Understanding the Mathematics of Personal Finance


You won’t see the term up-front costs in your mortgage contract. I’m using the term as a catchall for all the costs and fees involved in starting up a mortgage loan. This includes points. Points are a start-up fee that many lenders charge for giving you the loan. One point represents 1% of the loan amount. Then there are appraisal fees, paperwork fees, various state and county taxes, and so on. While you as the borrower certainly want to scrutinize every one of these items and make sure you’re getting the best deal available (i. e., the lowest amount of up-front money necessary to get yourself the loan), for my purposes, I’m just going to sum them all into up-front costs.

Many lenders will offer you several deals, for example, “6% plus 3.5 points, or 7% plus 2 points.” It is therefore not obvious which is the best deal, and it’s neces­sary to do some calculations before making your choice.

Some costs are tax deductible and therefore the loan actually costs you less than it appears when you take the loan. Without knowing your tax bracket, I can’t take this “discount” into account. If your tax bracket[12] is, say, 30%, then you can estimate that you’ll save almost 30% on the deductible costs.

Suppose you want all the costs folded into the loan. Assume that there are $10,000 in up-front costs. I’ll go back to my first example, the one that’s loaded onto the “Basic” tab on the spreadsheet we’ve been using. Just to recap, in case you’ve changed the data and don’t want to bother reloading the spreadsheet from my website, you are taking a $200,000 loan at 6.00% interest, to be paid back (amortized) over 240 monthly payments (20 years). The monthly payment is $1,432.86.

If you want your lender to fold the $10,000 that you need to come up with into the loan itself, then, from the lender’s point of view, you’re getting a $210,000 loan.

Using the same spreadsheet, if you change the $200,000 principal to $210,000, the regular monthly payment goes up to $1,468.68 (or, by scaling, just note that $210,000 is 5% greater than $200,000 and multiply $1,432.86 by 1.05). Write this number down, or just enter it into a vacant cell on the spreadsheet so that you can refer to it.

Change the principal in the spreadsheet back to $200,000. The regular payment drops back to $1,432.86. Start increasing the interest rate by small amounts until the payment equals to $1,468.68. At an interest rate of approximately 6.31%, this goal is accomplished. This is the effective interest rate on your loan of $200,000 when the $10,000 extra is absorbed by the lender.


Fixed interest, fixed payment mortgage loans are no different from the basic loans described in Chapter 3. The problems below will, therefore, concentrate on variations in calculation brought about by different types of mortgage loans. Solving these problems, on the other hand, will require a working ability to solve the simpler problems.

For problems 1-4, start with a basic mortgage loan of $325,000 taken in January 2010, payable monthly for 20 years with an APR of 5.80%.

1. Your up-front costs for getting your loan were 3 points and $450.00 in fees. You didn’t have this money available, so you had these costs folded into the loan. What’s your effec­tive interest rate?

2. Since you are short on cash, you make interest-only payments on the above loan for the first 3 years. How much are these payments, what is your balance at the end of 3 years, and what is your new regular payment to amortize the loan?

3. Continuing with the above situation, after 5 years (Pmt Nr 60), interest rates drop and you are offered a free refinancing of the loan at 5.00%. You’d like to pull some cash out for other purposes and you’re “used to” paying $2,577.66 each month. Your new mortgage period is again 20 years, but now it’s 20 years from the start of your new mortgage. How much is the new mortgage for and how much cash do you pull out?

4. I bought a house that was appraised at $425,000 on January 1, 2000. I got an 80% mortgage as a fixed 30-year mortgage at 5.00%. On January 1, 2007, I decided to take a second mortgage because I need some cash. I find that my house has appreciated 3% a year. My lender will give me a second mortgage for the difference between 80% of the appraised value and my equity. The second mortgage is at 6.2%. The second mortgage will be fully paid off at the same time that the first mortgage is paid off.

What is the principal of the second mortgage and what are my total monthly payments (first + second mortgages combined)? Assume 0 up-front costs for both mortgages.


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