If you receive $223 each month for 12 months and the discount rate is 0.11, what is the present value?

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BlackTom AI · Accepted Answer

Question restatement: You receive 223 dollars every month for 12 months, with a discount rate of 0.11. What is the present value (PV) of these payments?

Step 1: Identify the cash flow pattern. This is a level annuity-immediate scenario with 12 monthly payments of 223.

Step 2: Convert the annual discount rate to a monthly period rate. If the given discount rate is annual (0.11), the corresponding monthly rate is i = 0.11 / 12 ≈ 0.0091666667.

Step 3: Use the present value of an ordinary annuity formula for monthly payments: PV = Pmt × [(1 − (1 + i)^−n) / i], where Pmt = 223, i ≈ 0.0091666667, and n = 12.

Step 4: Compute the discount factor component. First calculate (1 + i)^−n = (1.0091666667)^−12. This requires either a calculator or iterative computation; approximately, (1.0091666667)^12 ≈ 1.1156, so its inverse is about 0.896. Therefore, 1 − (1 + i)^−n ≈ 1 − 0.896 = 0.104.

Step 5: Divide by the monthly rate i. Compute 0.104 / 0.0091666667 ≈ 11.34 (approximately, depending on rounding during the intermediate steps).

Step 6: Multiply by the monthly payment to obtain PV. PV ≈ 223 × 11.34 ≈ 2530.42 (approximately, exact value depends on the precision of the intermediate calculations).

Notes on precision and alternative interpretations:
- If a different interpretation of the discount rate is intended (for example, a different compounding convention or treating 0.11 as a monthly rate), the PV result would change accordingly.
- The calculation shown uses a standard monthly rate conversion from an annual rate and applies the ordinary annuity formula. Small differences in rounding during the exponentiation step can shift the final PV by a few dollars.

Summary of the approach: Convert annual rate to a per-period rate, apply the present value of an ordinary annuity formula, compute the annuity factor, and multiply by the monthly payment to obtain the present value.

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If you receive $223 each month for 12 months and the discount rate is 0.11, what is the present value?

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