Loan amortisation

In depth

Building a schedule

Write $B_0$ for the amount borrowed and $B_k$ for the balance after payment $k$. Each row of the schedule applies three steps: interest $I_k = r \times B_{k-1}$; principal repaid $P_k = C - I_k$; new balance $B_k = B_{k-1} - P_k$. Combining them, $B_k = B_{k-1}(1 + r) - C$: the balance grows by a period’s interest and the payment is taken off. In a spreadsheet each row is one copy of these three formulas, which is how most schedules are built.

A tidy consequence: with a level payment, the principal part grows by the factor $(1 + r)$ every period, $P_k = P_1(1 + r)^{k-1}$. For the brewer loan $432.94 \times 1.006 \approx 435.54$, the second month’s principal. The interest part falls by the same dollar amount that the principal part rises, because the two always add up to the payment.

Lenders usually round the payment and each month’s interest to the cent and let the last payment absorb the difference, as in the table above. A schedule kept that way can differ by a cent or two from the formulas below, which use the unrounded payment ($576.939452…$). Its 48 payments add up to $27,693.11, against $27,693.09 by formula. Neither is wrong; they round at different points.

Payments, interest and principal for each year are sums of the monthly rows. Fictional figures.

Interest falls from $1,553.09 in the first year to $262.60 in the last: more than 40% of all the interest is paid in the first year. The longer the loan, the more pronounced this is; on a 30-year mortgage, the subject of the mortgages lesson, early payments are mostly interest.

The balance at any date: two methods

You do not need the whole schedule to find the balance after $k$ payments. Two methods give it directly, and they always agree.

• Prospective (looking forward): the balance is the present value, at the loan rate, of the payments still to come. After $k$ of $n$ payments, $n - k$ remain: $B_k = C\,[1 - (1 + r)^{-(n-k)}] \div r$. For the brewer loan after 24 payments: $576.939452 \times 22.289933 \approx 12{,}859.94$.
• Retrospective (looking back): the balance is what was borrowed, grown at the loan rate, minus the payments made, each grown at the loan rate to the same date: $B_k = \text{PV}(1 + r)^k - C\,[(1 + r)^k - 1] \div r$. For the brewer loan: $24{,}000 \times 1.154387 - 576.939452 \times 25.731215 \approx 27{,}705.295 - 14{,}845.353 \approx 12{,}859.94$.

They agree because the loan was fair at the start: the payments were worth exactly the amount borrowed. Moving that equality forward $k$ periods splits it into the payments already made and the payments still due, and the balance is the gap between them. The prospective method is quicker when you know the payment and the remaining term; the retrospective method works even when payments have been irregular, as long as you grow each one to the same date. The lender’s rounded schedule shows $12,859.95, a cent more, for the reason given above.

The balance also gives the interest paid over any stretch of time: the payments made, minus the fall in the balance. In the second year Lantern would pay $12 \times 576.939452 - (18{,}629.807 - 12{,}859.942) \approx 6{,}923.273 - 5{,}769.865 \approx 1{,}153.41$ dollars of interest. Businesses need that figure for their accounts, where interest is an expense and repaid principal is not.

Interest-only versus amortising: why total interest misleads

On Lantern’s Coastline note the interest is 1.5% a quarter on $10,000 for eight quarters: $8 \times 150 = 1{,}200$ dollars in total. Suppose the same $10,000 at the same 6% had been amortised over the same eight quarters. The quarterly payment would be $10{,}000 \times 0.015 \div (1 - 1.015^{-8}) \approx 1{,}335.84$ dollars, and the total interest only about $686.72.

The interest-only note is not dearer. Both loans charge exactly 1.5% a quarter on whatever is owed, and at 1.5% the payments of each are worth exactly $10,000 today. The note costs more interest because Lantern keeps the whole $10,000 for two years instead of handing it back bit by bit. Total interest grows with how much is borrowed and for how long, not only with the rate. To compare the cost of two loans, compare their rates (as effective annual rates, including fees) or the present value of their payments at a common rate; never their total interest alone.

Interest-only loans suit borrowers who expect a large inflow at the end, or who want low payments now; their risk is the balloon payment, which must be repaid or refinanced on a fixed date whatever conditions are then. Lantern must find $10,000 on 1 March 2028.

Paying early: prepayment

A prepayment repays principal earlier than the schedule requires. Because interest is charged on the balance, every extra dollar of principal saves interest at the loan rate for the rest of the loan. Suppose Lantern paid $676.94 a month on the brewer loan, $100 more than required. The formula for the number of payments gives $n = -\ln(1 - 24{,}000 \times 0.006 \div 676.94) \div \ln 1.006 \approx 39.98$: 39 payments of $676.94 and a 40th of about $664.66, eight months sooner. Total interest would fall from about $3,693.09 to about $3,065.30, a saving of about $627.80.

Prepaying is like investing at the loan’s rate with no risk: worth doing if the money has no better use that earns more, but not if it leaves the business short of cash. Read the loan agreement first. Some loans charge a prepayment penalty; the Consumer Financial Protection Bureau, for example, describes mortgages whose terms charge a fee if the borrower pays off all or a large part of the loan early, typically within the first few years.

Refinancing

Refinancing replaces an existing loan with a new one, usually to get a lower rate, and uses the new loan to pay off the old balance. It is worth doing when the present value of the payment savings exceeds the costs of switching, such as an arrangement fee. Discount the monthly savings at the new loan’s rate: they are as certain as the payments themselves. The quick check is the break-even time, the fee divided by the monthly saving, which ignores interest but shows how long the borrower must keep the new loan for the switch to pay.

Short, small balances rarely repay the costs. If, after 24 payments, another lender offered to refinance Lantern’s $12,859.94 balance at 6.0% for the remaining 24 months for a $150 fee, the new payment would be about $569.96, saving only about $6.98 a month. The savings are worth about $157.47 at 0.5% a month, barely more than the fee, and the break-even time is more than 21 months. Worked example 3 shows a refinancing that does pay.

Fees and the APR of a closed-end loan

An up-front fee raises the cost of a loan even though the rate and the payments stay the same. If Coastline charged a 1.5% arrangement fee, $360, deducted from the loan, Lantern would receive $23,640 but still make 48 payments of $576.94. The rate that makes those payments worth $23,640 is about 0.6653% a month: 12 times that is about 7.98%, against the 7.2% quoted. This is the same search for an implied rate as in the annuities lesson.

Assumptions and limits

• The rate is fixed for the life of the loan. Variable-rate loans need the payment or the schedule recalculated when the rate changes; the mortgages lesson outlines adjustable rates.
• Payments are made on time and on schedule. Late payments add interest and often fees; prepayments shorten the loan or lower later payments, depending on the agreement.
• The payment period matches the compounding period. If not, convert the rate to the rate per payment period first.
• Taxes are ignored. For a business, interest is usually a tax-deductible expense, which lowers the after-tax cost of borrowing; F05 Corporate Finance takes this up.