If you receive $199 each quarter for 4 quarters and the discount rate is 0.02, what is the present value?

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BlackTom AI · Accepted Answer

Question restatement: You receive 199 dollars at the end of each quarter for 4 quarters, and the discount rate is 0.02 per quarter. What is the present value of these payments?

First, recognize this is a standard present value of an ordinary annuity with periodic payment P = 199, interest per period i = 0.02, and n = 4.

Approach 1: Use the closed-form annuity formula. The present value is
PV = P * [1 − (1 + i)^(-n)] / i.
Plugging in the numbers: PV = 199 * [1 − (1.02)^(-4)] / 0.02.
To approximate (1.02)^(-4), compute (1.02)^4 ≈ 1.082432, then take the reciprocal: (1.02)^(-4) ≈ 0.92385.
Thus, 1 − (1.02)^(-4) ≈ 0.07615.
Divide by i = 0.02: 0.07615 / 0.02 ≈ 3.8075.
Multiply by the annuity payment: 199 * 3.8075 ≈ 757.69.
So the present value is about 757.7 dollars (rounded).

Approach 2: Check with a term-by-term discounting (sanity check). The present value is the sum of each payment discounted back one period at 2% per period:
PV = 199 / 1.02 + 199 / (1.02)^2 + 199 / (1.02)^3 + 199 / (1.02)^4.
Calculating each term roughly:
- 199 / 1.02 ≈ 195.10
- 199 / (1.02)^2 ≈ 191.18
- 199 / (1.02)^3 ≈ 187.49
- 199 / (1.02)^4 ≈ 183.76
Sum ≈ 757.53.
These two methods yield very close results, reinforcing the approximate PV around 757.5–757.7.

Regarding the provided answer of 790.78: this value does not align with the standard PV calculation for these inputs. It could result from using a different timing (e.g., payments at the beginning of each quarter, i.e., an annuity due, which would increase PV but typically by a factor of (1 + i) = 1.02, bringing it closer to about 773–774, still not 790.78) or from using a different discount rate per period or a different number of periods. With i = 0.02 per quarter and 4 payments, the conventional present value is around 757.6–757.9.

Final takeaway: The present value, using the standard ordinary annuity formula with the given inputs, is approximately 757.7. If you obtained 790.78, recheck the timing of payments (end vs. beginning), the rate per period, or whether the calculation inadvertently used a different formula or rounding. The math above shows the correct framework and a consistent result near 757.6–757.7.

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If you receive $199 each quarter for 4 quarters and the discount rate is 0.02, what is the present value?

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