Loan and Overdraft Repayment (Amortisation)

When a loan (or a bank overdraft) is repaid in equal periodic instalments, the process is called amortisation. Each instalment is partly interest on the amount still outstanding and partly repayment of principal; as the outstanding balance falls, the interest portion of each instalment shrinks and the principal portion grows, even though the total instalment stays the same.

The size of the equal instalment is found by treating the loan as the present value of the future instalments: the sum borrowed today, $V$, is exactly the present value of the stream of $n$ equal repayments. Using the capital-recovery form of §3:

where $P$ is the equal instalment, $V$ the amount borrowed, $i$ the rate per repayment period and $n$ the number of instalments. For a monthly-repaid loan, $i$ is the monthly rate and $n$ the number of months; for a half-yearly overdraft, $i$ is the half-yearly rate and $n$ the number of half-years — the rate and the count must always match the repayment period, not the year.

The amortisation (repayment) schedule. The best cross-check on a computed instalment is to build the repayment schedule and confirm the final balance closes to zero:

1. Interest for the period $=$ opening balance $\\times\\, i$.
2. Principal repaid $=$ instalment $-$ interest.
3. Closing balance $=$ opening balance $-$ principal repaid.

Carrying this table down to the last instalment must leave a closing balance of (approximately) zero — any small residue is only rounding of the instalment to the nearest paisa. Worked Examples 7 and 8 each solve for the instalment by formula and then verify it by running the full schedule to a zero balance. …