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Calculating the Cost of a Mortgage Loan

Continued

by Arnold Kling

Selecting a time horizon

Now, suppose that we have a choice between our ten percent mortgage with no points and a mortgage that charges 2 points with a rate of 8.5 percent. The table of payments for the mortgage with points is:
time period      receipts        payments 
  0             $100,000          $2,000
  1                 0            $56,462
  2                 0            $56,461

If we discount the payments at 10 percent, then the total discounted value of the payments is $2,000 + $51,329 + $46662 = $99,991, which makes this a slightly less expensive mortgage than the 10 percent mortgage.

(Using this method, loan origination fees, such as the cost of an appraisal and a credit report, can be treated exactly like discount points. In contrast, the APR ignores these fees as long as they are "customary." So two lenders who charge different fees can quote identical APR's.)

Next, however, suppose that we shorten the time horizon so that we pay back the loan after one year. We make a payment of $108,500 at that time, and the total discounted value of payments is $2,000 + $98636 = $100,636. At the shorter time horizon, the 8.5 percent loan with two points up front is more expensive than the 10 percent loan with no points.

Be careful interpreting this example, because the magnitudes are distorted by the simplicity of the two-year mortgage term. However, the basic lesson is that the time horizon affects the relative cost of mortgages when the discount rate is applied. The example correctly illustrates that a low-rate, high-point mortgage can be better at a longer time horizon but worse at a shorter time horizon.

The time horizon is the length of time that you expect to retain the mortgage. Choosing a ten-year time horizon does not mean that you will rule out a 30-year fixed rate mortgage, but it keeps you from over-estimating the advantage of the old stand-by.

The possiblity that you might move within ten years is not the only reason for choosing a shorter time horizon. You may expect your financial situation to change dramatically within ten years. If your income rises substantially, you may be able to pay off a mortgage sooner than 30 years; if you face college tuition expenses in five years, you may need to refinance. These are considerations that would lead to choosing a time horizon of 10 years or less.

Interest rate scenario

In order to compare a fixed-rate mortgage with an adjustable-rate mortgage, you need to select an interest rate scenario. If you are indifferent to the risks of an adjustable-rate mortgage(ARM), then you might assume that interest rates remain constant. If you are extremely cautious about the risk of an ARM, then you might select a scenario in which rates rise by 3 percentage points and remain at those levels. If you are moderately cautious, you might select a scenario in which rates rise by 1 percentage point and then remain there.

It is important to remember that the interest rate on your ARM may rise by a different amount than the general interest rate increase in the scenario. Many ARMs start with low "teaser" rates, so that even in the scenario where market interest rates do not change your rate is likely to go up. Conversely, in the scenario where rates rise by 3 percent, your ARM will not necessarily go up by 3 percentage points when it first adjusts. Many ARMs have adjustment caps of 2 percent, so that your rate will adjust upward in stages under the high-rate scenario.

In our example, suppose that we have an ARM that is linked to the one-year rate. The hypothetical current value of the one-year rate is 7.8 percent, with a margin of 2.5 percent. That means that the rate on the ARM would be calculated as 10.3 percent if it were fully adjusted today. However, the ARM has an initial teaser rate of 9.5 percent, with no points. Moreover, it has a cap that says that the rate cannot go up by more than 2 percentage points.

Here is what the rate will be on our ARM next year under three scenarios:


scenario             index   margin    cap    rate 

constant rates         7.8    2.5      11.5   10.3
rates up 1 percent     8.8    2.5      11.5   11.3
rates up 3 percent    10.8    2.5      11.5   11.5*

*the rate would be 13.3, but it is limited by the cap

Payments under the three scenarios would be

time period     rates constant   rates up 1   rates up 3    
 
  1               $57,231          $57,231      $57,231
  2               $57,653          $58,175      $58,280

Discounting these payments at a 10 percent rate gives a total of $99,675 in the constant-rate scenario, $100,107 in the scenario with rates up 1 percent, and $100,193 in the scenario where rates go up by 3 percent. The ARM is a winner when rates stay constant but it loses to the 10 percent fixed rate otherwise.

Summary

You pick a discount rate, time horizon, and an interest rate scenario. Then, you can project the payments on a mortgage, including up-front points and fees. (See the mortgage analysis worksheet for instructions on creating a spreadsheet to project the payments on a mortgage.) Next, you can calculate the exact cost and choose the optimal mortgage. You can draw conclusions of the form "the best available mortgage today for a time horizon of n years, a scenario where interest rates go up by x percent, and a discount rate of z percent, is program X." In almost all cases, your answer will not depend on your choice of discount rate, as long as the rate is in the ballpark of the mortgage rates that are under consideration. The time horizon and interest rate scenario will prove to be more significant.

If you've gotten this far, then you probably will enjoy working with The Intelligent Mortgage Agent.


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