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.
(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:
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,
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?
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?