If you receive $36 each quarter for 5 years and the discount rate is 0.04, what is the future value?

Question

If you receive $36 each quarter for 5 years and the discount rate is 0.04, what is the future value?

BlackTom AI · Accepted Answer

We start by identifying the cash flows and the parameters given in the problem.
- You receive 36 dollars every quarter for 5 years. That gives you a total of 5*4 = 20 payments (n = 20).
- The discount rate is 0.04. In typical time-value-of-money problems like this, a quoted annual rate is converted to a per-period rate by dividing by the number of periods per year. Since payments are quarterly, the periodic interest (or discount) rate is i = 0.04 / 4 = 0.01 (1% per quarter).

Next, recall the formula for the future value of an ordinary annuity (payments at the end of each period): FV = P * [((1 + i)^n − 1) / i], where P is the payment amount.
- Here, P = 36, i = 0.01, n = 20.
- Compute (1 + i)^n = (1.01)^20. This is approximately 1.2200 (since (1.01)^20 ≈ 1.220).
- Then (1.01)^20 − 1 ≈ 0.2200.
- Divide by i: 0.2200 / 0.01 = 22.00.
- Finally multiply by P: 22.00 * 36 = 792. So the ordinary annuity future value is about 792.

If instead the payments are made at the beginning of each period (an annuity due), you would multiply the ordinary annuity value by (1 + i) to account for the extra compounding in each period:
- FV_due ≈ 792 * 1.01 ≈ 799.92, i.e., about 800.

It’s also worth checking an alternative interpretation of the discount rate:
- If the 0.04 were the per-quarter rate (not typical, but possible), then i = 0.04 and n = 20. The factor ((1.04)^20 − 1)/0.04 ≈ (2.191123 − 1)/0.04 ≈ 29.778. Then FV ≈ 36 * 29.778 ≈ 1,072.0, which is much larger than 792 or 800.

Comparing these results to the value given in the problem (828): none of the standard interpretations (ordinary annuity with annual rate split quarterly yielding ≈792, or annuity due ≈800, or a quarterly-per-rate interpretation ≈1,072) produce exactly 828. The 828 figure might reflect a different convention (e.g., a different compounding assumption, a different timing of payments, or a miscalculation).

In summary, under the common interpretation with a 4% annual rate compounded quarterly and end-of-period payments, the calculated future value is about 792 (ordinary annuity) or about 800 (annuity due). The 828 value does not align with these standard calculations, suggesting a possible mismatch in assumptions or a misprint.

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If you receive $36 each quarter for 5 years and the discount rate is 0.04, what is the future value?

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