A 30-year annuity is arranged to pay off a loan taken out today at a 5% annual effective interest rate. The first payment of the annuity is due in ten years in the amount of 1,000. The subsequent payments increase by 500 each year.

Calculate the amount of the loan

• 58,283
• 61,197
• 64,021
• 64,257
• 69,211

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

Solution: D

The amount of the loan is the present value of the deferred increasing annuity:

[[math]] (1.05)^{-10}\left[500\ddot{a}_{\overline{{{300}}}|0.05}+500(I\ddot{a})_{\overline{{{300}}}|0.05}\right] = (1.05^{-10})(500)\Bigg[\ddot{a}_{\overline{{{300}}}|0.05}+\frac{\ddot{a}_{\overline{{{300}}}|0.05}-30(1.05)^{-30}}{0.05/1.05}\Bigg]=64,257. [[/math]]

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