Practice Question
You are going to pay $800 into an account at the beginning of each of 20 years (first payment at \( t = 0 \)). The account will then be left to compound for an additional 20 years. At the end of the 41st year, you will begin receiving a perpetuity from the account. If the account pays 14%, how much each year will you receive from the perpetuity? (Round to the nearest $1,000)
Step 1: Calculate the future value of the annuity due at year 20
- \( PMT = 800 \)
- \( r = 0.14 \)
- \( n = 20 \)
- Because payments are at the beginning of each period, use the annuity due formula:
\[
FV = PMT \times \left(\frac{(1 + r)^n - 1}{r}\right) \times (1 + r)
\]
\[
FV = 800 \times \left(\frac{(1.14)^{20} - 1}{0.14}\right) \times 1.14 \approx 800 \times 91.0229 \times 1.14 = 800 \times 103.7651 = 83,012.08
\]
Step 2: Compound the value forward 20 more years to year 40
- Use future value of a lump sum formula:
\[
FV_{40} = 83,012.08 \times (1.14)^{20} \approx 83,012.08 \times 13.7432 = 1,140,575.85
\]
Step 3: Use the perpetuity formula to calculate the annual payment
- At year 41, this amount generates a perpetuity payment:
\[
PMT = PV \times r = 1,140,575.85 \times 0.14 = 159,680.62
\]
Conclusion: The annual amount you will receive from the perpetuity is approximately $160,000 (rounded to the nearest $1,000).