Savings Formula RULE

The amount that is accumulated

For the loan situation, the “uniform deposit” becomes the monthly payment:

EXAMPLE 3 Buying a House

Suppose that you buy a house with a $100,000 loan to be paid off over 30 years in equal monthly installments. Suppose that the interest rate for the loan is 6.00%. How much is your monthly payment?

Imagine changing the setup slightly so that instead of making monthly payments, you are supposed to pay off the entire principal and interest at the end. Meanwhile, you make payments to a savings fund that you’re building up to pay off the loan, and the savings fund earns the same rate of interest that the loan costs. The interest rate of 6.00% on the loan is compounded monthly, so the monthly rate is 0.5%. At the end of 30 years, the principal and interest on the loan would (by the compound interest formula) amount to

On the other hand, saving each month for 30 years at 6.00% interest compounded monthly, we know from the savings formula that you will accumulate

To make just the right amount to pay off the loan exactly, we need to solve the equation

for , getting as the monthly payment.

Now, $600,000+ is not what you will be paying over the course of the loan of $100,000! It is only an intermediate amount that we come across in calculating your payment. The total of your loan payments will be (“only”) — on a loan of just $ 100,000. (Usually, the bank will round up the regular monthly payment to the next nearest cent, with the consequence that the very last payment will be slightly less than the usual monthly payment.)

It is useful to note that for a 30-year mortgage at 6%, the monthly payment is almost exactly 0.6% of the amount of the loan—here, the payment is $599.55 and 0.6% of $100,000 is $600. Hence, for lower interest rates, the monthly payment will be a lower percentage of the amount of the loan, though the relationship is not proportional.

What is the monthly payment on a 30-year mortgage for $100,000 if the interest rate is 3.5% per year?

Amortization Payment Formula RULE

A conventional loan amount

can be paid off by uniform payments at the end of each compounding period in the amount

EXAMPLE 4 Repaying Your Student Loan

The standard repayment option for federal direct student loans is repayment over 10 years with a minimum monthly payment of $50 at the end of each month, beginning six months after you graduate, and a minimum monthly payment of $50. For the $5500 student loan of Example 1 (page 910), if you didn’t pay the interest over the 51 months, what will your monthly payments be on the $6502.83 that you will owe at the start of repayment?

With the amortization payment formula, it’s easy to figure out your monthly payment. We have , monthly interest rate , and months for the payback. We find the payment as

So your monthly payment will be $66.74. (That is for this loan; you may owe still more per month for loans for your other years in college.) Hence, over the lifetime of the loan, you will pay almost . We say “almost,” because your last payment will differ by a few cents. Of the total, is interest during the 10-year payback period, plus the $ 1002.83 interest accrued during the deferment and grace periods (calculated in Example 1), for a total of $2508.80 in interest, on an original loan principal of $5500.

In fact, the amount you receive is reduced by an origination fee of 1.073% of the loan amount, or ; so altogether, you will pay almost 50% as much in interest and fees as the original principal. If you were to stretch your payments over more years (permissible in some circumstances), an even greater proportion would be interest.

You can check these amounts, and those for your own loans, at studentloans.gov/ myDirectLoan/mobile/repayment/repaymentEstimator.action and (with more options and details) at www.finaid.org/calculators/loandiscountanalyzer.phtml.

Suppose that you graduate with the average amount of student loans to repay, $25,000. What would your monthly payment be if the interest rate on all the loans is 4.29% and the repayment term is the standard 10 years?

EXAMPLE 5 Buying a Car

You decide to buy a new Wheelmobile car. After a down payment, you need to finance (borrow) an additional $12,000. After you compare interest rates offered by the car dealership, local banks, and your credit union, the lowest monthly payment you can find is for an 84-month loan at 5.9% from the credit union, compounded monthly. What is your monthly payment?

We have , monthly interest rate , and . Using the amortization payment formula, we have

How much interest do you pay? You make payments totaling , so the interest is .

If you had bought a Plushmobile instead, with $24,000 to finance, you would have borrowed twice as much, and your monthly payment would have been twice as much.

What would be your monthly payment on the $12,000 loan, and the total interest paid over the term of the loan, if you got a 60-month loan instead, at the advertised rate of 2.99%?

EXAMPLE 6 How Much Car Can You Afford?

Suppose that you feel comfortable with a monthly car payment of $200, you don’t want to pay on the car for more than four years, and you can get financing at 1.99%. How large a loan can you get?

We have not taken into account the costs of insurance, registration, parking, gasoline, or tolls.

Suppose that you can afford $250 per month for the car payment (apart from insurance, gas, etc.), you can get a 0% car loan, and you are willing to pay over 7 years. How much car can you afford? (Careful! The amortization payment formula does not apply. Why not?)

EXAMPLE 7 A 30-Year Mortgage on a Median-Priced Home

Suppose that you are a family with the U.S. median household income of about $53,000 and you want to buy a median-priced home for $205,000, with a 30-year fixed-rate, federally insured mortgage at 3.25% (the data are for the beginning of 2015). Recall that the median (discussed in Section 5.4, pages 196-200) means that half are below and half are above. Suppose that you can make a down payment of only $7000 (just above the minimum of 3.5% required), plus pay closing costs of about $4000. Can you afford such a home?

Mortgage lenders have “affordability” guidelines that suggest that a family cannot afford to spend more than 31% of its monthly income on housing. Thus, by these guidelines, you can afford per month.

What is the monthly payment on the loan? The principal is , the monthly interest rate is , and months. The amortization payment formula gives a monthly payment of

919

Well, that sounds good. Unfortunately, there is more to the mortgage than just the amount needed to amortize the loan. Your payment will also have to cover real estate taxes, mortgage insurance, and homeowner’s insurance on the property. On a $198,000 home, these may add $500 or more to the monthly payment, which will then total about $1362.

It’s really close! So, the median household may just barely be able to afford the median-priced home (depending on the area of the country), even while interest rates I remain at record lows.

Equity DEFINITION

Equity is the amount of principal of a loan that has been repaid.

EXAMPLE 8 Home Equity

My wife’s parents sold their house in rural Minnesota to move to the town where we live. They had bought their house in 1980 with a 30-year mortgage for $100,000 at an 8% interest rate. After 22 years, how much equity did they have in the house—that is, how much of the principal had been repaid? And how much did they still owe on the house?

What may shock you (and disappointed them) is that when they sold their house in May 2002—after 269 months of payments, almost exactly three-quarters of the 30 years of the mortgage—they had only $50,000 in equity (hence, they still owed $50,000 on the house) but had already paid $147,000 in interest. Three-quarters of their payments had gone to interest.

We can use the amortization payment formula to determine just how much equity they had after 269 months of payments, but first we need to Í determine their monthly payment. We see , months, and monthly interest, getting .

Now we use the formula again, this time “in reverse.” Knowing and , we find out how much of the loan would have been paid off by the remaining payments of $733.76:

This is how much my parents-in-law had yet to pay, so their equity was . (The above formula would not apply if they had made larger or additional payments.)

Is it true for any mortgage that after three-quarters of the term of the loan, only half of the loan will be paid off?

We bought our house in 1992. We were offered a choice between a 30-year fixed-rate mortgage at 8.375% and a 30-year ARM at 6.875% whose rate could be raised (or lowered) by up to 2% every year. When we asked, we were also quoted slightly lower rates for corresponding 15-year mortgages.

We were planning to stay in the house much longer than the median of 5 years, and we were concerned that inflation might force the ARM considerably higher. Also, we did not want the obligation of the higher payments of a 15-year mortgage, in case our circumstances changed (such as through job loss or death). Some loans provide for penalties for paying off the loan early, but in our case (thanks to state law), there was no penalty for making extra payments (if we could afford them).

We chose the 8.375% fixed-rate 30-year mortgage (and made some extra payments). People in other circumstances, or with a different tolerance for risk, would no doubt have decided otherwise. Had we been sure then that interest rates would not go higher in the 1990s, we would have gone for the ARM. But hindsight is always better than foresight. Homeowners with mortgage interest rates such as ours later refinanced at much lower prevailing rates, near 5% for a fixed-rate 30-year mortgage.

Currently, about one-third of borrowers take ARMs rather than fixed-rate mortgages. Newer mortgage “products” include interest-only mortgages and shared appreciation mortgages (SAMs). With an interest-only ARM, payments are (just slightly) lower than for a conventional 30-year mortgage, but you accumulate no equity (at least, not by paying off the loan; the market value of the house may rise). After five to seven years, you start also paying off the principal—which means that your payments go up then. In some such loans, the interest rate—and your pay- ments—fluctuate as frequently as every month.

In a SAM, interest payments are lower or nonexistent, but the lender receives a portion of any appreciation (rise in value) when the house is sold. In a nationally reported instance in 2003, a single mother received a no-interest SAM loan to finance the $30,000 down payment on a $223,000 house in Pleasanton, California, through the city’s affordable housing program. Four years later, she sold the house for $385,000, and the “affordable housing” lenders got 60% of the $162,000 appreciation, or $97,000. She herself realized $65,000 (minus the cost of the sale) but complained bitterly, saying that she would have been better off to have put the loan on her credit card! Critics have termed SAMs an urban form of sharecropping.

Late 2007 saw the development and widening consequences of what has become known as the “mortgage crisis.” To understand what that was, why it took place, and how it will have widespread effects for some time, you need to know what happens when you get a mortgage to buy a house compared with what used to happen.

In the “good” old days, you would go to a local bank (or savings and loan, or credit union). If you proved “creditworthy”—meaning that after careful consideration, the personnel felt that you could repay the loan—the bank would lend you its money, raised from its depositors. The interest rate depended on your credit rating and down payment. You paid back the loan at a fixed rate of interest, usually over 30 years. If interest rates went down, you could refinance the loan at a lower rate by taking out a new loan to pay off the old one; if rates went up, your payments would not rise.

Meanwhile, the value of your house usually went up 5% to 10% per year, your income went up, and your payments stayed the same. (In fact, if you have only 10% equity in your house and it goes up in value 10% in one year, you have made 100% on your investment! This kind of “leveraging” can make real estate investments very profitable—so long as prices keep rising.)

What changed? Efforts to extend home ownership to a wider proportion of the population resulted in banks making more “subprime” loans (loans to people with poorer credit histories who are less likely or able to repay them), with lower down payments but higher rates of interest (because of the greater risk). Some of those loans were “predatory"—lending at high rates of interest to people who could not possibly make the payments.

Certainly, real estate speculation played a role, as did greed. House prices doubled from 2000 to 2006 (the “housing bubble”), but median income adjusted for inflation remained stagnant—and then fell with increasing unemployment, making it increasingly hard to afford houses. Banks countered with ARMs and interest-only mortgages. They also maximized their income with upfront charges (“origination fee” plus “points” paid by the buyer to lower the interest rate); and at the same time, they minimized their risk by immediately “flipping” the mortgages (selling them to bigger banks or other investors, to whom buyers would then make their payments). All these factors led to banks making more loans that were riskier.

What went wrong? Interest rates rose, and people with ARMs saw their payments rise beyond their ability to pay. At the same time, the “pyramid” of housing prices could not continue with the higher mortgage rates, so housing prices fell. When your house becomes “underwater"—worth less than the balance remaining on the mortgage—you might be better off just walking away from it (especially because you build up almost no equity in the first few years of the mortgage). Mortgage defaults lowered the value of investments in bundles of mortgages.

Houses are worth perhaps hundreds of billions of dollars less than a few years ago, and big investment banks still have mortgages on their hands that are worth hundreds of billions of dollars less than they paid for them. That means that for some time to come, banks have less money to lend and can (indeed, must) demand more creditworthy clients and higher rates of interest.

How does all this affect you? Financial institutions have less money to lend for any purpose—buying a home, buying a car, starting a business, etc.—hence the “credit crunch” following the mortgage crisis. We can only hope that the worst will be over before you are in the market for a house, car, or business loan.

Where you won’t see a direct effect is in the Consumer Price Index (CPI) (discussed in Section 19.5, on pages 785-786, and Section 21.6, on pages 891-893), which is based on the rental value of houses, not their prices. If the doubling of housing prices in 2000-2006 had been taken into account, the CPI would have risen by 5% per year rather than 3%.