Actuarial Factors Documentation

Transcription

1 Actuarial Factors Documentation Version Description of Change Author Date 1.00 Initial Documentation Douglas Hahn Dec 22, Corrected error in guaranteed pension Douglas Hahn Jan 6, 2017 Platinum Pro Ltd. # Stewart Green SW, Calgary, AB T3H 3C8 (403)

2 Overview This document describes the technical background behind our actuarial factors software. We have tried to follow the book Life Contingencies by Chester Wallace Jordan, Jr. We have included references from the book. The references will look something like this: LC Page 49 (2.29b) LC stands for Life Contingencies. The page number is followed by (2.29b) which is a reference to a formula in the book. (Second edition published 1975). The concept behind this software is to value pensions that are payable in the future. For example, if a pension is $1,000 a year payable annually, you obviously need more money than just $1,000. If the person is younger (like 40), you will probably need more money, but if the person is older (like 80), you will need less money. And, if the money you set aside is expected to earn 1% return on investment, you will need much more money than if the money is expected to return 10% a year. Actuarial Factors Page 1

3 Base Factor q s Our program bases its factors on a table of q s. Here are two tables that we used. We selected CSO58MaleOnly because this is a table that is listed in the Life Contingencies book. Using this table, we could verify that the factors that resulted from our software agreed with the factors in the book. MortalityTable Age Male_q Female_q CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly Actuarial Factors Page 2

4 MortalityTable Age Male_q Female_q CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly Actuarial Factors Page 3

5 MortalityTable Age Male_q Female_q CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly CSO58MaleOnly GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static Actuarial Factors Page 4

6 MortalityTable Age Male_q Female_q GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static Actuarial Factors Page 5

7 MortalityTable Age Male_q Female_q GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static Actuarial Factors Page 6

8 MortalityTable Age Male_q Female_q GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static GAM94Static Now, the Life Contingencies do not list q s to the decimal accuracy that we needed, so we keyed the l s into a spreadsheet and used them to calculate the q s using the following formula: 1958 CSO Age l d q 0 10,000,000 70, ,929,200 17, ,911,725 15, ,896,659 14, ,882,210 13, ,868,375 13, ,855,053 12, ,842,241 12, In the following formulas: x = age of annuitant d x = l x l x+1 q x = d x l x Both formulas are derived from formulas in LC Page 10. Actuarial Factors Page 7

9 d x = l x q x d x l x = q x q x = d x l x And l x+1 = l x d x l x+1 + d x = l x d x = l x l x+1 The GAM94 Static table came from the Society of Actuaries. Here s the reference that we used: This table has q s to the accuracy that we need, so we used them unaltered. The remainder of the life functions are based on the q s. Actuarial Factors Page 8

10 Expected Number of Survivors - l s The base number of lives is set to 10,000,000. l 0 = 10,000,000 The remaining l s are calculated from the q s using the following formula: l x = l x 1 q x 1 l x 1 This formula is derived from the following formulas found in LC Page 10. And l x+1 = l x d x d x = l x q x Putting the two equations together, we get l x+1 = l x l x q x l x+1 = l x q x l x l x = l x 1 q x 1 l x 1 The book also designates a method of interpolating l s on LC Page 19. l x+t l x t(l x l x+1 ) for integral x and 0 t 1. Actuarial Factors Page 9

11 Number of Expected Deaths in the Coming Year d s In the equation to follow: x = age of the annuitant LC Page 8 (1.2) d x = l x l x+1 Actuarial Factors Page 10

12 Probability that an Annuitant Will Survive p Probability that an annuitant age x will survive one year LC Page 9 (1.3) p x = l x+1 l x And probability that an annuitant age x will survive n years LC Page 9 (1.4) p x n = l x+n l x Actuarial Factors Page 11

13 Discount Rate If an interest rate is i, then the discount rate for one year is: And for multiple years (n), it is: v 1 1 = (1 + i) 1 v n 1 = (1 + i) n This formula works well for fractions of years. Actuarial Factors Page 12

14 Value of a Guaranteed Pension a There is no formula in the book that defines the formula for a guaranteed pension because there is no life component to it, however, we can extract the necessary formulas from other sources. n = guarantee in years m = number of payments per year e = Euler s number i = interest rate v=1/(1+i) (Excuse the slight variation on the equation, MS Word won t do the side bar properly). Arrears Formula The basic annual formula is derived from the following equation: a n = v 1 + v 2 + v v n Which can be changed to: v a n = v 2 + v 3 + v v n+1 And if you take the first formula and subtract the second formula, you get: a n v a n = v 1 v n+1 (1 v) a n = v 1 v n+1 a n = v1 v n+1 (1 v) If you have more than one payment per year (m payments per year), the formula would be: a (m) = 1 n m v 1 m + 1 m v 2 m + 1 m v 3 m m vm n m Which can be changed to: v 1 m a (m) = 1 n m v 2 m + 1 m v 3 m + 1 m v 4 m m v(m n)+1 m And if you take the first formula and subtract the second formula, you get: Actuarial Factors Page 13

15 a (m) vma 1 (m) = 1 n n m v 1 m 1 m v(m n)+1 m Simplifying both sides: (1 vm) 1 a (m) = v 1 m v (m n)+1 m n m 1 a (m) = v( m ) v (m n+1 m ) n m (1 v ( 1 m ) ) Advance Formula Paying in advance makes a formula that is similar: a n = v1 + v 2 + v v n Which can be changed to: v a n = v2 + v 3 + v v n+1 And if you take the first formula and subtract the second formula, you get: a n v a n = v1 v n+1 (1 v) a n = v1 v n+1 a n = v1 v n+1 (1 v) If you have more than one payment per year (m payments per year), the formula would be: Which can be changed to: a (m) = 1 n m v 1 m + 1 m v 2 m + 1 m v 3 m m vm n m v 1 m a (m) = 1 n m v 2 m + 1 m v 3 m + 1 m v 4 m m v(m n)+1 m And if you take the first formula and subtract the second formula, you get: Actuarial Factors Page 14

16 (m) vma 1 (m) = 1 m v 1 m 1 m v(m n)+1 m a n n Simplifying both sides: (1 vm) 1 a (m) = v 1 m v (m n)+1 m n m 1 a (m) = v( m ) v (m n+1 m ) n m (1 v ( 1 m ) ) a n = 1 v n m (1 v ( 1 m ) ) Continuous Formula The continuous formula requires that we use an integral: Since Then a n = Which can be evaluated to: And again using: n a n = n (1 + i) t dt t=0 x y y ln (x) = e (1 + i) t dt = t=0 a n = a n = n e t=0 e n ln (1+i) ln (1 + i) 0 t ln (1+i) 1 ln (1 + i) e n ln (1+i) ln (1 + i) n (1+i) 1 e n ln a n = ln (1 + i) dt Actuarial Factors Page 15

17 x y y ln (x) = e Results in: a n = 1 (1 + i) n ln (1 + i) Actuarial Factors Page 16

18 Commutation Functions D and N These commutation functions will be used for determining other actuarial factors. In the following formulas: x = age of annuitant w = maximum age of the mortality table t = years of the annuity D x = v x l x w x N x = D x+t t=0 We also need some functions for continuous calculations: And D x = (D x + D x+1 ) 2 N x = (N x + N x+1 ) 2 Actuarial Factors Page 17

19 Annuity with No Guarantee a First, the present value of an annuity payable in arrears without a guarantee payable annually. In the following formulas: x = age of the annuitant m = number of annuity payments per year LC Page 40 (2.2) a x = N x+1 D x Payable in arrears more frequently than annually (m times a year). LC Page 46 (2.19) a (m) x = N x+1 + m 1 D x 2m Annuity payable annually in advance. LC Page 42 (2.6) a x = N x D x Payable in advance more frequently than annually (m times a year). LC Page 47 (2.22) a (m) x = N x m 1 D x 2m Payable continuously. LC Page 50 (2.31) a x = N x D x Actuarial Factors Page 18

20 Deferred Annuity a An annuity that is payable at some time in the future (n years), if the annuitant is alive at that time. In the following formulas: x = age of the annuitant m = number of annuity payments per year n = number of guaranteed years Payable annually in arrears. LC Page 41 (2.4) n ax = N x+n+1 D x Payable in arrears more frequently than annually (m times a year). LC Page 46 (2.20) n a (m) x = N x+n+1 + m 1 D x 2m Annuity payable annually in advance. LC Page 42 (2.7) n a x = N x+n D x Payable in advance more frequently than annually (m times a year). LC Page 47 (2.23) (m) n a x = N x m 1 D x 2m Payable continuously. LC Page 50 (2.33) m a x = N x+n D x Actuarial Factors Page 19

21 Annuity with a Guarantee a Again, excuse the slight variation in the equation. The formula is: guaranteed pension value plus deferred annuity value. There are no Life Contingencies references for these formulas, I developed them logically. Arrears Advance Continuous a x:n = a n + a x:n = a n + a x:n = a n + a x n a x n a x n Actuarial Factors Page 20

22 Joint p This is the probability that two lives will survive n years. That they will both survive n years. LC Page 192 (9.1) n p x1 x 2 = np x1 np x2 Actuarial Factors Page 21

23 Joint and Last Survivor a This pension is payable in the full amount during the guarantee period. After the guarantee period, the full amount is paid for if the primary annuitant survives. After the guarantee period is complete and the primary annuitant has died, if the joint annuitant is alive a fraction of the pension (j) is paid to the joint annuitant. We have no reference from the Life Contingencies book for this calculation. The idea behind the factor is to value the life annuity for the primary annuitant with a guarantee and then add the joint annuitant value. The joint annuitant value is N/D, but each year must be multiplied by the probability that the primary annuitant has died. In the following formulas: x = age of the primary annuitant y = age of the joint annuitant w = maximum age in the mortality table j = Joint pension fraction (between 0 and 1), the fraction payable to the joint annuitant following the primary annuitant s death Arrears Advance Continuous n = Guaranteed annuity years w y a x:y:n = a x:n + t=n+1 D y+t (1 tp x ) j w y D y a x:y:n = a x:n + t=n D y+t (1 tp x ) j w y D y a x:y:n = a x:n + t=n D y+t (1 tp x ) j D y Actuarial Factors Page 22

24 Joint Pension a This pension is payable in the full amount during the guarantee period. While both annuitants are alive, the full amount is paid. If either annuitant dies, the pension is reduced. For example, for a 60% joint pension, the primary annuitant is paid 60% and joint annuitant is paid 40% while they are both alive. If the joint annuitant dies, the primary annuitant is still paid 60%, but the 40% for the joint annuitant is not paid. If the primary annuitant dies, and the joint annuitant is still alive, 40% of the pension is paid to the joint annuitant. Here again, we have no reference to the Life Contingencies book. The idea is that the guaranteed pension is valued, then the primary annuitant deferred annuity is valued and multiplied by j, followed by the joint annuitant deferred annuity is valued and multiplied by (1 j). Arrears Advance Continuous a x:y:n = a n + j a x:y:n = a n + j a x:y:n = a n + j n ax + (1 j) n a y n a x + (1 j) n a y n a + (1 j) a x y n Actuarial Factors Page 23

25 Accuracy We checked our calculations using the CSO 58 mortality table and 3% interest rate LC Page 333 and following. The differences have been highlighted in red. Here are our results for: a x Age LC Book Our Software Difference Actuarial Factors Page 24

26 Age LC Book Our Software Difference Actuarial Factors Page 25

27 Age LC Book Our Software Difference We double checked the table of l s around the ages that show the errors and we didn t find any keying errors. We would guess that the errors are minor differences that can be ignored. Our software was also verified by spreadsheet calculations. In these calculations, we had agreement to eight decimal places. Technically, the factors should only be accurate to seven significant figures. The Life Contingencies book does not provide tables of annuities with guarantees, advance single life, continuous single life, and joint factors, so we were not able to do comprehensive tests against the book. Actuarial Factors Page 26