You set the accuracy requirements for your approximation. This really means
that the job you are doing sets the requirements. Different jobs will have dif-
ferent requirements. For example, most financial mathematics approximation
jobs do not require particularly great precision. Making rockets to circle the
planet Jupiter will have great precision requirements. Determining these pre-
cision requirements might be the hardest part of the actual approximation job.
Sometimes the accuracy requirements are stated legally. For example, the
Municipal Securities Rulemaking Board (MSRB), one of the regulatory author-
ities for the municipal securities industry, requires that the yield equivalent of
the dollar price must be shown on the customer confirmation, accurate to
within .05% of yield, if the bond was traded at a dollar price. The MSRB has
devoted a whole section of a rule to accuracy requirements. You should always
make sure that any legal or regulatory requirements have been met in calcu-
lation of approximate numbers. Check with the rule publications, or your com-
pliance officer, for the requirements. A book like this is not a good source,
because the rules are subject to change.
Suppose that after 10 years, a $100 investment has grown to $160. What was
the annual return?
We need to find an interval that contains the 160 final value.
We try 3%, and find the value is 134.39, so
We try 4%, and find the value is 148.02, so
We try 5%, and find the value is 162.08, and we have our desired
interval, [4, 5].
We bisect the interval, and
Try 4.5%, and find the value is 155.30, so we bisect the interval [4.5, 5],
Try 4.75%, and find the value is 159.05, so we bisect the interval [4.75,
5], and
Try 4.875, and find the value is 160.96,
We decide we are close enough, so we bisect the interval [4.75, 4.875]
and obtain 4.8125 as the approximate interest rate. We are within
.0625 of the correct answer.
36 Compound Interest

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