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Reducing Balance Loans
Key Terms
- A reducing balance loan charges interest on the remaining (outstanding) balance each period
- Each repayment covers the interest for that period plus reduces the principal
- Recurrence relation: Vn+1 = Vn(1 + r) − P
- r = interest rate per period (annual rate ÷ 12 for monthly)
- P = regular repayment amount
- V0 = initial loan amount (principal)
- The loan is repaid when Vn = 0
Recurrence relation:
Vn+1 = Vn(1 + r) − P
Interest charged in period n:
In = r × Vn−1
Principal reduction in period n:
ΔVn = P − In
Vn+1 = Vn(1 + r) − P
Interest charged in period n:
In = r × Vn−1
Principal reduction in period n:
ΔVn = P − In
Worked Example: A $20 000 loan has an annual interest rate of 6% (0.5% per month). Monthly repayments are $500. Find the balance after 3 months.
r = 0.005 per month
V0 = 20 000
V1 = 20 000(1.005) − 500 = 20 100 − 500 = 19 600
V2 = 19 600(1.005) − 500 = 19 698 − 500 = 19 198
V3 = 19 198(1.005) − 500 = 19 293.99 − 500 = $18 793.99
r = 0.005 per month
V0 = 20 000
V1 = 20 000(1.005) − 500 = 20 100 − 500 = 19 600
V2 = 19 600(1.005) − 500 = 19 698 − 500 = 19 198
V3 = 19 198(1.005) − 500 = 19 293.99 − 500 = $18 793.99
Hot Tip: The interest portion of each repayment decreases over time (because the balance falls), while the principal portion increases. Early in a loan, most of your repayment is interest — later, most goes to principal.
How Reducing Balance Loans Work
In a reducing balance loan, the interest charged each period is calculated on the current outstanding balance — which gets smaller as you make repayments. This is different from flat-rate loans (where interest is calculated on the original amount).
The Repayment Process
Each repayment P is split into two components:
- Interest component: I = r × current balance
- Principal component: P − I (the amount by which the balance actually reduces)
As the balance decreases, the interest component per payment decreases, and more of each payment reduces the principal.
Using the Recurrence Relation
Vn+1 = Vn(1 + r) − P
Start with V0 = loan amount. Iterate until Vn ≤ 0 (loan paid off).
For monthly repayments: use monthly rate r = annual rate / 12.
Total Cost of a Loan
Total amount paid = P × number of repayments
Total interest = Total amount paid − Principal
Strategy: Build a repayment table showing Vn, interest charged, principal paid, for each period. This makes it easy to answer questions about specific periods or the total interest paid.
Mastery Practice
- Fluency A $15 000 loan charges 4% per annum (monthly rate 0.333%). Monthly repayments are $400. Find the balance after 2 months.
- Fluency A loan balance is $8 200. The monthly interest rate is 0.5%. A payment of $350 is made. What is the balance after the payment?
- Fluency For a loan with V0 = 10 000, monthly rate 0.4%, and P = 300, write out the recurrence relation and find V1.
- Fluency In the first month of a $25 000 loan at 6% p.a. monthly, the repayment is $600. How much of this payment goes to interest, and how much reduces the principal?
- Understanding A $30 000 car loan charges 7.2% p.a. (0.6% per month). Monthly repayments are $700. Complete a repayment table for the first 4 months showing: balance at start, interest charged, principal paid, balance at end.
- Understanding Two loans each have a $10 000 principal and a 5-year term. Loan A charges 5% p.a. and Loan B charges 8% p.a. Without calculating exact repayments, explain which loan will cost more in total interest and why.
- Understanding A borrower is partway through a reducing balance loan. The balance is $5 000 and the monthly rate is 0.75%. The monthly repayment is $400. How many more months until the loan is paid off? Use iteration.
- Understanding A home loan of $400 000 is taken out at 4.8% p.a. (monthly rate 0.4%). Monthly repayments are $2 100. Find the balance after 3 months and the total interest paid in those 3 months.
- Problem Solving A $12 000 personal loan is taken out at 9% p.a. (monthly rate 0.75%). The borrower makes monthly payments of $350.
- (a) Complete a table for months 1–5.
- (b) After 5 months, the borrower increases repayments to $500. Find the balance at the end of month 6.
- Problem Solving A borrower takes out a $20 000 loan at 6% p.a. (0.5% per month) and wants to pay it off in exactly 48 months.
- (a) Use the recurrence relation iteratively to find the approximate monthly repayment that results in V48 ≈ 0 (try values near $470).
- (b) Calculate the total interest paid over the life of the loan.