Understanding the Mathematics of Personal Finance
Comparing Loans
In the 1981 movie Absence of Malice, Paul Newman played an honest businessman who is related to some organized crime figures. For various plot reasons, a newspaper publishes a story about him that contains true statements that are assembled to lead to a false impression/conclusion about his involvement in organized crime. The story ruins his reputation and his business. Most of the movie is involved with how he, using the same misleading techniques, gets even. Several times in the movie, somebody looks at a collection of facts pointing to a conclusion and asks, of the conclusion, “Is it true?” The answer is always, “No, but it’s accurate.”
Unfortunately, accurate but not true information is not just a clever ploy in an old movie plot. It is still with us today. Incorrect manipulation of loan details, although mathematically absolutely correct (accurate), nevertheless can lead to erroneous conclusions (but not true). The fake balance calculation that I have shown in Chapter 5 is a good example of this. Also, many times, the error comes from improper comparison of loans.
Before launching into calculations for comparing loans, I have to consider first why one loan is, in some sense, better than another loan. I’ll limit myself to loans that are repaid with monthly payments such as a home mortgage loan or an automobile loan. There are many possible reasons for preferring one loan to another. If I’m short on cash, then the loan that offers the lowest possible payments looks best to me. I should consider how the lender will react if I’m able to increase my payments sometime in the future and what will happen if I’m able to pay off this loan early. The ability to pay off a loan early doesn’t require that I win a lottery or that a long  lost great aunt dies and leaves me a fortune. Interest rates may drop significantly, and I might want to look at refinancing opportunities, possibly with a different lender.
If I know that I’m coming into a lot of money sometime relatively soon (I’m getting a signing bonus with a football team at the start of next season, or my trust fund pays me $1,000,000 when I reach age 25), then a low prepayment penalty is important.
Probably, the most common comparison criterion is the ultimate total cost of a loan. The total cost must be calculated either as the present value of the loan or the future value of the loan. That is, loans must be compared using their present values or their future values as of the same date. Present and future values are a little tricky
Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by
Lawrence N. Dworsky
Copyright © 2009 John Wiley & Sons, Inc.
to assess because there must be some assumptions as to what interest rates will be at different times in the future. As Yogi Berra is reputed to have said, “It’s very hard to make predictions, particularly about the future.”
I ’ ll propose several scenarios taking changing interest into account. In most cases, the better loan in a comparison will stay the better loan even if interest rates move around a bit.
Let’s start with the most straightforward case: a mortgage loan with no upfront costs and a fixed interest rate for a fixed payment period. If two loans (for the same principal, of course) are amortized in the same number of payments, then clearly the loan with the lower interest rate must be the less expensive loan to take. On the other hand, what happens when two loans having the same principal have different interest rates and different payment periods, not to mention different savings bank (or CD) rates to use for the present value calculation?
The spreadsheet Ch8LoanPV. xls on my website will help to study this question. The first tab, labeled Basic, is very similar to the Mortgages spreadsheet on the website. The differences are that I eliminated the Month and Year entries, because all I want to track here are the number of payments, and that I added present value calculations.
The rate I used for the present value calculations is called the PV Rate. The variable Rate is the loan rate. In the example that’s preloaded on this spreadsheet, I used Nr Pmts = 180; Principal = $350,000; Rate = 6.00%; and PV Rate = 3.00%. Tot PV is the sum of the present value of all the payments, that is, the present value of the loan, which in this example is equal to $427,682.80.
As I look at the present value of various loans, I’ll keep the principal fixed at $350,000 for this example. The exact number for the principal doesn’t matter, because everything will scale with this number. I want to vary the Rate, the PV Rate, and the NR Pmts. I’ll create a table (Table 8.1), and then I’ll discuss making a loan decision based on the data in this table.
The first thing that pops out at me when I look at Table 8.1 is that I’ve created a monster. It’s just impossible to draw any general conclusions about these loans from this table. There are simply too many variables. Let’s see if some common sense can narrow things down a bit.
When you look for a home loan, you usually get offers involving different interest rates, different payoff periods, and different upfront costs. I’ll talk about factoring in upfront costs shortly. For now, let’s consider the loan interest rate and the payoff period as “negotiable variables.” You want the loan with the lowest PV that has monthly payments that you can handle.
Assume that interest on savings is about 3.50%. This lets me shrink the table significantly.
Table 8.2 is the rows of Table 8.1 that have a savings rate of 3.50%. I typed these items into a blank spreadsheet and then took advantage of the spreadsheet’s ability to sort a table so that I’ve listed the items in increasing order of PV.[20]
Table 8.1 Examples of Present Value of a Simple Mortgage Loan

Table 8.2 
Subset of Table 8.1 for Savings Rate = 
3.50% Sorted by PV 

Nr Pmts 
Loan rate (%) 
Savings rate (%) 
PV ($) 
Monthly payment ($) 
120 
6.00 
3.50 
392,950 
3,886 
120 
7.00 
3.50 
410,958 
4,064 
120 
8.00 
3.50 
429,431 
4,246 
240 
6.00 
3.50 
432,359 
2,508 
360 
6.00 
3.50 
467,309 
2,098 
240 
7.00 
3.50 
467,885 
2,714 
240 
8.00 
3.50 
504,783 
2,928 
360 
7.00 
3.50 
518,558 
2,329 
360 
8.00 
3.50 
571,920 
2,568 
Looking at Table 8.2, the lowest PV and the highest PV are easy to guess; the shortest term loan at the lowest interest rate results in the lowest PV, while the longest term loan at the highest interest rate results in the highest PV. Generally, shorter term loans are better than longer term loans, and lower loan interest rates are better than higher loan interest rates. However, the actual order of all of the rows in the table isn’t as obvious. It’s the combination of both factors (loan interest rate and loan term) that produces the PV, not any single factor. If I had produced a table with smaller increments in both of these factors, there would be many more notso  obvious results.
As the person about to take this mortgage loan, you now have a decision to make. The lowest PV loan (the top line in Table 8.2) is the best deal in terms of your actual total cost for the loan. What if you’re uncomfortable, however, about making the (approximately) $3,900 monthly payment?
Table 8.2 gives you the information you need. The fifth line from the top shows you the loan with the lowest monthly payments. You didn’t need this analysis to tell you that—the payment amount was certainly specified when the loan was offered to you. Nevertheless, let’s assume that these nine loans were actually offered to you.
Rounding off to the nearest hundred dollars, the $2,100 monthly payment loan is the lowest payment loan. If you can afford $2,500 a month, go up the list to the fourth loan and take this loan. For no reason should you take the $2,300 a month loan (seventh on the list)—it will ultimately cost you more than the $2,100 a month loan.
Putting it all together, your best choice is the $3,900 a month loan, if you can afford it. Next best is the $2,500 a month loan and, finally, the $2,100 a month loan. If you can’t afford to pay $2,100 a month, you’ll have to go shopping for some other loans or, unfortunately, give up the idea of taking this mortgage loan—at least until interest rates drop.