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# The rule of 72 for compound interest

## Video transcript

in the last video we talked a little bit about compounding interest and our example was calm interest that compounds out annually not continuously like we would see in a lot of banks but I 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 I mean if you have a reasonable calculator you can solve for some of these things if you know how to do it but it's very different it's nearly impossible to actually do it in your head for example at the last at the end of the last video we said hey if I have \$100 and if I'm compounding at 10 percent a year that's where this one 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 a 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 I say this oh sorry this should be log base 10 of 1.1 and I say this because most calculators have a log base 10 function and this and this are equivalent and I've proven it in other videos so in order to say how long does it take to double my money at 10 percent a year you'd have to put that in your calculator and let's try it out let's try it out right here so we're going to have 10 2 and we're going to take the logarithm of that just 0.3 / / o to open parentheses just to be careful divided by 1.1 and the logarithm of that and we close the parentheses is equal to seven point two seven years so roughly seven point three years so this is roughly equal to seven point three years and 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 I want to show you is a is a is a rule to approximate this question how long does it take for you to double your money and that rule this 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 so let's say I have 10% 10% compounding annually compounding compounding annually 10% interest compounding annually using the rule of 72 I say how long does it take for me to double my money I literally take 72 I take 72 that's why it's called a 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 divided by 10 and I get seven point two it was annual so seven point two years if this was 10% compounding monthly it would be seven point two months so I got seven point two years which is pretty darn close to what we got by doing all of that fancy math similarly let's say that I'm compounding let's do another problem let's say I'm I have I'm compounding six let's say 6% compounding annually compounding annually it's like that well using the rule of 72 I just take 72 divide it by the six and I get six 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 log basse anything really of to / well this is where we get the doubling our money from the two means two times our money / base log base of whatever this is ten of in this case instead of a one point one is going to be one point oh six so you can already see it's a little bit more difficult get our calculator out so we have to log of that / 1.06 log of that is equal to eleven point eight nine so about eleven point nine so when you do all the fancy math we got eleven point nine so once again you see this is a pretty good approximation and this math this math is much much much simpler than this math and I think most of us can do this in our head 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 okay here's the different interest rates this is the actual time it would take to double so I'm I'm actually using this formula right here to figure out the actual the precise amount of time it'll take to double let's say for this is in years if we're compounding annually so if you had one percent it will take you seventy years to double your money at 25 percent it'll only take you a little over three years to double your money so this is the actual this is the correct this is the correct and I will and I'll do this in blue this is the correct number right here so this is actual right there that right there is the actual and I've plotted it here - if you look at the blue line that's the actual so I didn't plot all of them if you started I think I started it maybe four percent so if you look at four percent it takes you seventeen point six years to double your money so four percent it takes seventeen point six years to double your money so that's that dot right there on the blue at five percent it takes you at 5% it takes you 14 years to double your money and so this is also giving an appreciation 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 so it takes you you double your money twice as fast so it really is really important especially for think about doubling your money or even tripling your money for that matter now in red in red over here I said what is the rule of 72 predict this is what the rule is so if you just take 72 and divide it by one percent you get 72 if you take 72 divided by four you get 18 rule of 74 rule of 72 says it'll take you 18 years to double your money at a four percent interest rate when their actual answer is 17 point seven years so it's pretty close so that's what's in red right there that's what's in red right there and you can see so I've plotted it here the curves are pretty close for low interest rates for low interest rates so that's these interest rates over here the rule of 72 the rule of 72 slightly slightly overestimates 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 and just if you have to think about you know is gee is 72 really the best number well this is kind of what I did if I just 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 well when I graphed here 4 percent all the way to 25 percent 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 I have a I have a I don't know a 4 well I already did that let's say I have a a 9 percent 9 percent annual compounding how long does it take for me to my double my money well 72 divided by 9 is equal to 8 years it'll take me 8 years double my money and the actual answer if this is using this is the approximate answer using the rule of 72 the actual answer nine percent is eight point zero four years so once again in our head we were able to do a very very very good approximation