An adjustable rate mortgage (ARM) is a mortgages in which the interest rate is typically fixed for a few initial years but varies based on certain index such as the LIBOR, federal funds rate, etc. during the rest of the mortgage term. A borrower’s monthly repayment obligations increases when the market interest rates are high and vice versa.

A mortgage is an amortizing loan i.e. a loan which requires the borrower to make equal periodic payments which comprise of both interest payment and a principal payment. The mortgage principal balance reduces with each payment. Outstanding balance on a mortgage equals the present value of the remaining periodic payments discounted at the mortgage interest rate.

In a fixed-rate mortgage, the interest rate is fixed for the mortgage term. The monthly payments are also fixed and hence more manageable. Further, the borrower is sure that the outstanding balance on the mortgage will fall with each periodic payment. However, this is not the case in the adjustable-rate mortgage. While the periodic payments maybe fixed for say initial two years on an ARM, they may rise or fall in future depending on the market interest rates. The periodic interest payments and the outstanding balance of the mortgage increases (decreases) with increase (decrease) in the market interest rates.

An adjustable-rate mortgage contract specifies many important terms such as the adjustment frequency, rate ceiling, rate adjustment cap, etc.

## Example

You bought a house for $600,000 on 1 January 20X5 paying 10% of your own savings and financing the rest with a 15-year mortgage 5/1-ARM that required interest at 3.5% per annum compounded and paid quarterly. Afterwards, the rate is adjusted quarterly to a benchmark rate plus 2.5%. 5/1-ARM means that the initial fixed interest rate will prevail in the initial five years. Afterwards, it will be adjusted one a year. The calculations below are based on the formula for present value of an ordinary annuity. The quarterly payments that you must make in the initial five years amount to$11,607 as worked out below:

$$\text{PMT}=\frac{\text{\600,000}\times(\text{1}-\text{10%})}{\frac{\text{1}-{(\text{1}+\frac{\text{3.5%}}{\text{4}})}^{-\text{15}\times\text{4}}}{\frac{\text{3.5%}}{\text{4}}}}=\text{\11,607}$$

You find that the quarterly payment is quite affordable. After the initial five years, the outstanding balance on your mortgage equals the present value of your quarterly mortgage payments for the next 10 years:

$$\text{PV}=\text{\11,607}\times\frac{\text{1}-{(\text{1}+\frac{\text{3.5%}}{\text{4}})}^{-\text{10}\times\text{4}}}{\frac{\text{3.5%}}{\text{4}}}=\text{\390,301}$$

By the end of the fifth year, the central bank has increased the interest rates drastically. The benchmark rate hovers around 6%. Your bank adjusts the mortgage rate to to 7% (i.e. index rate of 6% plus the spread of 1%). At the new rate, your quarterly payment works out to $13,450: $$\text{PMT}=\frac{\text{\390,301}}{\frac{\text{1}-{(\text{1}+\frac{\text{7%}}{\text{4}})}^{-\text{10}\times\text{4}}}{\frac{\text{7%}}{\text{4}}}}=\text{\13,650}$$ Your quarterly interest payment has increased by 15% which has stretched your finances. The higher the increase in market interest rates, the more pronounced will be the payment shock. If the increase in rates is coupled with a drop in real estate prices, you mightn’t be able to refinance and may have to face foreclosure. In a parallel universe, however, the central banker believes in quantitative easing. The benchmark rate has dropped to just 1% and the rate on your mortgage is just 2%. Your monthly payment will be just$10,710:

$$\text{PMT}=\frac{\text{\390,301}}{\frac{\text{1}-{(\text{1}+\frac{\text{2%}}{\text{4}})}^{-\text{10}\times\text{4}}}{\frac{\text{2%}}{\text{4}}}}=\text{\10,710}$$

An adjustable rate mortgage is a double-edged sword. It is sweet when interest rates fall and lethal when they rise. However, the initial lower fixed interest rates might cause you to overestimate your periodic payment appetite.