Five deposits of 100 are made into a fund at two-year intervals with the first deposit at the beginning of the first year. The fund earns interest at an annual effective rate of 4% during the first six years and at an annual effective rate of 5% thereafter.

Calculate the annual effective yield rate earned over the investment period ending at the end of the tenth year.

• 4.18%
• 4.40%
• 4.50%
• 4.58%
• 4.78%

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

Solution: C

Convert the two annual rates, 4% and 5%, to two-year rates as 1.04² -1 = 0.0816 and 1.05²-1 = 0.1025. The accumulated value is

[[math]] 100\ddot{S}_{\overline{{{3}}}|{0.0816}}(1.05)^{4}+100\ddot{S}_{\overline{{{2}}}|0.1025} [[/math]]

With only five payments, an alternative approach is to accumulate each one to time ten and add them up. The two-year yield rate is the solution to [math]100 \ddot{s}_{\overline{5}|i} = 659.269[/math]. Using the calculator, the two-year rate is 0.093637. The annual rate is 1.093637^0.5 -1 = 0.04577 which is 4.58%.

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