A loan of $9,500 at 6.9% interest per annum, compounding monthly on the reducing balance, is repaid over 6 years with equal monthly payments. The amount of the monthly repayment is closest to:

Question

BlackTom AI · Accepted Answer

Use the standard loan (annuity) repayment formula for equal monthly payments on a reducing balance: R = PV * i / (1 - (1+i)^(-n)).

1. Relevant knowledge points and concepts
- For a loan repaid by equal monthly payments with monthly compounding, use the annuity-immediate formula above.
- PV = present value (loan amount) = $9,500.
- Annual nominal rate 6.9% compounded monthly => monthly rate i = 0.069/12 = 0.00575.
- Total number of monthly payments n = 6 years × 12 = 72.

2. Complete reasoning and step-by-step calculation
- Compute (1 + i)^n:
  i = 0.00575, so 1 + i = 1.00575.
  (1.00575)^72 = e^{72·ln(1.00575)}. Using a series expansion for ln(1+x) and exponentiation gives (1.00575)^72 ≈ 1.51104.
- Compute the discount factor 1 - (1+i)^(-n):
  (1+i)^(-n) = 1 / 1.51104 ≈ 0.66186.
  Hence 1 - (1+i)^(-n) ≈ 1 - 0.66186 = 0.33814.
- Compute the payment factor i / (1 - (1+i)^(-n)):
  i / (1 - (1+i)^(-n)) ≈ 0.00575 / 0.33814 ≈ 0.0170048.
- Monthly repayment R:
  R = 9500 × 0.0170048 ≈ $161.51 (rounded to two decimal places).

3. Intermediate steps shown above include calculation of i, n, (1+i)^72, the discount factor, and the payment multiplier.

4. Why this answer is correct
- The chosen formula exactly models a reducing-balance loan with monthly compounding and equal monthly payments. Plugging in the correct i and n yields the computed payment. Numerical rounding yields $161.51, matching option b.

5. Why other options are incorrect
- a. $155.20 is too low — it would correspond to a materially smaller interest factor or a longer term.
- c. $170.30 is too high — it overestimates the monthly burden given the 6-year term and 6.9% p.a.
- d. $164.90 is slightly high — it does not match the precise annuity calculation above.

Conclusion: The correct monthly repayment is approximately $161.51, so the correct choice is b. $161.51.

A loan of $9,500 at 6.9% interest per annum, compounding monthly on the reducing balance, is repaid over 6 years with equal monthly payments. The amount of the monthly repayment is closest to:单项选择题

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