Algorithms for Life Contingencies

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    Richard Purvey
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    Computer Algorithms For Nine Of The Standard Algorithmic Calculations Currently Included In The Actuarial Life Contingencies Syllabus

    In each case, make the first line of the algorithm:

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

    For example, for a Makeham with A equal to 0.00022, B equal to 2.7*10^(-6) and c equal to 1.124, make the first line of the algorithm:

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

    And then;

    If you want the algorithm to calculate an n-year temporary curtate life expectancy for (x) when executed then add on the following:

    20 INPUT x,n

    30 ANSWER=0

    40 FOR r=1 TO n

    50 ANSWER=ANSWER+FNS(x+r)

    60 NEXT r

    70 ANSWER=ANSWER/FNS(x)

    80 PRINT ANSWER

    90 END

    Or if you want the algorithm to calculate the actuarial present value of an n-year temporary immediate life annuity, with a discount factor of v, of 1 per year payable once a year for (x) when executed then add on the following:

    20 INPUT x,n,v

    30 ANSWER=0

    40 FOR r=1 TO n

    50 ANSWER=ANSWER+v^r*FNS(x+r)

    60 NEXT r

    70 ANSWER=ANSWER/FNS(x)

    80 PRINT ANSWER

    90 END

    Or if you want the algorithm to calculate the actuarial present value of an n-year temporary life annuity due, with a discount factor of v, of 1 per year payable once a year for (x) when executed then add on the following:

    20 INPUT x,n,v

    30 ANSWER=0

    40 FOR r=0 TO n-1

    50 ANSWER=ANSWER+v^r*FNS(x+r)

    60 NEXT r

    70 ANSWER=ANSWER/FNS(x)

    80 PRINT ANSWER

    90 END

    Or if you want the algorithm to calculate the actuarial present value 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, when executed then add on the following:

    20 INPUT x,n,v

    30 ANSWER=0

    40 FOR r=0 TO n-1

    50 ANSWER=ANSWER+v^r*FNS(x+r)

    60 NEXT r

    70 ANSWER=ANSWER/FNS(x)

    80 ANSWER=1-v^n*FNS(x+n)/FNS(x)-(1-v)*ANSWER

    90 PRINT ANSWER

    100 END

    Or if you want the algorithm to calculate the actuarial present value 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, when executed then add on the following:

     

    20 INPUT x,n,v

    30 ANSWER=0

    40 FOR r=0 TO n-1

    50 ANSWER=ANSWER+v^r*FNS(x+r)

    60 NEXT r

    70 ANSWER=ANSWER/FNS(x)

    80 ANSWER=1-(1-v)*ANSWER

    90 PRINT ANSWER

    100 END

    Or if you want the algorithm to calculate the actuarial present value 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) when executed then add on the following:

    20 INPUT x,n,m,v

    30 ANSWER=0

    40 FOR r=1 TO m*n

    50 ANSWER=ANSWER+v^(r/m)*FNS(x+r/m)

    60 NEXT r

    70 ANSWER=ANSWER/(m*FNS(x))

    80 PRINT ANSWER

    90 END

    Or if you want the algorithm to calculate the actuarial present value 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) when executed then add on the following:

    20 INPUT x,n,m,v

    30 ANSWER=0

    40 FOR r=0 TO m*n-1

    50 ANSWER=ANSWER+v^(r/m)*FNS(x+r/m)

    60 NEXT r

    70 ANSWER=ANSWER/(m*FNS(x))

    80 PRINT ANSWER

    90 END

    Or if you want the algorithm to calculate the actuarial present value 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, when executed then add on the following:

    20 INPUT x,n,m,v

    30 ANSWER=0

    40 FOR r=0 TO m*n-1

    50 ANSWER=ANSWER+v^(r/m)*FNS(x+r/m)

    60 NEXT r

    70 ANSWER=ANSWER/(m*FNS(x))

    80 ANSWER=1-v^n*FNS(x+n)/FNS(x)-m*(1-v^(1/m))*ANSWER

    90 PRINT ANSWER

    100 END

    Or if you want the algorithm to calculate the actuarial present value 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, when executed then add on the following:

     

    20 INPUT x,n,m,v

    30 ANSWER=0

    40 FOR r=0 TO m*n-1

    50 ANSWER=ANSWER+v^(r/m)*FNS(x+r/m)

    60 NEXT r

    70 ANSWER=ANSWER/(m*FNS(x))

    80 ANSWER=1-m*(1-v^(1/m))*ANSWER

    90 PRINT ANSWER

    100 END

     

    I have written the above algorithms in BBC BASIC.

     

    Richard Purvey, February 2022.  Proud of having lived at 15/5 Marytree House, 12 Craigour Green, Edinburgh until the age of 12.

     

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