An annuity provides level payments of 1000 every six months for a fixed period. Using an annual effective interest rate of i, the future value of this annuity at the time of the last payment is 19,549.25 and the present value of this annuity at the time of the first payment is 7,968.89.

Calculate i.

• 7.4%
• 8.5%
• 15.4%
• 17.0%
• 17.7%

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

Solution: E

Let n be the number of payments and let j be the interest rate per half-year. Because the given values are n – 1 half-years apart,

[[math]] 7,968.89(1+j)^{n-1}=19,549.25. [[/math]]

Also,

[[math]] 7.968.89=1,000\ddot{a}_{\overline{n}|}=1,000(a_{\overline{n-1}|})+1)=1,000\left(\frac{1-\nu^{n-1}}{j}+1\right)=1.000\left(\frac{1-7,968.89\cdot19.549.25}{j}+1\right) [[/math]]

Then,

[[math]] j={\frac{1-7,968.89/19.549.25}{1,968.89}}=0.085 [[/math]]

for [math]i = (1.85)^2 -1 = 0.1772 = 17.7\%[/math]

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