Suppose you deposited $1,000 at the end of each month (starting next month) for 5 years.

With an annual interest rate of 6% APR, compounded monthly, what is the present value of these payments?

Round to a number with two decimals, do not include any commas or dollar signs. That is, $123,456.1234 should be input as 123456.12

Question

Suppose you deposited $1,000 at the end of each month (starting next month) for 5 years.

With an annual interest rate of 6% APR, compounded monthly, what is the present value of these payments?

Round to a number with two decimals, do not include any commas or dollar signs. That is, $123,456.1234 should be input as 123456.12

BlackTom AI · Accepted Answer

The present value of an ordinary monthly annuity is given by PV = PMT * [1 - (1 + i)^(-n)] / i.

1) Relevant concepts:
- PMT = 1,000 (payment at the end of each month).
- APR 6% compounded monthly ⇒ monthly rate i = 0.06 / 12 = 0.005.
- Number of payments n = 5 years × 12 months = 60.
- This is an ordinary annuity (payments at period end), so the standard annuity-immediate formula applies.

2) Step-by-step reasoning and calculations:
- Compute the monthly rate: i = 0.06/12 = 0.005.
- Compute (1 + i)^n = (1.005)^60. Using a calculator gives (1.005)^60 ≈ 1.348797925 (rounded).
- Invert to get (1 + i)^(-n) ≈ 1 / 1.348797925 ≈ 0.741372045.
- Compute the bracket: 1 - (1 + i)^(-n) ≈ 1 - 0.741372045 = 0.258627955.
- Divide by i: 0.258627955 / 0.005 = 51.725591.
- Multiply by PMT: PV = 1,000 × 51.725591 ≈ 51,725.591.

3) Intermediate steps shown:
- i = 0.005
- (1.005)^60 ≈ 1.348797925
- (1.005)^-60 ≈ 0.741372045
- (1 - (1.005)^-60) ≈ 0.258627955
- /i ⇒ annuity factor ≈ 51.725591
- PV ≈ 1,000 × 51.725591 ≈ 51,725.59 (rounded to two decimals)

4) Why this is correct:
The formula used is the standard present-value formula for an ordinary annuity (payments at period end). It discounts each monthly payment back to time 0 at the monthly effective rate (0.5%), summing those discounted values in closed form.

5) Why other interpretations would be wrong:
- Treating payments as an annuity-due (payments at the beginning of each month) would require multiplying the result by (1 + i) and would yield a larger PV.
- Using annual compounding or the nominal APR directly without converting to a monthly rate would misstate the discounting and produce an incorrect answer.
- Using simple interest rather than compound interest is incorrect because the problem specifies compounding.

6) Conclusion:
Applying PV = 1,000 * [1 - (1.005)^-60] / 0.005 yields a present value of approximately 51,725.56, which matches the provided correct answer.

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