If you receive $186 each quarter for 16 years and the discount rate is 0.05, what is the future value?

Question

If you receive $186 each quarter for 16 years and the discount rate is 0.05, what is the future value?

BlackTom AI · Accepted Answer

Question restatement: You receive 186 dollars every quarter for 16 years, with a discount rate of 0.05. What is the future value?

Step 1: Identify the cash flow details and the per-period interest rate. The payments are quarterly, so the payment period is a quarter. The annual discount rate is 0.05, so the per-quarter rate i = 0.05 / 4 = 0.0125 (1.25%).

Step 2: Determine the number of periods. There are 16 years of quarterly payments, so n = 16 × 4 = 64 periods.

Step 3: Recognize the formula for the future value of an ordinary annuity (payments at the end of each period): FV = PMT × [((1 + i)^n − 1) / i]. Here, PMT = 186, i = 0.0125, n = 64.

Step 4: Compute the compound factor (1 + i)^n. Calculate (1.0125)^64. This value captures how much a single quarterly payment grows over 64 quarters at 1.25% per quarter. An accurate computation gives a value around 2.2 (more precisely, close to 2.2–2.22 depending on rounding).

Step 5: Substitute into the annuity FV formula. Using the approximate (1.0125)^64 ≈ 2.214, we get:
- ((1.0125)^64 − 1) ≈ 1.214
- Divide by i: 1.214 / 0.0125 ≈ 97.12
- Multiply by PMT: 97.12 × 186 ≈ 18,060.

Step 6: Interpret the result. The computed future value is in the neighborhood of about 18,000 dollars, depending on the exact precision of the (1.0125)^64 calculation. Small changes in the rounding of the exponentiation step can shift the final figure by a few tens to a few hundred dollars.

Common pitfalls and alternatives:
- If you incorrectly used an annual compounding with quarterly payments, or if you used i = 0.05 per period, the result would be far different (and typically much larger or smaller) because the per-period rate and the number of periods would no longer align with the quarterly structure.
- If you treated this as an annuity due (payments at the beginning of each period) rather than an ordinary annuity, you would multiply the ordinary annuity value by (1 + i), yielding a slightly larger number, but still governed by the same basic growth mechanics.

In summary, applying the quarterly rate of 1.25% over 64 periods to an ordinary annuity with payments of 186 leads to a future value around 18 thousand dollars, with the exact figure depending on the precision of the exponential calculation.

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If you receive $186 each quarter for 16 years and the discount rate is 0.05, what is the future value?

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