Understanding the Mathematics of Personal Finance

Not all mortgage loans commit to a certain rate of interest for the life of the loan. The ones that don’t commit to this are called, reasonably enough, ARMs.

An ARM will typically offer an attractive low rate for a certain period of time (e. g., 5 or 10 years) after which time the rate can change. Just how much it can change, how often it can change, and what factors will be used to calculate a change will be specified in the contract. Also, there are laws that limit just how aggressively the lender can change the interest rate. Typically, the new mortgage rates will be tied to various federal government interest rates.

An ARM can be a very attractive way to get into a new home with modest monthly payments. The initial ARM rates are typically lower than fixed interest mortgage loan rates because the lender does not have to worry as much about how interest rates might vary in the future and therefore does not have to conservatively “cover his or her bets.”

On the other hand, if you are just managing to make your mortgage loan pay­ments and then your interest rate jumps, say, 1% on a \$300,000 loan balance, you suddenly could be looking at approximately \$250 per month higher payments. Add this to the fact that your real estate taxes have almost definitely gone up every year and you might find yourself unable to make your payments.

The ARM tab in my spreadsheet lets you examine various changing rate sce­narios. Comparing the ARM tab to the Basic tab, you’ll see that the rate entry in the Basic tab has been replaced by a Rate column in the ARM tab. If you enter the initial Rate at the top of this column (cell H2), this rate is immediately propagated down the column. In Table 4.3, I repeated the previous example (with the original payment schedule restored). Five years into the loan (at payment number 60), the mortgage rate changes to 7% and then at payment number 72 it changes to 8%. These numbers are put in by first entering 7% in the Rate column at payment number 60 and then entering 8% in the Rate column at payment number 72. The new rates propagate

Table 4.3 ARM Amortization Table

 Pmt Nr Month Year Rate (%) Balance (\$) Payment (\$) Interest (\$) Tot Int/ Year(\$) 57 4 2014 6.00 171,533.42 1,432.86 860.53 3,459.14 58 5 2014 6.00 170,958.23 1,432.86 857.67 4,316.81 59 6 2014 6.00 170,380.16 1,432.86 854.79 5,171.60 60 7 2014 7.00 169,799.20 1,432.86 851.90 6,023.50 61 8 2014 7.00 169,263.49 1,526.20 990.50 7,014.00 62 9 2014 7.00 168,724.66 1,526.20 987.37 8,001.37 63 10 2014 7.00 168,182.68 1,526.20 984.23 8,985.60 64 11 2014 7.00 167,637.54 1,526.20 981.07 9,966.66 65 12 2014 7.00 167,089.23 1,526.20 977.89 10,944.55 66 1 2015 7.00 166,537.71 1,526.20 974.69 974.69 67 2 2015 7.00 165,982.98 1,526.20 971.47 1,946.16 68 3 2015 7.00 165,425.01 1,526.20 968.23 2,914.39 69 4 2015 7.00 164,863.78 1,526.20 964.98 3,879.37 70 5 2015 7.00 164,299.28 1,526.20 961.71 4,841.08 71 6 2015 7.00 163,731.49 1,526.20 958.41 5,799.49 72 7 2015 8.00 163,160.39 1,526.20 955.10 6,754.59 73 8 2015 8.00 162,630.69 1,617.44 1,087.74 7,842.32 74 9 2015 8.00 162,097.45 1,617.44 1,084.20 8,926.53

down the Rate column automatically. In Table 4.3, or directly on the spreadsheet, you see that the original \$1,432.86 monthly payment jumps to \$1,526.20 at the first - rate jump and then to \$1,617.44 at the second-rate jump.

Some ARM mortgages have the annoying attribute that any payment larger than the original monthly payment might trigger a change in the APR. In other words, the original initial guaranteed fixed rate period of the loan ends immediately.

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