The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

Jennifer knows she will be getting a paycheck for $500 at the end of the week, but she

needs money now. She takes out a loan against this paycheck, borrowing as much as

she can pay off with her check when it arrives.

When you fi le your federal income taxes, you fi nd that you are owed a refund of

$737.15. Your tax preparer offers the option of getting your money right away, instead

of waiting for the IRS to process your return and send your refund check. In exchange

for agreeing to sign over your refund check when it arrives, the preparer agrees to lend

you $707.15 today.

The Cedar Junction Central School District is scheduled to receive a state aid

check for $72,500 on April 1. The district needs the funds by mid-February to meet

expenses, but unfortunately the state is unwilling to make the payment early. In order

to avoid a cash crunch, the district borrows $71,342 from First Terrapin National

Bank, to be repaid on April 1 when the state aid arrives. The amount borrowed is

based on the $72,500 that the district will have available for repayment when the aid

check arrives.

In each of these cases, a loan is being made, and so it is reasonable for the lenders to

expect to be paid some “rent” on their money. In the previous chapter, we have described

that rent as interest, and thought of it as being added on to the principal. Looking at

things as principal 
 interest  maturity value, we might describe the tax refund loan

example as:

$707.15


 $30.00

$737.15

While there is nothing wrong with this description, it doesn’t really represent things quite

the way they are happening. The figures are correct, but this mathematical description

suggests that you decided to borrow $707.15, added on $30 interest, and then arrived at

the $737.15 maturity value as a consequence. But that isn’t what happened at all. A truer

representation of the situation might be to turn the description around:

$737.15

 $30.00

$707.15

This second representation is financially and mathematically equivalent to the first, but it

presents a different way of looking at things. It suggests that you knew you could repay

$737.15, gave up $30.00 “interest,” and arrived at the $707.15 “principal” as a conse-

quence. This is much truer to what actually happened.

These examples demonstrate an alternative way of looking at the time value of money.

With this new way of looking at things, we will need some new terminology:

Definitions 2.1.1

A loan that is made on the basis of a fi xed maturity value is called a discount loan. The

lender subtracts an amount, called the discount, from the maturity value, and pays the

result, called the proceeds, to the borrower.

If we were looking at the tax refund example as interest, we would call the $707.15 the

principal and the $30 the interest, and add them together to obtain the $737.15 maturity

value. Looking at it as discount, we start with the $737.15 maturity value, subtract the $30

discount, and arrive a $707.15 in proceeds.

Most of the rest of our terminology remains the same, with one notable exception. Term,

maturity date, and maturity value all still mean the same thing. Recall though that the

face value of a simple interest note has heretofore always been the same as its principal.

The “size” of the loan is the amount most likely to be shown prominently on the note’s

face, hence the term. When a promissory note is based on discount, however, the size of the

2.1 Simple Discount 57