Understanding the Mathematics of Personal Finance

Present Value

Suppose you told me that you had

• walked into a television store,

• handed the store owner $900, and

• walked out with a television that sold for $1,000.

Assuming your story was true, I would conclude either that you were an amazing negotiator or that you were dealing with a very unintelligent (and soon to be out of business) store owner.

But what if your story was true (i. e., all the facts presented were absolutely correct), but something was omitted? Let’s retell the story with an omitted step: You

• walked into a television store,

• handed the store owner $900,

• came back 2 years later, and

• walked out with a television that sold for $1,000.

This is no longer an interesting story. If the store owner had taken your money, put it in a savings bank account at (approximately) 5% interest and waited 2 years, he or she would have $1,000 and would certainly be willing to hand over a $1,000 television set.

The more common way of doing this is for you to put your $900 in your bank for 2 years, withdraw your total $1,000, and go buy your $1,000 television. Financially, there is no difference between the two approaches. Note that this is not buying on credit. I’ve discussed this in another chapter.

This simple story exemplifies a basic concept of financial dealings: Nothing makes sense unless you include the effects of time when comparing costs, prices, bank account values, and so on. In the story above, you are loaning the store owner $900 for 2 years and applying the interest on this loan, along with your original $900, to purchase the television.

Let me present a slightly different example. Suppose today is January 1. You want to buy a used car and take possession of it today. Unfortunately, you don’t have

Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by

Lawrence N. Dworsky

Copyright © 2009 John Wiley & Sons, Inc.

enough money to pay fully today, so you agree to pay $1,000 today and $1,000 on each upcoming January 1 for the next 4 years (a total of five 1,000 payments). What did this car cost you?

The simple answer, $5,000, is incorrect. In the television purchase story above, didn’t you buy a $1,000 television 2 years from now for only $900 in your hand today? Using the same logic, doesn’t the $1,000 payment 2 years from today only cost you approximately $900 today?

The repeated reference to today in the above paragraph is the key to the concept of present value. You’re taking possession of the car today. The present value of the car is what it would cost if you could pay cash for it today. Since you will be paying for the car in five payments, each at a different time in the future, you need to figure out what each of these payments is worth today. (Remember, the $1,000 payment 2 years from today is only worth approximately $900 today.) The correct way to do this is to figure out how much money you would have to deposit into a savings bank on January 1 so that if you withdrew $1,000 from the savings bank every subsequent January 1, the savings account would have the exact amount of money to make these payments—there would be no money left after the fifth payment, and the account wouldn’t run out of money before making the fifth payment.

This is an absolutely fair way of doing things. Once you ’ ve calculated this present value you could put it in the bank and make the annual payments, or you could just give this present value to the car seller, and he or she in turn could put it in the bank and extract $1,000 every year for 5 years. By agreeing to sell the car for this payment plan, the car seller has effectively agreed that the sum of the present values of all the payments is exactly today’s selling price of the car.

I’tl go through this present value calculation slowly here just to show what’s happening, then I’ll show how to do it a bit more efficiently on a spreadsheet. Finally, I’ll show some online calculators that make it all very simple.

Assume that my savings bank is paying 5% interest, compounded annually. Annual compounding is not very typical, but it lets me go through the calculations without burying you in pages of numbers. On the spreadsheet to follow, I’ll use a more typical monthly compounding.

Start with the first payment: $1,000 today is worth $1,000 today. This is a very trivial thing to say, but it ’ s worth saying to help point out the contrast to future payments.

Next, look at the second payment: $1,000 a year from today. If I put $1 in the bank today, a year from today I would have $1 + 5% of $1 = (1.05) x $1 = $1.05. Going the other way, if I want to have $1,000 a year from today, I need to deposit $1,000/1.05 = $952.38.

For the third payment, I need to look at two compounding intervals. I’ve already looked at the result of one compounding interval, that is, $1,000 2 years from today is worth $952.38 1 year from today, which in turn is worth $952.38/1.05 = $907.03 today (this is the number that I approximated as $900 for the television set example).

Table 7.1 shows the present value of each of the five payments. The longer you can wait before making a payment, the less that payment costs you. Since each of

the five numbers in Table 7.1 is a present value, it makes sense to add them up and to calculate your actual purchase price for the car, which is $4,545.95.

The above calculation also demonstrates what actually happens when you buy a car on time. If the purchase price of the car is $4,546 and you agree to buy it “on time” with annual payments at an annual percentage rate (APR) of 5% (compounded annually), then your annual payment would be $1,000.