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Cost Calculation of a Mortgage Loan

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You would like to know whether to take a 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 which allows 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 you borrow (receive) $100,000 today, at a 10 % interest rate. The terms of the mortgage are you 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 The Present 0 $57,619
Two Years from The Present 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 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. For the current discount rate, click here.
The table below shows the payments discounted at alternative rates.
Time Period
Net Cost
Discounted Payments(10%)
Discounted Payments (20%)
If the payments are discounted at 10%, 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%, 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% per year, then I can make a profit by borrowing at 10% per year. More typically, I cannot invest at such a high rate, soI 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 making the net present value of the loan equal to zero (in our example, 10% is the APR).
You should select a fixed discount rate with which to evaluate mortgages. You might choose a number like 5%, which is close to but not higher than the rate on most mortgages. Alternatively, you might choose the rate currently quoted on zero-point 30-year fixed-rate mortgages.

Selecting a time horizon

Now, suppose you have a choice between a 10% mortgage with no points and a mortgage chargeing 2 points with a rate of 8.5%. 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 you discount the payments at 10%, then the total discounted value of the payments is $2,000 + $51,329 + $46,662 = $99,991, which makes this a slightly less expensive mortgage than the 10% 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." Two lenders who charge different fees can quote identical APR's.)
Next, however, suppose you shorten the time horizon so you pay back the loan after one year. You make a payment of $108,500 at that time, and the total discounted value of payments is $2,000 + $98,636 = $100,636. At the shorter time horizon, the 8.5% loan with two points up front is more expensive than the 10% 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 the time horizon affects the relative cost of mortgages when the discount rate is applied. The example correctly illustrates 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 you expect to retain the mortgage. Choosing a ten-year time horizon does not mean 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 possibility 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 considerations would lead you 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 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 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%, your ARM will not necessarily go up by 3 percentage points when it first adjusts. Many ARMs have adjustment caps of 2%, so your rate will adjust upward in stages under the high-rate scenario.
In our example, suppose we have an ARM linked to the one-year rate. The hypothetical current value of the one-year rate is 7.8%, with a margin of 2.5%. This means the rate on the ARM would be calculated as 10.3% if it were fully adjusted today. However, the ARM has an initial teaser rate of 9.5%, with no points. Moreover, it has a cap -- the rate cannot go up by more than 2 percentage points.
Here is what the rate will be on your ARM next year under three scenarios:
Scenario Index Margin Cap Rate
--------------- ------- ---------- ----- ------
Constant rates 7.8 2.5 11.5 10.3
Rates up 1% 8.8 2.5 11.5 11.3
Rates up 3% 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% rate gives a total of $99,675 in the constant-rate scenario, $100,107 in the scenario with rates up 1%, and $100,193 in the scenario where rates go up by 3%. The ARM is a winner when rates stay constant but it loses to the 10% fixed rate otherwise.


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%, and a discount rate of y%, is program Z." 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 under consideration. The time horizon and interest rate scenario will prove to be more significant.
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