If you read my article, Can You Solve this Math Problem, than you know I enjoy math. Numbers have always been my friend, and I have a few to share with you today. So let’s get to it.

**Rule of 72**

Many of you have heard of the Rule of 72, but let’s review just in case. To estimate the time it will take to double your money, divide 72 by the expected growth rate, expressed as a percentage. For example, if you expect to earn 10% per year on a $10,000 investment, it will double to $20,000 in about 7.2 years (72 / 10).

Now here is a neat way to use the Rule of 72 to determine annual growth rate. Let’s assume that in year 1 a company earned $2 per share, and by year 8 it was earning $8 per share. What was the annual EPS growth rate? Using the Rule of 72 makes answering this question easy. First, how many times did the EPS double over the eight year period? It doubled once from $2 to $4 and a second time from $4 to $8. Doubling twice in eight years means that the EPS doubled once every four years. Using the Rule of 72, we know that to double in 4 years the EPS must have grown at an annual compound rate of 18 (72 / 4). So the company’s EPS have grown at an annual rate of 18% over the past 8 years.

By the way, you can do the same thing to determine the growth rate of your salary (if that’s your thing). And you can use the Rule of 72 to determine, at a given inflation rate, how long it will take for your money to buy half of what it can by today (depressing).

**The Rules of 71, 70 and 69.3**

These rules are for us math geeks. They do the same thing as the Rule of 72, but are considered more accurate depending on the interest rate and compounding period (e.g., continuous, daily, annually). The rule of 71 is the most accurate when dealing with annual compounding. And the rule of 69.3 is more accurate for continuous or daily compounding. The Rule of 70 comes in because who wants to divide an interest rate into a number like 69.3? Ok, some do, but many don’t.

**The Rules of 114 and 144**

The Rules of 114 and 144 take the Rule of 72 to the next level. Rule of 114 can be used to determine how long it will take an investment to triple, and the Rule of 144 will tell you how long it will take an investment to quadruple. For example, at 10% an investment will triple in about 11 years (114 / 10) and quadruple in about 14.5 years (144 /10).

There is an important implication to the Rules of 72, 114 and 144. Notice that the numbers don’t double? That is, while it takes the interest rate divided into 72 to double, the interest rate divided into 144 doesn’t triple, it quadruples! That’s the power of compounding. And what’s the moral of this story–Save early and save often.

**The Rules of 1.5 and 1,080,000**

Here is where we calculate how long it will take you to be a millionaire. Let’s start with the Rule of 1.5, also known as Felix’s Corollary. This rule states that for a stream of investments (we’ll assume annual investments) where the number of years times the interest equals 72 (the Rule of 72 is back!), the ending value will equal approximately 1.5 times the amount invested. For example, investing $10,000 per year for 8 years at 9% interest (8 * 9 = 72), the value of the investments at the end of year 8 will equal about $120,000 ($10,000 * 8 * 1.5).

We can now use this information to create a How Long Will It Take You To Be A Millionaire calculator (the Rule of 1,080,000). Using Felix’s Corollary, all we need to do is figure out how long it will take you to save $720,000 at a given interest rate. Why $720,000? Because 720,000 times 1.5 equals 1,080,000 (which explains why I didn’t use 1,000,000). Trust me, this is easier than it looks.

For example, over 8 years to save $720,000 you need to save $90,000 per year. And at 9% annual interest, you would accumulate $1,080,000 over this 8 year period. Now I know must of us don’t have $90,000 per year to save, which is why most of us won’t accumulate a million dollars in 8 years. So let’s stretch it out to 16 years. Now what do we need to save to be a millionaire, again assuming a 9% rate of return? Well, using our friend the Rule of 72, we know that whatever we have saved over the first 8 years will double over the next 8 years because 72 divided by our interest rate of 9% equals 8.

So we can break the 16 year savings period into 3 equal portions: (1) what we save the first 8 years; (2) the doubling of this amount over the next 8 years; and (3) what we save the second 8 years. So 720,000 divided by 3 equals 240,000, which is what needs to be saved each of the two 8 year periods, or $30,000 per year. That comes out to $2,500 per month, which is doable for some.

If you want to estimate what it will take to be a millionaire in 24 years, just divide 720,000 by 7 (a question about this 7 comes at the end) and then again by 8. So, 720,000 divided by 8 equals 90,000 divided by 7 equals about $12,800. Thus, investing just over $1,000 per month at 9% interest over 24 years will make you a millionaire.

So the question for this last example is where does the number 7 come from? For 16 years we divided by 3, so why for 24 years are we dividing by 7? Leave a comment to let us know what you think. And finally, if the Rules of 1.5 and 1,080,000 on a Monday morning are just too much to take, you can always check out this Millionaire Calculator.

**Topics:**Investing

Get an HP12C calculator…!!!

Math is cool, well sometimes when it can save you time or make you money or help you out in some other way. I had never heard of these math rules before. Big eye opener!

Too many rules of thumbs. All your rules of thumb are exact at only one interest rate. and most of the time the math is not too difficult.

one equation:

Number of years = ln(# of multiples)/ln(1+int rate per unit)/ units

For instance, to double at 10% compounded annually (10% = .1)

annual ==>unit =1

Quarterly ==>unit = 4 (per year)

Monthly ++> unit = 12 (per year)

Time = ln(2)/ln(1+.1) = 7.3 years

To double at 10% compounded quarterly

Time = ln(2)/ln(1+.1/4)/4 = 7.02 years

to triple at 8% compounded monthly

time = ln(3)/(ln(1+.08/12)/12 = 13.78 Years

Without compounding:To triple at 8%

P+P*I*T= 3P, P=Principle, I = Interest Rate per time increment, t=time

3=1+IT or T=2/I or T=2/.08=25 years Compounding cuts the time almost in half.

Nice.

Even logarithms to base 10 will do.

log (how many times you want to multiply your investment)/ log (1+ R/100) = Number of years it will take.

I use capital R for the rate of interest. So if the rate is 10 per cent, the denominator in the equation will be log (1.1)

Most smartphone calculators these days have the log function. Why waste time figuring out approximations when these formulae will give you the exact figure!!!

My only question is where I can earn 9 and 10% interest? Very interesting numbers though, thanks for sharing.

Intresting facts, are you sure?

Hi. Glad you found Felix’s Corollary and the Millionaire’s Estimation handy.

This is to answer your question “Where does the 7 come from?” when you have three 8-year periods at 9%:

The amount accumulated in the first 8 years (principal and interest) will double in the second eight years, and double again in the third eight years. So that’s doubling twice (or 4 times the original), which accounts for 4 of the 7 in question.

The amount accumulated in the second eight years (principal and interest) will double during the third eight years. So that’s another 2 of the 7.

And finally, the amount accumulated in the third eight years (principal and interest) makes up the 7th of the 7.

Hope that helps!

– Felix, of “Felix’s Corollary”

“You can earn roughly 50% on your money, by spreading a planned investment over N annual payments at a rate of R%, if NxR = 72” – JF

Another little trick will let you calculate your savings assuming a steady contribution plus compound interest.

Say you want to save $4k per year for 30 years and earn 10% interest. How much money will you have at the end? The trick is that adding $4k per year is like getting 10% interest on $40k. So we can set the base principle to $40k and ignore the yearly addition of $4k. Then subtract the magical $40k at the end.

$40k * 1.1^30 – $40k = $657k

hmm! logarithms are magically delicious!

All I can say is….

whoa.

My husband has never shared some of these with me – and he’s securities licensed! Great information, thanks!

Holy cow. My brain nearly exploded from all the numbers. Although I can see some of those methods saving time if you get used to using them, I think my feeble mind will stick with the rule of 72!