One of the most common uses of time value formulas
is the calculation of constant payments on a loan that has a fixed initial
balance, fixed interest rate and fixed time of repayment. The fixed payment
over time has a principal component and an interest component. The ** principal
**component of the loan payment is that portion that reduces the loan
balance. As payments are made the principal component increases and the
interest component decreases.

The need for the specific fixed payment formula arises in figuring out mortgage and car loan payments. These loans have the elements needed to apply the formula: a fixed loan balance, a fixed interest rate and a fixed repayment period. Here is the fixed payment formula:

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One point to emphasize is that most installment loans compute interest in arrears. The above formula assumes this. What does this mean? If you borrow a lump sum and you are going to make interest and principal payments monthly, you do not make the first payment until after you have used the funds for the first month.

Example. You close on a
house purchase on the 1^{st} of October, using the proceeds of a
mortgage loan. Your first monthly payment will be due on November 1^{st}.
This makes sense because only after having use of the funds for the month
should you owe interest on those funds.

There is something else interesting going on with this formula. The “n” in most cases will be one month. But we all know that not all months in a year do not have the same number of days. Also, not all years have the same number of days. You might then expect that you would pay more interest expense in months that have more days than in months that have fewer days. Likewise, in leap years you would expect to pay one more day’s interest than in non-leap years. But these differences are not reflected in the formula which assumes that all months and years are of the same duration.

Using the calendar days in the month as a guide, fully seven of the months have 31 days, leading to what could be considered an under collection of interest using the standard fixed payment formula, four are neutral with 30 days, and only one with 28 or 29 days could lead to an over collection of interest. This problem has not escaped the attention of commercial lenders. To create an equalization of interest payments that adjusts for the idiosyncrasies of the calendar, commercial lenders will adjust nominal annual interest rates by multiplying this rate by the ratio of 365/360 and applying this adjusted rate in its computation of payments using the standard formula. This adjustment is generally made only for commercial loans, not for home mortgages.

Now the idea of an amortizable loan, one in which
the same fixed predictable payment is due every month, with a payment that
brings the loan balance to zero over the fixed term of the loan is one of the most
unsung economically revolutionary ideas ever developed. I do not believe it is
an exaggeration to claim that without the introduction of amortizable home
mortgages the United States would not have developed a large, relatively
comfortable middle class of homeowners.
Interestingly, the 30-year fixed interest mortgage was first introduced
on a widespread basis as part of the New Deal programs in the 1930s.

This introduction of the 30-year fixed interest mortgage loan combined with the use of mortgage insurance has greatly increased the percentage of households that own their own home. Think of it. How would it be possible for ordinary wage earners to buy and own a home if they had to pay cash for it? Only the wealthiest could afford to buy homes under these circumstances. And without amortization of the principal of these loans the middle-class’ largest asset, its equity in their homes, could not have been accumulated

Now in today’s mortgage markets borrowers have a wide variety of choices. They can choose 15 or 30-year fixed interest rate mortgages, which means that the balance is paid off in 15 or 30 years at a fixed interest rate. The short term and long-term impacts of each mortgage option are significant. Look at this example.

Assume you seek to purchase a $300,000 home and you can put 20% down, $60,000. This means that you will need a $240,000 mortgage. Assume you can obtain a 2.5% fixed interest, 30-year amortization mortgage or a 2.1% fixed interest 15-year amortization mortgage. This table summarizes the key differences in these mortgage options.

As you can see there is a significant difference in the monthly payments -- the 30-year loan calls for over $600 less monthly payment than the 15-year loan. But what you save in monthly payments costs you quite a bit more in the long run. If you keep your home for thirty years, you will pay a total of $341,348 in principal and interest under the 30-year mortgage option versus $279,989 in total payments under the 15-year mortgage option. This means that you would pay $101,384 in interest under the 30-year option and only $39,989 in interest under the 15-year option.

Now because there are no pre-payment penalties you could pay a little bit extra in principal every month under the 30-year option and see a pretty sizable impact on the total payments made and the interest expense incurred. The table above shows the impact of paying an extra $100 principal every month on a 30-year mortgage. As you can see the extra payment reduces the total payments, total interest expense and payback period.

In the above example it was assumed that the borrower was going to stay in the home for another thirty years. This is an unlikely prospect. Most U.S. homeowners will keep their homes for a far shorter period. The overall median ownership period in the U.S. is 13 years. But this length of ownership varies considerably by region. Rust belt and rural home ownership lengths tend to be much longer than sunbelt or urban and suburban ownership lengths.

In addition to fixed interest mortgage rates, lenders also offer adjustable-rate mortgages. These tend to offer a fixed interest rate for the first term of the loan and then allow interest rate adjustments after the first fixed term. The first fixed terms tend to be offered in increments of 3, 5 or 7 years. Usually, the amortization[b1] period is 30 years. Because there is less interest rate risk to lenders the initial term interest rates offered are usually less than fixed interest rate 15 or 30-year mortgages. The difference in interest rates between these adjustable interest loans and fixed interest loans has become less dramatic in recent years due to the overall steep decline in all forms of mortgage interest rates.

Now you may assume that if you have a choice between paying cash for a home or getting a mortgage that it is always a better idea to use cash. Similarly, you might think it is always a better idea to pay down your mortgage as soon as soon as possible to reduce the amount of interest expense you will incur. Certainly, from a psychological point of view it would seem better to have less debt rather than more debt. But will it always make economic sense to prefer less mortgage to more mortgage?

Perhaps surprisingly the answer is no. If the cash you would use as an alternative to having a mortgage or paying it down more quickly can be invested at a higher yield than the interest rate on the mortgage, then it makes sense to have and keep the mortgage.

Example. Suppose you have a 3% interest rate mortgage. Suppose further that you have enough income and assets to increase the amount of principal payment you make each month on the mortgage. Does it make economic sense to increase your payment? The answer is no if the rate of return on the cash you plan to use to make the extra principal payments is greater than 3%. Suppose that your surplus cash yields a return of say 6%. Then not only does it not make sense to retire the 3% mortgage loan early, it would also make sense to increase the 3% loan as much as possible as long as you could generate the 6% return on the new borrowings. Utilizing the spread between the cost of borrowing and the investment return from the use of the borrowed funds is called arbitrage.

Copyright 2018 Michael Sack Elmaleh