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Simulating a Worst Case Social Security Model
System of Interest
For the purpose of this simulation a simplified worst case social security model was
created for a developing countrys social security program. The program simulates
how long it would take this new social security program to go bankrupt, if it earns
no interest on its capital, and its only source of paying out benefits is members
monthly contributions and its initial capital of $750,000.
This model only deals with paying into the social security for the purpose of
retirement benefits. Persons pay into the social security program at one of three
possible earning levels monthly. This would lead to one of three possible payouts of
benefits monthly for persons 60 years or older. Simulating the point of bankruptcy
begins at the third part of the program when the organization has built up capital
over a period of time, which is defined within the simulation as the contribution
only period without paying out any benefits. At the end of the contribution only
period it begins to pay out benefits to anyone in the simulation eligible to receive
benefits which by definition is a members who is 60 years or older.
Events and System VariablesEvents: a new person joins the work force, payment of benefits, and receiving
contributions
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System State variables:
c- Social Security capital,
n- Maximum number of persons in the system at that point in time
t- Time
Discrete Event Simulation
The simulation represents a discrete event simulation. The events all take place
chronologically, and represent state changes at a particular point in the simulation.
The simulation begins with the creation of a fixed number of persons, and the time
at which a new person joining the workforce is generated from an exponential
distribution to move time forward. This step is followed by generating more times
persons joining the workforce, payment of benefits or receiving contribution; an
increment of one since time in the simulation is measured in months. Each event
moves time forward until the capital of the social security becomes negative; the
point of termination in the simulation.
Worst Case Social Security Model
The simulation represents a Poisson Process. Each inter arrival time of a person to
the workforce is independent of the person before them and it occurs at a rate
specified within the simulation.
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The simulation is a terminating simulation. The simulation ends when the social
security office becomes bankrupt. This event occurs when the amount of cash paid
out exceeds the amount currently held by the social security office.
The following simplification assumptions were made for the purpose of the model
or imposed by restrictions of the program used to simulate the model:
The maximum number of persons the system can accommodate is 90000persons.
There are only three levels of contributions and benefit payments. The simulation time variables are all measured in months. The simulations values of capital, benefits and payments all in $100,000. Persons join the workforce at a random age between 18 and 60 and only
leave the workforce at retirement.
Persons live to a life expectancy based on their gender; 70 years for malesand 76 years for females.
The population ratio for males to females is approximately 1:1.
The input distribution was exponential and was chosen based on data provided by
the Social Security the simulation was modeled on.
Purpose of performing simulation
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The purpose of the simulation is to calculate how long it takes for a simulation to go
bankrupt. From this initial question the rate of joining the workforce, return on
capital and the contribution period were all varied individually while holding
everything else constant to determine which variable extended the length of time it
takes to go bankrupt. An average of the time taken to go bankrupt was calculated for
all these variations to the base case.
The model in its current inception models an existing system, in a simplified version
of its worst case scenario. The variation to individual variables in the simulation
(rate of joining the workforce, contribution only periods and return on capital) also
allow recommendations to be made in extended the life of the social security even
under simplified worst case conditions.
Programing the Worst Case Scenario Simulation
The simulation used was written by the user using the programming language C++.
The program can be divided into three parts. The first part involves generating the
initial population the social security would start with. The second part of the
program involved simulating the contribution only period in which persons
between the age of 18 and 60 paid contributions based on their income level. During
this period persons between the age of 18 and 60 at a specified exponential monthly
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rate joined the social security and begin to pay contribution. Persons over the age of
60 do not receive benefits during the period of contribution only. The third part of
the program involves persons joining the workforce, paying contributing, and
paying benefits to persons age 60 or older based on their income level, until they
reach their life expectancy based on their gender. Each part was validated by using
cout statements and the data generated was observed for irregularities. Based upon
these observations all dollar amounts in the simulation are in $100,000 and the
Mersenne Twister was used instead of C++ standard rand generator.
Running Simulation
Base case conditions
Starting capital = $750,000
Persons join the social security at a rate of a 150 new persons a month
No return on capital
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Simulation Runtime (seconds)
No return on capital (Base Case) 5.47
1 percent return on capital monthly 8.79
$5 million return on capital monthly 26.2
Rate of joining 50 persons a month 1.26
Rate of joining 100 persons a month 3.03
Rate of joining 150 persons a month 5.47
Rate of joining 200 persons a month 9.03
Rate of joining 250 persons a month 13.64
60 contribution only periods 5.47
120 contribution only periods 9.10
150 contribution only periods 11.10
180 contribution only periods 13.63
210 contribution only periods 15.65
Termination of Simulation
All simulations performed terminate at the point which the capital held by the social
security plus the contribution is less than the benefits paid out for a particular
month.
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Capital + Contributions
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Table showing results of variation of the return on capital for the social security
monthly only.
Graph shows the difference in the length of time the social security goes bankruptwith a variation in the return on capital
Rate at which
persons join the
workforce
lower
limit
upper
limit variance
standard
error xbar execution time per run
50 139.988 141.452 14.0824 0.375266 140.72 1.26100 148.114 149.606 14.6267 0.382448 148.86 3.03
150 153.752 155.208 13.9491 0.373485 154.48 5.47
200 159.505 160.795 10.9369 0.330709 160.15 9.03
250 164.16 165.36 9.47717 0.30785 164.76 13.64
0
50
100
150
200
250
300
350
400
450
1 611
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
Month(no additional capital)
(5 million)Month(1percent reurn) Month
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Table showing results of variation in rate at which persons join the social security
monthly only.
Graph shows the difference in the length of time the social security goes bankruptwith a variation in the rate at which persons join the social security monthly
Variation in
contribution
lower
limit
upper
limit variance
standard
error xbar execution time per run
60 153.752 155.208 13.9491 0.373485 154.48 5.47
120 213.556 214.564 6.68623 0.258519 214.06 9.10
150 244.58 245.342 4.04798 0.201196 244.95 11.10
180 277.196 277.944 3.68192 0.191883 277.57 13.63
0
20
40
60
80
100
120
140
160
180
200
1 5 913
17
21
25
29
33
37
41
45
49
53
57
61
65
69
73
77
81
85
89
93
97
rate 50 Month
rate 100 Month
rate 150 Month
rate 200 Month
rate 250 Month
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210 309.716 310.384 2.93687 0.171373 310.15 15.65
Table showing results of variation in contribution only period.
Graph shows the difference in the length of time the social security goes bankruptwith a variation in the contribution only period
Results for Original Question
0
50
100
150
200
250
300
350
1 611
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
contributon period
120(Months)
contribution period 60
(months)
contribution period 150
(months)
contribution period 180
(months)
contribution period 210
(months)
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The original question asked when creating this model was; how long would a worst
case simplified social security model take to go bankrupt. After repeating the
original simulation 10,000 times under the base case conditions the following table
contains the results of the experiment.
lower
limit
upper
limit variance
standard
error xbar execution time per run
154.947 155.085 12.4408 0.3502715 155.016 5.63
All data generated for results except the base 10,000 case can be found in the
appendix.
Suggestions for Improving the Simulation
An exponential death rate for persons between the age of 18 and 60 should bedeveloped. A death rate for persons over the age of 60 should only be
developed if a normal case was to be developed. Since a worst case model
means the social security runs out of money; it is normally due to persons
maximize their benefits by living to their life expectancy or greater. Contributions and benefits should be determined as a percentage of a
persons income. Persons should be allowed to leave the work force temporary or permanently
for reasons other than death. The simplification assumptions should be removed one by one for improved
results.
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Reference page
Arbez, Birta. Modeling and Simulation Exploring Dynamic System Behaviour. Ottawa: Springer, 2007.
Law, Averill M. and W.David Kelton. Simulation Modeling & Analysis. McGraw-Hill, 1991.
Ross, Sheldon M. Simulation. San Diego: Academic Press, 2006.
Severance, Frank L. System Modeling and Simulation An Intoduction. New York: Wiley, 2001.
AppendixData generated by simulation and used to create the graphs
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Observation
contribution
period 120
contribution
period 60
contribution
period 150
contribution
period 180
contribution
period 210
1 213 156 243 275 308
2 218 156 244 277 309
3 213 153 242 279 312
4 216 167 243 278 310
5 211 165 244 276 309
6 215 155 247 278 310
7 211 158 247 277 312
8 216 153 245 279 310
9 213 152 244 278 310
10 213 159 243 280 313
11 212 153 246 277 311
12 213 154 246 281 310
13 215 149 242 280 312
14 215 162 247 277 312
15 211 156 246 279 310
16 217 154 250 281 309
17 214 149 244 277 313
18 210 150 245 278 313
19 208 157 249 279 313
20 215 161 243 279 311
21 213 153 248 278 311
22 220 157 246 280 307
23 214 151 243 276 312
24 212 157 246 275 308
25 213 146 245 277 311
26 217 152 243 276 307
27 217 153 245 274 312
28 214 151 246 275 308
29 219 154 244 274 307
30 216 148 243 277 310
31 215 151 247 277 309
32 213 160 245 276 311
33 213 155 245 279 304
34 216 156 248 276 310
35 218 153 245 278 309
36 214 149 244 280 309
37 211 155 248 278 309
38 214 149 244 275 310
39 213 156 243 282 310
40 214 157 245 280 311
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41 212 155 247 280 307
42 214 156 243 276 311
43 216 156 246 281 311
44 214 159 245 278 307
45 214 155 242 276 310
46 215 160 243 277 308
47 213 154 245 280 310
48 214 156 245 277 309
49 212 156 247 277 311
50 217 153 242 282 310
51 218 158 244 277 311
52 210 149 245 278 312
53 213 154 243 275 312
54 213 154 245 277 308
55 210 152 244 276 311
56 214 155 241 279 311
57 218 158 246 280 309
58 214 151 244 273 311
59 217 158 245 280 310
60 216 156 248 277 312
61 210 153 244 276 311
62 212 151 243 277 312
63 215 157 247 276 310
64 214 149 247 277 311
65 215 156 244 278 310
66 218 157 244 280 309
67 209 153 246 276 313
68 216 152 245 276 310
69 211 153 245 280 310
70 208 150 248 277 307
71 217 151 246 280 310
72 213 151 244 276 312
73 215 154 245 278 308
74 218 155 243 275 310
75 213 151 247 274 31176 216 155 244 275 309
77 215 159 249 279 307
78 214 154 246 282 308
79 217 152 242 277 311
80 213 155 245 279 309
81 218 151 245 279 313
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82 221 157 243 278 309
83 213 152 244 276 310
84 214 149 243 276 311
85 213 151 244 278 310
86 208 158 240 276 310
87 210 151 250 276 307
88 213 153 244 277 313
89 210 152 243 276 309
90 216 156 244 278 309
91 214 152 244 277 311
92 214 156 246 277 308
93 212 161 244 279 310
94 212 151 248 277 312
95 216 158 242 280 310
96 214 162 249 278 311
97 216 155 249 275 310
98 215 151 245 277 309
99 215 156 244 279 310
100 214 161 247 276 312
Observation
rate 50
Month
rate 100
Month
rate 150
Month
rate 200
Month
rate 250
Month
1 142 151 156 160 164
2 142 147 156 167 168
3 142 145 153 162 161
4 138 152 167 164 167
5 143 153 165 162 161
6 139 153 155 159 164
7 144 144 158 162 162
8 141 151 153 163 164
9 146 154 152 162 161
10 133 155 159 162 160
11 142 149 153 167 163
12 140 149 154 159 166
13 143 155 149 164 167
14 143 151 162 162 165
15 138 147 156 155 170
16 142 142 154 160 165
17 139 145 149 162 167
18 140 149 150 158 166
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19 146 154 157 159 164
20 139 144 161 156 167
21 140 141 153 154 168
22 140 151 157 161 171
23 141 152 151 156 162
24 143 152 157 161 173
25 138 152 146 157 165
26 143 148 152 155 171
27 137 147 153 158 164
28 141 155 151 156 166
29 146 152 154 160 163
30 139 142 148 162 164
31 131 143 151 160 165
32 141 150 160 161 169
33 137 147 155 162 166
34 133 146 156 159 159
35 149 146 153 165 166
36 139 150 149 172 163
37 141 153 155 160 166
38 139 150 149 156 162
39 137 148 156 159 164
40 133 148 157 155 164
41 140 152 155 157 162
42 142 147 156 163 172
43 144 144 156 161 164
44 136 145 159 158 167
45 141 150 155 158 161
46 142 148 160 161 163
47 145 149 154 163 162
48 143 151 156 166 162
49 146 146 156 162 166
50 144 145 153 157 161
51 140 148 158 158 169
52 142 150 149 162 163
53 140 152 154 162 15754 143 141 154 160 162
55 148 152 152 164 160
56 142 145 155 159 165
57 136 149 158 155 168
58 143 145 151 165 166
59 133 151 158 157 167
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60 152 147 156 159 164
61 145 145 153 160 162
62 142 146 151 158 163
63 144 149 157 158 166
64 144 141 149 160 162
65 142 152 156 160 166
66 135 154 157 162 162
67 142 147 153 167 169
68 136 143 152 164 161
69 143 144 153 159 161
70 141 154 150 152 171
71 145 144 151 157 165
72 144 148 151 162 166
73 141 148 154 161 168
74 140 146 155 162 163
75 141 150 151 160 164
76 140 144 155 157 163
77 141 150 159 157 169
78 146 152 154 159 158
79 141 157 152 161 166
80 139 147 155 164 163
81 140 149 151 152 161
82 146 146 157 161 165
83 141 156 152 159 165
84 133 154 149 155 162
85 137 151 151 161 166
86 143 148 158 160 169
87 145 151 151 161 170
88 137 153 153 165 163
89 137 148 152 158 169
90 135 152 156 161 168
91 140 148 152 162 162
92 133 144 156 164 167
93 145 142 161 159 166
94 141 151 151 156 16495 138 151 158 158 164
96 139 148 162 160 165
97 141 154 155 159 164
98 140 154 151 162 165
99 141 149 156 160 164
100 136 156 161 161 165
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Observation no return on capital 5 million 1percent return
1 156 398 205
2 156 410 1983 153 400 196
4 167 399 207
5 165 402 213
6 155 399 211
7 158 406 217
8 153 402 206
9 152 404 210
10 159 406 211
11 153 396 201
12 154 408 201
13 149 399 205
14 162 410 202
15 156 414 199
16 154 411 200
17 149 399 210
18 150 404 219
19 157 403 205
20 161 403 225
21 153 410 199
22 157 401 192
23 151 410 209
24 157 412 217
25 146 403 208
26 152 399 213
27 153 399 200
28 151 407 211
29 154 403 213
30 148 407 211
31 151 399 204
32 160 411 203
33 155 402 204
34 156 407 198
35 153 402 204
36 149 399 198
37 155 403 202
38 149 401 200
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39 156 408 206
40 157 410 207
41 155 404 200
42 156 410 210
43 156 409 203
44 159 398 205
45 155 405 211
46 160 398 210
47 154 404 199
48 156 410 206
49 156 400 211
50 153 404 198
51 158 409 199
52 149 404 203
53 154 393 198
54 154 410 202
55 152 411 202
56 155 407 194
57 158 414 198
58 151 399 203
59 158 402 209
60 156 404 205
61 153 400 201
62 151 403 204
63 157 403 210
64 149 398 202
65 156 394 212
66 157 385 195
67 153 404 204
68 152 398 208
69 153 397 197
70 150 410 207
71 151 403 201
72 151 404 211
73 154 400 20474 155 412 198
75 151 404 200
76 155 407 212
77 159 409 208
78 154 402 204
79 152 394 211
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80 155 408 206
81 151 403 217
82 157 408 203
83 152 394 200
84 149 398 220
85 151 408 208
86 158 401 206
87 151 412 206
88 153 397 216
89 152 403 205
90 156 399 210
91 152 405 197
92 156 407 203
93 161 398 208
94 151 408 190
95 158 408 215
96 162 405 210
97 155 399 211
98 151 399 209
99 156 397 199
100 161 394 206