Mortgage Tables

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Calculating Payments or the Interest Rate from a Mortgage Table

We mentioned earlier that before computers, bankers used to use mortgage tables to calculate monthly payments. I've included a complete set of tables (interest rate 0% to 20% in 0.05% increments) for determining the payment per $1,000 of principal. You sometimes see abbreviated versions of these mortgage tables with values per $1000 of principal in real estate magazines, to help you determine how much house you can afford. What makes it possible to reduce a fairly complex calculation to a simple table is that the complex part remains constant for a given interest rate and number of months. In other words, you only have to compute the big mess once to figure out the relationship between the amount of the mortgage principal and the monthly payment for a particular interest rate on a fixed rate mortgage. So referring back to our mortgage formula again, imagine that the messy part of the equation was replaced with a value from a table, so the equation now reads:

M = P [value from table] / 1,000

Where the value from the table is our familiar mess:

Value from table = [ i(1 + i)n ] / [ (1 + i)n - 1]

So how many interest rates should be calculated for a useful table? Since the calculation of a monthly payment based on the principal you do yourself doesn't include the various fees and charges that show up at closing, there's no point in trying to be super accurate when estimating your expenses. We carried the calculation out to enough significant digits that it should be within a penny on mortgages up to one million dollars, but that doesn't mean you have to keep all the digits yourself when trying to get a ballpark figure for affordability.

The table on the following page can be used to estimate your monthly payment, per thousand dollars of loan mortgage principal, for interest rates between 4.00% and 5.95%. We put fifteen year and thirty year mortgages in the same table for in case you want to print and keep a copy in your wallet or on the fridge while you're house shopping.

Interest

15 Year

30 Year

Interest

15 Year

30 Year

4.00%

7.39688

4.77415

5.00%

7.90794

5.36822

4.05%

7.42196

4.80302

5.05%

7.93401

5.39882

4.10%

7.44709

4.83198

5.10%

7.96013

5.42950

4.15%

7.47227

4.86103

5.15%

7.98629

5.46026

4.20%

7.49750

4.89017

5.20%

8.01251

5.49111

4.25%

7.52278

4.91940

5.25%

8.03878

5.52204

4.30%

7.54811

4.94871

5.30%

8.06509

5.55305

4.35%

7.57349

4.97812

5.35%

8.09145

5.58414

4.40%

7.59892

5.00761

5.40%

8.11787

5.61531

4.45%

7.62440

5.03719

5.45%

8.14433

5.64656

4.50%

7.64993

5.06685

5.50%

8.17083

5.67789

4.55%

7.67551

5.09661

5.55%

8.19739

5.70930

4.60%

7.70114

5.12644

5.60%

8.22400

5.74079

4.65%

7.72682

5.15637

5.65%

8.25065

5.77236

4.70%

7.75254

5.18638

5.70%

8.27735

5.80400

4.75%

7.77832

5.21647

5.75%

8.30410

5.83573

4.80%

7.80414

5.24665

5.80%

8.33090

5.86753

4.85%

7.83002

5.27692

5.85%

8.35774

5.89941

4.90%

7.85594

5.30727

5.90%

8.38464

5.93137

4.95%

7.88191

5.33770

5.95%

8.41158

5.96340

Since the table was prepared filled in hand, let's check the accuracy against some of the mortgage calculations we've made previously. Our first mortgage calculation was for a $100,000 fifteen year mortgage at 5%, and the monthly payment came out to $790.79. Using the formula

M = P [value from table] / 1,000

= $100,000 (7.90794) / 1,0000

= $790.794

Where the 4/100 of a penny is more accuracy than we need.

For our $187,000 thirty year mortgage at 5.5%, we computed the payment as $1081.77, and the table formula gives:

= $187,000 (5.67789) / 1,000

= $1061.77

We could have save the "divide by 1,000" step at this stage by filling the mortgage table with number that were already divided by 1,000, but you can do that yourself before calculating by just moving the decimal place three places to the left, since,

5.67789 / 1,000 = 0.0056789

and so on. Despite being highly competitive, mortgage are generally quoted in 5/100 increments, as those used in the table. But if you wanted to determine the monthly payment on a $187,000 thirty year mortgage at 5.53%, you can come very close by interpolation. For example, the closest interest rate in the table below 5.53% is 5.50%, for which we've already calculated the monthly payment to be $1061.77. The closest interest rate above 5.53% is 5.55%, for which the payment works out to:

=187,000 (5.70930) / 1,000

= $1067.64

We can use the formula for linear interpolation with a fixed rate mortgage when the bracketing interest rates are so close together, since the error will be small. The formula, which looks worse than it is, gives:

M = Mlow + (i - ilow) (Mhigh - Mlow) / (ihigh - ilow)

Where the low refers to the lower interest rate and payment, and high refers to the higher. Sticking our numbers in the formula give

= $1061.77 + ( 5.53 - 5.50) ($1067.64 - $1061.77) / (5.55 - 5.50)

= $1061.77 + (0.03) ($5.97) / (0.05)

= $1061.77 + $3.58

= $1065.35

Now let's compute the monthly payment for the same mortgage at 5.5%:

i = 5.53% / # months per year

= 0.053 / 12

= 0.0046083

(1 + i)n = (1+ 0.0046083)360

= (1.0046083)360

= 5.2340

Replacing the (1 + i)n in our formula with 5.2340 we get:

M = P [ i (5.2340)] / [5.2340 - 1] or

= P [0.0046083 x 5.2340] / 4.2340

= P [0.024120] / 4.2340

= P (0.0056967)

And on our $187,000 loan at 5.53%, the payment would be:

= $187,000 (0.0056967)

= $1065.28

So our interpolated answer using values from the mortgage table, $1065.35, came out seven cents ($0.07) per month too low, because the compound interest formula isn't a linear function, but the estimate is more than good enough for determining what you can afford.