If you receive $197 each month for 21 years and the discount rate is 0.03, what is the present value?

Question

If you receive $197 each month for 21 years and the discount rate is 0.03, what is the present value?

BlackTom AI · Accepted Answer

We need to compute the present value of an ordinary annuity with monthly payments.

- Step 1: Identify the parameters. The monthly payment (PMT) is 197 dollars. The total number of payments is 21 years × 12 months/year = 252 payments. The annual discount rate is 0.03, so the monthly rate i is 0.03 / 12 = 0.0025.

- Step 2: Use the present value of an ordinary annuity formula for monthly payments:
  PV = PMT × [1 − (1 + i)^−n] / i, where n is the number of payments and i is the monthly rate.

- Step 3: Compute (1 + i)^−n. Here, (1 + i) = 1.0025 and n = 252. We estimate (1.0025)^252 ≈ exp(252 × ln(1.0025)) ≈ exp(252 × 0.0024969) ≈ exp(0.629) ≈ 1.876. Therefore (1 + i)^−n ≈ 1 / 1.876 ≈ 0.533.

- Step 4: Compute the bracketed term: 1 − (1 + i)^−n ≈ 1 − 0.533 = 0.467.

- Step 5: Divide by i: 0.467 / 0.0025 ≈ 186.8.

- Step 6: Multiply by PMT: 186.8 × 197 ≈ 36,435.6.

- Step 7: Conclusion. The present value is approximately 36,435.59 dollars, based on the given monthly payment, term, and monthly discount rate. Note that small rounding differences in the exponent calculation can lead to slight variations in the final figure.

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If you receive $197 each month for 21 years and the discount rate is 0.03, what is the present value?

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