A stochastic differential game for optimal investment of an insurer with regime switching

We introduce a model to discuss an optimal investment problem of an insurance company using a game theoretic approach. The model is general enough to include economic risk, financial risk, insurance risk, and model risk. The insurance company invests its surplus in a bond and a stock index. The interest rate of the bond is stochastic and depends on the state of an economy described by a continuous-time, finite-state, Markov chain. The stock index dynamics are governed by a Markov, regime-switching, geometric Brownian motion modulated by the chain. The company receives premiums and pays aggregate claims. Here the aggregate insurance claims process is modeled by either a Markov, regime-switching, random measure or a Markov, regime-switching, diffusion process modulated by the chain. We adopt a robust approach to model risk, or uncertainty, and generate a family of probability measures using a general approach for a measure change to incorporate model risk. In particular, we adopt a Girsanov transform for the regime-switching Markov chain to incorporate model risk in modeling economic risk by the Markov chain. The goal of the insurance company is to select an optimal investment strategy so as to maximize either the expected exponential utility of terminal wealth or the survival probability of the company in the ‘worst-case’ scenario. We formulate the optimal investment problems as two-player, zero-sum, stochastic differential games between the insurance company and the market. Verification theorems for the HJB solutions to the optimal investment problems are provided and explicit solutions for optimal strategies are obtained in some particular cases.     

MDP algorithms for portfolio optimization problems in pure jump markets

We consider the problem of maximizing the expected utility of the terminal wealth of a portfolio in a continuous-time pure jump market with general utility function. This leads to an optimal control problem for piecewise deterministic Markov processes. Using an embedding procedure we solve the problem by looking at a discrete-time contracting Markov decision process. Our aim is to show that this point of view has a number of advantages, in particular as far as computational aspects are concerned. We characterize the value function as the unique fixed point of the dynamic programming operator and prove the existence of optimal portfolios. Moreover, we show that value iteration as well as Howard’s policy improvement algorithm works. Finally, we give error bounds when the utility function is approximated and when we discretize the state space. A numerical example is presented and our approach is compared to the approximating Markov chain method.      

Optimal Asset-Liability Management for an Insurer Under Markov Regime Switching Jump-Diffusion Market

This paper considers an asset-liability management problem under a continuous time Markov regime-switching jump-diffusion market. We assume that the risky stock’s price is governed by a Markov regime-switching jump-diffusion process and the insurance claims follow a Markov regime-switching compound poisson process. Using the Markowitz mean-variance criterion, the objective is to minimize the variance of the insurer’s terminal wealth, given an expected terminal wealth. We get the optimal investment policy. At the same time, we also derive the mean-variance efficient frontier by using the Lagrange multiplier method and stochastic linear-quadratic control technique.     

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 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.     

Dynamic trading with uncertain exit time and transaction costs in a general Markov market

This paper investigates a dynamic trading problem with transaction cost and uncertain exit time in a general Markov market, where the mean vector and covariance matrix of returns depend on the states of the stochastic market, and the market state is regime switching in a time varying state set. Following the framework proposed by Gârleanu and Pedersen (2013), the investor maximizes his or her multi-period mean–variance utility, net of quadratic transaction costs capturing the linear price impact where trades lead to temporary linear changes in prices. The explicit expression for the optimal strategy is derived by using matrix theory technique and dynamic programming approach. Finally, numerical examples are provided to study the effects of transition cost and exit probability on the wealth process, the trading strategy, turnover rate and the total transaction cost.     

Investment decisions when utility depends on wealth and other attributes

The problem of optimal investment under a multivariate utility function allows for an investor to obtain utility not only from wealth, but other (possibly correlated) attributes. In this paper we implement multivariate mixtures of exponential (mixex) utility to address this problem. These utility functions allow for stochastic risk aversions to differing states of the world. We derive some new results for certainty equivalence in this context. By specifying different distributions for stochastic risk aversions, we are able to derive many known, plus several new utility functions, including models of conditional certainty equivalence and multivariate generalisations of HARA utility, which we call dependent HARA utility. Focusing on the case of asset returns and attributes being multivariate normal, we optimise the asset portfolio, and find that the optimal portfolio consists of the Markowitz portfolio and hedging portfolios. We provide an empirical illustration for an investor with a mixex utility function of wealth and sentiment.     

Optimal capital growth with convex shortfall penalties

The optimal capital growth strategy or Kelly strategy has many desirable properties such as maximizing the asymptotic long-run growth of capital. However, it has considerable short-run risk since the utility is logarithmic, with essentially zero Arrow–Pratt risk aversion. It is common to control risk with a Value-at-Risk (VaR) constraint defined on the end of horizon wealth. A more effective approach is to impose a VaR constraint at each time on the wealth path. In this paper, we provide a method to obtain the maximum growth while staying above an ex-ante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall. The effect of the path VaR condition and shortfall penalties is a lower growth rate than the Kelly strategy, but the downside risk is under control. The asset price dynamics are defined by a model with Markov transitions between several market regimes and geometric Brownian motion for prices within a regime. The stochastic investment model is reformulated as a deterministic programme which allows the calculation of the optimal constrained growth wagers at discrete points in time.     

Optimal investment strategies for general utilities under dynamic elasticity of variance models

This paper studies the optimal investment strategies under the dynamic elasticity of variance (DEV) model which maximize the expected utility of terminal wealth. The DEV model is an extension of the constant elasticity of variance model, in which the volatility term is a power function of stock prices with the power being a nonparametric time function. It is not possible to find the explicit solution to the utility maximization problem under the DEV model. In this paper, a dual-control Monte-Carlo method is developed to compute the optimal investment strategies for a variety of utility functions, including power, non-hyperbolic absolute risk aversion and symmetric asymptotic hyperbolic absolute risk aversion utilities. Numerical examples show that this dual-control Monte-Carlo method is quite efficient.     

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.     

Lifetime asset allocation with idiosyncratic and systematic mortality risks

This paper considers a lifetime asset allocation problem with both idiosyncratic and systematic mortality risks. The novelty of the paper is to integrate stochastic mortality, stochastic interest rate and stochastic income into a unified framework. An investor, who is a wage earner receiving a stochastic income, can invest in a financial market, consume part of his wealth and purchase life insurance or annuity so as to maximize the expected utility from consumption, terminal wealth and bequest. The problem is solved via the dynamic programming principle and the Hamilton–Jacobi–Bellman equation. Analytical solutions to the problem are derived, and numerical examples are provided to illustrate our results. It is shown that idiosyncratic mortality risk has significant impacts on the investor’s investment, consumption, life insurance/annuity purchase and bequest decisions regardless of the length of the decision-making horizon. The systematic mortality risk is largely alleviated by trading the longevity bond. However, its impacts on consumption, purchase of life insurance/annuity and bequest as well as the value function are still pronounced, when the decision-making horizon is sufficiently long.     

Portfolio Optimization in Discontinuous Markets under Incomplete Information

We consider the problem of maximization of expected utility from terminal wealth for log and power utility functions in a market model that leads to purely discontinuous processes. We study this problem as a stochastic control problem both under complete as well as incomplete information. Our contribution consists in showing that the optimal strategy can be obtained by solving a system of equations that in some cases is linear and that a certainty equivalence property holds not only for log-utility but also for a power utility function. For the case of a power utility under incomplete information we also present an independent direct approach based on a Zakai-type equation.      

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.     

Around the Life Cycle: Deterministic Consumption-Investment Strategies

We study a classical continuous-time consumption-investment problem of a power utility investor with deterministic labor income with the important feature that the consumption-investment process is constrained to be deterministic. This is motivated by the design of modern pension schemes of defined contribution type where, typically, the savings rate is constant and the proportional investment in growth stocks is a function of age or time-to-retirement, a so-called life-cycle investment strategy. We derive and study the optimal behavior corresponding to the optimal product design within this realistic family of products with deterministic decision profiles. We also propose a couple of suboptimal deterministic strategies inspired from the optimal stochastic strategy and compare the optimal stochastic control, the optimal deterministic control, and these suboptimal deterministic controls. The conclusion is that only little is lost by constraining to deterministic strategies and only little is lost by implementing the suboptimal simple explicit strategies rather than the optimal one we derive.     

Optimal investment for insurers

We consider a stochastic model for the wealth of an insurance company which has the possibility to invest into a risky and a riskless asset under a constant mix strategy. The total insurance claim amount is modeled by a compound Poisson process and the price of the risky asset follows a geometric Brownian motion. We investigate the resulting integrated risk process and the corresponding discounted net loss process. This opens up a way to measure the risk of a negative outcome of the integrated risk process in a stationary way. We provide an approximation of the optimal investment strategy, which maximizes the expected wealth under a risk constraint on the Value-at-Risk.     

Consumption–investment optimization with Epstein–Zin utility in incomplete markets

In a market with stochastic investment opportunities, we study an optimal consumption–investment problem for an agent with recursive utility of Epstein–Zin type. Focusing on the empirically relevant specification where both risk aversion and elasticity of intertemporal substitution are in excess of one, we characterize optimal consumption and investment strategies via backward stochastic differential equations. The superdifferential of indirect utility is also obtained, meeting demands from applications in which Epstein–Zin utilities were used to resolve several asset pricing puzzles. The empirically relevant utility specification introduces difficulties to the optimization problem due to the fact that the Epstein–Zin aggregator is neither Lipschitz nor jointly concave in all its variables.     

Optimal lifetime consumption and investment under a drawdown constraint

We consider the infinite-horizon optimal consumption-investment problem under a drawdown constraint, i.e., when the wealth process never falls below a fixed fraction of its running maximum. We assume that the risky asset is driven by the with constant coefficients. For a general class of utility functions, we provide the value function in explicit form and derive closed-form expressions for the optimal consumption and investment strategy.      

Correspondence between lifetime minimum wealth and utility of consumption

We establish when the two problems of minimizing a function of lifetime minimum wealth and of maximizing utility of lifetime consumption result in the same optimal investment strategy on a given open interval O in wealth space. To answer this question, we equate the two investment strategies and show that if the individual consumes at the same rate in both problems—the consumption rate is a control in the problem of maximizing utility—then the investment strategies are equal only when the consumption function is linear in wealth on O, a rather surprising result. It then follows that the corresponding investment strategy is also linear in wealth and the implied utility function exhibits hyperbolic absolute risk aversion.      

Optimal portfolios with a positive lower bound on final wealth

We consider the determination of optimal portfolios under a lower bound on the final wealth. Possible applications range from capital guarantee strategies over life insurance investment where part of the benefit is a guaranteed return on capital to continuous-time mean-variance problems with a strictly positive lower bound. Our solution method consists of transforming the original problem into a portfolio problem without a positive lower bound but a transformed utility function and a modified initial wealth.     

HARA utility maximization in a Markov-switching bond–stock market

We present a flexible multidimensional bond–stock model incorporating regime switching, a stochastic short rate and further stochastic factors, such as stochastic asset covariance. In this framework we consider an investor whose risk preferences are characterized by the hyperbolic absolute risk-aversion utility function and solve the problem of optimizing the expected utility from her terminal wealth. For the optimal portfolio we obtain a constant-proportion portfolio insurance-type strategy with a Markov-switching stochastic multiplier and prove that it assures a lower bound on the terminal wealth. Explicit and easy-to-use verification theorems are proven. Furthermore, we apply the results to a specific model. We estimate the model parameters and test the performance of the derived optimal strategy using real data. The influence of the investor’s risk preferences and the model parameters on the portfolio is studied in detail. A comparison to the results with the power utility function is also provided.