Rule of 72

Key takeaways

Rule of 72 belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Rule of 72 to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rule of 72 from memory before moving on to harder problems.

Reference excerpt

In finance, the rule of 72, the rule of 70 and the rule of 69.3 are methods for estimating an investment's doubling time. The rule number (e.g., 72) is divided by the interest percentage per period (usually years) to obtain the approximate number of periods required for doubling. Although scientific calculators and spreadsheet programs have functions to find the accurate doubling time, the rules are useful for mental calculations or when only a basic calculator is available. These rules apply to exponential growth and are therefore used for compound interest as opposed to simple interest calculations. They can also be used for decay to obtain a halving time. The choice of number is mostly a matter of preference: 69 is more accurate for continuous compounding, while 72 works well in common interest situations and is more easily divisible. There are a number of variations to the rules that improve accuracy. For periodic compounding, the exact doubling time for an interest rate of r percent per period is

t = ln ⁡ ( 2 ) ln ⁡ ( 1 + r / 100 ) ≈ 72 r , {\displaystyle t={\frac {\ln(2)}{\ln(1+r/100)}}\approx {\frac {72}{r}},}

where t is the number of periods required. The formula above can be used for more than calculating the doubling time. If one wants to know the tripling time, for example, replace the constant 2 in the numerator with 3. As another example, if one wants to know the number of periods it takes for the initial value to rise by 35%, replace the constant 2 with 1.35.

Using the rule to estimate compounding periods To estimate the number of periods required to double an original investment, divide the most convenient "rule-quantity" by the expected growth rate, expressed as a percentage.

For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum, the rule of 72 gives 72/9 = 8 years required for the investment to be worth $200; an exact calculation gives ln(2)/ln(1+0.09) = 8.0432 years. Similarly, to determine the time it takes for the value of money to halve at a given rate, divide the rule quantity by that rate.

To determine the time for money's buying power to halve, financiers divide the rule-quantity by the inflation rate. Thus at 3.5% inflation using the rule of 70, it should take approximately 70/3.5 = 20 years for the value of a unit of currency to halve. To estimate the impact of additional fees on financial policies (e.g., loading and expense charges on variable universal life insurance investment portfolios), divide 72 by the fee. For example, if the Universal Life policy charges an annual 3% fee over and above the cost of the underlying investment fund, then the total account value will be cut to 50% in 72 / 3 = 24 years, and then to 25% of the value in 48 years, compared to holding exactly the same investment outside the policy.

Choice of rule The value 72 is a convenient choice of numerator, since it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. It provides a good approximation for annual compounding, and for compounding at typical rates (from 6% to 10%); the approximations are less accurate at higher interest rates. 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. For lower annual rates than those above, 69.3 would also be more accurate than 72. For higher annual rates, 78 is more accurate.

Note: The most accurate value on each row is in bold.

History An early reference to the rule is in the Summa de arithmetica (Venice, 1494. Fol. 181, n. 44) of Luca Pacioli (1445–1514). He presents the rule in a discussion regarding the estimation of the doubling time of an investment, but does not derive or explain the rule, and it is thus assumed that the rule predates Pacioli by some time.

A voler sapere ogni quantità a tanto per 100 l'anno, in quanti anni sarà tornata doppia tra utile e capitale, tieni per regola 72, a mente, il quale sempre partirai per l'interesse, e quello che ne viene, in tanti anni sarà raddoppiato. Esempio: Quando l'interesse è a 6 per 100 l'anno, dico che si parta 72 per 6; ne vien 12, e in 12 anni sarà raddoppiato il capitale. (emphasis added). Roughly translated:

In wanting to know of any capital, at a given yearly percentage, in how many years it will double adding the interest to the capital, keep as a rule [the number] 72 in mind, which you will always divide by the interest, and what results, in that many years it will be doubled. Example: When the interest is 6 percent per year, I say that one divides 72 by 6; 12 results, and in 12 years the capital will be doubled.

Derivation

Periodic compounding For periodic compounding, future value is given by:

F V = P V ⋅ ( 1 + r / 100 ) t {\displaystyle FV=PV\cdot (1+r/100)^{t}}

where P V {\displaystyle PV} is the present value, t {\displaystyle t} is the number of time periods, and r {\displaystyle r} stands for the interest rate per time period. The future value is double the present value when:

F V = 2 ⋅ P V {\displaystyle FV=2\cdot PV}

which is the following condition:

( 1 + r / 100 ) t = 2 {\displaystyle (1+r/100)^{t}=2\,}

This equation is easily solved for t {\displaystyle t} :

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Rule of 72

Start with the simplest possible case. Write down what Rule of 72 claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rule of 72 before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rule of 72 ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rule of 72

In research: Rule of 72 appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rule of 72 in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rule of 72 is common in secondary-school and first-year university syllabi. It links to neighbouring topics Debt, Exponentials, Interest, so understanding it makes those chapters shorter.
In everyday life: Look for Rule of 72 outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Rule of 72 in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rule of 72 means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Rule of 72 out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rule of 72 in simple terms?

In finance, the rule of 72, the rule of 70 and the rule of 69.3 are methods for estimating an investment's doubling time. The rule number (e.g., 72) is divided by the interest percentage per period (usually years) to obtain the approximate number of periods required for doubling.

Why does Rule of 72 matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rule of 72?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rule of 72.