Continuous compounding and the rule of 72

The amount of interest one makes of course depends on the amount of principal, but from rewriting the compound interest equation as

A/P = (1 + r/m)^(mt)

it is manifest that the time it takes money to double (or increase by another factor) is independent of the original principal. This suggests the question of how long it takes money to double. we shall return to this question later.

Continuous compounding

If one takes a given interest rate for a fixed period and increases the frequency of compounding, the ratio by which the pricipal is increased keeps getting larger. For example:

(1 + .05)^10 = 1.63
(1 + .05/4)^(4×10) = 1.64
(1 + .05/12)^(12×10) = 1.65
(1 + .05/365)^(365×10) = 1.65

However, as the frequency of compounding increases, the value never increases above 1.65 (actually never above 1.6487 ...). 1.6487 = e^(.05×10). In general increasing the frequency of compounding approaches the limit:

e^(rt)

for the factor by which the principal is increased. Therfore continuous compounding is defined by the formula

A = Pe^(rt) or
P(t) = P(0)e^(rt)

which can be rewritten to give the present value as:
P = A/e^(rt) = Ae^(-rt) or
p(0) = P(t)/e^(rt) = P(t)e^(-rt).

The effective annual yield is given as
e^r - 1

Exercise: How much will $4000 be worth in 8 years with 5% interest, continuously compounded?
How much money must you deposit now to have $10,000 in 12 years at 8% continuously compounded?
What is the effective annual yield of 7% compounded continuously?

Rule of 72

We now return to the question of how long it takes money to double. We can calculate the time for various interest rates with annual, quarterly, monthly and daily compounding (the left hand quantities are greater than or equal to 2, but use the lowest time consistent with an integral number of years or quarters or months or days):
(1.02)^35 = 2
(1 + .02/4)^(4×34.75) = 2
(1 + .02/12)^(12×34.58) = 2
(1 + .02/365)^(365×34.54) = 2
(1.04)^18 = 2
(1 + .04/4)^(4×17.5) = 2
(1 + .04/12)^(12×17.33) = 2
(1 + .04/365)^(365×17.27) = 2
(1.12)^7 = 2
(1 + .12/4)^(4×6) = 2
(1 + .12/12)^(12×5.83) = 2
(1 + .12/365)^(365×5.76) = 2

All these times are close to .72/r, this is the rule of 72: divide 72 by the interest rate to get the number of years required to double. For high interest rates with infrequent compounding the time is greater than .72/r, but for most interest rates and frequencies of compounding the time is less. With continuous compounding,the time to double is .69/r because the natural log of 2 is .69, but the rule of 72 is left over from days when compounding was infrequent, and it was nice to have a number it waseasy to divide numbers into.

CompetencyHow much money will one have in 7 years if he deposits $2000 in the bank at 8% interest compounded continuously?
How much money must one deposit in the bank at 8% interest compounded continuously inorder to have $2000 seven years from now?
Use the rule of 72 to estimate how long it will take money to double at 3% interest; at 6% interest.
How long will it take money to double with continuous compounding at 3% interest? at 6% interest?

Reflection:

Challenge:

May2003

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campbell@math.uni.edu