If you receive $99 each quarter for 8 years and the discount rate is 0.1, what is the present value?

Question

If you receive $99 each quarter for 8 years and the discount rate is 0.1, what is the present value?

BlackTom AI · Accepted Answer

To approach this problem, I first identify the cash flows and the discount rate context.
We have a level payment of 99 dollars every quarter for a total of 8 years, which amounts to 8 × 4 = 32 quarterly payments.
The stated discount rate is 0.1, which I will interpret as an annual rate of 10%. When valuing quarterly payments, we should convert this annual rate to an appropriate quarterly rate. A common conversion is to divide by 4, giving a quarterly rate r = 0.10 / 4 = 0.025 (2.5%).
With an ordinary annuity (payments at the end of each quarter), the present value formula is:
PV = P × [1 − (1 + r)^−n] / r,
where P = 99, r = 0.025, n = 32.
Plugging in the numbers, I compute (1 + r)^−n = (1.025)^−32. Estimating (1.025)^32 via natural logs: ln(1.025) ≈ 0.02469, so 32 × 0.02469 ≈ 0.790, and e^0.790 ≈ 2.204. Therefore (1.025)^−32 ≈ 1/2.204 ≈ 0.454.
Then 1 − (1.025)^−32 ≈ 1 − 0.454 = 0.546. Dividing by r = 0.025 gives 0.546 / 0.025 ≈ 21.84. Multiplying by the periodic payment 99 yields PV ≈ 21.84 × 99 ≈ 2162.16.
Thus, the present value is approximately 2162 dollars, assuming end-of-period payments, a 10% annual rate converted to a 2.5% quarterly rate, and 32 total payments.
If a different convention were used—for example, treating the 10% as a different compounding basis or using an annuity due (payments at the beginning of each quarter)—the result would differ. Specifically, an annuity due would multiply the ordinary annuity value by (1 + r) ≈ 1.025, giving a slightly higher PV around 2215.26, still not matching 2411.76.
The provided answer 2411.76 would imply a different rate conversion, timing, or payment schedule than the standard interpretation applied here. In absence of those details, the calculation above follows the typical interpretation and yields about 2162.16.

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If you receive $99 each quarter for 8 years and the discount rate is 0.1, what is the present value?

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