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This paper studies the optimal investment problem for an insurer in an incomplete market. The insurer's risk process is modeled by a Lévy process and the insurer is supposed to have the option of investing in multiple risky assets whose price processes are described by the standard Black-Scholes model. The insurer aims to maximize the expected utility of terminal wealth. After the market is completed, we obtain the optimal strategies for quadratic utility and constant absolute risk aversion (CARA) utility explicitly via the martingale approach. Finally, computational results are presented for given raw market data.

Recently, the problem of optimal investment for an insurer has attracted a lot of attention, due to the fact that the insurer is allowed to invest in financial markets in practice. In the meantime, this is also a very interesting portfolio selection problem in the finance theory. Some important early work is done by Browne [

In most of these papers, the optimal investment for an insurer is considered in a complete market. But the real market is always incomplete; that is, the number of risky assets (stocks) is strictly less than the dimension of the underlying Brownian motion. Hence, it is necessary to find an optimal strategy in an incomplete market. However, the traditional martingale method cannot be used directly in an incomplete market. To overcome the problem of incompleteness, many researchers have developed different ways to handle the problem. For instance, Karatzas et al. [

The optimal investment problem for an insurer in an incomplete market is studied in the present paper. For this case, the traditional martingale method is problematic. Thus, we first complete the market via the approach proposed by Zhang [

An optimal strategy of CARA utility maximizing in an incomplete market is also obtained in Wang [

This paper is structured as follows. In Section

The insurer can invest in a financial market consisting of one riskless bond with price

Suppose the initial reserve is

The insurer is allowed to invest in those

A trading strategy

The set of all admissible trading strategies is denoted by

In this section, we “complete” the incomplete market described in (

Reducing the dimension of the Brownian motion.

For each

Creating independent component Brownian motions from the correlated ones.

Let

Suppose that the insurer has a utility function

Furthermore, for any

If there exists a strategy

Such sufficient conditions to the optimal investment are well known in the martingale approach; refer to [

By (

It is well known that maximizing the expected quadratic utility is equivalent to finding a mean-variance efficient strategy while CARA utility is commonly used. Thus, we will work out the optimal strategies explicitly for these two utilities in the next section.

First, we introduce some notations and a martingale representation theorem that will be used. Let

One can consult Proposition 9.4 in [

For any local (resp., square-integrable) martingale

In this subsection, we consider problem (

Let

Suppose

Conversely, suppose that there exists

In what follows, we will solve FBSDE (

Conjecturing the form of solution.

Put

Let

Verifying the conjecture in Step

Let

Finally, by Proposition

Let

In this subsection, we consider problem (

Conjecturing the form of

Put

For any stopping time

On the other hand, by (

Verifying that

Substituting (

Verifying that

It is clear that

By Proposition

Let

In this section, we present some remarks and conduct computational experiments for the optimal investment problems of this paper according to the real market data. The objective of these computational tests is to contrast the insurer’s optimal strategies in an incomplete market with those in a complete market.

According to Theorems

For quadratic utility, as shown in (

When no claim occurs at time

When a claim occurs at time

In terms of (

Suppose there are 12 underline Brownian motions, 12 stocks for investment in a complete market, and 10 stocks in an incomplete market. The universe of assets are chosen from the 12 industry categories of finance.cn.yahoo.com in 2009 (see Tables

Stocks in a complete market.

Industry | Company | Code |
---|---|---|

Aerospace and defense | Honeywell Intl. | HON |

Automanufactures | Toyota Motor Corp. ADR | TM |

Biotechnology and drug manufacturers | Johnson & Johnson | JNJ |

Chemicals | EI DuPont de Nemours & Co. | DD |

Communication equipment | Qualcomm | QCOM |

Computer software | Microsoft | MSFT |

Discount | Wal-Mart Stores Inc. | WMT |

Diversified computer systems | Hewlett-Packard Co. | HPQ |

Major integrated oil and gas | BP Plc. | BP |

Semiconductor-broad line | Intel Corp. | INTC |

Telecom services | AT & T | T |

Utilities (gas and electric) | Duke Energy Corporation | DUK |

Stocks in an incomplete market.

Industry | Company | Code |
---|---|---|

Aerospace and defense | Honeywell Intl. | HON |

Automanufactures | Toyota Motor Corp. ADR | TM |

Biotechnology and drug manufacturers | Johnson & Johnson | JNJ |

Chemicals | EI DuPont de Nemours & Co. | DD |

Communication equipment | Qualcomm | QCOM |

Computer software | Microsoft | MSFT |

Diversified computer systems | Hewlett-Packard Co. | HPQ |

Major integrated oil & gas | BP Plc. | BP |

Semiconductor-broad line | Intel Corp. | INTC |

Utilities (gas & electric) | Duke Energy Corporation | DUK |

Transaction cost for quadratic utility.

We easily observe that the optimal strategies in an incomplete market incurs lower transaction cost than those in a complete market. Because there are fewer stocks in an incomplete market, than in a complete market, the number of stocks whose investment amount will be changed is smaller in an incomplete market. Thus the transaction cost for fewer stocks is low.

For the optimal strategy of CARA utility, (

The experimental procedure is the same as in Section

In Figure

Comparison of the average total dollar amount invested in stocks in a complete market and an incomplete market.

Figure

Relative differences of the average total dollar amount invested in stocks between the two markets.

Given a sequence of portfolios

Average transaction cost for CARA utility in the two markets.

In this paper, we consider the optimal investment problem for an insurer in an incomplete market. After transforming the incomplete market into a complete one, we solve the problem via the martingale approach. From the computational experiments, we find that an insurer’s optimal investment strategy in an incomplete market incurs lower transaction cost than the one in a complete market. The amount invested in stocks in an incomplete market is more than that in a complete market. For CARA utility, the computational results also show that the incomplete market produces the same effects on the optimal strategies of insurers with different risk aversion coefficients.

See Tables

The authors are very grateful to reviewers for their suggestions and this research was supported by the National Natural Science Foundation of China (Grant no. 11201335) and the Research Projects of the Social Science and Humanity on Young Fund of the Ministry of Education (Grant no. 11YJC910007). J. Cao would like to acknowledge the financial support from Auckland University of Technology during his sabbatical leave from July to December 2009 and thank the Department of Mathematics at Tianjin University for hospitality to host him in October 2009.