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? ›
The Rule of 72 is a calculation that estimates the number of years it takes to double your money at a specified rate of return. If, for example, your account earns 4 percent, divide 72 by 4 to get the number of years it will take for your money to double.
How to do the compounding Rule of 72? ›
Do you know the 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.
How to calculate continuous compounding? ›
Continuous Compounding Formula = P * erf
where, P = Principal amount (Present Value) t = Time. r = Interest Rate.
How to calculate continuously compounded return? ›
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? ›
Hence, the doubling time is simply 70 divided by the constant annual growth rate. For instance, consider a quantity that grows consistently at 5% annually. According to the Rule of 70, it will take 14 years (70/5) for the quantity to double.
How to do a compounding calculation? ›
What is the compound interest formula, with an example? Use the formula A=P(1+r/n)^nt. For example, say you deposit $5,000 in a savings account that earns a 3% annual interest rate, and compounds monthly. You'd calculate A = $5,000(1 + 0.03/12)^(12 x 1), and your ending balance would be $5,152.
How do you solve for compounding periods? ›
The second way to calculate compound interest is to use a fixed formula. The compound interest formula is ((P*(1+i)^n) - P), where P is the principal, i is the annual interest rate, and n is the number of periods.
What is the 8 4 3 rule of compounding? ›
The 8-4-3 investment rule provides you with a clear path for your mutual fund investments. It tells you that if you stay invested for long, the magic of compounding comes in and you make more money in less time as your investment age.
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.
Continuous compounding is used to show how much a balance can earn when interest is constantly accruing. This allows investors to calculate how much they expect to receive from an investment earning a continuously compounding rate of interest.
How is the continuous compounding formula derived? ›
The formula for continuous compounding is derived from the compound interest formula, and it involves using the mathematical constant 'e. ' PV (Present Value): The initial investment amount. i (Interest Rate): The stated annual interest rate.
What is the formula for the forward rate of continuous compounding? ›
The formula for the forward rate: f(i, j) = jS(j) − iS(i) j − i . f(T,T + ∆T) = S(T) + T ∂S ∂T . f(T) > S(T) if and only if ∂S/∂T > 0.
What is the formula for continuous compounding using LN? ›
Continuously compounded return = ln(S2/S1), where ln is the logarithmic function. Annual effective rate, also called the “APY” (annual percentage yield) in the United States, is a standardized way of expressing rates with different nominal rates and compounding frequencies.
What is the continuous compounding 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.
What are three things the Rule of 72 can determine? ›
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.
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.
