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,
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max 0.9
ω
= andω
min =0.4 have been shown to give good conversion,ϕ
1 andϕ
2 are two constants known as the acceleration coefficients, and 0≤ϕ ϕ
1, 2 ≤ , which 2 control the relative proportion of cognition and social interaction in the swarm (Shi and Eberhart, 1998). Valuest
max and t indicate the maximum and current iteration numbers and we sett
max to be 1500.Algorithm
The standard PSO algorithm to maximize function
f
:ℜ → ℜ is presented N below:1. Set the iteration number t to be zero, and initialize swarm S of N-dimensional particles pi0; each component
p is randomly initialized to a value in the initial
ij0 domain of the swarm, an interval [p
min,p
max]. Since the particles are already randomly distributed, velocities of particles are initialized to the zero vector 0T. 2. Evaluate performance f p( it) of each particle.3. Compare the personal best of each particle to its current performance, and set xit to be the better performance for
1 1
1
, if ( ) ( ) , if ( ) ( )
t t t
t i i i
i t t t
i i i
x f p f x
x p f p f x
− −
−
⎧ ≤
= ⎨⎩ > .
4. Set the global best %xt∈{x1t,x2t,L,xst f x( )}%t =max{ ( ), (f x1t f x2t),L, ( )}f xst to the position of the particle with the best performance within the entire swarm.
When a local best PSO is implemented, set the neighborhood best
%Qit { i (%Qit) max{ ( jt)}
x ∈ Q f x = f x , ∀ ∈ =
x
jQ
i {x
i k− t,L,x
it,L,x
i k+ t} , k is the number of nearest neighborhoods.5. Change the velocity vector for each particle according to equation (a1).
6. Let t = t + 1.
7. Go to Step 2, and repeat until convergence or
t
=t
max =1500.Effectiveness of PSO
Effectiveness of PSO has been recognized to be more efficient than other algorithms in solving complex non-linear and multi-modal functions with multi- variables (Huang, 2009). Complex functions in Huang (2009) originate from Schwefel (1981), Yao and Liu (1996), and Vesterstrom and Thomsen (2004).
Appendix Table 1 presents these functions.
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Table A1: High Dimension Complex Functions
Function list (n=50) Constrains Minimal value Remark
(
1 2)
Table A2: Notations and Values of Asset Models’ Parameters
Description Notation Value
CIR Interest Rate Model
Mean reverting speed κ 0.25
Long term interest rate µ 0.04
Volatility of interest rate σ r 0.03
Interest Rate Adjusted Geometric Brownian Model
Risk premium π s 0.07
Average jump size as proportion of
the real estate index β Uniform(-0.5,0.5)
Alternative Investments Expected return of high-return and
high risk investments µ χhh 0.15
Volatility of high-return –high-risk
investments σ χhh 0.5
Correlation Matrix
Specific correlation matrix after
Cholesky decomposition
R
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Table A3: Actuarial Assumption of Twenty-Year Term Life Insurance
Insured’s Age
Mortality Rate of age a
At the Beginning of Policy Year
Surrender Value
Commission Rate
Fixed Expense
Variable Cost Rate
a q( )ad term k Bs k,term Lcm k,term λkterm Lvcostterm
30 0.0009790 1 N/A 62.40% 420 0.001
31 0.0010055 2 N/A 22% 126 0.001
32 0.0010481 3 1,359 14.6% 126 0.001
33 0.0011075 4 2,758 8.0% 126 0.001
34 0.0011826 5 4,169 8.0% 126 0.001
35 0.0012712 6 5,566 8.0% 126 0.001
36 0.0013711 7 6,923 8.0% 126 0.001
37 0.0014807 8 8,212 8.0% 126 0.001
38 0.0015989 9 9,410 8.0% 126 0.001
39 0.0017291 10 10,491 8.0% 126 0.001
40 0.0018749 11 11,422 5.0% 126 0.001
41 0.0020407 12 11,981 5.0% 126 0.001
42 0.0022297 13 12,284 5.0% 126 0.001
43 0.0024446 14 12,282 5.0% 126 0.001
44 0.0026795 15 11,918 5.0% 126 0.001
45 0.0029268 16 11,141 5.0% 126 0.001
46 0.0031784 17 9,910 5.0% 126 0.001
47 0.0034268 18 8,198 5.0% 126 0.001
48 0.0036671 19 5,986 5.0% 126 0.001
49 0.0039091 20 3,263 5.0% 126 0.001
50 N/A 20* N/A N/A N/A N/A
1. The death benefit is $1,000,000. The policy is issued to a 30 year-old male, and the annual premium the insured is expected to pay at the beginning of each surviving year is $4,200 under the policy crediting rate of 4%.
2. We assume a fixed rate of surrenders in each policy year at 5% for term life insurance.
3. Notation 20* is used to denote the end of policy year 20.
4. Policies surrendered at the beginning of the first, second, and last policy year have no surrender value. Neither mortality nor expenses apply when these policies mature. We denote all these values as N/A.
5. The variable cost is assumed to be 0.1%.
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Table A4: Actuarial Assumption of Twenty-Year Endowment
Insured’s Age
Mortality Rate of age a
At the Beginning of Policy Year
Surrender Value
Commission Rate
Fixed Expense
Variable Cost Rate
a qa( )d ed k Bs k,ed Lcm k,ed λked Lvcosted
30 0.0009790 1 N/A 62.40% 4,530 0.001
31 0.0010055 2 8,161 27.00% 1,359 0.001
32 0.0010481 3 39789 20.60% 1,359 0.001
33 0.0011075 4 73,767 14.00% 1,359 0.001
34 0.0011826 5 110,192 13.00% 1,359 0.001
35 0.0012712 6 149,173 12.00% 1,359 0.001
36 0.0013711 7 190,831 10.00% 1,359 0.001
37 0.0014807 8 235,294 10.00% 1,359 0.001
38 0.0015989 9 282,707 10.00% 1,359 0.001
39 0.0017291 10 333,223 10.00% 1,359 0.001
40 0.0018749 11 387,004 7.00% 1,359 0.001
41 0.0020407 12 437,655 7.00% 1,359 0.001
42 0.0022297 13 490,342 7.00% 1,359 0.001
43 0.0024446 14 545,163 7.00% 1,359 0.001
44 0.0026795 15 602,227 7.00% 1,359 0.001
45 0.0029268 16 661,664 7.00% 1,359 0.001
46 0.0031784 17 723,620 7.00% 1,359 0.001
47 0.0034268 18 788,259 7.00% 1,359 0.001
48 0.0036671 19 855,760 7.00% 1,359 0.001
49 0.0039091 20 926,314 7.00% 1,359 0.001
50 N/A 20* 1,000,000 N/A N/A N/A
1. The death benefit and survival benefit is $1,000,000. The policy is issued to a 30 year-old male, and the annual premium payable at the beginning of each surviving year is $45,300 under the policy crediting rate of 4%.
2. We assume a fixed surrender rate in each policy year at 7% level for endowment.
3. Notation 20* is used to denote the end of policy year 20.
4. A policy surrendered at the beginning of the first policy year has no surrender value. Neither mortality nor expenses apply when the policy matures. We denote all these values as N/A.
5. The variable cost is assumed to be 0.1%.
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Table A5: Actuarial Assumption of Twenty-Year Pure Endowment
Insured’s Age
Mortality Rate of age a
At the Beginning of Policy Year
31 0.0010055 2 27,707 28.00% 1,071 0.001
32 0.0010481 3 57,552 21.60% 1,071 0.001
33 0.0011075 4 89,652 15.00% 1,071 0.001
34 0.0011826 5 124,130 14.00% 1,071 0.001
35 0.0012712 6 161,121 13.00% 1,071 0.001
36 0.0013711 7 200,769 6.00% 1,071 0.001
37 0.0014807 8 243,227 6.00% 1,071 0.001
38 0.0015989 9 288,660 6.00% 1,071 0.001
39 0.0017291 10 337,247 6.00% 1,071 0.001
40 0.0018749 11 389,178 5.00% 1,071 0.001
41 0.0020407 12 438,092 5.00% 1,071 0.001
42 0.0022297 13 489,259 5.00% 1,071 0.001
43 0.0024446 14 542,825 5.00% 1,071 0.001
44 0.0026795 15 598,957 5.00% 1,071 0.001
45 0.0029268 16 657,831 5.00% 1,071 0.001
46 0.0031784 17 719,633 5.00% 1,071 0.001
47 0.0034268 18 784,554 5.00% 1,071 0.001
48 0.0036671 19 852,785 5.00% 1,071 0.001
49 0.0039091 20 924,525 5.00% 1,071 0.001
50 N/A 20* 1,000,000 N/A N/A N/A
1. The survival benefit is $1,000,000. The policy is issued to a 30 year-old male, and the annual premium payable at the beginning of each surviving year is $35,700 under the policy crediting rate of 4%.
2. We assume a fixed surrender rate in each policy year at 7% level for pure endowment.
3. Notation 20* is used to denote the end of policy year 20.
4. A policy surrendered at the beginning of the first policy year has no surrender value. Neither mortality nor expenses apply when the policy matures. We denote all these values as N/A.
5. The variable cost is assumed to be 0.1%.
Table A6: Promising Asset Allocation and Leverage Ratio before PSO Converges
Leverage Ratio = 20 Objective Value = 0.2443
Simulation Paths Insolvencies = 20
Volatility of Annual Rate of Return on Equity = 0.07215
Asset Weights (%)
Asset Class / Time 1-5 6-10 11-15 16-20
(A) Stock 11.16 16.23 61.35 9.94
(B) Real Estate 3.83 29.14 9.64 2.83
(C) Alternative Investment 16.87 12.37 3.94 39.37 (D) Risk-Free Zero-Coupon –Bonds 68.15 42.25 25.08 47.85 Total Risky Assets = (A)+(B)+(C) 31.85 57.75 74.92 52.15 Total Fix Income Securities = (D) 68.15 42.25 25.08 47.85