Life Contingencies Proofs, UDD Assumption APVs

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    Richard Purvey
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    1. Actuarial Life Contingencies, Proof of that Relationship, Under the UDD Assumption, Between the mthly and Once a Year Life Annuity Due APVs

     

    Given that:

     

    apv_(x)_n_life annuity due of 1 per year payable m times per year = sum (k=0…m*n-1) (v^(k/m) * (k/m)_p_x)/m

     

    and

     

    apv_(x)_n_life annuity due of 1 per year payable once a year = sum (j=0…n-1) v^j * j_p_x, show that, under the UDD assumption,

     

    apv_(x)_n_life annuity due of 1 per year payable m times per year = (v^(-1)-1-m*(v^(-1/m)-1))/(m*(v^(-1/m)-1)*m*(1-v^(1/m))) * (v^n * n_p_x – 1) + (v^(-1)-1)*(1-v)/(m*(v^(-1/m)-1)*m*(1-v^(1/m))) * apv_(x)_n_life annuity due of 1 per year payable once a year.

     

    Method

     

    sum (k=0…m*n-1) (v^(k/m) * (k/m)_p_x)/m can be expressed as sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * (j + r/m)_p_x)/m)

     

    Since under the UDD assumption, (j + r/m)_p_x = j_p_x + (r/m) * ((j + 1)_p_x – j_p_x), under the UDD assumption, sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * (j + r/m)_p_x)/m) = sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * (j_p_x + (r/m) * ((j + 1)_p_x – j_p_x)))/m)

     

    sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * (j_p_x + (r/m) * ((j + 1)_p_x – j_p_x)))/m)

     

    = sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * j_p_x)/m + sum (r=0…m-1) (v^(j + r/m) * (r/m) * (j + 1)_p_x)/m – sum (r=0…m-1) (v^(j + r/m) * (r/m) * j_p_x)/m)

     

    = sum (j=1…n) (sum (r=0…m-1) (v^(j – 1 + r/m) * (r/m) * j_p_x)/m) + sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * j_p_x)/m – sum (r=0…m-1) (v^(j + r/m) * (r/m) * j_p_x)/m)

     

    = – sum (r=0…m-1) (v^(0 – 1 + r/m) * (r/m) * 0_p_x)/m + sum (r=0…m-1) (v^(0 – 1 + r/m) * (r/m) * 0_p_x)/m + sum (j=1…n) (sum (r=0…m-1) (v^(j – 1 + r/m) * (r/m) * j_p_x)/m) + sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * j_p_x)/m – sum (r=0…m-1) (v^(j + r/m) * (r/m) * j_p_x)/m)

     

    = – sum (r=0…m-1) (v^(0 – 1 + r/m) * (r/m) * 0_p_x)/m + sum (j=0…n) (sum (r=0…m-1) (v^(j – 1 + r/m) * (r/m) * j_p_x)/m) + sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * j_p_x)/m – sum (r=0…m-1) (v^(j + r/m) * (r/m) * j_p_x)/m)

     

    = – sum (r=0…m-1) (v^(0 – 1 + r/m) * (r/m) * 0_p_x)/m + sum (r=0…m-1) (v^(n – 1 + r/m) * (r/m) * n_p_x)/m + sum (j=0…n-1) (sum (r=0…m-1) (v^(j – 1 + r/m) * (r/m) * j_p_x)/m) + sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * j_p_x)/m – sum (r=0…m-1) (v^(j + r/m) * (r/m) * j_p_x)/m)

     

    = – v^0 * 0_p_x * sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m + v^n * n_p_x * sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m + sum (j=0…n-1) (sum (r=0…m-1) (v^(j – 1 + r/m) * (r/m) * j_p_x)/m) + sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * j_p_x)/m – sum (r=0…m-1) (v^(j + r/m) * (r/m) * j_p_x)/m)

     

    = (- v^0 * 0_p_x + v^n * n_p_x) * sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m + sum (j=0…n-1) (sum (r=0…m-1) (v^(j – 1 + r/m) * (r/m) * j_p_x)/m) + sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * j_p_x)/m – sum (r=0…m-1) (v^(j + r/m) * (r/m) * j_p_x)/m)

     

    = (v^n * n_p_x – 1) * sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m + sum (j=0…n-1) (sum (r=0…m-1) (v^(j – 1 + r/m) * (r/m) * j_p_x)/m) + sum (j=0…n-1) (sum (r=0…m-1) (v^(j + r/m) * j_p_x)/m – sum (r=0…m-1) (v^(j + r/m) * (r/m) * j_p_x)/m)

     

    = (v^n * n_p_x – 1) * sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m + sum (j=0…n-1) (sum (r=0…m-1) (v^(j – 1 + r/m) * (r/m) * j_p_x)/m + sum (r=0…m-1) (v^(j + r/m) * j_p_x)/m – sum (r=0…m-1) (v^(j + r/m) * (r/m) * j_p_x)/m)

     

    = (v^n * n_p_x – 1) * sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m + sum (j=0…n-1) v^j * j_p_x * (sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m + sum (r=0…m-1) (v^(r/m))/m – sum (r=0…m-1) (v^(r/m) * (r/m))/m)

     

    sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m, after using an Online Summation Calculator, is found to equal (m * v^(1/m) – v^(1/m) + v^(1/m – 1) – m)/(m^2 * (v^(1/m) -1)^2), which, after multiplying both the numerator and denominator by v^(-1/m), clearly equals (v^(-1)-1-m*(v^(-1/m)-1))/(m*(v^(-1/m)-1)*m*(1-v^(1/m))).

     

    sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m + sum (r=0…m-1) (v^(r/m))/m – sum (r=0…m-1) (v^(r/m) * (r/m))/m, after using an Online Summation Calculator, is found to equal (v – 1)^2 * v^(1/m – 1)/(m^2 * (v^(1/m) -1)^2), which, after multiplying both the numerator and denominator by v^(-1/m), clearly equals (v^(-1)-1)*(1-v)/(m*(v^(-1/m)-1)*m*(1-v^(1/m))).

     

    And so, = (v^n * n_p_x – 1) * sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m + sum (j=0…n-1) v^j * j_p_x * (sum (r=0…m-1) (v^(- 1 + r/m) * (r/m))/m + sum (r=0…m-1) (v^(r/m))/m – sum (r=0…m-1) (v^(r/m) * (r/m))/m)

     

    = (v^(-1)-1-m*(v^(-1/m)-1))/(m*(v^(-1/m)-1)*m*(1-v^(1/m))) * (v^n * n_p_x – 1) + (v^(-1)-1)*(1-v)/(m*(v^(-1/m)-1)*m*(1-v^(1/m))) * sum (j=0…n-1) v^j * j_p_x

     

    = (v^(-1)-1-m*(v^(-1/m)-1))/(m*(v^(-1/m)-1)*m*(1-v^(1/m))) * (v^n * n_p_x – 1) + (v^(-1)-1)*(1-v)/(m*(v^(-1/m)-1)*m*(1-v^(1/m))) * apv_(x)_n_life annuity due of 1 per year payable once a year.

     

    An Interesting Note;

     

    lim v→1 (v^(-1)-1-m*(v^(-1/m)-1))/(m*(v^(-1/m)-1)*m*(1-v^(1/m))), after using an Online Limits Calculator, is found to equal (m-1)/(2*m).

     

    lim v→1 (v^(-1)-1)*(1-v)/(m*(v^(-1/m)-1)*m*(1-v^(1/m))), after using an Online Limits Calculator, is found to equal 1.

     

    Richard Purvey, proud of having lived at 15/5 Marytree House, Edinburgh, until the age of 12.

     

    September 2022.

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