Math for Grownups (9 page)

P
= principal (the amount of the loan)

M
= monthly payment

r
= monthly interest rate

n
= number of months in the loan

Even though you’re only working with a couple of variables here, you may get tripped up in a couple of places if you’re not paying attention. First,
r
is the
monthly
interest rate, and Naima was quoted a
yearly
rate. Therefore, she needs to divide 4.25% by 12 to get
r
.

r
= yearly mortgage rate / 12

r
= 4.25% / 12

r
= 0.0425 / 12 (Convert the percent to a decimal.)

r
= 0.0035

The number of months in the loan, or
n
, is another potential pitfall. Naima has a 30-year mortgage, but she needs to come up with the number of months:

n
= number of years • 12

n
= 30 • 12

n
= 360

Here’s what Naima knows:

M
= $1,250

r
= 0.0035

n
= 360

It’s an ugly formula, for sure, but Naima buckles down and gives it a shot. She plugs in her variables first:

Remembering the order of operations (PEMDAS, or Please Excuse My Dear Aunt Sally), Naima knows that she must deal with what’s in parentheses first. So, she gives her full attention to the fraction part of the formula. Looking closer, she adds within the smaller parentheses.

Now she looks for exponents. Only a math prodigy can find the 360th power of a number without a calculator, but last year Naima gave her scientific calculator to her younger brother, who is now taking Algebra II. She grabs her laptop and finds a free online calculator by searching for “free scientific calculator.” A couple of clicks later, she has more information for her formula:

Now the calculations are pretty simple. She subtracts in the numerator (that’s the top number) and then multiplies in the denominator (that’s the bottom number).

Naima can finish the last two operations quickly. She divides the numbers in the parentheses and then multiplies.

P
= 1,250 • 204.6911

P
= $255,863.87

The loan formula shows that Naima can afford the monthly payments on a house priced at $255,863.87. This is based on what she can spend each month on a mortgage payment, as well as the interest rate her lender can offer. But is her answer right? Well, not quite.

Remember that Naima has a down payment of $17,450. Using the loan formula, she found out that she can afford a house priced at $255,863.87, but that doesn’t take her down payment into consideration. Actually, she can afford to buy a house priced at $255,863.87 + $17,450, or $273,313.87.

Quiet living, here she comes.

You may be looking for the simple life, but you’re not going to find it in your monthly mortgage payment. That’s because the monthly check that you write to your mortgage company includes more than just a payment toward principal and interest.

A monthly mortgage payment usually has four parts. These parts are called PITI, for Principal, Interest, Taxes, Insurance.

• The
principal
is the amount that you borrowed. In the early years of the mortgage, you’ll pay only a little bit of the principal each month, and most of your payment will go toward interest. That’s because the bank wants its cut as soon as possible. As the mortgage ages, the principal portion of your monthly payment grows. (But don’t worry; if you have a fixed-rate mortgage, your monthly payment will stay the same.)

• And then there’s a little something called
interest
. In return for the loan, you agree to pay the lender (probably a bank or mortgage company) a percent of the price of the house each month. This interest is compounded—which means the interest earns interest.
     (Note that the formula Naima used includes only principal and interest. The taxes and insurance have to be figured in later, because they vary so widely from place to place.)

The word
mortgage
was probably coined in the fourteenth century. It’s an Old French word that literally means “death pledge.”

But that doesn’t mean that a mortgage will kill you.
Au contraire
. With a mortgage, the debt dies when it’s paid. In other words, you can get out of a mortgage in one of two ways—by paying it off or by refinancing with another loan.

The beauty of renting is that someone else takes care of maintenance and repairs. So who’s responsible for those costs when you buy a house? That would be you.

The water heater breaks? Either you replace it or you take cold showers. A giant icicle brings down the gutters along the entire east side of your roof? You’ll be paying the gutter guys. A family of birds takes up residence in your bathroom fan? You guessed it. You’re in charge of evicting your feathered friends and installing a new fan.

Real estate experts estimate that these costs will average 1% to 2% of the value of the home each year. So if the home you’re planning to buy costs $189,000, you can expect to pay between $1,890 and $3,780 for maintenance and repairs each year.

Unless, of course, you bought that fixer-upper. In that case, you’re probably on the hook for a whole lot more.

• Property or real estate
taxes
also make your monthly payment add up. State and local governments charge these taxes to pay for everything from new roads to schools. The tax bill is due on an annual or semiannual basis, but your lender may offer to pay these for you each year, which enables you to spread this cost month to month—and adds the cost to your monthly mortgage payment.

• Finally, there’s
insurance
. Your homeowner’s insurance can be rolled into the monthly mortgage payment. Your lender will almost certainly require you to carry homeowner’s insurance in order to qualify for the loan. And if you didn’t put down at least 20%, you’ll also pay private mortgage insurance (PMI).

Okay, houses are expensive. That’s not difficult to understand. But you might be shocked by how much you pay for a home over the life of the mortgage.

That’s because the interest on a mortgage is
compounded
. In other words, you pay interest on the interest.

Here’s how it works, using some very simple numbers. Let’s say you’ve borrowed $1,000 at 10% interest that is compounded monthly. And let’s also say that your lender has agreed that you can pay $150 each month to pay off that loan. In the first month of the loan, you will owe $1,000 plus 10% interest, or $100. So, the total you will owe in that month is $1,100.

But that’s not what your lender expects you to pay that month. Remember, your monthly payment is $150. You make that payment, which means you still owe $950 (because $1,100 - $150 = $950).

The next month rolls around, and you need to make a payment. But do you owe $950? Nope. That’s because interest has been calculated again. Ten percent of $950 is $95, so your total debt has risen again: $950 + $95 = $1,045. After making your $150 payment, you owe $895 (because $1,045 - $150 = $895).

And so it goes every single month. The lender adds interest, and you make a payment. But you are always paying interest on the interest.

Drew bought a house almost 30 years ago. In fact, he’s about to make the very last payment of $608—the same check he’s been writing for the last 359 months. His original mortgage was $120,358. Just for kicks, he’d like to know how much he paid in interest over the life of the loan.

Each month, $20 of the payment goes to cover taxes, so Drew’s mortgage lender has received $588 per month for 30 years. (Drew decided not to include his home insurance in his monthly mortgage payment, so the insurance part of PITI is not a factor.) What was the total that Drew paid to his lender? He can use the total mortgage formula to find out:

T
=
Mn

T
is the total of the loan payments

M
is the monthly payments

n
is the total number of payments over the life of the mortgage

Remember, there are 12 months in a year, and Drew made a mortgage payment each and every month for 30 years. So, he made a total of 360 payments.

T
= $588 • 360

T
= $211,680

Because Drew originally borrowed $120,358, how much interest has he paid over the life of the mortgage?

$211,680
-
$120,358
=
$91,322

Would someone please get Drew a chair? He looks a little woozy.

If Drew had taken out a 15-year loan, he’d have paid it off long, long ago. And he wouldn’t have paid as much interest. The shorter the term of the loan, the less interest is paid on interest.

You know that the longer the mortgage, the smaller the monthly payment. But the length of the mortgage also affects the total amount that you’ll pay.

Here’s how it works mathematically.

Let’s say that Drew bought his house using a 15-year mortgage with a 4.5% fixed-rate. And remember, he bought the house for $120,358.

Using a trustworthy online calculator, Drew finds that his monthly payment on a 15-year mortgage would have been $920.73. This payment is higher than the payment he owed with his 30-year mortgage, but that’s because he wouldn’t be taking so long to pay it off.