If you receive $170 each 6 months for 7 years and the discount rate is 0.02, what is the future value?

Question

If you receive $170 each 6 months for 7 years and the discount rate is 0.02, what is the future value?

BlackTom AI · Accepted Answer

Question restatement: You receive 170 dollars every 6 months for 7 years. The discount rate is 0.02. What is the future value?

Step 1 – Identify key values and timing
- Payment amount (PMT): 170
- Payment frequency: semiannually (every 6 months)
- Number of years: 7, so total number of payments: 7 years × 2 payments/year = 14 payments
- Discount rate: 0.02. In time value questions, this rate must be aligned with the period of the payments. The ambiguity is whether 0.02 is per year, per half-year, or some other convention.

Step 2 – Translate the rate to the correct period
- If 0.02 is an annual rate and payments are semiannual, the per-half-year rate would typically be 0.02/2 = 0.01 per 6-month period.
- If 0.02 is already the semiannual rate (rare, but possible in some problems), then the per-period rate would be 0.02.
- If 0.02 were a different convention (e.g., effective quarterly, etc.), the per-period rate would need to be adjusted accordingly.

Step 3 – Choose the appropriate future value formula by timing
- If payments are made at the end of each period (ordinary annuity), the future value at the end of 14 periods is FV = PMT × [((1 + i)^n − 1) / i], where i is the per-period rate and n = 14.
- If payments are made at the beginning of each period (annuity due), the FV would be multiplied by (1 + i) to account for one extra period of compounding, giving FV = PMT × [((1 + i)^n − 1) / i] × (1 + i).

Step 4 – Compute under plausible interpretations
- Interpretation A: i = 0.01 (i.e., 1% per 6-month period), n = 14
  FV = 170 × [((1.01)^14 − 1) / 0.01] ≈ 170 × [0.1454 / 0.01] ≈ 170 × 14.54 ≈ 2472. (If annuity due, multiply by 1.01 to get ≈ 2499.)
- Interpretation B: i = 0.02 (i.e., 2% per 6-month period), n = 14
  FV = 170 × [((1.02)^14 − 1) / 0.02] ≈ 170 × [0.319 / 0.02] ≈ 170 × 15.95 ≈ 2711.5. (If annuity due, multiply by 1.02 to get ≈ 2765.7.)
- Interpretation C: Some problems treat the given 0.02 as a nominal rate convertible differently, or they may compute a different target time (e.g., end of year 7 vs end of each half-year). In such cases, you would adjust i and/or n accordingly and re-calculate the FV with the same formula structure.

Step 5 – Compare with the provided numeric result and discuss plausibility
- The value 2538.86 lies between the results from Interpretations A and B, suggesting that the intended interpretation may involve a per-period rate between 1% and 2% or a different timing convention (for example, a mix of annuity due vs ordinary annuity, or a nonstandard discount convention).
- If the rate was slightly lower than 2% per half-year but higher than 1%, or if the problem intended a half-year compounding with a misaligned convention, you could land near 2538.86. Small changes in i or the timing (end vs beginning of period) can shift the FV by hundreds of dollars over 14 periods.

Step 6 – Key takeaway for solving
- Clearly establish: (a) the period length and number of periods, (b) the exact per-period interest rate corresponding to the given annual (or other) rate, and (c) whether payments are at the end or beginning of each period.
- Apply the ordinary annuity FV formula or the annuity due variant, then compute carefully, preferably with a calculator or software to avoid arithmetic errors.

Note on missing context
- The problem statement does not explicitly declare whether the 0.02 rate is per half-year or per year, nor whether payments are made at the end or the beginning of each period. Those choices materially affect the numeric result. If you have access to the source, check the convention used for i (per period) and the timing of payments to align the calculation with the intended interpretation.

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If you receive $170 each 6 months for 7 years and the discount rate is 0.02, what is the future value?

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