If you receive $122 each month for 12 months and the discount rate is 0.08, what is the future value?

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BlackTom AI · Accepted Answer

Question restatement: You receive 122 dollars every month for 12 months, with a discount rate of 0.08. What is the future value (FV) after the 12th payment?

Step 1: Identify the cash flow pattern and the FV formula. You have an ordinary annuity (payments at end of each month) of N = 12 payments of P = 122. The monthly interest rate i needs to be derived from the annual discount rate 0.08. Different conventions give different i:
- Convention A: monthly rate as i = 0.08/12 ≈ 0.0066667 (simple division). Use FV = P * [((1+i)^N - 1) / i].
- Convention B: effective monthly rate from annual effective rate: i = (1+0.08)^(1/12) - 1 ≈ 0.006433. Use the same FV formula.
- Convention C: some problems use annual compounding within the year and then project to year end, which can yield another result, but is less common for monthly payments.

Step 2: Compute FV under Convention A (i = 0.08/12).
- (1+i)^N = (1.0066667)^12 ≈ 1.0827
- ((1+i)^N - 1) = 0.0827
- ((1+i)^N - 1) / i ≈ 0.0827 / 0.0066667 ≈ 12.405
- FV ≈ 122 * 12.405 ≈ 1511. 
Rounding differences can shift this around a bit; you might see values near 1511 to 1512 or 1513 depending on precision.

Step 3: Compute FV under Convention B (i ≈ 0.006433 from effective monthly rate).
- (1+i)^N = (1.006433)^12 ≈ 1.08
- ((1+i)^N - 1) = 0.08
- ((1+i)^N - 1) / i ≈ 0.08 / 0.006433 ≈ 12.430
- FV ≈ 122 * 12.430 ≈ 1517.

Step 4: Compare results and check the given figure. The value 1523.58 is higher than the typical results from these common conventions by about 6 to 12 dollars. This discrepancy suggests either:
- a different compounding convention (e.g., monthly rate treated differently), or
- rounding at an intermediate step, or
- payments assumed at the beginning of the month (an annuity due), which would increase FV.

Step 5: Explore the annuity-due variant briefly. If payments are at the start of each month (annuity due), FV can be larger because each payment accrues one additional period of interest. The FV would be FV_due = FV_ordinary * (1 + i). Using Convention A’s i ≈ 0.0066667, FV_due ≈ 1511 * 1.0066667 ≈ 1521, which is closer to 1523.58 but still not exact due to rounding.

Step 6: Noting common sources of error or confusion. When solving these, ensure you consistently apply:
- the correct monthly rate from the chosen convention,
- whether payments are end-of-period (ordinary annuity) or beginning-of-period (annuity due), and
- precise rounding at each step or using a calculator with sufficient precision.

Step 7: Summary guidance for solving similar problems. Pick a clear convention for i, decide if the annuity is ordinary or due, apply FV = P * [((1+i)^N - 1) / i], and then, if needed, multiply by (1 + i) for annuity due. Recompute with tight precision to minimize rounding drift. If your result differs from a provided answer like 1523.58, check which convention was assumed in the source and align your calculation to that convention.

Note: Because the original question provided only the numeric result 1523.58 without stating the convention, there isn’t a single unique path to that exact figure here. Both the chosen monthly rate convention and the annuity type (ordinary vs due) substantially influence the final FV.

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If you receive $122 each month for 12 months and the discount rate is 0.08, what is the future value?

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