Calculating Adjustable Rate Mortgage (ARM) Payments

Adjustable rate mortgages present a little more complicate procedure for calculating mortgage payments. This complication results from two factors. First, the interest rate is allowed to adjust periodically (usually annually), and second, the initial rate is usually set so low that even if interest rates do not increase, the payments will usually increase for the first three or four years. When working with an ARM mortgage, the following terms are important:

• The initial rate on the mortgage. This is often referred to as the teaser rate since it entices people into accepting this type of a loan.
• The index rate is the interest rate indicated by the index the bank uses. Sometimes this index is the prime rate.
• The profit margin is the amount of interest that is to be added to the index rate. This profit margin is often listed in basis points. A basis point is 1/100 of a percent. Thus, if the lender was charging 250 basis points profit, the lender would add 2.5% to the index rate. In the early years of the loan, this creates a hidden rate. The hidden rate is the rate the loan will eventually increase to if there is no change in the index rate. It is the sum of the index rate and the base rate. Whenever the annual rate charged is less than the hidden rate, the interest rate charged will increase the following year.
• The annual cap is the maximum amount of increase (or decrease) in the interest rate from the previous year's rate. For example, if last year's interest rate was 10% and the annual cap was 1.5%, the maximum interest rate that could be charged is 11.5%.
• The lifetime cap is the maximum amount of increase (or decrease) in the interest rate from the initial rate on the mortgage.

Let's try it with a problem
Suppose you wish to find the initial monthly payment necessary to amortize a \$100,000 mortgage for 30 years at 6.5% nominal interest. Suppose that this loan is an ARM loan and that the 6.5% is the teaser rate, the current index is 5.5%, the profit margin is 325 basis points (3.25%), the annual cap is 1.5%, and the lifetime cap is 6%. Assuming that the index rate did not change during the loan's 30 year term, what would the payment be for years one through three?

Now let's compute the problem

1. First, you calculate the initial mortgage payment just as any other loan.

• Enter the proper payments per year. You first must type in the appropriate number (eg. monthly would be 12, quarterly would be 4, etc.), then push the orange SHIFT key , then the P/YR key .

In our problem,
• First we must enter the appropriate payments per year by pushing 12, then push the orange SHIFT key , then the P/YR key . We use 12 since the loan requires monthly payments.

• The order of the next three steps is not important, but I recommend that you follow across the financial tour of your calculator from left to right. If you do this, then the next step would be to enter in the proper number of years. You first type the appropriate number, then push the orange SHIFT key , then the xP/YR key.

In our problem, next we will enter the number of years by pushing 30, then the orange SHIFT key , then the xP/YR key.

• Now enter the appropriate interest rate per year. This is done by entering the appropriate annual interest rate as a whole number, not as a decimal (the calculator will convert it to decimal automatically), then pressing the I/YR key.

In our problem, now we will enter the interest rate per year by pushing 6.5 and then pressing the I/YR key. An interest rate of 6.5 was used since it was the initial (teaser) rate.

• Next,enter the amount of the loan (what the bank gives you) key.

In our problem, we enter the Present Value by pressing 100000, then the PV key.

• Finally all that is necessary is push the PMT key.

In our problem, we press the PMT key to compute the monthly payment.

The display then shows -632.07 if the display was set to show two decimal places. Note that the answer is negative. This indicates that the direction of the cash flow is out. (This is logical if we will be getting \$100,000 today, we must pay \$632.07 each month for the next thirty years.)

2. Since the hidden rate of 8.75% (5.5% + 3.25%) is greater than the 6.5% that was charged the first year, the interest rate will increase the next year, that payment would be calculated as follows:

• The first step is to calculate the balance on the mortgage after the previous year's payments. (This assumes that the interest rate will adjust annually. If it adjusts more or less frequently than that, simply calculate the balance after the last payment.) This is accomplished by entering the number of years that remain on the mortgage, hitting the orange SHIFT key  key, then xP/YR key, and then the PV key.

In our problem, we enter the number of years that remain on the mortgage, 29 in this case, then push the orange SHIFT key  key, the xP/YR key, and then the PV key. The display shows 98,882.27.

• Next, enter the new interest rate per year and then press the I/YR key.

In our problem,
• enter the new interest rate per year, 8 (6.5% + the annual cap of 1.5%), and then press the I/YR key.

• Finally, all that is necessary is push the PMT key. This is the payment for the second year. If the hidden rate is still higher than the current rate, then these three steps will be repeated.
In our problem,
• all that is necessary is push the PMT key. The display shows -731.68, the payment for the second year.

Since the hidden rate, 8.75% is still higher than the current rate, these three steps will be repeated to compute the payment for the third year.

Now, the first step is to calculate the balance on the mortgage after the previous year's payments. This is accomplished by entering the number of years that remain on the mortgage, now 28 years, hitting the orange SHIFT key  key, the xP/YR key, and then the PV key. The display shows 97,980.15.

Next, enter the new interest rate per year, 8.75, and then press the I/YR key. (Since 8% + the annual cap of 1.5% exceeds the hidden rate, only the hidden rate is used. Without any changes in the index rate, the loan interest rate will remain at 8.75% for the remaining term.)

Finally, all that is necessary is push the PMT key. The display shows -782.57, the payment for the third year (and remaining years if the index rate does not change). Send mail to Dr. Jerry Belloit with questions or comments about this web site.