Mortgages

In depth

The payment formula and the schedule

With loan $\text{PV}$, monthly rate $r$ and $n$ monthly payments, the payment is $C = \text{PV} \times r \div [1 - (1 + r)^{-n}]$: the annuity formula of module 3 solved for the payment. Each month the interest is the opening balance times $r$, the principal repaid is $C$ minus that interest, and the new balance is the old balance minus the principal repaid. The first three months of Jonah’s 30-year schedule:

Interest = previous balance × 0.065 ÷ 12. Calculated with the unrounded payment; figures rounded to the cent.

The principal repaid grows by the factor 1 + r each month, so the balance falls slowly at first and quickly at the end. Jonah would have repaid only 6.4% of the loan after five years and 15.2% after ten; the halfway point comes in month 257, about 21 years in. The chart in “Explore” shows each year’s payments split into interest and principal.

The balance after k years

The loan-amortisation lesson gave two equivalent methods. Prospectively, the balance after $k$ payments is the present value of the payments still to come: $B_k = C \times [1 - (1 + r)^{-(n - k)}] \div r$. Retrospectively, it is what the loan would have grown to, less what the payments made would have grown to. After ten years Jonah’s 30-year balance is $1{,}820.36 \times [1 - 1.00541667^{-240}] \div 0.00541667 \approx 244{,}155.24$ dollars. This is the figure a buyer needs when planning to sell or refinance: the sale must repay it.

Points, break-even and the APR

The simple break-even (cost ÷ monthly saving) ignores two things that pull in opposite directions. The lower rate also pays the balance down a little faster, so a seller owes slightly less; and a dollar of saving in year five is worth less than a dollar of points paid today. A sounder test values every cash flow of the points loan at the rate Jonah would otherwise pay, 6.5%. The no-points loan is worth exactly $288,000 at 6.5% for any holding period, by construction. The points loan, kept $k$ months and then repaid, costs:

If that is less than $288,000, the points were worth buying. Valuing at 6.5% treats the alternative to paying points as borrowing the $4,320 at the loan rate, or using it to prepay the loan. If the cash would otherwise earn less in savings, discounting at that lower rate makes the points look better; the rate is a judgement, so state it. For Jonah it is about $614.50 more than $288,000 if he keeps the loan four years, and about $3,107.38 less if he keeps it ten. The crossover comes at about 58 months, close to the simple estimate here because the two omitted effects nearly cancel. With higher rates or a larger rate cut, they need not.

Treating the $4,320 of points as the only finance charge, the APR of Jonah’s points loan is the rate at which 360 payments of $1,749.92 are worth what he actually receives, $288{,}000 - 4{,}320 = 283{,}680$ dollars. By trial, or with a spreadsheet’s RATE function, that is about 6.27%, against 6.5% for the no-points loan. The APR spreads the points over the full 30 years, so it flatters points for anyone who will not keep the loan that long: kept for four years, the same loan costs about 6.56% a year. Compare APRs between loans of the same type and term, and then ask how long you will really keep the loan.

Extra payments

Prepayment, paying principal ahead of schedule, saves interest because every extra dollar stops being charged interest from that month onward. If the borrower keeps paying a fixed larger amount $C$, the number of months needed solves $\text{PV} = C \times [1 - (1 + r)^{-n}] \div r$ for $n$, using logarithms as in the annuities lesson:

Adding $250 a month to Jonah’s 30-year payment, so $C = 2{,}070.36$: $r \times \text{PV} \div C = 1{,}560 \div 2{,}070.36 \approx 0.753494$, and $n = -\ln(0.246506) \div \ln(1.00541667) \approx 1.400367 \div 0.005402 \approx 259.23$. He would make 259 full payments and a smaller 260th, finishing in 21 years 8 months instead of 30 years. Running the schedule shows total interest of about $248,697.14, a saving of about $118,630.99.

Before planning extra payments, check the loan for a prepayment penalty. The CFPB explains that not all mortgages have one; where a loan does, it typically applies when the whole balance is paid off within a set period, usually three or five years, for example by selling or refinancing, and small extra principal payments normally do not trigger it.

A fixed payment in an inflating world

A fixed-rate mortgage payment is a fixed nominal amount: $1,820.36 in 2027 and $1,820.36 in 2056. With inflation, later payments are made in dollars that buy less. Measured in today’s dollars with the method of the previous lesson, the payment made at the end of year $t$ is worth $C \div (1 + i)^t$. At 3% inflation (an assumption for illustration):

1,820.36 ÷ 1.03^t, rounded to the cent. Treats each year’s payments as if made at the end of the year.

The real burden falls further if pay rises with prices. On a fictional salary of $92,000 a year, $1,820.36 is about 23.7% of Jonah’s gross monthly pay at the start; if his pay rises 3% a year, it is about 17.7% ten years later. In real terms the loan costs less than its quoted rate: the 6.5% compounded monthly is an effective annual rate of about 6.697%, a real interest rate of about $1.06697 \div 1.03 - 1 \approx 3.59\%$ at 3% inflation. With a fixed rate, the lender bears the risk that inflation turns out higher than expected; the borrower bears the risk that it turns out lower.

Adjustable-rate mortgages in outline

An adjustable-rate mortgage (ARM) moves the interest-rate risk the other way. Its rate is fixed for an initial period (five years in worked example 4) and then resets at regular intervals. The CFPB explains the reset: the new rate is an index, an interest rate that moves with market conditions and that the lender chooses when you apply, plus a margin, a number of percentage points set in the loan agreement that does not change after closing. Index plus margin is called the fully indexed rate, and the actual new rate is that figure subject to the loan’s rate caps.

• Initial adjustment cap: how far the rate can move at the first reset; the CFPB says two or five percentage points is common.
• Subsequent adjustment cap: how far it can move at each later reset; most commonly one or two points.
• Lifetime cap: how far it can move in total over the life of the loan; most commonly five points.

At each reset the payment is recalculated with the payment formula: the current balance, the new monthly rate and the months remaining. Worked example 4 shows the jump a borrower can face at the first reset, and why the right question about an ARM is not “what is the starting payment?” but “what is the highest payment the caps allow, and could I afford it?”