constraints

In this paper, the optimal investment strategies for minimizing the probability of lifetime ruin under borrowing and short-selling constraints are found. The investment portfolio consists of multiple risky investments and a riskless investment. The investor withdraws money from the portfolio at a constant rate proportional to the portfolio value. In order to find the results, an auxiliary market is constructed, and the techniques of stochastic optimal control are used. Via this method, we show how the application of stochastic optimal control is possible for minimizing the probability of lifetime ruin problem defined under an auxiliary market.     

A bridge from ruin theory to credit risk

The structural model uses the firm-value process and the default threshold to obtain the implied credit spread. Merton’s (J Finance 29:449–470, 1974) credit spread is reported too small compared to the observed market spread. Zhou (J Bank Finance 25:2015–2040, 2001) proposes a jump-diffusion firm-value process and obtains a credit spread that is closer to the observed market spread. Going in a different direction, the reduced-form model uses the observed market credit spread to obtain the probability of default and the mean recovery rate. We use a jump-diffusion firm-value process and the observed credit spread to obtain the implied jump distribution. Therefore, the discrepancy in credit spreads between the structural model and the reduced-form model can be removed. From the market credit spread, we obtain the implied probability of default and the mean recovery rate. When the solvency-ratio process in credit risk and the surplus process in ruin theory both follow jump-diffusion processes, we show a bridge between ruin theory and credit risk so that results developed in ruin theory can be used to develop analogous results in credit risk. Specifically, when the jump is Logexponentially distributed, it results in a Beta distributed recovery rate that is close to market experience. For bonds of multiple seniorities, we obtain closed-form solutions of the mean and variance of the recovery rate. We prove that the defective renewal equation still holds, even if the jumps are possibly negative. Therefore, we can use ruin theory as a methodology for assessing credit ratings.      

A risk-sensitive stochastic control approach to an optimal investment problem with partial information

We consider an infinite time horizon optimal investment problem where an investor tries to maximize the probability of beating a given index. From a mathematical viewpoint, this is a large deviation probability control problem. As shown by Pham (in Syst. Control Lett. 49: 295–309, 2003; Financ. Stoch. 7: 169–195, 2003), its dual problem can be regarded as an ergodic risk-sensitive stochastic control problem. We discuss the partial information counterpart of Pham (in Syst. Control Lett. 49: 295–309, 2003; Financ. Stoch. 7: 169–195, 2003). The optimal strategy and the value function for the dual problem are constructed by using the solution of an algebraic Riccati equation. This equation is the limit equation of a time inhomogeneous Riccati equation derived from a finite time horizon problem with partial information. As a result, we obtain explicit representations of the value function and the optimal strategy for the problem. Furthermore we compare the optimal strategies and the value functions in both full and partial information cases.

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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.     

On a Class of Renewal Risk Processes

Abstract

In this paper I show how methods that have been applied to derive results for the classical risk process can be adapted to derive results for a class of risk processes in which claims occur as a renewal process. In particular, claims occur as an Erlang process. I consider the problem of finding the survival probability for such risk processes and then derive expressions for the probability and severity of ruin and for the probability of absorption by an upper barrier. Finally, I apply these results to consider the problem of finding the distribution of the maximum deficit during the period from ruin to recovery to surplus level 0.     

“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 accuracy of phase-type approximations of heavy-tailed risk models

Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a pre-specified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.     

Comparison of Option Prices in Semimartingale Models

In this paper we generalize the recent comparison results of El Karoui et al. (Math Finance 8:93–126, 1998), Bellamy and Jeanblanc (Finance Stoch 4:209–222, 2000) and Gushchin and Mordecki (Proc Steklov Inst Math 237:73–113, 2002) to d-dimensional exponential semimartingales. Our main result gives sufficient conditions for the comparison of European options with respect to martingale pricing measures. The comparison is with respect to convex and also with respect to directionally convex functions. Sufficient conditions for these orderings are formulated in terms of the predictable characteristics of the stochastic logarithm of the stock price processes. As examples we discuss the comparison of exponential semimartingales to multivariate diffusion processes, to stochastic volatility models, to Lévy processes, and to diffusions with jumps. We obtain extensions of several recent results on nontrivial price intervals. A crucial property in this approach is the propagation of convexity property. We develop a new approach to establish this property for several further examples of univariate and multivariate processes.     

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.     

Ruin Probabilities for Insurance Models Involving Investments

In this paper we study the ruin problem for insurance models that involve investments. Our risk reserve process is an extension of the classical Cramér-Lundberg model, which will contain stochastic interest rates, reserve-dependent expense loading, diffusion perturbed models, and many others as special cases. By introducing a new type of exponential martingale parametrized by a general rate function, we put various Cramér-Lundberg type estimations into a unified framework. We show by examples that many existing Lundberg-type bounds for ruin probabilities can be recovered by appropriately choosing the rate functions.     

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.