In the last video, we talked a little bit about compounding
interest. Our example was interest that compounds
annually, not continuously like we would
see in a lot of banks.
I'll really just wanted to let you understand that although
the idea is simple.
Every year you get 10% of the money that you started off
with that year.
And it's called compounding because the next year you get
money not just on your initial deposit, but you also get
money or interest on the interest from previous years.
That's why it's called compounding interest. And
although that idea is pretty simple, we saw that the math
can get a little tricky.
If you have a reasonable calculator, you can solve for
some of these things if you know how to do it.
But it's nearly impossible to actually do it in your head.
For example, at the end of the last video, we said if I have
a hundred dollars.
And if I'm compounding at 10% a year, that's where this 1
comes from, how long does it take for me to double my money
and end up with this equation?
And to solve that equation, most calculators don't have at
log base 1.1.
And I've shown this in other videos.
This you could also say, x is equal to log base 10 of 2,
divided by log base 1.1 of 2.
This is another way to calculate log base 1.1 of 2.
This should be log base 10 of 1.1.
I say this because most calculators have
a log base 10 function.
And this and this are equivalent.
I've proven it in other videos.
So in order to say how long does it take to double my
money at 10% a year?
You'd have to put that in your calculator.
And let's try it out.
Let's try it out right here.
We're going to have, 2, and we're going to take the
logarithm of that, 0.3, divided by 1.1, and the
logarithm of that.
We close the parentheses.
Is equal to 7.27 years.
Roughly 7.3 years.
So this is roughly equal to 7.3 years.
As we saw in the last video, this is not necessarily
trivial to set up, but even if you understand the math here,
it's not easy to do this in your head.
It's literally almost impossible to do in your head.
So what I want to show you is a rule to
approximate this question.
How long does it take for you to doubles your money?
And that rule is called the rule of 72.
Sometimes it's the rule of 70 or the rule of 69.
But rule of 72 tends to be the most typical one, especially
when you're talking about compounding over
set periods of time.
Maybe not continuous compounding.
Continuous compounding you'll get closer to 69 or 70.
But I'll show you what I mean in a second.
So to answer that same question, let's say I have 10%
compounding annually.
Using the rule of 72, I say how long does it take for me
to double my money?
I literally take 72, that's why it's called the rule of
72, I divide it by the percentage.
So the percentage is 10.
It's decimal representation is 0.1.
But it's 10 per 100 percentage.
So 72 two divided by 10.
And I get 7.2, it was annual, so 7.2 years.
If this was 10% compounding monthly, it
would be 7.2 months.
So I got 7.2 years, which is pretty darn close to what we
got by doing all that fancy math.
Similarly, let's say that I'm compounding--
let's do another problem.
Let's say I have 6% compounding annually.
Using the rule of 72, I just take 72 divided by the 6.
And I get 6 goes into 72, 12 times.
So it'll take 12 years for me to double my money, if I'm
getting 6% on my money compounding annually.
Let's see if that works out.
So we learned last time, the other way to solve this would
literally be, we would say x, the answer to this, should be
close to log base anything of 2 divided by-- this is where
we get the doubling our money from, the 2 means 2 times our
money-- divided by log base whatever this is 10 of.
In this case instead of 1.1 it's going to be 1.06.
So you can already see it's a little bit more difficult.
Get our calculator out.
So we have 2, log of that, divided by 1.06, log of that,
is equal to 11.89.
So about 11.9.
So when you do all the fancy math, we got 11.9.
So once again you see this is a pretty good approximation,
and this math is much simpler than this math.
And I think most of us can do this in our heads.
So this is actually a good way to impress people.
And just to get a better sense of how good this number 72 is,
what I did is I plotted on a spreadsheet.
I said, OK, here's the different interest rates.
This is the actual time it would take to double.
So I'm actually using this formula right here to figure
out the precise amount of time it'll take to double.
Let's say this is in years if we're compounding annually.
So if you're at 1%, it'll take you 70 years
to double your money.
At 25% it'll only take you a little over 3 years to double
your money.
This is the correct-- and I'll do this in blue--
number right here.
So this is actual.
And I plotted it here, too.
If you look at the blue line, that's the actual.
I didn't plot all of them.
I think I started at maybe 4%.
So if you look at 4%, it takes you 17.6 years
to double your money.
So 4%, it takes 17.6 years to double your money.
So that's that dot right there on the blue.
At 5% percent, it takes you 14 years to double your money.
And so this should also give you an appreciation that every
percentage really does matter when you're talking about
compounding interest. When it takes 2%, it takes you 35
years to double your money.
1% takes you 70 years.
You double your money twice as fast. It it really is
important, especially if you think about doubling your
money, or even tripling your money for that matter.
Now in red, I said what does the rule of 72 predict?
So if you just take 72 and divided it by 1%, you get 72.
If you take 72 divided by 4, you get 18.
Rule of 72 says it'll take you 18 years to double your money
at a 4% interest rate, when the actual
answer is 17.7 years.
So it's pretty close.
So that's what's in red right there.
So I've plotted it here.
The curves are pretty close.
For low interest rates, so that's these interest rates
over here, the rule of 72 slightly over estimates how
long it'll take to double your money.
And as you get to higher interest rates, it slightly
underestimates how long it'll take you to double your money.
If you had to think about is 72 really the best number?
Well this is kind of what I did.
If you just take the interest rate and you multiply it by
the actual doubling time.
And here you get a bunch of numbers.
For low interest rate 69 works good.
For very high interest rates 78 works good.
But if you look at this, 72 looks like a pretty good
approximation.
You can see it took us pretty well all the way from 4% all
the way to 25%.
Which is most of the interest rates most of us are going to
deal with for most of our lives.
So hopefully you found that useful.
It's a very easy way to figure out how fast it's going to
take you to double your money.
Let's do one more, just for fun.
Let's say I have a 9% percent annual compounding.
And how long does it take for me to double my money?
Well 72 divided by 9 is equal to 8 years.
It'll take me 8 years to double my money.
And the actual answer-- this is the approximate answer
using the rule of 72-- 9% is 8.04 years.
So once again, in our head we were able to do a very good
approximation.