Lucas opens a bank account with 1000 and lets it accumulate at an annual nominal interest rate of 6% convertible semiannually. Danielle also opens a bank account with 1000 at the same time as Lucas, but it grows at an annual nominal interest rate of 3% convertible monthly. For each account, interest is credited only at the end of each interest conversion period.

Calculate the number of months required for the amount in Lucas’s account to be at least double the amount in Danielle’s account.

• 276
• 282
• 285
• 286
• 288

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

Solution: E

Let n = years. The equation to solve is

[[math]] \begin{array}{l}{{1000(1.03)^{2n}=2(1000)(1.0025)^{12n}}}\\ {{2n\ln1.03+\ln1000=12n\ln1.0025+\ln2000}}\\ {{m=23.775.}}\end{array} [[/math]]

This is 285.3 months. The next interest payment to Lucas is at a multiple of 6, which is 288 months.

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