The first payment of a five-year annuity is due in five years in the amount of 1000. The subsequent four annual payments increase by 500 each year. The annual effective interest rate is i.

Determine which of the following formulas gives the present value of the annuity.

• [[math]]\quad v^6\left[500 a_{5 \mid i}+500(I a)_{5 \mid i}\right][[/math]]
• [[math]]\quad v^6\left[500 \ddot{a}_{\left.5\right|_i}+500(I \ddot{a})_{\left.5\right|_i}\right][[/math]]
• [[math]]v^5\left[500 a_{\left.5\right|_i}+500(I \ddot{a})_{\left.5\right|_i}\right][[/math]]
• [[math]]\quad v^5\left[500 \ddot{a}_{5 \mid i}+500(I \ddot{a})_{\left.5\right|_i}\right][[/math]]
• [[math]]\quad v^5\left[1000 \ddot{a}_{\left.5\right|_i}+500(I \ddot{a})_{5 \mid i}\right][[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

Using time 5 as the first reference point, then bringing that value back to time 0:

[[math]] v^5[500\ddot{a}_{\overline{5}|i} + 500(I\ddot{a})_{\overline{5}|i}] [[/math]]

This combines a five-year level annuity-due of 500 plus an increasing annuity-due starting with 500 and increasing by 500.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.