- Continuous compounding
- 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 anotherfactor) is independent of the original principal. This suggests the question ofhow long it takes money to double. we shall return to this questionlater.
If one takes a given interest rate for a fixed period and increases thefrequency of compounding, the ratio by which the pricipal is increased keepsgetting 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 increasesabove 1.65 (actually never above 1.6487 ...). 1.6487 = e^(.05×10). Ingeneral increasing the frequency of compounding approaches the limit:
e^(rt)
for the factor by which the principal is increased. Therfore continuouscompounding 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?
We now return to the question of how long it takes money to double. We cancalculate the time for various interest rates with annual, quarterly, monthlyand 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 quartersor 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 theinterest rate to get the number of years required to double. For high interestrates with infrequent compounding the time is greater than .72/r, but for mostinterest rates and frequencies of compounding the time is less. With continuouscompounding,the time to double is .69/r because the natural log of 2 is .69, butthe rule of 72 is left over from days when compounding was infrequent, and itwas nice to have a number it waseasy to divide numbers into.
CompetencyHow much money will one have in 7 years if he deposits $2000 inthe bank at 8% interest compounded continuously?
How much money must one deposit in the bank at 8% interest compoundedcontinuously 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
returntoindexcampbell@math.uni.edu
FAQs
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.
Does the rule of 72 account for compounding? ›
The Rule of 72 is a simplified formula that calculates how long it'll take for an investment to double in value, based on its rate of return. The Rule of 72 applies to compounded interest rates and is reasonably accurate for interest rates that fall in the range of 6% and 10%.
What is the rule for compounded continuously? ›
The continuous compounding formula is nothing but the compound interest formula when the number of terms is infinite. This formula says, when an amount P is invested for the time 't' with the interest rate is r% compounded continuously, then the final amount is, A = P ert.
What is the rule of 70 in continuous compounding? ›
The rule of 70 calculates the years it takes for an investment to double in value. It is calculated by dividing the number 70 by the investment's growth rate. The calculation is commonly used to compare investments with different annual interest rates.
How do you calculate compounded continuously? ›
Continuous Compounding Formula = P * erf
where, P = Principal amount (Present Value) t = Time. r = Interest Rate.
What does the Rule of 72 have to do with compound interest? ›
It's an easy way to calculate just how long it's going to take for your money to double. Just take the number 72 and divide it by the interest rate you hope to earn. That number gives you the approximate number of years it will take for your investment to double.
What is the 8 4 3 rule of compounding? ›
The rule of 8-4-3 when it comes to compounding indicates a style of investment that accelerates growth with time. Initially, a corpus doubles within 8 years through an average annual return of 12% subsequently another doubling happens for the same period after another 4 years following its initial setting up.
What happens if interest is compounded continuously? ›
What is Continuously Compounded Interest? Continuously compounded interest is interest that is computed on the initial principal, as well as all interest other interest earned. The idea is that the principal will receive interest at all points in time, rather than in a discrete way at certain points in time.
What happens when something is compounded continuously? ›
Compounded continuously means that interest compounds every moment, at even the smallest quantifiable period of time. Therefore, compounded continuously occurs more frequently than daily.
How to calculate apy for continuous compounding? ›
How Is APY Calculated? APY standardizes the rate of return. It does this by stating the real percentage of growth that will be earned in compound interest assuming that the money is deposited for one year. The formula for calculating APY is (1+r/n)n - 1, where r = period rate and n = number of compounding periods.
Continuous compounding adds more interest, so it is better for investors, whereas discrete compounding adds less. However, all forms of compounding are better for investors than simple interest, which only calculates interest on the principal amount.
What is the continuous compounding rule of 69? ›
What Is Rule Of 69? Rule of 69 is a general rule to estimate the time that is required to make the investment to be doubled, keeping the interest rate as a continuous compounding interest rate, i.e., the interest rate is compounding every moment.
Why do we use the rule of 70 instead of the Rule of 72? ›
The rule of 72 is best for annual interest rates. On the other hand, the rule of 70 is better for semi-annual compounding. For example, let's suppose you have an investment that has a 4% interest rate compounded semi-annually or twice a year. According to the rule of 72, you'll get 72 / 4 = 18 years.
How do you manually calculate continuous compounding? ›
The formula for continuous compound interest is A = P × e^rt, where 'A' is the amount of money after a certain amount of time, 'P' is the principle or the amount of money you start with, 'e' is Napier's number (approximately 2.7183), 'r' is the interest rate (always represented as a decimal), and 't' is the amount of ...
Is compounded continuously the same as annually? ›
Continuous compounding is similar in concept to annual compounding, except the compounding periods are infinitely small. Although the annual compounding formula can be easily modified to accommodate smaller periods, the number of compounding periods used for continuous compounding would be infinitely numerous.
What is the formula for continuous compounding in Excel? ›
To calculate the future value (FV) with continuous compounding, use the formula: FV = PV x e(i x t), where PV is the present value, “i” is the interest rate, “t” is the time in years, and “e” is the mathematical constant. With our revamped Full Stack Development Program: master Node.
What are the rules of compounding? ›
The Rule of 72 is a heuristic used to estimate how long an investment or savings will double in value if there is compound interest (or compounding returns). The rule states that the number of years it will take to double is 72 divided by the interest rate.
Does compounding apply to debt? ›
If you have a debt that uses compound interest, the amount you owe will grow each time the interest compounds and your payments will get larger over time. For that reason, it is wise to pay down compounding debts as quickly as you can.
What is the most accurate rate of compound Rule of 72 rule of 70 rule of 69 rule of 71? ›
Choice of rule
For continuous compounding, 69 gives accurate results for any rate, since ln(2) is about 69.3%; see derivation below. Since daily compounding is close enough to continuous compounding, for most purposes 69, 69.3 or 70 are better than 72 for daily compounding.
What is the limitation of Rule 72? ›
It is not an exact value and can only provide a general estimate of the time required to double the investment. If the interest rate changes due to some factor, the Rule of 72 becomes null and void. The Rule of 72 does not apply to changing interest rate investments or basic interest investments.
