Coaching Centre
Reducing-balance loans
How a reducing-balance loan works month by month, building a loan table with interest and repayments, the recurrence relation for the balance, and total repayments and total interest.
Car loans, personal loans and home loans are almost all reducing-balance loans. Every period the bank charges interest on what you still owe, and then your repayment comes off. As the balance shrinks, so does the interest, which means more of each repayment goes towards the loan itself. It starts slowly and then speeds up.
One month of a loan
Each period follows the same three steps:
- Interest $=$ balance $\times$ interest rate per period.
- Add the interest to the balance.
- Subtract the repayment.
Here $B_n$ is the balance owing after $n$ periods, $r$ is the interest rate per period and $M$ is the repayment. The starting balance $B_0$ is the amount borrowed.
Kai borrows $\$10\,000$ for a used car at $12\%$ p.a. (an example rate), with interest calculated monthly. He repays $\$500$ a month. Complete the first four rows of the loan table.
| Month | Opening balance ($\$$) | Interest ($\$$) | Repayment ($\$$) | Closing balance ($\$$) |
|---|---|---|---|---|
| $1$ | $10\,000.00$ | $100.00$ | $500$ | $9600.00$ |
| $2$ | $9600.00$ | $96.00$ | $500$ | $9196.00$ |
| $3$ | $9196.00$ | $91.96$ | $500$ | $8787.96$ |
| $4$ | $8787.96$ | $87.88$ | $500$ | $8375.84$ |
The repayment is the same each month, but the interest falls, so the amount paid off the loan rises: $\$400$, then $\$404$, then $\$408.04$, then $\$412.12$.
Step through Kai's loan one month at a time and check the table. Then press Pay it off. How many months does it take, and how much interest is paid altogether?
Drop the repayment to $\$100$. Why does the balance never go down?
Total repaid and total interest
When every repayment is the same, the totals are quick to find:
A $\$20\,000$ personal loan at $9\%$ p.a. is repaid with $60$ monthly repayments of $\$415.17$. Find the total interest.
The balance graph of a loan is a curve that starts flat and gets steeper: early repayments are mostly interest, later ones are mostly the loan itself.
Your turn
A loan of $\$25\,000$ is charged interest at $8.4\%$ p.a., calculated monthly. Repayments are $\$1200$ a month. Find the balance after $3$ months.
Show the working
$r = 0.084 \div 12 = 0.007$. Month 1: $25\,000 \times 1.007 - 1200 = \$23\,975$. Month 2: $23\,975 \times 1.007 - 1200 = \$22\,942.83$. Month 3: $22\,942.83 \times 1.007 - 1200 \approx \$21\,903.42$.
A loan of $\$8000$ is charged $9\%$ p.a. interest, calculated monthly. Find the interest for the first month, in dollars.
$r = 0.09 \div 12 = 0.0075$. Interest $= 8000 \times 0.0075 = \$60$.
For the same loan, the repayment is $\$400$ a month. Find the balance after the first repayment.
$8000 + 60 - 400 = \$7660$.
A loan has balance $B_n$ after $n$ months, monthly rate $r$ and monthly repayment $M$. Which recurrence relation is correct?
Interest is added first (multiply by $1 + r$), then the repayment is subtracted.
A $\$20\,000$ loan is repaid with $60$ monthly repayments of $\$415.17$. Find the total interest paid.
$415.17 \times 60 = \$24\,910.20$. Interest $= 24\,910.20 - 20\,000 = \$4910.20$.
A loan has a balance of $\$9196$ at the start of a month. Interest is $12\%$ p.a. calculated monthly and the repayment is $\$500$. Find the balance at the end of the month.
Interest $= 9196 \times 0.01 = \$91.96$. Balance $= 9196 + 91.96 - 500 = \$8787.96$.
Why does the interest charged on a reducing-balance loan fall each month?
The rate and the repayment stay the same. Interest is a percentage of the balance, and the balance falls after every repayment.
A home loan balance is $\$300\,000$ at $6\%$ p.a., calculated monthly. What is the smallest monthly repayment that stops the balance from growing?
The repayment must at least cover one month's interest: $300\,000 \times 0.06 \div 12 = \$1500$.
A loan of $\$15\,000$ is charged $7.2\%$ p.a. interest calculated monthly, with repayments of $\$2000$ a month. Find the balance after $2$ months, to the nearest cent.
$r = 0.006$. Month 1: $15\,000 \times 1.006 - 2000 = \$13\,090$. Month 2: $13\,090 \times 1.006 - 2000 = \$11\,168.54$.
Exam style: A loan of $\$25\,000$ is charged $8.4\%$ p.a. interest calculated monthly, with repayments of $\$1200$ a month. Find the balance owing after $3$ months, to the nearest cent.
$r = 0.007$. $B_1 = 25\,000(1.007) - 1200 = \$23\,975$. $B_2 = 23\,975(1.007) - 1200 = \$22\,942.83$. $B_3 = 22\,942.83(1.007) - 1200 \approx \$21\,903.42$.
Exam style: Jordan borrows $\$30\,000$ for a car and repays $\$980$ a month for $3$ years. How much interest does Jordan pay?
Number of repayments $= 3 \times 12 = 36$. Total repaid $= 980 \times 36 = \$35\,280$. Interest $= 35\,280 - 30\,000 = \$5280$.