The Rule of 70 might sound daunting. But, have no fear. This rule might actually be one of the easiest math formulas out there. It is true that some may try and complicate this rule. So you may hear arguments that you should be using the Rule of 69 or the Rule of 72. That’s why we’d like to prove to you that the concepts are so similar you can just stick with remembering one very simple rule. That rule is the Rule of 70.
What is the Rule of 70?
This rule refers to the method of estimating how long it would take for you to double something. This something most often refers to your money. You can do this as long as you are given a level of variable growth. That’s why you might hear the Rule of 70 also referred to as the “doubling time”. For this formula to be used correctly, your variable growth must continue at a consistent rate.
The result of this calculation is the number of time periods it will take, given the specified growth rate. The Rule of 70 is often used to find the number of years it takes to double something. But, the time periods could be anything depending on the growth rate you are calculating.
Calculating Doubling Time
Alright, now we get to the fun part! As we just mentioned the Rule of 70 formula is pretty easy and looks something like this:
70 / r = t
where r = the growth rate and t = time
Since the variables may change depending on who you talk to, we could also write:
70 / growth rate = # of time periods it takes to double
Applying the Rule
To put this into practice, let’s say you are half-way to retirement and do not want to work anymore. You can invest your current savings into Bitcoin. This investment will earn 400 basis points. This works out to a rate of 4%.
According to the Rule of 70, all we have to do to figure out how long it would take for you to double your money is to apply this formula.
To do this, we would take 70 and divide it by your 4% rate. This would give you:
70 / 4 = 17. 5 years
Therefore, as long as the interest rate remains constant throughout this whole time period, you would be able to successfully retire in just 17.5 years.
A Friendly Reminder
When applying this formula, we just need to keep in mind the growth rate must be written as a whole number. So, r or our growth rate should be written as 4 and not as 0.04.
Furthermore, it is important for us to note that because the interest rate is growing at 4% a year that our final answer would be in years as well. If for example, our investment was growing at 4% a month, it would only take 17.5 months for our investment to double.
The Origins of 70
For those who consider themselves more mathematically inclined, you might be wondering where the number 70 actually came from and why it usually gives us a pretty accurate answer. To break it down we have to first go back to the traditional future value formula we often see in finance.
If it’s been a while, the formula looks something like this:
FV = PV (1 + r)t
where FV = the future value of your investment and PV = the present value of your investment
Since we want to find the amount of time it would take to double our current value (PV), we could say that 2 multiplied by our PV would give us our future value. Then, we could rewrite this formula to look something like this:
2PV = PV (1 + r)t
After rewriting this to reflect the doubled amount, we would likely notice that the variable PV occurs on both sides of the equation. Next, we can apply the basic rules of algebra and divide both sides by the variable “PV”.
The resulting formula would look like:
PV = (1 + r)t
Next, we could then take the natural log of both sides of the equation since we are trying to solve for an exponent. The formula would further be adjusted to now look something like this:
t = ln (2) ln ( 1+ r)
We can now solve for the ln (2) which works out to approximately 0.70. Since the exact number of 0.693147, you might also hear people use the rule of 69 or 69.3. This is because this is a more accurate value derived from the natural logarithm of 2. Don’t worry about this right now. The Rule of 70 is used more frequently anyways.
With ln (2) calculated our formula now looks like:
t = 70 ln ( 1+ r)
Finally, we can simplify, ln (1 + r) to approximately equal r. This would then give us the doubling formula we saw earlier:
t = 70 / r
Validating the Accuracy
As you might have noticed, towards the end of solving this equation we made a couple of simplifications that may presumably affect the accuracy of our answers. To test if this should be of concern to us, we decided to test this theory.
Let’s look at one more example. Say we are now looking at a growth rate or 4%, compounded annually.
If we substitute 4% back into our formula, we would then get,
t = ln (2) ln (1 + 0.04)
t = ln (2) ln (1.04)
Calculating this value would give us:
t = 13.86
So we could conclude that at a growth rate of 4% it would take 13.86 years for us to double our money.
Since that probably required us to do some thinking and pull out a calculator, let’s try our simplified rule of 70.
70 / r = t
If we substitute our values in, we would get:
70 / 5 = 14
As a result, we can conclude that the rule of 70 is pretty accurate. After all, the error was only 0.14 or just under 2 months. This isn’t horribly far off. Since the formula saved us a significant amount of time (and thinking) we can confidently state that the doubling time holds some accuracy.
Variations of the Rule
It can be hard to remember which formula is best for which situation, so if you can remember the rule of 70 you are likely in a pretty good place already.
However, when it comes to investing it is pretty hard to find a formula that encompasses all the possible scenarios. That’s why you may also hear about the Rule of 69 or the Rule of 72. Both rules are applied very similarly, but better represent different scenarios you might face. For example, the Rule of 72 is good for scenarios with larger growth rates. On the contrary, if the interest rate is smaller or is compounded on a more continuous process, the rule of 69 might be a slightly better bet.
The Rule of 72
Depending on who you talk to, you may hear some die-hard “Rule of 72” fans. Before you ask, yes they do exist. Although the rule is very similar to what we have just covered, let’s consider why.
In situations of annual compounding, especially, when the compounding rates are between 6% and 10% or other “normal” rates the Rule of 72 may prevail.
Sorry just one more example, we promise! Say our interest rate is actually 6%. If we calculate this using our proof from earlier, we would get:
t = ln (2) ln (1.06)
t = 11.9 years
If we used the rule of 70 we would’ve gotten:
70 / 6 = 11.67 years (which isn’t bad!)
However, using the Rule of 72 would have been even better, giving us an answer of:
72 / 6 = 12 years
This is also likely a little bit closer to math that we can solve in our heads.
Consider, as interest rates increase it is believed that this rule actually becomes less and less accurate. However, the Rule of 72 is very known for its convenience. This is because 72 has more factors than 69.3 or 70 and includes factors such as 1, 2, 3, 8, 9, 12, 18, 24, 36 and 72.
If you happen to be a banking specialist by trade, you might choose the Rule of 72 since it is a little bit easier to explain and execute for those who aren’t largely concerned about the variations of this rule.
The Rule of 69
When compounding happens on a more continuous rate, such as on a daily basis, the Rule of 69 will likely give you more accurate results. For those who are newer to the compounding game, continuous compounding refers to a growth rate in which interest is earned in infinitely small periods of time. Interest is then earned on the interest accumulated and works out to a higher amount than the interest that is compounded less frequently.
It is important to remember that an investment with continuous compounding grows faster than an investment with simple interest, so the typical formulas that would have been used or usually not quite so accurate.
Why 69 though? Well as the natural log of 2, we can simplify that for smaller growth rate values we can approximate that the natural logarithm is actually just equal to r. For improved accuracy, this value can also be calculated using 69.3 but this kind of defeats the purpose of using a formula that can be solved using mental math.
However, if you are good at dividing numbers then you can also use the formula:
69.3 / interest rate (percentage) = time in number of periods
Even if we are looking at instances of continuous compounding, it is easiest to consider the nearest integer with many possible factors. Hence, why we often refer to the Rule of 70.
Why do I need this rule?
Arguably our new golden rule (The Rule of 70) is a great tool to help us improve our mental math. It also helps to reduce the time it takes to compute complex formulas. Consequently, this shows that compounding growth is one of the most powerful tools when it comes to our savings. Even small changes can result in big changes (like doubling our money) over time.
You might just so happen to be someone who is about to invest your life savings. If this is the case, you probably have to decide between a few different investment options. While you should also be assessing the risk associated with each, you will likely also compare how fast your money will be growing. This is significantly easier since this formula allows you to compare how fast each investment will allow you to double your money. If you are pressed for time you can also consider what adjustments need to be made so you can reach your goal.
Knowing your growth rate, you can now determine the length of time it takes to double the population of a given country, your investment, a country’s GDP or the number of orange trees you are growing in your backyard. Yes, you could use it for just about everything.
Okay, but what if we know that something doubled over a given period of time. Could we figure out what the growth rate is? We’re glad you asked. Especially, since the answer to this question is also a resounding yes.
We may have lied before, this is actually the last formula you will see in this post. Say we actually only have 10 years to double our investment, at what interest rate would we need to be earning to reach our goal?
Looking back at our original formula:
70 / r = t
We can easily adjust this to instead solve our growth rate (r).
The formula would then look something like:
70 / t = r
In our example, we could then determine that
70 / 10 = 7 % growth rate
This means if our investment compounded annually, (since we specified we had 10 years to wait) we could need our money to grow at a 7% rate.
Limitations of the Rule
Okay, this rule sounds pretty good but remember that isn’t a full-proof. Remember that any growth rates are estimates and do not account for fluctuations in rates (which are very normal and expected in the real world) and do not take into account the inability to forecast the unforeseeable events in the future.
Especially in the world of finance, growth rates are not constant. What was a 4% growth rate one day may quickly drop to a more expected rate of 0.5%. With that in mind, when rates change the original use of the rule of 70 may prove to be slightly inaccurate. Therefore if the growth rate is anticipated to vary drastically, the rule of 70 may not be the best tool at your disposal.