1997 update: The ideas in this article are embodied in The Intelligent Mortgage Agent
You would like to know whether to take that three-year adjustable-rate mortgage or go with a 30-year fixed rate. You read books and newspaper columns that talk about "pros and cons," but we're not talking about stripes vs. polka-dots here. Shouldn't there be a precise answer?
In fact, there is a mathematical approach that will allow you to choose the optimal mortgage. This article will explain the approach, which depends on three factors:
For those of you who already are familiar with the discount rate, this section will set the stage for the rest of the article.
Let us start with a simple example. Suppose that I borrow (receive) $100,000 today, at a 10 percent interest rate. The terms of the mortgage are that I pay it back in two annual payments of $57,619.05 each. This can be summarized in a table as follows:
time period receipts payments the present $100,000 0 one year from now 0 $57,619 two years from now 0 $57,619
Many people would answer $15,238, which is the difference between the payments and the receipts. Unfortunately, that answer is equivalent to thinking that a nickel is worth more than a dime because it is bigger. The payments in years one and two have to be discounted back to the present.
The table below shows the payments discounted at alternative rates.
time period receipts payments discounted discounted payments(10%) payments (20%) 0 $100,000 0 0 0 1 0 $57,619 $52,381 $48,016 2 $57,619 $47,619 $40,013 Total $100,000 $115,238 $100,000 $88,029 Net Cost -- $15,238 0 ($11,971)
Incidentally, the Annual Percentage Rate calculation is designed to solve for the discount rate that makes the net present value of the loan equal to zero (in our example, 10 percent is the APR). I do not think it is as valuable a tool for comparing mortgages as the approach that I am developing here.
I recommend that you select a fixed discount rate with which to evaluate mortgages. You might choose a number like 7 percent, which is close to but not higher than the rate on most mortgages. Alternatively, you might choose the rate currently quoted on zero-point thirty-year fixed-rate mortgages.
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