Practice Question
Assume you are to receive a 20-year annuity with annual payments of $50. The first payment will be received at the end of Year 1, and the last payment will be received at the end of Year 20. You will invest each payment in an account that pays 10 percent. What will be the value in your account at the end of Year 30?
Answer +
7,427.830
Explanation +
Step 1: Calculate Future Value of Annuity at Year 20
The future value of an ordinary annuity is given by:
\[ FV = C \times \left( \frac{(1 + r)^t - 1}{r} \right) \]Substitute the values:
\[ FV_{20} = 50 \times \left( \frac{(1.10)^{20} - 1}{0.10} \right) \] \[ FV_{20} = 50 \times \left( \frac{6.7275 - 1}{0.10} \right) = 50 \times 57.275 = 2,863.75 \]Step 2: Compound Annuity Forward to Year 30
Now grow the amount for 10 more years at 10%:
\[ FV_{30} = FV_{20} \times (1 + r)^{10} \] \[ FV_{30} = 2,863.75 \times (1.10)^{10} = 2,863.75 \times 2.5937 = 7,427.83 \]Final Answer
The value in your account at the end of Year 30 will be $7,427.83.