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In this example we are going to show how to calculate the liability and the predicted cash flows for a retirement benefit for an individual in a pension fund. A determinist and a stochastic version of the calculus for a pension scheme that includes a regular retirement income, a disability income, a death pension benefit and a life insurance will be presented below.

 

A

B

C

1

Pension liability example:

 

 

2

 

 

 

3

Retired contribution

0.05

 

4

Worker contribution rate over gross salary

0

 

5

Worker contribution rate over net salary (gross - social security)

0.1

 

6

Administrative costs

0.1

 

7

Current gross salary (annual)

100000

 

8

Current age

30

 

9

Retirement age

65

 

10

Growth (real) rate of worker salary

0.03

 

11

Growth (real) rate of retired pension

0

 

12

Disability entrance table

80

 

13

Life table normal life

6

 

14

Life table disabled life

2

 

15

Current Social Security reference benefit (annual)

30000

 

16

Growth (real) rate of Social Security reference benefit

0.01

 

17

Is normal

TRUE

 

18

Forward (real) interest rate curve

0.05

 

19

Life table dependent

1

 

20

Expected inflation rate

0.05

 

21

Life insurance benefit (% over salary)

0.083333333

 

22

Age at wedding

27

 

23

Children maximum age to receive benefits

20

 

24

Partner age

28

 

25

First child age

1

 

26

Second child age

-2

 

27

Third child age

131

 

28

Fourth child age

131

 

29

Fifth child age

131

 

30

Extra worker contribution

0

 

31

Limiting age to become disabled

130

 

32

Retirement salary conversion rate

1

 

33

Dependent benefit conversion rate

1

 

34

 

 

 

35

Actuarial liability (deterministic)

-613131.82

=sActuarialBenefitsLiability($B$3,$B$4,$B$5,$B$6,$B$7,$B$8,$B$9,sActuarialFlatRateVector($B$10),

sActuarialFlatRateVector($B$11),sMxElemMult(sActuarialqxVector($B$12),

sActuarialdummyVector($B$31,1)),$B$13,$B$14,$B$15,sActuarialFlatRateVector($B$16),

$B$17,sActuarialFlatRateVector($B$18),$B$19,$B$20,$B$21,

$B$22,$B$23,$B$24,$B$25,$B$26,$B$27,$B$28,$B$29,$B$30,$B$32,$B$33)

36

Actuarial liability (stochastic)

-830719.63

=sActuarialRANDBenefitsLiability($B$3,$B$4,$B$5,$B$6,$B$7,$B$8,$B$9,sActuarialFlatRateVector($B$10),

sActuarialFlatRateVector($B$11),sMxElemMult(sActuarialqxVector($B$12),sActuarialdummyVector($B$31,1)),

$B$13,$B$14,$B$15,sActuarialFlatRateVector($B$16),$B$17,sActuarialFlatRateVector($B$18),

$B$19,$B$20,$B$21,$B$22,$B$23,$B$24,$B$25,$B$26,$B$27,$B$28,$B$29,$B$30,$B$32,$B$33)

37

 

 

 

As the actuarial liability presented in cell B36 is random, that is, it varies each time that the spreadsheet is recalculated, a Monte Carlo simulation study can be run using the Simulation Toolbox. The graph below shows the result of 1,000 samples.

As you should note, as we are expecting the average of the stochastic liability is very close to the one calculated using the determinist formula.

Now, the same kind of analysis can be made to get the projected cash flows. The charts below were built with the functions “sActuarialCashFlows” and “sActuarialRANDCashFlows” that receives the same inputs that the functions “sActuarialLiabilities” and “sActuarialRANDLiabilities”.

The stochastic chart was built using the time series simulation procedure, as explained here.

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