Optimal portfolio choice for an insurer with loss aversion

The problem of optimal investment for an insurance company attracts more attention in recent years. In general, the investment decision maker of the insurance company is assumed to be rational and risk averse. This is inconsistent with non fully rational decision-making way in the real world. In this paper we investigate an optimal portfolio selection problem for the insurer. The investment decision maker is assumed to be loss averse. The surplus process of the insurer is modeled by a Lévy process. The insurer aims to maximize the expected utility when terminal wealth exceeds his aspiration level. With the help of martingale method, we translate the dynamic maximization problem into an equivalent static optimization problem. By solving the static optimization problem, we derive explicit expressions of the optimal portfolio and the optimal wealth process.     

Stochastic differential portfolio games for an insurer in a jump-diffusion risk process

We discuss an optimal portfolio selection problem of an insurer who faces model uncertainty in a jump-diffusion risk model using a game theoretic approach. In particular, the optimal portfolio selection problem is formulated as a two-person, zero-sum, stochastic differential game between the insurer and the market. There are two leader-follower games embedded in the game problem: (i) The insurer is the leader of the game and aims to select an optimal portfolio strategy by maximizing the expected utility of the terminal surplus in the “worst-case” scenario; (ii) The market acts as the leader of the game and aims to choose an optimal probability scenario to minimize the maximal expected utility of the terminal surplus. Using techniques of stochastic linear-quadratic control, we obtain closed-form solutions to the game problems in both the jump-diffusion risk process and its diffusion approximation for the case of an exponential utility.     

Optimal strategies for hedging portfolios of unit-linked life insurance contracts with minimum death guarantee

In this paper, we are interested in hedging strategies which allow the insurer to reduce the risk to their portfolio of unit-linked life insurance contracts with minimum death guarantee. Hedging strategies are developed in the Black and Scholes model and in the Merton jump-diffusion model. According to the new frameworks (IFRS, Solvency II and MCEV), risk premium is integrated into our valuations. We will study the optimality of hedging strategies by comparing risk indicators (Expected loss, volatility, VaR and CTE) in relation to transaction costs and costs generated by the re-hedging error. We will analyze the robustness of hedging strategies by stress-testing the effect of a sharp rise in future mortality rates and a severe depreciation in the price of the underlying asset.     

离散时间单位连结人寿保险合同的局部风险最小对冲策略

单位连结人寿保险合同是保险利益依赖于某特定股票的价格的保险合同。当保险公司发行这样的保险合同后，保险公司将面临金融和被保险人死亡率两类风险。因此这样的保险合同相当对不完全金融市场上的或有索取权，不能利用自我融资交易策略复制出。本提出利用不完全市场的局部风险最小对冲方法对冲保险的风险，我们在离散时间的框架下给出了局部风险最小对冲策略。     

离散时间单位连结人寿保险合同的局部风险最小对冲策略

单位连结人寿保险合同是保险利益依赖于某特定股票的价格的保险合同 .当保险公司发行这样的保险合同后 ,保险公司将面临金融和被保险人死亡率两类风险 .因此这样的保险合同相当于不完全金融市场上的或有索取权 ,不能利用自我融资交易策略复制出 .本文提出利用不完全市场的局部风险最小对冲方法对冲保险者的风险 .我们在离散时间的框架下给出了局部风险最小对冲策略 .     

Assessing the solvency of insurance portfolios via a continuous-time cohort model

This paper evaluates the solvency of a portfolio of assets and liabilities of an insurer subject to both longevity and financial risks. Liabilities are evaluated at fair-value and, as a consequence, interest-rate risk can affect both the assets and the liabilities. Longevity risk is described via a continuous-time cohort model. We evaluate the effects of natural hedging strategies on the risk profile of an insurance portfolio in run-off. Numerical simulations, calibrated to UK historical data, show that systematic longevity risk is of particular importance and needs to be hedged. Natural hedging can improve the solvency of the insurer, if interest-rate risk is appropriately managed. We stress that asset allocation choices should not be independent of the composition of the liability portfolio of the insurer.     

Mean–variance asset–liability management: Cointegrated assets and insurance liability

The cointegration of major financial markets around the globe is well evidenced with strong empirical support. This paper considers the continuous-time mean–variance (MV) asset–liability management (ALM) problem for an insurer investing in an incomplete financial market with cointegrated assets. The number of trading assets is allowed to be less than the number of Brownian motions spanning the market. The insurer also faces the risk of paying uncertain insurance claims during the investment period. We assume that the cointegration market follows the diffusion limit of the error-correction model for cointegrated time series. Using the Markowitz (1952) MV portfolio criterion, we consider the insurer’s problem of minimizing variance in the terminal wealth, given an expected terminal wealth subject to interim random liability payments following a compound Poisson process. We generalize the technique developed by Lim (2005) to tackle this problem. The particular structure of cointegration enables us to solve the ALM problem completely in the sense that the solutions of the continuous-time portfolio policy and efficient frontier are obtained as explicit and closed-form formulas.     

Optimal hedging strategies for multi-period guarantees in the presence of transaction costs: A stochastic programming approach

Multi-period guarantees are often embedded in life insurance contracts. In this paper we consider the problem of hedging these multi-period guarantees in the presence of transaction costs. We derive the hedging strategies for the cheapest hedge portfolio for a multi-period guarantee that with certainty makes the insurance company able to meet the obligations from the insurance policies it has issued. We find that by imposing transaction costs, the insurance company reduces the rebalancing of the hedge portfolio. The cost of establishing the hedge portfolio also increases as the transaction cost increases. For the multi-period guarantee there is a rather large rebalancing of the hedge portfolio as we go from one period to the next. By introducing transaction costs we find the size of this rebalancing to be reduced. Transaction costs may therefore be one possible explanation for why we do not see the insurance companies performing a large rebalancing of their investment portfolio at the end of each year.     

Optimal insurance with belief heterogeneity and incentive compatibility

People may evaluate risk differently in the insurance market. Motivated by this, we examine an optimal insurance problem allowing the insured and the insurer to have heterogeneous beliefs about loss distribution. To reduce ex post moral hazard, we follow Huberman et al. (1983) to assume that alternative insurance contracts satisfy the principle of indemnity and the incentive-compatible constraint. Under the assumption that the insurance premium is calculated by the expected value principle, we establish a necessary and sufficient condition for an optimal insurance solution and provide a practical scheme to improve any suboptimal insurance strategy under an arbitrary form of belief heterogeneity. By virtue of this condition, we explore qualitative properties of optimal solutions, and derive optimal insurance contracts explicitly for some interesting forms of belief heterogeneity. As a byproduct of this investigation, we find that Theorem 3.6 of Young (1999) is not completely true.     

Hedging unit‐linked life insurance contracts in a financial market driven by shot‐noise processes

We consider the risk‐minimizing hedging problem for unit‐linked life insurance in a financial market driven by a shot‐noise process. Because the financial market is incomplete, the insurance claims cannot be hedged completely by trading stocks and bonds only, leaving some risk to the insurer. The theory of ((pseudo) locally) risk‐minimization is applied after a change of measure. Then the risk‐minimizing trading strategies and the associated intrinsic risk processes are determined for two types of unit‐linked contracts represented by the pure endowment and the term insurance. Copyright © 2009 John Wiley & Sons, Ltd.     

Mean-variance portfolio selection for a non-life insurance company

We consider a collective insurance risk model with a compound Cox claim process, in which the evolution of a claim intensity is described by a stochastic differential equation driven by a Brownian motion. The insurer operates in a financial market consisting of a risk-free asset with a constant force of interest and a risky asset which price is driven by a Lévy noise. We investigate two optimization problems. The first one is the classical mean-variance portfolio selection. In this case the efficient frontier is derived. The second optimization problem, except the mean-variance terminal objective, includes also a running cost penalizing deviations of the insurer’s wealth from a specified profit-solvency target which is a random process. In order to find optimal strategies we apply techniques from the stochastic control theory.     

Pricing of long dated equity-linked life insurance contracts

This article adopts an approach to pricing of equity-linked life insurance contracts, which only requires the existence of the numéraire portfolio. An equity-linked life insurance contract is equivalent to a sum of the guaranteed amount and the value of an option on the equity index with some mortality risk attached. The numéraire portfolio equals the growth optimal portfolio and is used as numéraire or benchmark, where the real-world probability measure is taken as pricing measure. To obtain tractable solutions the short rate is modelled as a quadratic form of some Gaussian factor processes. Furthermore, the dynamics of the mortality rate is modelled as a threshold life table. The dynamics of the discounted equity market index or benchmark is modelled by a time transformed squared Bessel process. The equity-linked life insurance contracts are evaluated analytically.     

Bond portfolio management with repo contracts: the Italian case

In this paper we present a model for management of bond portfolio including financing and investment repo contracts. Different specifications are suggested in order to reduce the problem to a linear programming problem and to consider a self-financing portfolio. The models are tested on historical data assuming a technical time scale equal to the minimum length of the contracts in the portfolio. We also compared different operative strategies on a time horizon of one month. This revised version was published online in June 2006 with corrections to the Cover Date.     

Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets

In this paper, we investigate the optimal time-consistent investment–reinsurance strategies for an insurer with state dependent risk aversion and Value-at-Risk (VaR) constraints. The insurer can purchase proportional reinsurance to reduce its insurance risks and invest its wealth in a financial market consisting of one risk-free asset and one risky asset, whose price process follows a geometric Brownian motion. The surplus process of the insurer is approximated by a Brownian motion with drift. The two Brownian motions in the insurer’s surplus process and the risky asset’s price process are correlated, which describe the correlation or dependence between the insurance market and the financial market. We introduce the VaR control levels for the insurer to control its loss in investment–reinsurance strategies, which also represent the requirement of regulators on the insurer’s investment behavior. Under the mean–variance criterion, we formulate the optimal investment–reinsurance problem within a game theoretic framework. By using the technique of stochastic control theory and solving the corresponding extended Hamilton–Jacobi–Bellman (HJB) system of equations, we derive the closed-form expressions of the optimal investment–reinsurance strategies. In addition, we illustrate the optimal investment–reinsurance strategies by numerical examples and discuss the impact of the risk aversion, the correlation between the insurance market and the financial market, and the VaR control levels on the optimal strategies.     

Mean–variance asset–liability management with asset correlation risk and insurance liabilities

Consider an insurer who invests in the financial market where correlations among risky asset returns are randomly changing over time. The insurer who faces the risk of paying stochastic insurance claims needs to manage her asset and liability by taking into account of the correlation risk. This paper investigates the impact of correlation risk to the optimal asset–liability management (ALM) of an insurer. We employ the Wishart process to model the stochastic covariance matrix of risky asset returns. The insurer aims to minimize the variance of the terminal wealth given an expected terminal wealth subject to the risk of paying out random liabilities of compound Poisson process. This ALM problem then becomes a linear–quadratic stochastic optimal control problem with stochastic volatilities, stochastic correlations and jumps. The recognition of an affine form in the solution process enables us to derive the explicit closed-form solution to the optimal ALM portfolio policy, obtain the efficient frontier, and identify the condition that the solution is well behaved.     

Design of insurance contracts using stochastic programming in forestry planning

This work addresses a tactical planning problem faced by a forestry firm, deciding which timber units to harvest and what roads to build to obtain the greatest possible benefits. We include uncertainty in prices by means of utility theory. This enables solutions to be found that the firm finds preferable to those obtained when risk aversion is ignored and makes it possible to design insurance contracts that benefit the firm while also being attractive to an insurer. Two types of contract are designed; one dependent on the firm’s operating result and the other independent of it. Metrics are then developed to quantify the benefits conferred by a contract, demonstrating that the latter contract type dominates the former. These results are then illustrated by applying them to a simplified planning problem of a forest owned by the Chilean forestry operator Millalemu.     

Risikopreis in der (Rück-)Versicherung: Ein alternativer Ansatz zum Shortfall-Loading

We model the impact of a (re-)insurance transaction on the (re-)insurer share price. In a second step, we investigate under which conditions (for instance the minimum premium and the optimal investment strategy) this impact will have a positive effect on shareholder portfolio. The model presented here tries to combine, under simple hypotheses, the diversification of the risks within the (re-)insurer portfolio with the diversification of a given shareholder’s portfolio.     

Life insurance policy termination and survivorship

There has been some work, e.g. Carriere (1998), Valdez (2000b), and Valdez (2001), leading to the development of statistical models in understanding the mortality pattern of terminated policies. However, there is a scant literature on the empirical evidence of the true nature of the relationship between survivorship and persistency in life insurance. When a life insurance contract terminates due to voluntary non-payment of premiums, there is a possible hidden cost resulting from mortality antiselection. This refers to the tendency of policyholders who are generally healthy to select against the insurance company by voluntarily terminating their policies. In this article, we explore the empirical results of the survival pattern of terminated policies, using a follow-up study of the mortality of those policies that terminated from a portfolio of life insurance contracts. The data has been obtained from a major insurer which traced the mortality of their policies withdrawn, for purposes of understanding the mortality antiselection, by obtaining their dates of death from the Social Security Administration office. Using a representative sample of this follow-up data, we modeled the time until a policy lapses and its subsequent mortality pattern. We find some evidence of mortality selection and we consequentially examined the financial cost of policy termination.     

Robust equilibrium excess-of-loss reinsurance and CDS investment strategies for a mean–variance insurer with ambiguity aversion

This paper considers the robust equilibrium reinsurance and investment strategies for an ambiguity-averse insurer under a dynamic mean–variance criterion. The insurer is allowed to purchase excess-of-loss reinsurance and invest in a financial market consisting of a risk-free asset and a credit default swap (CDS). Following a game theoretic approach, robust equilibrium strategies and equilibrium value functions for the pre-default case and the post-default case are derived, respectively. For the ambiguity-averse insurer, in general the equilibrium strategies can be characterized by unique solutions to some algebraic equations. For the degenerate case with an ambiguity-neutral insurer, closed-form expressions of equilibrium strategies and equilibrium value functions are obtained. Numerical examples demonstrate that the consideration of model uncertainty and CDS investment improves the insurer’s utility. In this regard, our paper establishes theoretical and numerical support for the importance of ambiguity aversion, credit risk and their interplay in insurance business.     

Indifference pricing of a life insurance portfolio with risky asset driven by a shot-noise process

In this paper, we investigate the pricing problem for a portfolio of life insurance contracts where the life contingent payments are equity-linked depending on the performance of a risky stock or index. The shot-noise effects are incorporated in the modeling of stock prices, implying that sudden jumps in the stock price are allowed, but their effects may gradually decline over time. The contracts are priced using the principle of equivalent utility. Under the assumption of exponential utility, we find the optimal investment strategy and show that the indifference premium solves a non-linear partial integro-differential equation (PIDE). The Feynman–Kač form solutions are derived for two special cases of the PIDE. We further discuss the problem for the asymptotic shot-noise process, and find the probabilistic representation of the indifference premium. We also provide some numerical examples and analyze parameter sensitivities for the results obtained in this paper.