Where you are. The module’s first half has been about positions deliberately levered for trading. This lesson turns to the levered position most people actually hold, which is a mortgage, and to the schedule that governs it. It is also the input to the next two lessons, because a pool of these schedules is what gets packaged and sold.
Three years of payments and almost no progress
Borrow three hundred thousand for thirty years at five percent. The payment is about 1,610 a month, unchanging, for three hundred and sixty months.
After three years you have paid roughly 58,000. Ask how much of the house you now own, and the answer is about 12,000 - a fifth of what you paid. The other 46,000 was interest.
Nothing went wrong and nobody cheated you. It is the arithmetic of charging interest on a balance you have barely begun to reduce, and it is the single most surprising fact about the largest loan most people ever take.
The idea in one paragraph
An amortising loan is repaid by a level payment covering both interest and principal, sized so the balance reaches exactly zero on the final payment. Because interest is charged on the outstanding balance, and the balance is highest at the start, the early payments are mostly interest and buy very little ownership. As the balance falls, the interest portion falls with it, so more of each identical payment goes to principal - which reduces the balance faster, which reduces the interest further. The mix flips slowly and then decisively, and the crossover comes far later than intuition suggests. Nothing about the payment changes; only its composition does.
The flip, and when it happens
The exercise generates all 360 rows and finds the crossover: the month where principal first exceeds interest.
On a thirty-year loan at five percent, it arrives around month 154 - roughly year thirteen. For the first thirteen years, more than half of every payment is rent on money rather than purchase of the asset.
The total interest over the full term is roughly 280,000 on 300,000 borrowed: the loan very nearly doubles what the house costs, at a rate most people would describe as low.
Why an extra payment early is worth so much
Because a pound of principal paid today removes that pound from every future interest calculation. Early on, when the remaining term is long, one extra pound of principal saves interest for hundreds of months.
That is why overpayment calculators produce numbers that look like errors: a modest extra payment in the first years can cut years off the term. The same pound paid in year twenty-eight saves almost nothing, because there is almost no term left for it to save.
What this feeds
A mortgage is a stream of dated payments, which is exactly the object module 4’s bond lesson taught you to price. The difference is that each payment mixes interest and principal, and the borrower can prepay, which makes the stream uncertain in a way a bond’s is not.
Pool a few thousand of these streams, split the payments by a rule, and you have the next two lessons: the priority rule that allocates each collection, and the tranching that decides who absorbs losses.
Check yourself
1. The payment never changes. Why does the split between interest and principal change so much?
Because interest is charged on the outstanding balance, which falls with every payment. Less interest due means more of the same payment is left for principal, which reduces the balance faster still. The payment is fixed by design; its composition follows the balance.
2. After three years of a thirty-year loan, why is so little of the debt repaid?
Because the balance has barely moved, so almost the entire payment is still going to interest on it. The early payments are dominated by rent on the borrowed money, and only after the balance has fallen materially does the composition shift.
3. Why is an extra payment in year two worth far more than the same payment in year twenty-eight?
Because principal repaid removes that amount from every future interest calculation. In year two there are hundreds of months of interest still to be charged on it; in year twenty-eight there are almost none, so the same pound saves almost nothing.
4. Why does moving house every seven years leave a borrower with little ownership despite decades of payments?
Because they repeatedly pay the front of the schedule, which is the interest-heavy part, and leave before reaching the principal-heavy part. Restarting a thirty-year schedule resets the composition to its most interest-heavy point, so the pattern repeats rather than progressing.
Do this
Generate the full schedule and find the crossover.
python3 code/amortisation.pyThe starter provides the level payment and leaves you schedule: month by month, compute the interest on the current balance, treat the remainder of the payment as principal, reduce the balance, and record the row. The assertions check that the loan retires exactly at the final payment, that early payments are interest-dominated and late ones principal-dominated. Success prints three rows and the line reporting the crossover month and the total interest paid.
The completed version is in solutions/amortisation.py.
What you can now do. You can generate any amortising loan’s full schedule, explain the flip and why it comes so late, and say what an early overpayment is really buying. The next lesson takes thousands of these schedules and asks who gets each pound that arrives.