The Mathematics of Money

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

Finding Terms across Two or More Calendar Years

In all of the previous problems the loan date and maturity date have fallen within the same

year. But just as date numbers resetting at the start of each month caused us trouble before,

day of year numbers resetting at the start of each new year will cause problems when a

loan’s term crosses from one year to the next. The techniques we have used so far require

some adjustment when the dates do not fall in the same year.

For example, suppose a note is dated December 22, 2005, and matures on January 19,

2006. January 22 is day 356, while January 19 is day 19. So far we have found our terms
      by subtracting the loan date from the maturity date. But 19  356 is a negative number,
      which does not make any sense. Now it may be tempting to switch the numbers and
      subtract 19 from 356 to get a positive number. But while this switch will give a positive
      answer, it won’t give a correct positive answer. That should be obvious in this case, since
      subtracting 356  19  337 days, which is obviously much longer than the time between
      December 22 and January 19.
      The most straightforward way of handling this is to attack the problem directly at its
      source. We didn’t run into any trouble when everything stayed within the same year, so let’s
      break the term of this note up into pieces: the part in 2005 and the part in 2006. Counting the
      days in 2005 is no trouble: the year ends on day 365, and so the note exists for 365  356 
      9 days in 2005. Counting the days in 2006 is easy. Since the note runs from the start of the
      year until day 19, the number of days in 2006 is obviously 19. These two pieces together
      make up the whole term, and so putting them together, we find that the term of the note is
      9
      19  28 days.
      Visualizing this with a time line may be helpful:

12/22/05

9 days 19 days

2005 2006

End of ’05 1/19/06

A problem with a long term will further illustrate this approach.

Example 1.4.8 Find the term of a note dated June 7, 2005, that matured on March

15, 2007.

June 7 is day 158, and March 15 is day 74. Since they fall in different years, we split the term

of the note by year into the parts that fall in 2005, 2006, and 2007.

In 2005, the note runs from day 158 to the end of the year, day 365. Thus the note lives for

365  158  207 days in 2005.

The note’s term includes all of 2006, which is 365 days.

In 2007, the note runs from the start of the year until Day 74, so there are 74 days in 2007.

This can be represented by a time line as follows:

74 days

6/7/04

207 days 365 days

2005 2006 2007

3/16/06

Thus, the term of the note is 207 
 365 
 74  646 days.

It is not absolutely essential to draw the time lines, and some people find them unnecessary.

However, even if you did not find them necessary in these two examples, as we handle

more and more complicated problems you may find that drawing a time line is a very help-

ful tool to keep track of what is going on in a problem.

Finding Dates across Two or More Calendar Years

What about finding maturity dates and loan dates when the note crosses years? For example,

if I sign a 300-day note on November 1, 2006, I know that the maturity date cannot possibly

1.4 Promissory Notes 37