Compound interest and doubling your money

There are some basic rules of finance whose understanding is integral to the proper management of money.  The first rule has been to understand the concept of the time value of moneyInterest rates drive all things finance; from credit card debt and house buying to savings accounts and bond investments.  The concept of the time value of money is rooted in mathematics and can be cumbersome to commit to memory.  The good news is that there has long been a rule that is easy to commit to memory and serves as the 95% solution to figuring out how much an asset can make you or how much a loan can cost you.

The rule is called “The divide by 72 rule“.  If you already know of this rule, you’re ahead of many.  However, do you know why that rule works, or even whether it actually does or not.  Anything based math should not be taken on faith, but instead proved through relationships.  The 72 rule states:

To quickly calculate the time in years to double your money in an investment due to compounding interest payments, divide 72 by the annual interest rate.  For example an 8% interest rate would take (72/8) 9 years to double the money invested.

This can work with interest rates that are compounded annually or even in shorter terms, so long as the annual effective rate is used in the calculation.

Rate Rule of 72 Annual Delta %
1% 72.0 69.7 3.2%
6% 12.0 11.9 0.9%
11% 6.5 6.6 -1.5%
16% 4.5 4.7 -3.8%
21% 3.4 3.6 -6.1%
26% 2.8 3.0 -8.3%
31% 2.3 2.6 -10.5%
36% 2.0 2.3 -12.7%

>>>The above table shows a comparison between the rule of 72 and annually compounded interest rates.  Notice that the error is low for interest rates that we commonly calculate.  The formula for the time to double your money is Time=ln(2)/ln(1+rate), where the time is in years and the interest rate is expressed as a percentage.  The divergence at higher interest rates from the rule of 72 is cause by the rule of 72 actually being tied to the formula for continuously compounding interest.  This is where interest is always compounding upon itself.  The formula for doubling you money in this case in Time=ln(2)/rate.  This formula tracks perfectly with a rule of 69.3 as shown below.  The reason that 72 is used is because “head math” with 72 works easier as 72 has plenty of multipliers (1,2,3,4,6,8,12…and so on) that are common interest rates.

Rate Rule of 69.3 Annual Delta %
1% 69.3 69.3 0.0%
6% 11.6 11.6 0.0%
11% 6.3 6.3 0.0%
16% 4.3 4.3 0.0%
21% 3.3 3.3 0.0%
26% 2.7 2.7 0.0%
31% 2.2 2.2 0.0%

 

 

 

Retirement planning using cash flows

The question comes up often as to what amount is needed to retire.  Money in the bank and diversified assets are indeed important, but the critical measurement of what gets you buy on a daily, monthly, and yearly basis is your income.  For most, a retirement will be short lived if you live off of only the cash that you have on hand.

Ideally, your retirement income will come from multiple robust sources.  This will act as a hedge against having your income collapse in the event of adverse conditions.  Social Security may play a part in this, however, if it plays a substantial part for your retirement planning, you are likely short of what you truly desire for income in retirement.  So, how much is needed?

The current median income in the top quintile is currently $111,000 a year.  This means that this income is higher than 90% of all American households since it is in the middle of the top quintile (80% to 100%).  If your income is $111,000, first of all congrats on the hard work.  Second, your current lifestyle in retirement can be sustained on less.  The amount required is about 50% as work related expenses, commuting costs, and taxes at the new 50% income will decrease.

Now the question becomes, how is a retirement had at $55,500 per year?  To answer this there is a simple formula that FPO has developed as a starting point as well as a personal goal to shoot for.  This amount can be met by accumulating nine buckets of $500 extra income per month as described in the four part FPO series (1, 2, 3, 4) on creating extra income.  At least seven of these buckets must come from income generators in the first three parts, with two being permitted to be from part 4.  More than 2 from part 4 is welcome, however the effort to create additional savings would not be beneficial at the expense of achieving that seventh bucket.

>>> When broken down like this, seven buckets seems achievable, right?  Of course it is, but it is never explained like this.  Many financial experts frame a funded retirement as something that only the most disciplined can achieve, and that it is some sort of lofty goal.  In reality, it is easier than advertised when broken into chunks.  However, expect that it won’t happen overnight.  But aim for creating one of the seven every five years and you’re set.  Start at 25 and retire at 60.

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