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Part Two: A Promising Asset Allocation and Leverage Strategy for a Life Insurer by Simulation Optimization
INTRODUCTION
Managing investments is important for life insurers to ensure that funds are available to pay claims when they fall due, in the future. However, conflicts of interest between shareholders, regulators and policyholders make investment decisions difficult. Shareholders of insurers urge them to generate higher returns from investments and underwriting but regulators and policyholders ask them to maintain risk at acceptable levels. Life insurers thus have to manage divergent expectations emanating from both assets and liabilities sides. In this study, we re-think the classic asset allocation problem specifically for life insurers. We propose a non-linear simulation model with stochastic variables to derive promising asset allocations and leverage strategies.
Extant literature has covered the asset allocation problem quite extensively.
In financial literature, there are two categories of methods to address this problem.
One is the mean-variance analysis of Markowitz (1952), which suggests the efficient frontier representing the best portfolios in terms of return-risk tradeoff. The mean-variance analysis, however, is prone to two fundamental flaws: the single-period framework, and the inappropriate utility function assumed for the investor (Brennan et al., 1997). The solution to a static portfolio choice problem can be different from the solution to a multi-period dynamic problem (Campbell, 2000).
The other method to construct optimal portfolios originated from Merton (1971;
1990). The literature along this line formulates the asset allocation problem as a stochastic optimal control problem; solutions are characterized by Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDE) but it is difficult to get a closed-form solution from a high-dimensional PDE. Also, numerical solutions of PDE can be obtained only in rare cases. Cox and Huang (1989) made conceptual progress by showing that one can apply the Martingale representation theory to reduce the stochastic dynamic programming problem to a static problem in complete markets. However, few closed-form solutions have been available, except for the simplest cases, and complex hedging terms are difficult to evaluate numerically.
We need a powerful tool to integrate the asset allocation problem and leverage strategies for efficient asset and liability management by life insurers. A company-wide simulation model is one such tool (Browne, Carson and Hoyt, 1999;
Browne, Carson and Hoyt, 2001; Kaufmann et al., 2001; and Hardy, 1993, 1996).1 It is a “systemic approach” to financial modeling which projects financial results under a variety of possible scenarios, showing how outcomes might be affected by changing business, competitive and economic conditions.” The system starts with two fundamental equations, as follows:
1 A company-wide simulation system is often named as “Dynamic Financial Analysis” (DFA) system in the non-life insurance industry. What is called DFA in non-life insurance is also known as “Asset Liability Management” (ALM) in life insurance.
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respectively, at time t, and ∆(.) denotes the change of the variable. Equation (1) depicts the fundamental relations among financial variables at any given point of time;Equation (2) captures the dynamic relations among the variables across time. The system can specify models for values of individual asset and liability items at time t.
These models are supposed to reflect the stochastic nature of financial markets and insurance underwriting.2
A company-wide simulation model, though powerful, is merely a descriptive model. It only helps us understand the dynamics of, and complex interactions among, the elements of the system, and this system lacks optimization capability. In other words, a simulation model helps us to know which proposed strategy is better but is unable to determine what the optimal strategy is. It does not have the mechanism/algorithm to search for the optimum. We, therefore, have to make educated guesses on what the optimal strategy could be like, and employ the trial-and-error method to determine the right strategy. Trying all possible strategies to seek the optimum is infeasible due to the large number of decision variables. A simulation model without an optimization mechanism is, therefore, incapable of helping managers maximize shareholder value.
In this study, we apply the techniques of simulation optimization to address the asset allocation problem by simulating the system of a life insurer. Comparatively few researchers have used a company-wide simulation model for optimization of a life insurer’s asset allocation. We have found two articles in insurance and financial literature that focus on investment management for servicing participating policies with minimum guarantees. Iwaki and Yumae (2004) analyze trading strategies in a continuous time economy by utilizing the Martingale method. They derive an efficient frontier for the company, as well as trading strategies for efficient portfolios.
Consiglio, Saunders, and Zenios (2006) examine asset-liability management associated with single-premium participating policies with minimum guarantees.
In our company-wide simulation system for a life insurance company, we incorporate four types of assets and three types of insurance products. Assets include default-free zero-coupon bonds, stock index, real estate index, and alternative investment characterized by “high-return and high-risk.” Insurance products include 20-year non-participating term life insurance, endowment, and pure endowment. We assume that leverage represents the premiums received at the beginning of the first policy year, divided into the initial equity of shareholders. Without loss of generality, for simplicity, we assume no new business comes after the second policy year. The objective function of our simulation and optimization problem is maximization of the
2 The system’s major outcome is the insurer’s surplus/equity distribution at some point of time in the future. Managers can employ the simulated surplus distribution to make choices among alternative strategies. Life insurers can use a company-wide simulation system to assess asset allocation strategies by examining the impacts of alternative strategies on the surplus distribution over a target time horizon. The simulation system therefore can help managers make investment and business decisions in a comprehensive and robust way.
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立 政 治 大 學
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expected annual rate of return of equity minus the risk and insolvency penalty. We maximize this function through re-allocating investments in different assets every five periods.
We find the promising asset allocation and leverage strategies by particle swarm optimization (PSO). The PSO is a novel computational method that can solve difficult problems efficiently and reliably (Kendall and Su, 2005). Eberhart and Kennedy (1995) introduced PSO, which is based on the analogy of birds flocking and fish schooling. PSO has been shown to be powerful, easy to implement, rapid to converge, and computationally efficient (Poli, 2008). Dissimilar to evolutionary algorithms, such as GA, PSO considers parameters of crossover probability, mutation probability, and population size, and it is more implementable.
The promising leverage is 20 (i.e. the total premiums in the first year are twenty times the initial equity) and the corresponding asset allocation varies across periods. The objective value is 24.63% with 5 insolvencies in the simulation. We find no consistency in composition of the portfolio in terms of risky assets (stock, real estate and alternative investment) and fix income securities (default-free zero coupon bonds) at the time of different re-allocations. In the first and sixteenth periods, weights of fix income securities are higher than weights of risky assets. But it is the opposite in sixth and eleventh periods. Allocations for the first and the last re-allocation period are relatively conservative, when weights of fix income securities are 70.75% and 67.62%, respectively. At the sixth and the eleventh re-allocation, the insurer needs to hold more risky assts to improve the objective value. The ratio of risky assets to fix income securities can even go up to 2.43. However, weights of fix income securities at each re-allocation period are higher than 29% due to life insurers’ investment strategies being subject to penalty for violation of stipulated norms for asset allocation.
Among risky assets, weights of equities and real estate dominate alternative investment. Asset allocations show that equities have a prominent share in the investment portfolio in each period. At each re-allocation, weight of stocks is higher than 13%. Weight of real estate undergoes significant change after the fifth period, in the sixth period (rising from 4.78% to 28.15%), as well as after fifteenth, in the sixteenth period, when it plunges from 36.61% to 3.43%. The rationale is that the price of real estate follows a jump diffusion process with the average number of jumps being 0.1 per year. Alternative investment is not the first choice of investment because of the high risk it entails. Even in the sixth and the eleventh period, to improve the objective value, the insurer prefers stock and real estate rather than alternative investment since the risks of real estate (0.18) and stock (0.25) are smaller than alternative investment (0.5). When determining the composition of risky assets, insurers would concern more about risk than the return of each type of asset.
Another contribution of this study is that we investigate asset allocation strategies under different leverages. Insurance literatures indicate that capital structure (leverage) affects the risk-taking behavior of insurers. Michaelsen and Goshay (1967), Hammond et al. (1976), and Harrington and Nelson (1986) found some degree of support for the hypothesis that insurers with higher portfolio risk operate with lower leverage ratios (measured by the ratio of net premium written to equity, which is similar to its definition in our study). Cummins and Sommer (1996)
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indicated that property and liability (P&L) insurers prefer to operate at finite levels of leverage (capital to asset ratio) and risk to avoid bankruptcy cost under the cost-based hypothesis, as in Shrieves and Dahl (1992). Baranoff and Sager (2002; 2003; 2004) indicate support for finite risk hypothesis, that is, for life insurers, leverage (total liabilities to total assets) and the proportion of risky assets (stock) in portfolio are negatively interrelated. We assume two different leverages (12 and 16) to compare promising asset allocations under these two scenarios with our promising asset allocation and leverage strategies.
The results show that when leverage increases, insurers need to hold more fix income securities in the first period. In other words, the finite risk hypothesis holds conditionally in a multi-period asset allocation. When leverage is 12, the insurer re-allocates more investment to fix income securities in the first, sixth, and eleventh periods. But in the sixteenth period, the insurer re-allocates to risky assets in full to increase the objective value. When leverage is 16, the insurer holds more fix income securities only in the first and the sixth periods. From the 11th period onwards, the insurer increases weights of risky assets to increase the objective value. Asset allocation at these two points is different from the optimal because leverage strategies are not optimal in the simulation. Thus, the insurer’s investment decisions are far away from the promising strategies. Besides, volatility of annual rate of return of equity is decreasing as the leverage increases. Numbers of insolvencies in simulation paths exhibit no correlation with leverage. The objective value is then increasing, as the leverage increases, to reach closer to the promising strategy.
The remainder of this paper is structured as follows. Section 2 presents our simulation model, including the setting of asset and liability sides. Section 3 presents balance sheets of assets and liabilities for each period and formulates asset allocation and leverage strategy as a high-dimensional constrained optimization problem. Section 4 presents the results and exhibits how leverage affects asset allocation strategy. Section 5 presents conclusions.