A reducing loan and a final smaller payment

Selena borrows 200000 dollars from a bank for a home purchase on 1 January 2025. On day 1 of each month, 0.5% interest is added, first on 1 January 2025. On each last day she pays xx dollars, starting 31 January 2025.

(i) Show debt at end of nn months is 200000(1.005)n−200x(1.005n−1)200000(1.005)^n-200x(1.005^n-1). marks

(ii) If she pays 1500 dollars monthly on each last day, on what date does she fully repay, and what is the last repayment amount? marks

(iii) To fully repay in 10 years, how much monthly? marks

[2024/JPJC/Prelim/I/8]

Total marks: 9

Solution

(i)

Debt after n payments

Derive the debt after nn month-end payments.

Hint

  1. The repayment powers run from 0 to n−1.

Answer · (i)

Bn=200000(1.005)n−200x[(1.005)n−1]B_n=200000(1.005)^n-200x[(1.005)^n-1].

(ii)

Payoff date and final amount

For monthly payments of 1500 dollars, find the payoff date and final payment.

Explore this idea

Hint

  1. Multiply the residual debt by 1.005 before calculating the last repayment.

Answer · (ii)

Payoff 3131 May 20432043; last payment 407.70407.70 dollars.

Explore the idea

A negative scheduled balance signals a smaller last payment

This plots the formula assuming a full 1500 payment every month. Its negative value at 221 flags overpayment; actual debt stops at zero.

Payment countn = 220
Try an example
scheduled balance at n = 220
405.6723
Key idea

After 220 full payments the debt is 405.67234. One more interest charge gives the smaller final payment of 407.70.

Ideas to try
Compare the values on either side of the boundary. What makes one count qualify?
(iii)

Payment to finish within ten years

Find a monthly payment in whole cents that clears the loan within ten years.

Hint

  1. This is a lower bound: ordinary nearest-cent rounding can be insufficient.

Answer · (iii)

x≥1000(1.005)120/[(1.005)120−1]≈2220.410039x\ge1000(1.005)^{120}/[(1.005)^{120}-1]\approx2220.410039 dollars.

Whole-cent payment must be at least 2220.42 dollars; nearest-cent 2220.41 falls short.

Takeaway

Fix the order: interest first, repayment second

Ten years means 120 month-end payments; the payment must make that balance nonpositive.