Question

A 5-year ordinary annuity has a present value of \[1,000\]. If the interest rate is 8 percent, the amount of each annuity payment is closest to which of the following?
A.\[250.44\]
B.\[231.91\]
C. $ 181.62 $
D. $ 184.08 $
E. $ 170.44 $

Answer 1

Hint: In this question, we need to determine the amount of each annuity payment such that on the completion of 5 years of time, the annuity amount becomes \[1,000\] with the rate of interest at 8%. For this we will use the relation between the amount accumulated, each annuity amount, rate of interest and time period which is given by $ A = \dfrac{P}{r}\left[ {1 - {{(1 + r)}^{ - n}}} \right] $ where ‘A’ is the amount received at the end of the time period ‘n’, ‘P’ is the annuity amount deposited periodically and ‘r’ is the decimal equivalent of the rate of interest gained over the annuity amount deposited.

Complete step by step solution:
Let the total amount of annuity be P.
The total amount received after the completion of the interest period is the sum of the amount deposited and the interest over the deposited amount. Mathematically, $ A = P + I $ where, A is the total amount after the completion of the interest period, P is the amount deposited, and I is the interest on the deposited amount over the time period.
Now, following the formula for the annuity payment as $ A = \dfrac{P}{r}\left[ {1 - {{(1 + r)}^{ - n}}} \right] $ .
Here, the total amount accumulated after a time period of 5 years at 0.08 rate of interest is 1000.
So, substitute A=\[1,000\], r=0.08 and t=5 years in the formula
 $ A = \dfrac{P}{r}\left[ {1 - {{(1 + r)}^{ - n}}} \right] $ to determine each annuity amount deposited.
 $
  A = \dfrac{P}{r}\left[ {1 - {{(1 + r)}^{ - n}}} \right] \\
   \Rightarrow 1000 = \dfrac{P}{{0.08}}\left[ {1 - {{(1 + 0.08)}^{ - 5}}} \right] \\
   \Rightarrow P\left[ {1 - {{(1.08)}^{ - 5}}} \right] = 1000 \times 0.08 \\
   \Rightarrow P = \dfrac{{80}}{{1 - 0.68058}} \\
   \Rightarrow P = \dfrac{{80}}{{0.3194}} \\
   \Rightarrow P = 250.44 \;
  $
Hence, the amount of each annuity payment is $ 250.44 $ .
So, the correct answer is “Option A”.

Note: It is very important to note here that the question is asking for the amount of each annuity amount and not for one time deposited amount. So we have used the formula $ A = \dfrac{P}{r}\left[ {1 - {{(1 + r)}^{ - n}}} \right] $ instead of a simple interest formula.