proportional payment for the time in the period of death, i.e. a "loss" of payment for on average half a period. Conversely, for contracts costing an equal lumpsum and having the same internal rate of return, the longer the period between payments, the larger the total payment per year. Life assurance as a function of the life annuity The APV of whole-life assurance can be derived from the APV of a whole-life annuity-due this way: This is also commonly written as: In the continuous case, In the case where the annuity and life assurance are not whole life, one should replace the assurance with an n-year endowment assurance (which can be expressed as the sum of an n-year term assurance and an n-year pure endowment), and the annuity with an n-year annuity due. See also Actuarial science Actuarial notation Actuarial reserve Actuary Life table Present value References Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J., Chapter 4-5 Models for Quantifying Risk (Fourth Edition), 2011, By Robin J. Cunningham, Thomas N. Herzog, Richard L. London, Chapter 7-8 Applied mathematics Actuarial science