Understanding the Mathematics of Personal Finance


The present value of amounts of money in the future, as the above calculations show, depends on the available interest rate. If interest rates are very low, then the present value looks essentially like the simple sum of all the payments. As interest rates get higher, payments far in the future become pretty worthless.

Table 7.3 shows the total present value of a 24 monthly payment account with $100 being deposited each month for different interest rates. As you can see from the table, the present value falls as the rate of interest increases.

One factor that’s very easy to model on a spreadsheet but very difficult to predict in advance is just what interest rates will be a few years from now. As long as you can borrow money at a lower rate than you’re paying on a loan, you stand to profit from the difference in rates.

To illustrate this, let’s return to the simple example of the five-payment car loan shown in Table 7.1. You bought a car that’s worth $4,546 in trade for five annual payments of $1,000. This is a perfectly equitable deal so long as prevailing interest rates remain at 5%.

Assume that high inflation occurs 3 years into your 5-year deal and savings bank interest rates jump to 10%. While high inflation in general is a terrible thing for the economy, in the case of this particular scenario, you can consider it a windfall. The money remaining in your account to pay off your car is now earning interest at a faster rate than anticipated while the amount pulled out each year to make a car payment, $1,000, doesn’. change. This means that at the end of 5 years not only have you satisfied the payment agreement on your car, but that you’ve got money left in your bank account. On the other hand, if savings bank interest rates fall, your money is accruing interest more slowly than anticipated and there won’t be enough in the account to make the last payment(s). You’ll have to add some money out of your pocket to fulfill your obligation.

One last discussion point—the television sale example at the beginning of this chapter neglected some very important real-world considerations. The store owner borrowed money to buy the television set that he or she wants to sell to you. As long

Table 7.3 Present Value of the Example for Various Interest Rates

Interest rate (%)

Present value ($)















as that television is in his or her store, he or she cannot repay the loan and he or she is paying interest on the money. He or she really doesn’t want to wait 2 years to sell you the set. In addition, new television models are coming out and in order to stay competitive, the store owner must be continually “turning over” his or her inventory, replacing older models with new models. On top of everything else, he or she has the fixed costs of renting his or her store from a building owner, paying his or her employees (and himself) a salary, and so on. Coming up with a pricing strategy that lets the store owner make money in this very dynamic situation is not a trivial task. Studying this in detail is far too involved a task for us here, but I’d like to emphasize that if you did set out to study it, the only concepts involved are those of compound interest and present value.

I’ll present real-world situations where present value calculations are needed in the chapter on comparing loans and in the chapter on fixed annuities.

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