# The Rule of 72

Recently someone asked why the rule of 72 works. It’s interesting how we take things for granted and pass the information on to others who treat is as magic or something close.

While magic and miracles are part of my plan, I find it works better if I understand how they may come to pass and organize situations to make them possible.

### So the Rule of 72

The idea is we have a dollar compounding at some rate, i. We want to know how many years it takes for (1+i/100)^n to equal 2. We solve for that and find

n times LOG(1+i/100) = LOG(2) or n = LOG(2)/LOG(1+i)

We could look up a logarithm table or a spreadsheet and find

for i =1%, n = 69.6607 years and   Rule of 72 says 72.0 (too Long)

for i = 2%, n = 35.0028 years and Rule of 72 says 36.0 (too Long)

for i = 7%, n = 10.2448 years and Rule of 72 says 10.3 (too Long)

for i = 8% n =   9.0065   years and Rule of 72 says 9.0 (not long enough)

We know the rule gives an approximation that is accurate between give or take 3% for interest rates between 2% and 14%. (Adequate for most calculations) At 14 % the rule says it will take 5.143 years to double while the precise time is 5.290 years. About 7 weeks different.

If we want a 3% variation maximum, as rates rise you use a bigger number instead of 72. Using 75 instead of 72 will work  between 14% and 24%, 78 will work better between 23% and 34% and so on as rate are 10% higher add 3 to the rule. I have tested it up to 45%. Its an approximation remember.

Approximations are guidelines and for some reason we think about a time to double easier than an interest rate. That’s the real strength of the rule of 72.   We know 72/i = time to double approximately.

### Point of interest (1)

The last double is the one that matters. Suppose I can invest at 9% and so money will double every 8 years. In 40 years I will get 5 doubles. 2^5 = 32 so 32 times my money. \$1,000 is now \$32,000. At 32 years, just 4 doubles, I would have \$16,000.

The first four doubles added \$15,000 to my horde. The fifth double added \$16,000

You start early to make room for the last double in your time frame. If you start 8 years late you have to buy back the time. So start with twice as much money.

### Point of Interest (2)

If you let it compound at 25%, say the rate on an overdue credit card, the debt will be 10 times greater in 10.3 years. You can work it out. LOG(10)/LOG(1.25). Frightening.

### Point of Interest (3)

Maybe you are looking for the rate you want in a given period. Let’s say someone offers a 10-bagger in three years. LOG(1+i) = LOG(10)/3, and LOG(1+1) = 0.333333. The required rate of return is (10^.33333-1)% or 115% . How much would you have after 4 years at his rate? 21.5 times. Twenty-one baggers are rare and you can imagine why. 115% rates are nearly unknown.

### The Takeaway

Some basic arithmetic can give you insights into financial things that matter to you.

1. Don’t let credit cards compound
2. Don’t start saving late
3. Don’t believe rates higher than about 12% without solid evidence.

I help people have more retirement income and larger, more liquid estates.

Call in Canada 705-927-4770, or email don@moneyfyi.com

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