### More Interest Formulas

#### Continuous Compounding

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Single payment formulas for
continuous compounding are determined by taking the limit of compound interest
formulas as m approaches infinity, where m is the number of compounding periods
per year. Here “e” is the exponential constant (sometimes called Euler's
number).

With continuous compounding
at nominal annual interest rate r (time-unit, e.g. year) and n is the number of
time units we have:

F =
P e ^{r n} F/P

P =
F e ^{- r n} P/F

i_{a}
= e ^{r }- 1 Actual interest rate for the time unit

Example 1: If $100 is
invested at 8% interest per year, compounded continuously, how much will be in
the account after 5 years?

P =
$100

r =
8%

n =
5 years

F =
P e^{ r n} = ($100) e^{ (.08)(5)}

=
($100) e^{ 0.4} = ($100)(1.4918) = $149.18

Example 2: If $100 is
invested at 0.667% interest per month, compounded continuously, how much will
be in the account after 5 years?

P = $100

r = 0.666667%

n = 5 years * 12 months

F = P e^{ r n} =
($100) e^{ (.00666667)(60)}

= ($100) e^{ 0.4} =
($100)(1.4918) = $149.18

*Note* that the answers in the two examples
are the same because the interest is compounded continuously, the nominal rate
for the time unit used is consistent (in this case both are 8% for 12 months),
and the total time periods (5 years or 60 months) are the same. This is an
important aspect of continuous compounding.

### Interest Formulas

#### Continuous Compounding

Question 1

Question 2

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Question 1.

Suppose that a
savings account pays 6% annual interest, compounded continuously. How much must
be invested now to have $100,000 in the account 30 years from now?

Choose an answer by clicking on one of the letters below, or click on
"Review topic" if needed.

A P = $100,000 (P/F, 6%, 30) =
($100,000) (.1741) = $17,410

B P = $100,000 e ^{-(0.06)(30)}
= $100,000 e ^{- 1.8} = $16,530

C P = $100,000 e ^{- 0.06}
= $100,000 (0.9418) = $94,180

D P = $100,000 e ^{(0.06)(30)}
= $100,000 e ^{1.8} = $604,960

Review topic

Question 2.

If a loan shark
charges 80% interest, compounded continuously, what effective annual interest
rate is being charged?

Choose an answer by clicking on one of the letters below, or click on
"Review topic" if needed.

A i_{a} = e^{ 0.80}
- 1 = 1.2255 = 122.55 %

B i_{a} = [ 1 + (0.80 /
365) ] ^{365} - 1 = 1.2236 = 122.36 %

C i_{a} = 1 + 0.80 = 1.80 = 180 %

D i_{a} = e^{ - 0.80}
= 0.4493 = 44.93 %

Review topic