If you receive $51 each quarter for 3 years and the discount rate is 0.09, what is the future value?

Question

If you receive $51 each quarter for 3 years and the discount rate is 0.09, what is the future value?

BlackTom AI · Accepted Answer

We need to compute the future value of a level payment annuity with quarterly payments.

- Given: payments = 51 dollars every quarter for 3 years, so number of periods n = 3 years × 4 quarters/year = 12 payments.
- Rate assumption: the problem states a discount rate of 0.09. In many time-value-of-money problems, a nominal annual rate is used with quarterly compounding, so the rate per quarter is i = 0.09 / 4 = 0.0225 (2.25%). This yields a quarterly accumulation factor of 1 + i = 1.0225.
- Future value formula for an ordinary annuity (payments at the end of each period): FV = P × [((1 + i)^n − 1) / i], where P is the periodic payment.

Compute step-by-step:
1) Compute (1 + i)^n = (1.0225)^12. While a precise calculation would use a calculator, we can approximate using exponentials: ln(1.0225) ≈ 0.02225, so 12 × ln(1.0225) ≈ 0.2670, and e^0.2670 ≈ 1.306. Thus (1.0225)^12 ≈ 1.306.
2) Subtract 1: 1.306 − 1 = 0.306.
3) Divide by i: 0.306 / 0.0225 ≈ 13.6.
4) Multiply by the payment: 13.6 × 51 ≈ 693.6.

Rounding nuances can yield a result around 693.7. If a more exact calculator yields 693.71, that aligns with the same approach and rounding at the end.

Summary of reasoning: using a nominal annual rate of 9% with quarterly compounding gives i = 0.0225 per quarter, apply the future value of an ordinary annuity formula over 12 periods, and multiply by the quarterly payment of 51 to obtain approximately 693.7.

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If you receive $51 each quarter for 3 years and the discount rate is 0.09, what is the future value?

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