The length of time required for an investment to double in value at a fixed annual rate of return Show In finance, the Rule of 72 is a formula that estimates the amount of time it takes for an investment to double in value, earning a fixed annual rate of return. The rule is a shortcut, or back-of-the-envelope, calculation to determine the amount of time for an investment to double in value. The simple calculation is dividing 72 by the annual interest rate. ## Time (Years) to Double an InvestmentThe Rule of 72 gives an estimation of the doubling time for an investment. It is a fairly accurate measurement, and more so when using lower interest rates rather than higher ones. It is used for situations involving compound interest. A simple interest rate does not work very well with the Rule of 72. Below is a table showing the difference between the Rule of 72 calculation and the actual number of years required for an investment to double in value: ## Rule of 72 FormulaThe Rule of 72 formula is as follows: ## Example of the Rule of 72You are the owner of a coffee machine manufacturing company. Due to the large capital needed to establish a factory and warehouse for coffee machines, you have turned to private investors to fund the expenditure. You meet with John, who is a high net-worth individual willing to contribute $1,000,000 to your company. However, John is only willing to contribute the said amount on the presumption that he will get a 12% annual rate of return on his investment, compounded yearly. He wants to know how long it will take for his investment in your company to double in value. Using the Rule of 72: It will take approximately six years for John’s investment to double in value. ## Deriving the Rule of 72Let us derive the Rule of 72 by starting with a beginning arbitrary value: $1. Our goal is to determine how long it will take for our money ($1) to double at a certain interest rate. Suppose we have a yearly interest rate of “r”. After one year, we will get:
At the end of two years, we will get:
Extending this year after year, we get:
If we want to determine how long it takes to double our money, turning $1 into $2:
Step 1: $1 x (1+r)^n = $2 Step 2: (1+r)^n = $2 Step 3: ln((1+R)^n) = ln(2) (Taking the natural log of both sides) Step 4: n x ln(1+r) = .693 Step 5: n x r = 0.693 (Approximation that ln(1+r) = r) Step 6: n = .693 / r Step 7: n = 69.3 / r (Turning r into an integer rather than a decimal) Notice that after deriving the formula, we end up with 69.3, not 72. Although 69.3 is more accurate, it is not easily divisible. Therefore, the Rule of 72 is used for the sake of simplicity. The number 72 also provides more factors (2, 3, 4, 6, 12, 24…). ## Rules of 72, 69.3, and 69Rules of 69.3 and of 69 are also methods of estimating an investment’s doubling time. The rule of 69.3 is considered more accurate than the Rule of 72, but can be much more troublesome to calculate. Therefore, investors typically prefer to use a rule of 69 or 72 rather than the rule of 69.3. Comparing the doubling time for rules of 69, 69.3, and 72 to actual years: As you can see from the table above, the rule of 69.3 yields more accurate results at lower interest rates. However, as the interest rate increases, the rule of 69.3 loses some of its predictive accuracy. The Rule of 72 is a simple, helpful tool that investors can use to estimate how long a specific compound interest investment will take to double their money. ## More ResourcesThank you for reading CFI’s guide on the Rule of 72. Below are additional free resources from CFI: How many years does it take for an investment to double in value if it is invested at 6% if interest is compounded quarterly? if interest is compounded continuously? ** compound interest formula: A=P(1+r)^t, P=initial investment, r=interest rate per period, t=number of periods, A=amount after t periods. For continuous compounding: A=Pe^rt .. Quarterly compounding: A/P=2, r=.06/4=.015, t=quarters A=P(1+r)^t A/P=(1+.06/4)^t A/P=(1+.015)^t 2=(1.015)^t take log of both sides log(2)=t*log(1.015) t=log(2)/log(1.015) t≈46.55 qtrs≈11.64 years .. Continuous compounding: A/P=2, r=.06, t=years A=Pe^rt A/P=e^rt 2=e^rt 2=e^.06t take log of both sides ln2=.06t*lne lne=1 .06t=ln2 t=ln2/.06 t≈11.55 years
The internet is packed with calculators showing how much you need to save to reach a particular target, or what your investments may be worth after a certain time. But if you want to know how long it will take for your investments to double in value, a simple shortcut is the ‘rule of 72’. It works like this. You just divide 72 by the return your money is earning, to see the years it will take to double your money. In other words: 72/return = years to double. If you have $10,000 earning a return of 6%, it’s going to take 12 years to grow to $20,000 (72/6 = 12). Keep that money invested at the same rate, and it could be worth $40,000 in another 12 years! It doesn’t matter how much you have invested, the same rule applies. What’s handy about the rule of 72 is that it can also be used to work out how much your investments need to earn to double within a given number of years. Using the above example, if you wanted to double $10,000 in 8 years instead of 12, you’d need to earn returns of 9% annually. And this inevitably means taking on more risk. There’s nothing magic about the number 72. The formula derives from a for more complex equation for compounding returns. (If you’re interested in algebra, you can Google it). However, the key word here is ‘compounding’. The rule of 72 assumes you leave your original capital intact to let compounding work its magic. Dipping into an investment will re-set the time taken to double your money. It’s also important to note that investments rarely deliver the same return year after year. Returns tend to fluctuate, sometimes dramatically, even over short periods. So the rule of 72 offers a very rough guide only. The rule of 72 also applies to debt. Only this time it works against you. If you have, say, $5,000 in credit card debt at 16% and you don’t pay down the balance, in just 4.5 years the debt will have doubled to $10,000. Great for the bank, disastrous for you! This all highlights why it’s so important to look for a low rate on debt while exploring opportunities – within your tolerance for risk, for healthy returns on investments. If you need convincing on the latter, consider this. Right now you’ll be lucky to earn 1% on a savings account. Even if you do, it will take 72 years to double a $10,000 deposit to $20,000. That may be fine if you’re aged five, but it’s not much comfort for the rest of us. By taking on more risk and earning 7% annually, you could double that money in 10 years. As always, it’s about finding balance in the risk/return trade-off, and building a diverse portfolio that can help your money grow faster, while still keeping risk to a level you’re comfortable with. Effie Zahos is an independent Director of InvestSMART, money commentator at Canstar.com.au and Channel 9 Today Show. For more information on building your wealth click here: https://www.investsmart.com.au/what-we-offer/wealth-planning
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