simulating a worst case social security model

7/29/2019 Simulating a Worst Case Social Security Model

1/20

Kenji L Logie Modeling Simulation Final Project

Page 1 of20

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


2/20


Page 2 of20

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.


3/20


Page 3 of20

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


4/20


Page 4 of20

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


5/20


Page 5 of20

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


6/20


Page 6 of20

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





60 contribution only periods 5.47





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.


7/20


Page 7 of20

Capital + Contributions


8/20


Page 8 of20

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


9/20


Page 9 of20

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


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


10/20


Page 10 of20

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)


(months)


(months)


(months)


11/20


Page 11 of20

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


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.


12/20


Page 12 of20

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


13/20


Page 13 of20

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


14/20


Page 14 of20

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


15/20


Page 15 of20

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


16/20


Page 16 of20

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


17/20


Page 17 of20

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


18/20


Page 18 of20

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


19/20


Page 19 of20

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


20/20


Page 20 of20

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

simulating a worst case social security model

Documents