A mortgage worksheet that shows what two points of interest really cost

Interest rates feel like weather. They are out there somewhere, they change, they ruin your plans, and you nod along when the news calls it "the rate." Then you sit at a kitchen table with a pencil, divide a loan by a number of months, and the weather turns into a bill. This post is a worksheet. Get a pencil, a calculator and a cup of coffee. By the end you will be able to work out what any mortgage costs, and you will know why a gap of two percentage points in the rate is not two percent of anything. It is a mortgage payment that can change a family's whole budget.

Everything below is a hypothetical, labelled as one. I am not quoting any current rate, because rates change. I am teaching a method, and the method does not expire.

The set-up

Our hypothetical borrower takes out a $300,000 loan with a thirty-year fixed rate. That is 360 monthly payments. We will price it twice: once at 4 percent a year, once at 6 percent. Two points apart.

The standard payment formula looks scary on paper and is simple in practice:

Payment = Loan x r x (1 + r)^n, divided by ((1 + r)^n minus 1)

Here, r is the monthly interest rate, which is the yearly rate divided by 12, and n is the number of monthly payments. The part (1 + r)^n means: multiply (1 + r) by itself n times. A calculator with an exponent key will do it.

Worksheet, step by step

Step 1: Find r. At 4 percent, r = 0.04 divided by 12 = 0.0033333. At 6 percent, r = 0.06 divided by 12 = 0.005.

Step 2: Find (1 + r)^n for n = 360. At 4 percent, 1.0033333 raised to the 360th power is about 3.3135. At 6 percent, 1.005 raised to the 360th power is about 6.0226. Notice the difference already. Over thirty years, interest at 6 percent grows the loan's "weight" nearly twice as much.

Step 3: Compute the payment factor. This is r times (1 + r)^n, divided by ((1 + r)^n minus 1). At 4 percent: 0.0033333 times 3.3135 is 0.011045, divided by 2.3135 gives about 0.00477415 when you carry more decimal places. At 6 percent: 0.005 times 6.0226 is 0.030113, divided by 5.0226 gives about 0.0059955.

Step 4: Multiply by the loan. At 4 percent: $300,000 times 0.00477415 = $1,432.25 (rounded to the cent) a month. At 6 percent: $300,000 times 0.0059955 = $1,798.65 a month.

Step 5: Take the difference. $1,798.65 minus $1,432.25 = $366.40 a month, or $4,396.80 a year (366.40 times 12).

That is for principal and interest only. Hold on to that phrase, because it comes back.

The thirty-year total

Now multiply each payment by 360 months.

• At 4 percent: $1,432.25 times 360 = $515,610. Subtract the $300,000 borrowed, and the interest paid is $215,610.
• At 6 percent: $1,798.65 times 360 = $647,514. Subtract $300,000, and the interest is $347,514.

The difference in interest is $347,514 minus $215,610 = $131,904. If you carry the unrounded payments through all 360 months, the gap comes out within a couple of dollars of $131,906. The bank rounds each payment to the cent, so the rounded version is what a borrower would actually pay. Either way, the answer is the same story: two percentage points on a thirty-year loan costs a household about $132,000 in interest, on a loan of $300,000. That is 44 percent of the amount borrowed, which is $131,904 divided by $300,000. For a family, it is a college education, or a retirement account, or a decade of car payments.

The first year, where the money goes

Early payments are mostly interest. In the first month at 4 percent, the interest is $300,000 times 0.0033333 = $1,000.00, and the rest of the $1,432.25, which is $432.25, pays down the loan. At 6 percent, the first month's interest is $300,000 times 0.005 = $1,500.00, and only $298.65 pays down the loan.

Run it for twelve payments and the pattern holds. At 4 percent, the borrower pays about $11,904 in interest in year one and trims the balance to about $294,717. At 6 percent, the borrower pays about $17,900 in interest and trims the balance to about $296,316. After a year, the 6 percent borrower has paid down $3,684 of principal. The 4 percent borrower has paid down $5,283. After five years the balances are about $271,343 and $279,163. The higher the rate, the slower you build equity, and equity is how a family becomes wealthy through a house.

Flip the question: what can a payment buy?

Turn the worksheet around. Say a family can afford a payment of $1,432.25 a month. At 4 percent, that supports a $300,000 loan. At 6 percent, the same payment supports a loan of only about $238,887, because $1,432.25 divided by the 6 percent factor of 0.0059955 is about $238,887. The difference is $300,000 minus $238,887, or about $61,113 less house for the same monthly payment.

This is how a rise in rates takes a home out of reach without any change in the price. The sticker is the same, but the family's purchasing power just shrank by about a fifth. It is also how a rate rise slows the whole housing market: sellers do not want to cut prices, buyers cannot stretch, and little moves.

"Principal and interest" is not the whole bill

Most families pay a monthly sum that includes property taxes and homeowners insurance, collected by the lender in an escrow account. Add a hypothetical $3,600 a year in property tax and $1,500 a year in insurance. That is $5,100 divided by 12, which is $425 a month. The full payment becomes $1,432.25 plus $425 = $1,857.25 at 4 percent, and $1,798.65 plus $425 = $2,223.65 at 6 percent.

The gap stays $366.40, but taxes and insurance make the base bigger. In many places, property taxes are set by local governments that can raise bills with a vote the homeowner never made, and insurance premiums go up and down by region. Those are not trivial add-ons. When you hear "housing costs," remember that the interest rate is one input of several. Another input is the cost of land and building, which is the subject of zoning may be the biggest price control you never got to vote on.

"Two points" and "points" are different things

Careful with the words. A "point" in the headline of this post is a percentage point of the interest rate, 4 to 6. But when a lender talks about "discount points," a point is a fee equal to 1 percent of the loan, paid up front to buy a lower rate. On $300,000, one point is $3,000.

Here is a hypothetical to practise on, with an assumed trade that varies by lender, so treat the numbers as an illustration and not a quote. Suppose paying one point, $3,000, lowers a 6 percent rate to 5.75 percent. The monthly payment at 5.75 percent on $300,000 is about $1,750.72, which is $47.93 less than $1,798.65. The break-even is $3,000 divided by $47.93, which is about 62.6 months, or just over five years. If you will keep the loan longer than that, the point pays for itself. If you may move or refinance sooner, it does not. That is a decision about your own timeline, not about the interest rate, and the worksheet makes it visible.

Who really sets a mortgage rate?

Here is where many people go wrong. The Federal Reserve does not set your mortgage rate. It sets a target for an overnight rate between banks. A thirty-year mortgage is priced off longer-term rates, and the best benchmark is the yield on the ten-year Treasury note. Why the ten-year, when the loan runs for thirty? Because most mortgages are paid off or refinanced well before their thirtieth year, as people move, sell and refinance, so the effective life is much shorter than the stated one.

Lenders typically sell their loans, and most go into securities that investors buy. Those investors compare the return on a mortgage bond with the return on a safe Treasury. They want a premium over the Treasury yield for the risk that borrowers will repay early or default. So a mortgage rate is roughly the ten-year Treasury yield plus a spread, and the spread widens when investors are nervous or when the market is disrupted. The government's role is large. Fannie Mae and Freddie Mac, the two government-sponsored enterprises that guarantee and buy a big share of home loans, have been in federal conservatorship since September 2008, a "temporary" arrangement that is now very old. How to fix that is one of the longest-running arguments in conservative economic policy.

What does move the ten-year yield? Investors' expectations of where short-term rates will go, their expectations for inflation, and how much extra they demand to lend to a government that keeps borrowing. The Fed's policy rate feeds into that, which is why the Fed matters; see The Federal Reserve in plain English and the levers it really holds. But inflation expectations and the deficit matter too. A government that spends beyond its means, and a central bank that is seen as soft on prices, both push long-term rates up. That is the connection to Inflation is the one tax that never needs a vote in Congress and twelve questions about the national debt, answered straight. When the family budget is squeezed by rates, the cause is sometimes sitting in Washington's own books.

If you want to see how this played out at the extreme, look at How Paul Volcker broke inflation and why it hurt so much, where a mortgage payment on a $100,000 loan was more than double what it is at 6 percent.

The lock-in effect, with numbers

Here is something the worksheet explains that the cable shows rarely do. Picture a hypothetical homeowner who has a $200,000 balance on a loan at 4 percent. Her monthly principal and interest on a $200,000 loan of this size is $954.83. If she sells and takes a new $200,000 loan at 6 percent, the payment is $1,199.10. That is $244.27 more every month, or $2,931.24 a year, for the very same borrowing.

Now multiply her by millions of households. A family that would otherwise move for a better job, a bigger house or a closer school looks at that extra $2,931 a year and stays put. Fewer houses come onto the market, so prices stay firm, and buyers who must buy face both high prices and high rates. A pencil explains the frozen housing market better than most panel discussions. The solution is not to scold the owners. It is to increase supply, which comes back to land-use rules, and to keep inflation and borrowing from pushing rates up in the first place.

Fixed or adjustable, in words

A fixed-rate loan holds the rate for the whole term, so the payment of principal and interest never moves. An adjustable-rate mortgage starts with a rate that is fixed for an initial period and then resets periodically, up or down, according to a formula tied to a market index, within caps written into the contract. The initial rate is usually lower, which is the bait. The risk is that after the reset the payment can climb. The way to test it is to run the worksheet at the highest rate the contract's caps allow, not at the teaser rate. If you could not carry the payment at the cap, you are betting your home on the weather. A fixed loan is, in effect, insurance against rising rates, and like any insurance it costs something up front.

Read the Loan Estimate before you sign anything

Federal rules require a lender to give you a standardized form called the Loan Estimate within three business days of receiving your application. It lays out the interest rate, the monthly payment, the closing costs and the cash needed to close, on a page designed so you can compare offers from different lenders side by side. Use it. Take the rate and the loan amount from it, run the worksheet yourself, and see if the lender's monthly figure matches. Then compare the annual percentage rate, the APR, which folds many of the fees into a single yearly figure. A low advertised rate with high fees can cost more than a slightly higher rate with low fees, and only the arithmetic will tell you which.

Shop at more than one lender. The same borrower can be offered different rates by different institutions on the same day, and a quarter of a point on a $300,000 loan is worth the afternoon. You have just seen that it is worth about $48 a month in our example, and over a few years that is real money.

Your turn: the blank worksheet

Copy this and fill it in with your own numbers.

1. Loan amount: ______
2. Yearly interest rate: ______ percent. Divide by 12 to get r: ______
3. Number of monthly payments, n (360 for thirty years, 180 for fifteen): ______
4. Compute (1 + r)^n: ______
5. Payment factor = r x (1 + r)^n, divided by ((1 + r)^n minus 1): ______
6. Monthly payment = loan x factor: ______
7. Total paid = payment x n: ______. Interest = total minus loan: ______
8. Add monthly property tax and insurance: ______
9. Repeat with a rate one or two points higher, and subtract the first payment from the second.

For a fifteen-year loan, change n to 180 and redo it. On our $300,000 loan at 6 percent, the fifteen-year payment is about $2,531.57 a month, nearly $733 more than the thirty-year payment, but the total interest is far lower. That is the trade: a higher payment now for a house that is yours sooner.

The bottom line

• A two-point difference in the rate on a $300,000, thirty-year loan changes the payment from $1,432.25 to $1,798.65, a gap of $366.40 a month, and adds about $131,900 in total interest.
• A payment that supports a $300,000 loan at 4 percent supports only about $238,887 at 6 percent, so higher rates shrink your buying power even when prices stay the same.
• Add property taxes and insurance to every comparison. The rate is only one piece of the monthly bill.
• The Fed does not set mortgage rates. The ten-year Treasury yield, which reflects inflation expectations and government borrowing, is the better guide.
• Before you pay discount points, divide their cost by the monthly savings. If you will stay longer than that many months, they pay.