τ . This deduction (punishment) of the utility is implying the company’s dissatisfaction. The constant is chosen by the company to properly reflect the importance of excessive ruin probability.
k
23 For the search technique ES, we can easily set the risk level as our constraint to search the return under this specific risk. However, the grid search method does not allow us to specify the risks.
0
The entire utility function can convey the practical situation. That is when the company pursues the high profit and naturally lifts up the risk and return, it will result in the higher ruin probability and force the value of the utility lower. However, when the company avoids the high ruin probability of the risky assets, it usually gets lower returns. We use MPES to find the appropriate investment portfolio to make the utility reach the maximum. This means that the company can gain the greatest profit and lower the ruin probability at the same time.
θ
~C. Experiment Design
The experiments simulated investment of the insurance company for twenty-five years and the number of simulated paths is 10,000, which are the various situations of the return and loss of the markets. We preceded two main parts of experiments to compare the differences between the all sorts of searching methods and adjustment rules (see Table 1). The first one uses the grid search method to solve the problems and the other uses MPES. The parameters of the two main experiments are identical and the detail settings are listed in the appendix.
Table 1: Experiment design for the asset allocation strategies and search methods Asset Allocation Strategies
Search Method Adjusting Rule Adjusting Method Simplified Form Grid search method periodical
interval
rebalance rebalance
Grid-PR Grid-IR
MPES
periodicalWe used the grid search method as the benchmark to compare with the MPES because people usually adopt it to solve the searching problems when they do not have efficient methods. The grid search method used the combination of some specific figures to determine the allocation of investment portfolio. The number of the asset classes is five and each figure in the combination varies by 20%. All asset allocations are non-negative and sum to one, so it will produce 126 combinations24. Some examples of the combinations are such as (1, 0, 0, 0, 0), (0.8, 0.2, 0, 0, 0), (0.2, 0.6, 0, 0, 0.2), etc.
VI. MULTI-PHASE EVOLUTION STRATEGIES
In this section, we introduce the proposed evolutionary strategies algorithm, multi-phase evolutionary strategies (MPES). We will first introduce a basic evolutionary strategies algorithm and its application to generate the optimal asset allocation for the simulation models. Then, we will discuss the ideas and the procedure of MPES and explain how MPES is able to improve the performance of the basic evolutionary strategies.
Furthermore, we will discuss the performance of MPES for five complicated benchmark functions..
A. Evolution Strategies
The evolution strategies algorithm (ES) has been presented since 1970s (Rechenberg, 1973, Schwefel, 1981). It is a randomized search method that incorporates the nature of evolution into its processes. Evolutionary algorithms, unlike traditional optimization addition term is the result from [0.6, 0.2, 0.2, 0, 0] and [0.4, 0.4, 0.2, 0, 0]. The last two terms are the results from [1, 0, 0, 0, 0, 0] and [0.2, 0.2, 0.2, 0.2, 0.2].)
techniques, use “population” instead of single points to search and solve complex optimization problems. The population for the initial generation is usually generated randomly. From the members (parents) in the population, genetic operators are then used to produce offspring, and the favorable offspring, corresponding to the “survival of the fittest theory” in the biological world, are chosen to constitute the population for the next generation. The process continues for generations until a termination criterion is satisfied and a superior solution is acquired. A concise pseudo code of a basic evolution strategies algorithm, denoted as (µ,λ) ES, is presented as follows, where µ is the number of parents in the current population and λ is the number of offspring produced by the parents for the next generation, and λ is about seven times of µ.:
(µ,λ) ES Pseudo Code:
1. Initialize µ parents
2. For generation :=1 to n do
3. Recombination & Mutation from µ parents to produce λ offspring
4. Parent selection: Evaluate λ offspring and choose the best µ offspring to be new parent 5. End Do
6. Output the first best solution.
A more detailed description of the steps of the (µ,λ) ES algorithm is presented below (Nissen and Biethahn, 1995, and Back, 1996).
Step 1: Generate a population for the initial generation.
A population of µ solutions (members) is generated. Each solution is usually represented by a row vector consisting of two parts. The elements in the first part are the values of the decision variables (xj) considered in a given application, and the elements in the second part are the mutation step sizes (σj) corresponding to the decision variables in the first part. The decision variables in our application are dependent on the asset allocation strategies used. If the periodical reallocation strategy is used, the decision variables will be the proportions of the
asset allocations for the five types of risky asset in five periods, so there will be twenty-five decision variables. We randomly generate the proportions of the five asset allocations for each period from a uniform distribution with a range of [0, 1]. Note that the sum of the proportions of the five asset allocations in a period has to be equal to one, so the proportions for each period are simply normalized by summing up the proportions and dividing each proportion by the sum. Additionally, in each solution, all the mutation step sizes are set to 3.0 (Back, 1996).
Step 2: Apply recombination and mutation to the parents to produce λ offspring.
A pair of parents, A and B, is randomly chosen from the population, and recombination and mutation are applied to A and B to produce a child C. Discrete recombination is used to determine the first part, the decision variable values of child C. The value of each decision variable in C is randomly and equally chosen from the value of the same variable in A and B.
Intermediate recombination is used to determine the second part, the mutation step sizes of C.
The j-th mutation step size in C is simply determined by the average of the j-th mutation step size in A and B (σj (C) = 0.5 (σj(A) +σj(B)). The generated child C is then mutated by first modifying its mutation step sizes and then adding these step sizes to mutate the corresponding decision variables. Each mutation step size σj (C) is modified by the following equation:
σj’(C) =σj (C) exp(τ’ N(0,1) + τ Nj(0,1)), where N(0,1) is a standard-normally distributed random variable, and the values of τ and τ’ are set to 1.0 (Back, 1996) . And, each decision variable xj is mutated by the following equation:
(C)) (0,
N (C) x (C)
x′j = j + j
σ
j′ .When child C is generated, the simple normalization method is applied to the proportions of the asset allocations of each period to maintain its feasibility. The reproduction procedure is repeated until λ offspring are produced.
Step 3: Evaluate the λ offspring and choose the best µ offspring to constitute the population for the next generation.
The decision variables of a child are submitted to the simulation model, and the computational result is the fitness value of the child.
Step 4: Check the termination criterion. If the termination criterion is satisfied, stop; otherwise go to Step 2.
The most commonly used termination criterion is to set the fix generation for the entire process. This criterion is also used in this research.
B. Multi-Phase Evolution Strategies
Intensification and diversification are two principles in developing an optimization method. When we applied the basic ES algorithm to solve five benchmark functions, it was found that the algorithm usually converged within 100 iterations; the converged solutions were affected by randomly generated initial solutions, and the converged solutions were not always the optimal solutions. This may conclude that the basic ES algorithm was able to intensively converge to a local optimal solution; however, it was not able to diversely search large enough solution space. The idea behind the multi-phase evolution strategies is to improve the diversification capability of the basic ES algorithm. The basic ES algorithm is implemented