A 20-year arithmetically increasing annuity-due sells for 600,000 and provides annual payments. The first payment is X, and each payment thereafter is X more than the previous payment. A 25-year arithmetically increasing annuity-due provides annual payments. The first payment is X, and each payment thereafter is X more than the previous payment. The prices of the two annuities are calculated using a continuously compounded annual interest rate of 6%.

Calculate the price of the 25-year annuity.

• 667,026
• 668,707
• 750,000
• 779,336
• 782,712

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

Solution: D

With a continuously compounded annual interest rate of 6%,

The value of the first annuity is

[[math]] 600.000=X(I\ddot{a})_{\overline{{{20}}|}}=X\frac{\ddot{a}_{\overline{20}|}-20v^{20}}{d}=X\,{\frac{{\frac{1-e^{-0.2}}{1-e^{-0.06}}}-20e^{-1.2}}{1-e^{-0.06}}}=102.614X. [[/math]]

Hence, X = 600,000/102.614 = 5847.155. Then the value of the second annuity is

[[math]] 5847.155{\frac{\ddot{a}_{\overline{25}|}-25\nu^{25}}{d}}=5847.155{\frac{\frac{1-e^{-0.5}}{1-e^{-0.06}}-25e^{-1.5}}{1-e^{-0.06}}}=779.366. [[/math]]

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