The closed form for 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 closed form for 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 closed form for the actuarial 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 actuarial present value is a geometric series.

Where the survival function is simply an exponential distribution, the closed form for the actuarial 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 actuarial present value is a geometric series.

Where the survival function is known, standard APVs can be calculated completely accurately using either a computer program or a spreadsheet whether or not a closed form can be found for the APV, but only where a closed form can be found for the APV, as in the above examples, can it be calculated within only one row of a spreadsheet.

Finding a closed form expression for the APV of the (x+t)_m_n_v immediate life annuity or for the APV of the (x+t)_m_n_v life annuity due, which ever one of these two APVs was needed, and putting this into the dt integrals, would, however, vastly improve the accuracy of the calculation of the APV of an accrued age retirement benefit of the type shown in David Dickson’s book, Life Contingent Risks/section 9.6.2 (Career average earnings plans)/example 9.8/page 313.

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

Richard Purvey August 2024