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that PSO has been recognized for being as good as genetic algorithm (GA) and evolution strategy (ES) in solving high-dimensional and non-linear functions. The other is that PSO is more applicable than other optimization algorithms (i.e. PSO has been less explored and offers more potential operations resources). Details of PSO, including its formulation, algorithm, and effectiveness, are provided in the Appendix.
RESULTS
Promising Asset Allocation and Leverage
In the promising solution obtained from our simulation, leverage level is 20 and asset allocations vary across periods. The objective value is 24.63% with 7 insolvencies occurring in the simulation. There is no consistency in the composition of allocations, in terms of risky (stock, real estate and alternative investment) and risk-free (default-free zero coupon bonds) assets in re-allocations for different periods.
In 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 in the first and the last period are relatively conservative; weights of fix income securities are 70.75% and 67.62%, respectively. In sixth and eleventh periods, the insurer adjusts its portfolio to have more risky assets, to improve the objective value; the ratio of risky assets to fix income securities goes up to 2.43. Besides, weights of fix income securities after all re-allocations are higher than 29% because of intermediation and investment strategies being subject to penalty for volatility of equity and insolvency norms.10
[Insert Table 1 Here]
In Figure 1, we observe that for risky assets, weights of stock index plus real estate index are higher than alternative investment. In addition, the promising asset allocations show that equities play an important role in the investment portfolio in each period. After each re-allocation, weights of stocks are above 13%. However, weights of real estate index and alternative investment vary across periods. Weights of real estate experience significant change in the sixth period (rising from 4.78% to 28.15%), as well as the sixteenth period (declining 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 in any of the periods because of its high risk. Even in sixth and eleventh periods, to improve the objective value, the insurer prefers stocks and real estate over alternative investment since risks of real estate (0.18) and stock (0.25) are smaller than alternative investment (0.5). When determining the composition of risky assets, the insurer would be more concerned about the risk than the return of each asset.
[Insert Figure 1 Here]
Comparison of Leverage
10 We record the asset allocations and leverage level before PSO converges in Appendix (Table 6).
The leverage is same as the promising case, 20, and asset allocations are similar to the promising case except the last period. However, in this case, there is more insolvency with a higher volatility.
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We set two leverage levels, 12 and 16, and find the corresponding promising asset allocations. Especially, we examine the finite risk hypothesis of capital and risk in insurance literature. Promising asset allocations for leverages of 16 and 12 are shown in Tables 2 and 3. Compared to Table 1, the results show that when leverage increases, the insurer needs to hold more fix income securities in the first period. In other words, the finite risk hypothesis in insurance literature holds conditionally in a multi-period asset allocation.
[Insert Tables 2 and 3 Here]
When leverage is 12, the insurer holds more fix income securities in the first, sixth, and eleventh periods. But in sixteenth period and thereafter, the insurer invests in risky assets to the fullest extent, 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 eleventh period and thereafter, the insurer increases weights of risky assets to increase the objective value. Asset allocations in these two scenarios are different from the promising allocation because leverage strategies in the simulation are not optimal. Besides, volatility of annual rate of return of equity is decreasing as the leverage increases. Insolvencies in simulation paths exhibit no consistent correlation with leverage. Objective value increases as leverage increases.
In Figure 2, where leverage = 16, we observe that weights of risky assets increase gradually from the sixth period onwards. It reveals that the simulation optimization mechanism leads the insurer to improve the objective value by increasing weights of risky assets. This phenomenon is also shown for leverage = 12 in Figure 3.
[Insert Figures 2 and 3 Here]
In Figures 2 and 3, we find that the stock index is important for the life insurer.
Weights of stock index in each re-allocation are higher than 10.2%. Real estate index is also favorable for the insurer to improve the objective value. Weights of real estate index are higher than 11.37% in all periods. Besides, weights of stock index plus real estate index dominate weights of alternative investment. Even if leverage does not reach the promising level, the insurer is concerned more about the risk than the return while choosing risky assets.
CONCLUSIONS
This study presents a company-wide simulation model and optimization algorithm for analyzing asset allocation and leverage strategies for a life insurer selling traditional policies. The model allows the insurer to compare different leverage strategies to determine how to construct a promising asset allocation. The promising asset allocation and leverage strategies are derived from numerical calculations that consider leverage as an internal factor in asset allocation. Also, our results demonstrate how to compare different leverage strategies for specific promising asset allocations.
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TABLES AND FIGURES
Table 1: Promising Asset Allocation and Leverage
Leverage = 20
Objective Value = 0.2463
Simulation Paths Insolvencies = 7
Volatility of Annual Rate of Return on Equity = 0.0514
Asset Weights (%)
Asset Class / Time 1-5 6-10 11-15 16-20
(A) Stock 13.76 16.28 29.23 15.13
(B) Real Estate 4.78 28.15 36.61 3.43
(C) Alternative Investment 10.71 12.35 5.09 13.82 (D) Default-Free Zero-Coupon-Bonds 70.75 43.22 29.07 67.62 Total Risky Assets = (A)+(B)+(C) 29.25 56.78 70.93 32.38 Total Fix Income Securities = (D) 70.75 43.22 29.07 67.62
Table 2: Asset Allocation; Given Leverage = 16
Leverage = 16
Objective Value = 0.2390
Simulation Paths Insolvencies = 8
Volatility of Annual Rate of Return on Equity = 0.0543
Asset Weights (%)
Asset Class / Time 1-5 6-10 11-15 16-20
(A) Stock 10.20 21.03 35.68 18.25
(B) Real Estate 11.37 23.24 18.93 35.55
(C) Alternative Investment 16.79 5.23 10.46 44.09 (D) Default-Free Zero-Coupon-Bonds 61.64 50.50 34.92 2.10 Total Risky Assets = (A)+(B)+(C) 38.36 49.50 65.08 97.90 Total Fix Income Securities = (D) 61.64 50.50 34.92 2.10
Table 3: Asset Allocation; Given Leverage = 12
Leverage = 12
Objective Value = 0.2211
Simulation Paths Insolvencies = 5
Volatility of Annual Rate of Return on Equity = 0.0652
Asset Weights (%)
Asset Class / Time 1-5 6-10 11-15 16-20
(A) Stock 12.63 12.63 19.92 71.82
(B) Real Estate 12.63 16.46 12.55 12.47
(C) Alternative Investment 16.61 1.33 7.57 15.71 (D) Default-Free Zero-Coupon-Bonds 58.13 69.58 59.95 0.00 Total Risky Assets = (A)+(B)+(C) 41.87 30.42 40.05 100.00 Total Fix Income Securities = (D) 58.13 69.58 59.95 0.00
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0%
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20%
30%
40%
50%
60%
70%
80%
90%
100%
t=1 t=6 t=11 t=16
(A ) Stock (B) Real Es tate
(C) A lternative Inves tment (D) Default-Free Zero-Coupon-Bond
Figure 1: Composition of Promising Assets after Each Re-allocation under Optimal Leverage
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
t=1 t=6 t=11 t=16
(A ) Stock (B) Real Es tate
(C) A lternative Inves tment (D) Default-Free Zero-Coupon-Bond
Figure 2: Assets Composition after Each Re-allocation under Leverage = 16
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0%
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20%
30%
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50%
60%
70%
80%
90%
100%
t=1 t=6 t=11 t=16
(A) Stock (B) Real Es tate
(C) A lternative Inves tment (D) Default-Free Zero-Coupon-Bond
Figure 3: Assets Composition after Each Re-allocation under Leverage = 12
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In PSO, the particles are placed in the search space of some problem or function, and each evaluates the fitness at its current location. Each particle then determines its movement through the search space by combining some aspect of the history of its own fitness values with those of one or more members of the swarm, and then moving through the search space with a velocity determined by the locations and processed fitness values of other members, along with some random perturbations.
Members of the swarm that a particle can interact with are called its social neighborhood. Social neighborhoods of all particles together form a PSO social network.
Take a maximizing N-dimensional function f for example. Each particle is
N-dimensional, and is a potential optimum of f. Each particle has a memory of the
best solution that is found, called its personal best. A particle flies through the search space with a velocity which is dynamically adjusted according to its personal best and the best solution found by a neighborhood of particles.This is, thus, a sharing of information. Particles profit from discoveries and previous experiences of other particles during the exploration and search for higher objective function values. The first, called global best (gbest), connects all particles in the population to one another. The second, called local best (lbest), creates a neighborhood for each individual comprising it and its k nearest neighborhoods in the population.
Formulation
Let i indicate a particle’s index in the swarm. Then
S
={ ,p p
1 2,L,p
s} is a swarm of s particles. Each particle has a current position pi =(pi1,pi2,LpiN)T and flies through the N-dimensional search space ℜN with current velocity1, 2
( , , )T
i i i iN
v
=v v
Lv
which is dynamically adjusted according to its own previous best solution xi =(xi1,xi2,L,xiN)T and the current best solution %x of the entire swarm
i(gbest) or the particle’s neighborhood (lbest).
At iteration time t of the PSO, the velocity and particle updates are specified separately for each dimension j of the velocity and particle vectors. A particle
P
i will interact and move according to the follow equations:% is a constant known as the inertia weight which determines the speed of convergence,