FRS17 Case Study
by P Barber (April 2010)
Introduction
1. The requirements of FRS 17 necessitated the valuation of the Company’s unfunded pension obligations. Under a now obsolete scheme a selected number of people on leaving the Company had been granted a graceandfavour indexlinked pension, which was paid out of Company funds. In recent years the number of pensioners on this scheme has reduced to single digit levels, and annual pension payments amount to some £13,000 in total and a provision of £162,400 has been deemed sufficient to cover the outstanding obligation.
2. The process for calculating the value of the FRS17 provision has been in place for a number of years. It is based on a table of lifeexpectance, downloaded from the web, which differentiates between males and females and provides an expected number of years, based on the age of the pensioner. A segment of the table is shown below:
Life Expectancy 


Age 
M 
F 
66 
14.32 
18.21 
67 
13.7 
17.48 
68 
13.09 
16.76 
69 
12.5 
16.04 
70 
11.92 
15.35 
71 
11.35 
14.66 
" 
" 
" 
99 
2.34 
2.64 
100 
2.22 
2.48 
101 
2.11 
2.34 
102 
1.99 
2.2 
103 
1.89 
2.06 
104 
1.78 
1.93 
" 
" 
" 
115 
0.89 
0.89 
116 
0.83 
0.83 
117 
0.77 
0.77 
118 
0.71 
0.71 
119 
0.66 
0.66 
3. As can be seen from the table above, a male aged 99 years is expected to live for 2.34 years more, or until he is 101.34 years of age (99 years + 2.34 years). However, should he live until he is 100 then his life expectancy is 2.22 years and he is expected to live until he is 102.22 years of age (100 years + 2.22 years)
4. It was found that when using a model based on the above assumption that the calculated value of the provision altered little from year to year. This find was at odds with the expectation that:
Closing Provision = Opening Provision – amount expended during the year
5. An investigation was therefore initiated to devise an alternative model, and it is this model which is presented as a case study.
New Model
6. The search for am alternative model began on the web, and after some investigation an alternative source of life expectancy data was located at the Office of National Statistics. A segment of the table is shown below:
Interim
Life Tables, 








 












Period
expectation of life 








Office
for National Statistics  
Based on
data for the years 20062008 







 












Age 
Males 

Females  
x 


0 
0.005393 
0.005379 
100000.0 
537.9 
77.74 

0.004436 
0.004426 
100000.0 
442.6 
81.88 
1 
0.000402 
0.000402 
99462.1 
40.0 
77.16 

0.000347 
0.000347 
99557.4 
34.6 
81.24 
2 
0.000237 
0.000237 
99422.1 
23.6 
76.19 

0.000204 
0.000204 
99522.9 
20.3 
80.27 
3 
0.000165 
0.000165 
99398.6 
16.4 
75.20 

0.000163 
0.000163 
99502.6 
16.2 
79.28 
4 
0.000138 
0.000138 
99382.2 
13.7 
74.22 

0.000115 
0.000115 
99486.4 
11.4 
78.30 
5 
0.000135 
0.000135 
99368.5 
13.4 
73.23 

0.000099 
0.000099 
99475.0 
9.9 
77.30 
6 
0.000106 
0.000106 
99355.1 
10.5 
72.24 

0.000095 
0.000095 
99465.1 
9.4 
76.31 
7 
0.000102 
0.000102 
99344.6 
10.1 
71.24 

0.000078 
0.000078 
99455.7 
7.7 
75.32 
8 
0.000126 
0.000126 
99334.5 
12.5 
70.25 

0.000093 
0.000093 
99448.0 
9.3 
74.33 
9 
0.000120 
0.000120 
99322.0 
11.9 
69.26 

0.000075 
0.000075 
99438.7 
7.5 
73.33 
10 
0.000104 
0.000104 
99310.1 
10.4 
68.27 

0.000087 
0.000087 
99431.2 
8.6 
72.34 
11 
0.000115 
0.000115 
99299.7 
11.4 
67.28 

0.000092 
0.000092 
99422.6 
9.1 
71.34 
12 
0.000118 
0.000118 
99288.3 
11.8 
66.28 

0.000100 
0.000100 
99413.5 
9.9 
70.35 
13 
0.000157 
0.000157 
99276.5 
15.6 
65.29 

0.000119 
0.000119 
99403.6 
11.8 
69.36 
7. The field indicated in the table are:
m_{x }– the number of deaths, at the age x (last birthday) in the three year period to which the table relates, divided by the average population at that age over the same period.
q_{x} – the probability that a person aged x (exactly) will die before reaching age (x +1)
l_{x} – the number of survivors to (exact) age x per 100,000 live births.
d_{x} – the number dying between exact age x and (x+1) described similarly to l_{x}, that is d_{x} = l_{x} – l_{x+1}
e_{x} – the average life expectancy, that is the average number of years that those aged x (exact) will live beyond x
8. The new model assumes that a pension is paid in full for each year up to the year of death, and that no payment is made thereafter. Although FRS17 suggests that discounting of the cashflows should be applied, this is not attempted in this model, as there is no attempt to create a specific cash or deposit fund to cover these payments.
9. It is assumed that the value of the pension is increased by the rate of inflation. The pension payment is therefore modelled using the incremental equation:
Px+1 = Px (Inflation Factor)
Where:
Px = the pension paid in year x.
Px+1 = pension paid in the year following year x, and
Inflation factor is model using a BetaPERT Distribution, defined (in Oracle Crystal Ball) as follows:
10. The table below shows a segment of the calculation to determine the pension that might typically be paid to one individual over a period of twelve years. The table shows that a current pension of £7,395 might rise to £10,053 by year 8. In the model the calculation is repeated 10,000 to determine the range of outcomes that might apply. On each iteration and for each year a possible inflation factor is selected at random from the BetaPERT range that has been applied. In this model it has been assumed that the distribution of the inflation factors remains constant through the whole range of the model. However, a different inflation assumption could have been applied to each year, should this have been thought necessary.

Inflation 
Annual 
Year 
Forecast 
Pension 
0 

7,395 
1 
1.038726 
7,682 
2 
1.037561 
7,970 
3 
1.046469 
8,340 
4 
1.038355 
8,660 
5 
1.044785 
9,048 
6 
1.031487 
9,333 
7 
1.040164 
9,708 
8 
1.035507 
10,053 
9 
1.038252 
10,437 
10 
1.029655 
10,747 
11 
1.034493 
11,117 
12 
1.032799 
11,482 
11. When considering the liability, with regard to a pensioner for a specific year, there are three possibilities:
a) the pensioner survives, in which case the pension plus inflation is paid in that particular year, or
b) the pensioner dies in that year, in which case no payment is made in that particular year, or
c) the pensioner died in a previous year, in which case no payment will be made in that particular year.
12. The probability of death in a particular year is shown in column q_{x} of the Life Expectancy table, while the probability of survival is given by (1 – q_{x}).
13. In the example below the process is illustrated with reference to an 89 yearold, female pensioner. Appropriate details from the Life Expectancy table are reproduced below:
Age 
Female 
x 

89 
0.122312 
90 
0.138332 
91 
0.159600 
92 
0.178012 
93 
0.196646 
94 
0.214254 
95 
0.234928 
96 
0.254641 
97 
0.268735 
98 
0.298142 
99 
0.314155 
100 
0.339337 
14. The Life Expectancy table shows that for a female aged 89 years there is a 0.122312 chance of dying before the age of 90 years. However, we are interested in the probability of living, which is (1 – 0.122312 = 0.8777). This probability can be defined by a Yes/No distribution shown below:
15. During a simulation Oracle Crystal Ball produces a value of 1 on 87.7688% of occasions and a value of zero on 12.2312% of occasions. In the following year (age 90 years) Crystal Ball will produce, at random, a value of zero on 13.8332% of occasions and a value of 1 one 86.1668% of occasions. If a value of zero is selected in a particular year then the values of all subsequent years will also be set to zero. A segment of the simulation, illustrating three iterations, is shown in the Appendix. The result of a full tenthousand iteration simulation is shown below.
16. Oracle Crystal Ball has suggested that the result of the full simulation is best described by a Beta Distribution. However, with an AD statistic as high as 63.848 the fit is not particularly good. Unfortunately, Oracle Crystal Ball does not provide a PValue for the Beta Distribution.
17. The results of a simulation for all of the pensioners is shown below.
18. With Crystal Ball suggesting that the best fit to the data was a Beta Distribution, with an AD of 0.2053; this option was selected, and is shown in more detail below.
19. The validity of the above result was investigated further (see Appendix 2). It was found that AD statistic could be replicated in an Excel spread sheet, a result which indicated that:
a) The formula for the Beta Distribution available in Excel “BETADIST(x, Alpha, Beta, Minimum, Maximum)” was the same as that employed in Crystal Ball, and
b) That the Anderson Darling statistic was calculated in the same way.
Appendix 1 (page 1)

In
year 





Die =
0 
Once 
Inflation 
Inflated 
Possible 
Age 
Live =
0 
Dead =
0 
Factor 
Payment 
Payment 

(a) 
(b) 
(c) 
(d) 
(e) 




885 

89 
1 
1 
1.0417 
922 
922 
90 
1 
1 
1.0400 
959 
959 
91 
1 
1 
1.0301 
988 
988 
92 
1 
1 
1.0463 
1,033 
1,033 
93 
1 
1 
1.0332 
1,068 
1,068 
94 
1 
1 
1.0323 
1,102 
1,102 
95 
1 
1 
1.0292 
1,134 
1,134 
96 
1 
1 
1.0359 
1,175 
1,175 
97 
1 
1 
1.0234 
1,203 
1,203 
98 
1 
1 
1.0258 
1,234 
1,234 
99 
1 
1 
1.0328 
1,274 
1,274 
100 
1 
1 
1.0325 
1,316 
1,316 
101 
1 
1 
1.0301 
1,355 
1,355 
102 
1 
1 
1.0302 
1,396 
1,396 
103 
1 
1 
1.0363 
1,447 
1,447 
104 
1 
1 
1.0350 
1,497 
1,497 
105 
1 
1 
1.0406 
1,558 
1,558 
106 
0 
0 
1.0331 
1,610 
0 
107 
1 
0 
1.0293 
1,657 
0 
108 
1 
0 
1.0415 
1,726 
0 









Possible
Liability 
20,660 
a) Column (a) shows that in the first iteration, the options selected result in the pensioner surviving until she is 105 years of age, this is indicated by the unbroken series of ones appearing in column (a)
b) Column (b) monitors the values appearing in column (a) and once a value of zero has occurred ensures that all subsequent values appearing in column (b) are set to zero. “If(Value of (a) = 0, then set to 0, otherwise set to previous value of (b)”
c) Column (c) shows the inflation factors which have been selected.
d) The Inflation Factors from column (c) are used to calculate a possible value for the inflated pension, which appears in column (d). At the start of the simulation the pension is £885, by the time the pensioner reaches 90 years of age the pension is: £885 x 1.0417 x 1.04 = £959
e) The value shown in column (e) shows the value of pension that might be paid. This value is calculated by multiplying column (b) by column (e). For example at 105 years of age 1 x £1,558 = £1,558, while at 106 years of age 0 x £1,610 = 0. The total at the bottom of column (e) indicates that the FRS17 liability, based on all the selected assumptions made during the simulation, might be £20,660
Appendix 1 (page 2)

In
year 





Die =
0 
Once 
Inflation 
Inflated 
Possible 
Age 
Live =
0 
Dead =
0 
Factor 
Payment 
Payment 

(a) 
(b) 
(c) 
(d) 
(e) 




885 

89 
1 
1 
1.0353 
916 
916 
90 
1 
1 
1.0386 
952 
952 
91 
1 
1 
1.0365 
986 
986 
92 
1 
1 
1.0375 
1,023 
1,023 
93 
1 
1 
1.0343 
1,058 
1,058 
94 
1 
1 
1.0320 
1,092 
1,092 
95 
1 
1 
1.0336 
1,129 
1,129 
96 
1 
1 
1.0304 
1,163 
1,163 
97 
0 
0 
1.0341 
1,203 
0 
98 
0 
0 
1.0277 
1,236 
0 
99 
1 
0 
1.0443 
1,291 
0 
100 
1 
0 
1.0373 
1,339 
0 
101 
1 
0 
1.0336 
1,384 
0 
102 
1 
0 
1.0263 
1,420 
0 
103 
1 
0 
1.0301 
1,463 
0 
104 
0 
0 
1.0385 
1,520 
0 
105 
1 
0 
1.0341 
1,571 
0 
106 
1 
0 
1.0301 
1,619 
0 
107 
0 
0 
1.0336 
1,673 
0 
108 
0 
0 
1.0293 
1,722 
0 









Possible
Liability 
8,320 
Under the assumptions made in the second iteration the possible liability has been calculated at £8,320.
Appendix 1 (page 3)

In
year 





Die =
0 
Once 
Inflation 
Inflated 
Possible 
Age 
Live =
0 
Dead =
0 
Factor 
Payment 
Payment 

(a) 
(b) 
(c) 
(d) 
(e) 




885 

89 
1 
1 
1.0315 
913 
913 
90 
1 
1 
1.0395 
949 
949 
91 
1 
1 
1.0297 
977 
977 
92 
0 
0 
1.0401 
1,016 
0 
93 
0 
0 
1.0275 
1,044 
0 
94 
1 
0 
1.0409 
1,087 
0 
95 
1 
0 
1.0417 
1,132 
0 
96 
0 
0 
1.0362 
1,173 
0 
97 
1 
0 
1.0338 
1,213 
0 
98 
0 
0 
1.0358 
1,256 
0 
99 
0 
0 
1.0379 
1,304 
0 
100 
1 
0 
1.0425 
1,360 
0 
101 
0 
0 
1.0378 
1,411 
0 
102 
1 
0 
1.0292 
1,452 
0 
103 
1 
0 
1.0292 
1,495 
0 
104 
0 
0 
1.0238 
1,530 
0 
105 
0 
0 
1.0324 
1,580 
0 
106 
0 
0 
1.0336 
1,633 
0 
107 
1 
0 
1.0431 
1,703 
0 
108 
0 
0 
1.0328 
1,759 
0 









Possible
Liability 
2,839 
Under the assumptions made in the third iteration the possible liability has been calculated at £2,839
Appendix 2 (page 1)
1. It has
been found that on a number of occasions when Crystal Ball has fitted a Beta
Distribution to a dataset it has not been possible to replicate the calculation
of the Anderson Darling statistic in a spreadsheet using the parameters advised.
A request for information was made to Oracle but a response is outstanding still
outstanding a week later.
CB.Beta2(18.581, 760332, 2.78692, 9.12524, 12,
1000000,”Name”)
Note that the use of the term “Nameof” is not clear, as it
does not seem to feature in the Crystal Ball help menu. However, it is found
that if the Sampling preference is set to use the same sequence of random
numbers, then stepping the model creates a sequence of numbers of particular
values. Alternatively, the same sequence of numbers can be produced by using the
Beta Distribution as shown in section 18. above.