The Rule of 72 is a mathematical shortcut used to predict when a population, investment, or another growing category will double in size for a given rate of growth. It is also used as a heuristic device to demonstrate the nature of compound interest. It has been recommended by many statisticians that the number 69 be used, rather than 72, to estimate the results of continuous compounding rates of growth. Calculate how quickly continuous compounding will double the value of your investment by dividing 69 by its rate of growth.
The rule of 72 was actually based on the rule of 69, not the other way around. For non-continuous compounding, the number 72 is more popular because it has more factors and is easier to calculate returns quickly.
Key Takeaways
- The Rule of 72 is a heuristic for figuring out how long an investment will take to double in value.
- By dividing the number 72 by the expected annual rate of return, you can get a rough idea of how many years this will take.
- The number 72 itself is taken instead of the more accurate 69.3, or the natural logarithm of 2.
Continuous Compounding
In finance, continuous compounding refers to a growth rate with compounding periods that are infinitesimally small; the interest generated is calculated and compounded more than once per second, for example.
Because an investment with continuous compounding grows faster than an investment with simple or discrete compounding, standard time value of moneycalculations are ill-equipped to handle them.
Rule of 72 and Compounding
The rule of 72 comes from a standard compound interest formula:
VFuture=PV∗(1+r)nwhere:VFuture=FuturevaluePV=Presentvaluer=Interestraten=Numberofcompoundingperiods
This formula makes it possible to find a future value that is exactly twice the present value. Do this by substituting Vf = 2 and PV = 1:
2=(1−r)n
Now, take the logarithm of both sides of the equation, and use the power rule to simplify the equation further:
2ln20.693=(1−r)n∴=ln(1−r)n=n∗ln(1−r)∴≈n∗r
Since 0.693 is the natural logarithm of 2. This simplification takes advantage of the fact that, for small values of r, the following approximation holds true:
ln(1+r)≈r
The equation can be further rewritten to isolate the number of time periods: 0.693 / interest rate = n. To make the interest rate an integer, multiply both sides by 100. The last formula is then 69.3 / interest rate(percentage) = number of periods.
It isn't very easy to calculate some numbers divided by 69.3, so statisticians and investors settled on the nearest integer with many factors: 72. This created the rule of 72 for quick future value and compounding estimations.
Continuous Compounding and the Rule of 69(.3)
The assumption that the natural log of (1 + interest rate) equals the interest rate is only true as the interest rate approaches zero in infinitesimally small steps. In other words, it is only under continuous compounding that an investment will double in value under the rule of 69.
If you really want to calculate how quickly an investment will double for a given interest rate, use the rule of 69. More specifically, use the rule of 69.3.
Suppose a fixed-rate investment guarantees 4% continuously compounding growth. By applying the rule of 69.3 formula and dividing 69.3 by 4, you can find that the initial investment should double in value in 17.325 years.
FAQs
The Rule of 72 is a heuristic for figuring out how long an investment will take to double in value. By dividing the number 72 by the expected annual rate of return, you can get a rough idea of how many years this will take. The number 72 itself is taken instead of the more accurate 69.3, or the natural logarithm of 2.
What is the rule of 72 in continuous compounding? ›
You take the number 72 and divide it by the investment's projected annual return. The result is the number of years, approximately, it'll take for your money to double.
How do you calculate continuous compounding rate? ›
Continuous Compounding Formula = P * erf
where, P = Principal amount (Present Value) t = Time. r = Interest Rate.
How to do the compounding rule of 72? ›
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 formula for continuous compounding duration? ›
The formula for Continuous Compounding is A = P e^{rt}, where A is the future value, P is the principal, r is the annual interest rate, t is the time in years, and e is Euler's number.
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 many times a year is continuous compounding? ›
Continuous compounding means that there is no limit to how often interest can compound. Compounding continuously can occur an infinite number of times, meaning a balance is earning interest at all times.
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 ...
What is an example of a continuous compounded interest? ›
One example of continuous compounding in action is an account that earns interest at a rate of 14% per year, compounded monthly. The balance continually earns interest, which is added to the balance, and because there are 12 months in a year, the account balance increases by 1.17% each month.
What is the rule of 72 and how is it used? ›
The Rule of 72 is a simple way to determine how long an investment will take to double given a fixed annual rate of interest. Dividing 72 by the annual rate of return gives investors a rough estimate of how many years it will take for the initial investment to duplicate itself.
Using the rule of 72 allows you to have a solid idea of when your investment would double just from the investment rate. Very conveniently, the number 72 divides cleanly into 1, 2, 3, 4, 6, 8, 9 and 12, allowing for a quick and simple division problem instead of your usual compound interest problem.
What is the 8 4 3 rule of compounding? ›
Now, as per the 8-4-3 Rule: Year 1-8: With a compounded return of 12% on average, your investment might reach approximately Rs 8.36 lakh by the end of year 8. It considers both your monthly contributions and the returns generated. Years 9-12: The power of compounding kicks in.
How many is compounded continuously? ›
Continuously compounded interest is the mathematical limit of the general compound interest formula with the interest compounded an infinitely many times each year.
What is the discount factor formula for continuous compounding? ›
Sometimes this is referred to as discounting the amount x by the discount rate r, and the factor (always less than 1) by which we multiply x to obtain its present value is called the discount factor. Under the continuous compounding assumed above, the discount factor is e−rt.
What is the compounded daily formula? ›
How is daily compound interest calculated? Daily compound interest is calculated using the formula: A = P (1 + r / n)nt, where P is the principal amount, r is the annual interest rate, n is the number of compounding periods per year (365 for daily), and t is the time the money is invested, in years.
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.
How to double $2000 dollars in 24 hours? ›
The Best Ways To Double Money In 24 Hours
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dividing 72 by the interest rate will show you how long it will take your money to double. How many years it takes an invesment to double, How many years it takes debt to double, The interest rate must earn to double in a time frame, How many times debt or money will double in a period of time.
