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Chapter 4

Auto & Student Loans

Amortization, the length of a loan, and interest that accrues in school

A car loan asks a different question from a credit card. Not "what is this costing me" but "what do I owe every month for the next three years" — and the answer comes from one formula that never changes.

Chapter 4 · The idea

One payment, fixed for the term

An installment loan is repaid in equal payments. Each one covers the interest that has accrued since the last, and whatever is left over comes off the principal — so early payments are mostly interest and later ones are mostly principal, while the amount you hand over never changes.

M=P r(1+r)n(1+r)n−1M = P\,\frac{r(1+r)^n}{(1+r)^n - 1}
What you borrowed, after any down paymentPP
The interest rate per payment period, not per yearr=APR12r = \frac{\text{APR}}{12}
How many payments there are in totalnn
Chapter 4 · Worked problem

A $10,000 car, and what it costs each month

The problem
You buy a $10,000 used car. After putting $2,000 down in cash, you finance the rest at 6% annual interest for 36 months. What is the fixed monthly payment?
Working1 / 6
  1. The down payment is not borrowed, so it is not principalP=10000−2000=8000P = 10000 - 2000 = 8000

Thirty-six payments of $243.38 is $8,761.68 against $8,000 borrowed. The loan costs about $762 — which is the number to compare against, not the 6%.

Chapter 4 · Inside a payment

Where each payment goes

The payment is fixed; its ingredients are not. Interest is charged on what is still owed, so the first payment carries the most of it and the last almost none, and the share going to principal grows every month. A schedule that lists the split is called an amortization table, and the first two lines of one are enough to see the pattern.

Figure 6Every bar is one payment, the same height each month. The amber part is interest, the blue part is what came off the loan, and the green line is what is still owed. Lengthen the term and watch the amber grow.
The problem
The $8,000 loan at 6% over 36 months, payment $243.38. How much of the first payment is interest, how much is principal, and what is owed after it? And after the second?
The first two payments1 / 6
  1. Interest is the monthly rate on the balance8000×0.005=40.008000 \times 0.005 = 40.00

By the last payment the split has reversed: month 36 carries $1.21 of interest and the rest is principal. That reversal is the whole shape of the figure.

Chapter 4 · The term

A longer loan is a smaller payment and a bigger price

Dealers quote the monthly payment, and the easiest way to make a payment smaller is to make the loan longer. The rate does not change; the number of months it is charged for does.

The same $8,000 at 6%, over three yearsM=243.38,interest $762M = 243.38,\quad \text{interest } \$762
Over fourM=187.88,interest $1,018M = 187.88,\quad \text{interest } \$1{,}018
Over fiveM=154.66,interest $1,280M = 154.66,\quad \text{interest } \$1{,}280
Over sixM=132.58,interest $1,546M = 132.58,\quad \text{interest } \$1{,}546

From three years to six, the payment falls by $111 a month and the loan costs twice as much. A car loses value faster than a long loan repays it, which means that somewhere in year two the loan is larger than the car — and a car written off or sold in that window leaves a debt with nothing attached to it.

Chapter 4 · Student loans

Interest that runs while you are not paying

A student loan looks like the car loan with a different name on it, and for the repayment it is. The difference is what happens before repayment starts. On an unsubsidised loan, interest accrues from the day the money is disbursed — through four years of study and the six-month grace period after — and when repayment begins, the accrued interest is added to the principal. From then on you pay interest on the interest.

Capitalization
Accrued interest being folded into the principal. It happens once, at the start of repayment, and it is why a $20,000 loan can begin repayment as a $25,000 one.
The problem
Ben borrows $20,000 at 5.5% at the start of a four-year degree, unsubsidised. With the six-month grace period, interest accrues for four and a half years before a ten-year repayment begins. What is his monthly payment — and what would it have been had he paid the interest as it accrued?
Letting it accrue1 / 5
  1. Simple interest accrues through school and the grace period20000×0.055×4.5=495020000 \times 0.055 \times 4.5 = 4950
Paying the interest as it accrues1 / 3
  1. The interest each month while in school20000×0.05512≈91.67\frac{20000 \times 0.055}{12} \approx 91.67

Ninety-two dollars a month during school, against $54 a month more for ten years afterwards and about $1,500 more in total. The example simplifies — real loans arrive in yearly pieces, so less accrues — but the shape holds: interest paid before capitalization is paid once, and interest left to capitalise is paid on for a decade.