^{1}

^{*}

^{2}

^{*}

This paper considers an optimal life insurance for a household subject to mortality risk. The household receives wage income continuously, which could be terminated by unexpected premature loss of earning power. In order to hedge the risk of losing income stream, the household enters a life insurance contract. The household may also invest their wealth into a financial market. Therefore, the problem is to determine an optimal insurance/investment/consumption strategy. To reflect a real-life situation better, we consider an incomplete market where the household cannot trade insurance contracts continuously. We provide explicit solutions in a fairly general setup.

In the management of pension funds, a long term portfolio strategy taking into account a liability is one of the most significant issues. The main reason is the demographic changes in the developed countries: if the working-age population is enough to provide for old age, the liability is a minor issue in the portfolio management of pension funds. Since the life expectancy have increased in recent decades, it becomes insufficient to provide for old age. Furthermore the low birth rate continues and drives up this problem for decades. Thus pension funds face a challenging phase to construct long term portfolio strategies which hedge their liabilities.

A lot of pension funds except a few ones [

Therefore the aim of this paper is to propose a long term portfolio strategy which 1) involves an evaluation of a liability, 2) admits changes of the strategy at any time, and 3) is obtained in realistic time. To tackle this problem, we employ the LQG (Linear, Quadratic cost, Gaussian) control problem (see, e.g., Fleming and Rishel [

A continuous time stochastic control approach is one of the most popular method to obtain the suitable long term portfolio strategy. The literature about this approach is quite rich. The papers treating the management of pension funds are, for instance, as follows: Deelstra et al. [

The organization of the present paper is as follows. We introduce continuous time models of assets and a benchmark in Section 2. To fit in the LQG control problem, they are defined by the linear stochastic differential equations (SDEs). We mention that our portfolio strategy is represented by the amounts of assets. In Section 3, we define a criterion of the investment performance and provide the optimal portfolio strategy explicitly. Several numerical results are served in Section 4 Throughout the section the parameters related to the assets are determined by an empirical data provided by the Government Pension Investment Fund in Japan. The simulation using an artificial data are discussed in Section 4.1 and this result gives conditions that our optimal portfolio strategy works well. Section 4.2 provides a case study using an empirical estimations published by the Japanese Ministry of Health, Labour and Welfare. It demonstrates that our strategy is able to hedge the liability well.

In this section, we present mathematical models of assets and a benchmark. The market which we are considering consists of only one risk-free asset and -risky assets and we have -benchmark component processes.

Let be a filtered probability space be a -dimensional Brownian motion where and be a space of stochastic processes which satisfy

We denote price process of the risk-free asset, those of the risky assets and the benchmark component processes by, and respectively, where the asterisk means transposition. To fit in the LQG control problem, we assume that, and are governed by the following SDEs:

where, , ,

, and

are deterministic continuous functions and represents the maturity. Coefficients, and stand for the risk-free rate and the expected return rate of the -th asset and the volatility.

Let a class of portfolio strategy be the collection of -valued -adapted process which satisfies

be the amount of the risky asset held by an investor at time, and be the value of our portfolio at time. Then the amount of the risk-free asset held by the investor is represented by. Hence, is governed by

where. To emphasize the initial wealth and the control variable, we may write.

The solution of the SDE (4) is given as follow:

Moreover since, , and are continuous functions on and, is in:

where, and are constants.

We define the criterion of investment performance by

where and are constants, is a constant vector, and is a deterministic continuous function. Hence our investment problem is to find the control s.t.,. Since the performance criterion is represented by quadratic functions, our investment problem becomes the LQG control problem. We determine, and the parameters of to be able to regard and as a liability.

The optimal portfolio strategy is represented in the following form:

Theorem 1 We define the portfolio strategy as follows:

where and are solutions of following ordinary differential equations (ODEs):

Here we have written.

Then satisfies and,.

The proof of Theorem 1 is given in the appendix.

We note that has feedback terms of and. This implies that our optimal strategy has delays to catch up the the benchmark process. Hence the preferable situation applying our strategy is the case that does not fluctuate violently.

We apply our method to an empirical data provided by the Japanese organizations. This section is divided to two subsections according to the type of liabilities: an artificial liability and the liability constructed by the estimations published by the Ministry of Health, Labour and Welfare of Japan. The former one suggests the situation that our optimal strategy works well and the latter one demonstrates that our portfolio strategy is able to hedge the liability.

Before we move on the each subsection, we determine the common parameters in following subsections. The first task is to determine the parameters relating to the benchmark component processes. They consist of the income of a pension fund and his or her expense and thus and. We set the parameters constructing the benchmark process as follows:

Hence, the benchmark process is which represents a shortfall of the income and then we regard this shortfall as the liability. To discuss the performance of the strategy, we introduce a hedging error function of the -th sample path and its average as follows:

where is the -th sample path of and is the number of the sample paths. We set except as otherwise noted.

The next task is to determine the risk-free rate and the expected return rates and volatilities of risky assets. We invest the following four assets: indices of the domestic bond, the domestic stock, the foreign bond and the foreign stock; we number them sequentially. According to the estimations of return rate and volatilities by the Government Pension Investment Fund in Japan [

where the Cholesky decomposition of, a variance-covariance matrix of the assets:

We choose a money market account as the risk-free asset and we set.

In this subsection, we consider the following an artificial deterministic liability model:

i.e., we set and. We assume that our wealth coincides with the benchmark at the initial time:. We construct the optimal portfolio strategy over three decades, i.e.,. Then we determine the functions, and by solving the ODEs (7)-(9) numerically. and simulate paths of on according to Equations (1)-(3) using a standard Euler-Maruyama scheme with time-step.

The most significant issue it indicates is that the performance of the strategy is quite poor near the maturity.

To obtain the stationary solutions of the ODEs (7)-(9), we replace to a value large enough. We denote it by and set.

Results of simulations using the improved strategy are described as follows.

Figures 5 and 6 indicate that the performance near the maturity is improved and it does not depend on the sample paths. This result leads us to the conclusion that we should construct the strategy with the stationary solutions of the functions, and if they exists.

At the end of this subsection, we mention about our portfolio composition.

and high return asset. If is deficient in, the strategy increases the proportion of the domestic bond and the foreign stock.

According to the Japanese actuarial valuation published in 2009 [

We regard these estimations as and and simulate the three decades investments using our optimal strategy from 2040 when the shortfall of the pension fund starts to expand drastically. The following reasons support that this situation is a valid case study: 1) a phase expanding, the shortfall of the pension fund, is the most typical one expressing the demographic changes; 2) the behaviour of in this term meets the condition to apply our optimal strategy: is increasing in the entire region. Throughout this subsection we set the start point as the year 2040, i.e., and represent the year 2040 and the year 2055 respectively.

To construct the optimal strategy, we first calibrate, and to fit the estimations. Setting and as a numerical differentiation of the estimations is a simple method to accomplish the purpose. Since we are discussing the three decades portfolio, we determine. As suggested in Section

4.1, we set to obtain the stationary and. We are unable to expect the stationary because explicitly depends on. We assume that our wealth coincide with the benchmark at the initial time:. Then we simulate paths of on according to Equations (1)-(3) using a standard Euler-Maruyama scheme with time-step which means that we can rearrange our portfolio every quarter. Results of the simulations are as follows.

We are able to argue that our strategy hedges the shortfall well since