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Richard Purvey
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Computer programs to calculate single life (x) actuarial present values

Computer programs to calculate single life (x) actuarial present values, using a life table

FIRST COMPUTER PROGRAM

The computer program below will calculate and display the APV of an n-year temporary immediate life annuity, with a discount factor of v, of 1 per year payable once a year for (x) for the values x=z up to and including x=h, in steps of 1, using a life table in the form of x=xlowest up to and including x=xhighest, in steps of 1, with corresponding lx values l(xlowest), l(xlowest+1),…, l(xhighest-1) and l(xhighest).

With v equal to 1, the resulting APV will have the same numerical value as the curtate life expectancy.

Provided l(xhighest)=0, a high enough value for n will result in a whole life APV being calculated.

10 INPUT xlowest,xhighest

20 DIM l(xhighest)

30 FOR x=xlowest TO xhighest

40 INPUT l(x)

50 NEXT x

60 INPUT z,h

70 INPUT n,v

80 FOR x=z TO h

90 IF n>xhighest-x THEN n=xhighest-x

110 FOR r=1 TO n

130 NEXT r

160 NEXT x

170 GOTO 60

SECOND COMPUTER PROGRAM

The computer program below will calculate the following four APVs and display them in this order;

APV of an n-year temporary life annuity due, with a discount factor of v, of 1 per year payable once a year for (x)

APV of an endowment insurance, with a discount factor of v, of 1, where the death benefit is payable at the end of the year of death for (x), provided this occurs within n years

APV of a death benefit, with a discount factor of v, of 1 payable at the end of the year of death for (x), provided this occurs within n years

APV of a pure endowment, with a discount factor of v, of 1 payable after n years as long as (x) is still alive then

for the values x=z up to and including x=h, in steps of 1, using a life table in the form of x=xlowest up to and including x=xhighest, in steps of 1, with corresponding lx values l(xlowest), l(xlowest+1),…, l(xhighest-1) and l(xhighest).

Provided l(xhighest)=0, a high enough value for n will result in a whole life APV being calculated.

10 INPUT xlowest,xhighest

20 DIM l(xhighest)

30 FOR x=xlowest TO xhighest

40 INPUT l(x)

50 NEXT x

60 INPUT z,h

70 INPUT n,v

80 FOR x=z TO h

90 IF n>xhighest-x THEN n=xhighest-x

110 FOR r=0 TO n-1

130 NEXT r

190 NEXT x

200 GOTO 60

Computer programs to calculate single life (x) actuarial present values, using a survival function

FIRST COMPUTER PROGRAM

Making line 10

10 DEF FNS(U)=whatever the survival function, S(U), is,

for example,

10 DEF FNS(U)=EXP(-0.00022*U-2.7*10^(-6)*(1.124^U-1)/LN(1.124)),

and then adding the lines below, will result in a computer program which will calculate and display the APV of an n-year temporary immediate life annuity, with a discount factor of v, of 1 per year payable m times per year for (x) for the values x=z up to and including x=h, in steps of 1/m, using a survival function, S(U).

m normally equals 1, 2, 3, 4, 6 or 12.

With m equal to 1 and v equal to 1, the resulting APV will have the same numerical value as the curtate life expectancy.

Provided S(U) tends to 0 as U tends to infinity, a high enough value for n will result in a whole life APV being calculated.

20 INPUT z,h

30 INPUT m,n,v

40 FOR x=z TO h STEP 1/m

60 FOR r=1 TO m*n

80 NEXT r

110 NEXT x

120 GOTO 20

SECOND COMPUTER PROGRAM

Making line 10

10 DEF FNS(U)=whatever the survival function, S(U), is,

for example,

10 DEF FNS(U)=EXP(-0.00022*U-2.7*10^(-6)*(1.124^U-1)/LN(1.124)),

and then adding the lines below, will result in a computer program which will calculate the following four APVs and display them in this order;

APV of an n-year temporary life annuity due, with a discount factor of v, of 1 per year payable m times per year for (x)

APV of an endowment insurance, with a discount factor of v, of 1, where the death benefit is payable at the end of the 1/m year of death for (x), provided this occurs within n years

APV of a death benefit, with a discount factor of v, of 1 payable at the end of the 1/m year of death for (x), provided this occurs within n years

APV of a pure endowment, with a discount factor of v, of 1 payable after n years as long as (x) is still alive then

for the values x=z up to and including x=h, in steps of 1/m, using a survival function, S(U).

m normally equals 1, 2, 3, 4, 6 or 12.

Provided S(U) tends to 0 as U tends to infinity, a high enough value for n will result in a whole life APV being calculated.

20 INPUT z,h

30 INPUT m,n,v

40 FOR x=z TO h STEP 1/m

60 FOR r=0 TO m*n-1

80 NEXT r

140 NEXT x

150 GOTO 20

Richard Purvey May 2024

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