Mortgage Loan Cost - Information and Advice

"How to calculate the cost of a mortgage loan."

Calculating the Cost of a Mortgage Loan
by Arnold Kling

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:

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.
 

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 30-year fixed-rate mortgages.
 

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:
 

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 possibility 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:

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


Payments under the three scenarios would be:

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.

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.

Related Pages

Mortgage APR Rates - How mortgage APR rates are calculated - How to get the lowest mortgage APR rate with no points added and pay no lender fees.

Current Mortgage Rates - Information on current mortgage rates - How mortgage interest rates affect the housing market - Home loan borrowing tips.

Comparing Mortgage Terms - 15-year vs 30-year mortgage - Pick the right term for your mortgage loan based on what works best for you - 15-year loans have many advantages.

Biweekly Mortgage Reduction Program - Information and tips for implementing a biweekly mortgage reduction program using a payment schedule - Pay off your home loan ten years early.

Adjustable Rate Mortgages - Information on adjustable rate mortgages - Adjustable versus fixed rate mortgage - pros and cons of each type of home loan.

Interest Only Mortgage Loans - Get the scoop on interest only mortgage loans - Reduce your payments to a minimum level - Discover how to grow home equity without paying to create it.
 

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