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Continuous Compounding

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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

ia = 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

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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 ia = e 0.80 - 1 = 1.2255 = 122.55 %

B ia = [ 1 + (0.80 / 365) ] 365 - 1 = 1.2236 = 122.36 %

C ia = 1 + 0.80 = 1.80 = 180 %

D ia = e - 0.80 = 0.4493 = 44.93 %

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