If you receive $172 each quarter for 15 years and the discount rate is 0.11, what is the present value?

Question

If you receive $172 each quarter for 15 years and the discount rate is 0.11, what is the present value?

BlackTom AI · Accepted Answer

First, identify what is being asked: the present value of a series of equal payments received periodically. Here, you receive 172 dollars every quarter for 15 years, and the discount rate is 0.11.
To use the standard present-value of an ordinary annuity formula for periodic payments, convert the annual rate to the per-period rate and count the total number of periods.
Step 1: Determine the per-period rate. If 0.11 is an annual rate with quarterly compounding, the quarterly rate is 0.11 / 4 = 0.0275 (2.75% per quarter).
Step 2: Determine the number of periods. There are 15 years with quarterly payments, so n = 15 * 4 = 60 quarters.
Step 3: Apply the present-value of an ordinary annuity formula: PV = Pmt × [1 − (1 + i)^−n] / i, where Pmt = 172, i = 0.0275, n = 60.
Step 4: Compute (1 + i)^−n. Evaluate (1.0275)^60 and then take the reciprocal to get its negative power. Using natural-log approximation: ln(1.0275) ≈ 0.027134, so ln[(1.0275)^60] ≈ 60 × 0.027134 ≈ 1.62804, and (1.0275)^60 ≈ e^1.62804 ≈ 5.10. Therefore (1.0275)^−60 ≈ 1/5.10 ≈ 0.196.
Step 5: Compute the annuity factor: [1 − 0.196] / 0.0275 ≈ 0.804 / 0.0275 ≈ 29.26.
Step 6: Multiply by the payment amount: PV ≈ 172 × 29.26 ≈ 5032.72.
Thus, under these standard assumptions, the present value is approximately 5033 dollars (to the nearest dollar).
Notes and alternative interpretations:
- If 0.11 is an effective quarterly rate (not typical, but possible in some contexts), i_q would be 0.11, n = 60, and PV would be 172 × [1 − (1.11)^−60] / 0.11, which would yield a different value.
- If 0.11 is the nominal annual rate with a different compounding convention than quarterly, the per-period rate would change accordingly, leading to a different PV.
- The value 5273.36 corresponds to a different interpretation of the rate or timing (for example, a different per-period rate or whether payments are at the beginning or end of periods). If the problem specifies a particular compounding convention or timing (e.g., payments at the start of each quarter), you would adjust the formula to an annuity-due variant, which changes the result slightly.
In summary, with standard quarterly compounding of an 11% annual rate and ordinary annuity timing, the present value is about 5033.

If you receive $172 each quarter for 15 years and the discount rate is 0.11, what is the present value?简答题

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