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Calculating the Cost of a Mortgage Loan

by Arnold Kling
March 6, 1995

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:

  1. discount rate
  2. time horizon
  3. interest rate scenario

Selecting a discount rate

The discount rate (or present value calculation) is a basic concept in financial analysis. If you are not familiar with the discount rate, then this article will be difficult to digest right away.

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

What is the cost of this mortgage?

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)

If the payments are discounted at 10 percent, then the total discounted value of the payments matches the loan proceeds, and the net cost is zero. If the payments are discounted at 20 percent, then the discounted value of the payments is less than the receipts from the loan, for a gain (negative cost) of $11,971. If I could invest money at 20 percent per year, then I can make a profit by borrowing at 10 percent per year. More typically, I cannot invest at such a high rate, so that I will use a discount rate that is at or below the rate on the mortgage-- but definitely higher than zero!

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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