Recall the importance of interest rates
they determine the time value of money. A dollar today
is not worth a dollar in the future, even without any inflation! Remember that the cost of
something is what must be given up in order to get it. To get a dollar in one year I don't need to
give up a dollar today, not when I can put about 97 cents into the bank and, with 3% interest, get
$1 in a year's time.
Generally we use continuouslycompounded interest, so that an amount invested at a fixed
interest rate grows exponentially. Unless you've read the really fine print at the bottom of some
loan document, you probably haven't given much thought to the differences between the various
sorts of compounding
annual, semiannual, etc.
If $1 is invested and
grows at rate R then
annual compounding means I'll have
(1 + R) after one year.
If $1 is invested and
grows at rate R then
semiannual compounding means I'll have
after one year.
"
compounding 3 times means I'll have
after one year.
…
…
…
"
compounding m times means I'll have
after one year.
"
continuous compounding (i.e. letting
) means I'll have
e
R
after one year.
And note that sometimes we write e
R
; sometimes exp{R} if the stuff buried in the superscript is
important enough to get the full font size.
You might hope that that would be enough, but it's not. The real world is complicated so there
are lots of different ways to measure interest! This class will usually think of it in the
mathematically convenient continuous form, and use all of the rest of the definitions to translate.
If the interest is only compounded once per year, then the present value of some amount of
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 Spring '12
 Poniachek
 Time Value Of Money, Interest, Interest Rate, Options

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