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Wouldn’t it be good to always be able to sum the series form of a survival function based expected present value of a life annuity?
The present value of an m_n_v immediate annuity is (1-v^n)/(m*(v^(-1/m)-1)), because the series form of this present value is a geometric series.
The present value of an m_n_v annuity due is (1-v^n)/(m*(1-v^(1/m))), because the series form of this present value is a geometric series.
Where the survival function is simply an exponential distribution, the expected present value of the (x)_m_n_v immediate life annuity is (1-(v*EXP(-mu))^n)/(m*((v*EXP(-mu))^(-1/m)-1)), because the series form of this expected present value is a geometric series.
Where the survival function is simply an exponential distribution, the expected present value of the (x)_m_n_v life annuity due is (1-(v*EXP(-mu))^n)/(m*(1-(v*EXP(-mu))^(1/m))), because the series form of this expected present value is a geometric series.
Where the survival function is a more complicated one, such as a Makeham distribution, however, obtaining the function of x which results from summing the series form of the expected present value of the (x)_m_n_v immediate life annuity or the function of x which results from summing the series form of the expected present value of the (x)_m_n_v life annuity due, are, or, at least, seem to be, impossible tasks (I have tried many online algebraic summation solvers to no avail).
The two advantages of managing this would be;
1. the expected present values of the life annuities could be calculated using just one row of a spreadsheet (read, also, the first of the two notes below).
2. By putting which ever one of the two functions of x+t was needed into the dt integral, accurate calculations of expected present values of accrued age retirement benefits of the type shown in David Dickson’s book, Life Contingent Risks/section 9.6.2 (Career average earnings plans)/page 313 would be realised.
Notes
the expected present value of the (x)_m_n_v endowment insurance could also be calculated, in the cell immediately after the one which would calculate the expected present value of the (x)_m_n_v life annuity due, since they are connected by the equation, the expected present value of the (x)_m_n_v endowment insurance=1-m*(1-v^(1/m))*the expected present value of the (x)_m_n_v life annuity due.
m normally equals 1, 2, 3, 4, 6 or 12.
Richard Purvey July 2024
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