Summarise an amortised loan
An amortised loan is reduced by regular payments, followed by a final adjusted payment that clears the remaining balance. Find the final-period interest from the preceding balance before finding the final payment
\[F=B_{f-1}+I_f,\quad T=nA+F,\quad I_{\text{total}}=T-P_0\]
- Use the balance before the final period Calculate final-period interest from the balance left after the last regular payment \(\displaystyle B_{f-1}=948.74,\quad I_f=0.05\times948.74=47.437\rightarrow47.44\)
- Round final interest to cents Round the final-period interest to cents before adding it to the preceding balance \(\displaystyle I_f=47.437\rightarrow47.44\)
- Set the final payment to clear the balance The adjusted final payment equals the preceding balance plus the final-period interest \(\displaystyle F=B_{f-1}+I_f=948.74+47.44=996.18\)
- Count totals once Add the regular payments and the final payment once, then subtract the original principal to find total interest \(\displaystyle T=nA+F,\quad I_{\text{total}}=T-P_0\)
Worked example
A loan of $8000 is repaid with four regular annual payments of $2200. The annual interest rate is 5%, and interest is charged before each payment. The final payment is adjusted to clear the balance. Find the final payment and total interest
\[B_n=B_{n-1}+I_n-2200,\quad B_0=8000\]
- \(\displaystyle I_1=0.05\times8000=400,\quad B_1=8000+400-2200=6200\)The first posted interest is $400, leaving a balance of $6200 after the regular payment
- \(\displaystyle I_2=0.05\times6200=310,\quad B_2=6200+310-2200=4310\)The second interest amount is based on the preceding posted balance
- \(\displaystyle I_3=0.05\times4310=215.50,\quad B_3=4310+215.50-2200=2325.50\)The third regular row leaves a posted balance of $2325.50
- \(\displaystyle I_4=0.05\times2325.50=116.275\rightarrow116.28,\quad B_4=2325.50+116.28-2200=241.78,\quad I_f=0.05\times241.78=12.089\rightarrow12.09,\quad F=241.78+12.09=253.87,\quad I_{\text{total}}=4\times2200+253.87-8000=1053.87\)Round the posted interest, then calculate the adjusted final payment and subtract the principal from all payments
Common mistakes
Omitting final-period interest
✕\(\displaystyle F=B_{f-1}\)✓\(\displaystyle F=B_{f-1}+I_f\)
The final payment must cover both the remaining balance and the interest charged on that balanceForcing the final payment to equal the regular payment
✕\(\displaystyle F=A\)✓\(\displaystyle F=B_{f-1}+I_f\)
The final payment is adjusted to clear the balance after final-period interestCounting the final payment twice
✕\(\displaystyle T=(n+1)A+F\)✓\(\displaystyle T=nA+F\)
The total contains n regular payments and one adjusted final paymentDifficulty levels
introFind the final interest and adjusted final payment from a supplied amortisation table
standardFind final and total cash-flow quantities from a supplied table or model
challengeInfer a regular-payment count or compare total interest between two amortised models
examSelect a supported summary of final interest, final payment and total cash flow
Questions about this topic
Why is the final payment usually different from a regular payment?+
The final payment covers the exact remaining balance and the interest for the final period, so it need not equal the regular payment
How is total interest found from the payments?+
Add every regular and final payment, then subtract the original principal from that total
Why is final interest based on the preceding balance?+
Interest for a period is calculated on the balance at the start of that period, which is the balance left after the last regular payment
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, General Mathematics, chapter 8: Cash flow and amortisation
In this chapter
Iterate a first-order recurrence8A Cash-flow recurrence models: Model and interpret a periodic cash-flow recurrence8A Cash-flow recurrence models: Analyse balances and graphs of an affine cash-flow model8B Interpreting cash-flow models: Model and interpret a periodic cash-flow recurrence8B Interpreting cash-flow models: Analyse balances and graphs of an affine cash-flow modelConstruct and interpret an amortisation table8D Final payments and totals: Analyse balances and graphs of an affine cash-flow model8D Final payments and totals: Final payment, total cash flow and total interest8E Time value of money: Final payment, total cash flow and total interest8E Time value of money: Set up a time-value-of-money calculationSolve for present or future value with regular cash flows8F Time value of money inverses: Final payment, total cash flow and total interest8F Time value of money inverses: Set up a time-value-of-money calculationSolve a time-value-of-money parameterAnalyse a cash-flow model with changed conditions8H Interest-only loans: Interest-only loans and perpetuities8I Perpetuities: Interest-only loans and perpetuities