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Car loan interest rate and lump sum at year 3

Short answer

Loan amount Rs 12,00,000. Solving 12,00,000 = 25,000(P/A,i,60) gives monthly i ≈ 0.7629%, so nominal annual rate ≈ 9.15%. At end of year 3, the lump sum to clear dues is the present worth of remaining 24 payments: Rs 5,46,381.

Mr. Basnet purchases a car which costs Rs. 20,00,000. He pays 40% as down payment. Remaining amount will be paid on installment basis and he wishes to pay Rs. 25,000 per month for next five years. What annual interest rate will he be paying? At the end of 3rd year, what lump sum amount should he pay to clear all his dues?

Loan amount

  • Down payment = 40% × 20,00,000 = Rs 8,00,000
  • Loan P = 20,00,000 − 8,00,000 = Rs 12,00,000
  • Monthly installment A = Rs 25,000, n = 5 × 12 = 60 months
 +12,00,000                             (loan received)
 0    1     2     3   ...   59    60    (months)
      ↓     ↓     ↓         ↓     ↓
     25,000 each month (end of month)

(a) Annual interest rate

Equation: 12,00,000 = 25,000 (P/A, i, 60), so (P/A, i, 60) = 48.

Trial and error (monthly i):

i per month(P/A, i, 60)Compare with 48
0.70%48.8559too high
0.80%47.5042too low
0.7629%48.0000= 48

Interpolating between 0.70% and 0.80%: i = 0.70 + 0.10 × (48.8559 − 48)/(48.8559 − 47.5042) ≈ 0.763% per month.

  • Nominal annual rate = 12 × 0.7629% = 9.15% per year
  • Effective annual rate = (1.007629)^12 − 1 = 9.55%

(b) Lump sum at the end of year 3

After 36 installments are paid, 24 installments remain. The amount that clears the dues is the present worth of the remaining 24 payments at month 36:

  • (P/A, 0.7629%, 24) = 21.8553
  • Lump sum = 25,000 × 21.8553 = Rs 546,381

This is the outstanding balance just after the 36th installment is paid. If he has not yet paid the 36th installment, he pays 25,000 + 546,381 = Rs 571,381.

Answer: interest ≈ 9.15% per year (nominal); lump sum at end of year 3 ≈ Rs 546,381.

✓ Answer checked by Claude Opus 5.5 (high effort)

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