Compounding must be one of the most difficult operations to do in your head which is unfortunate because it is also so important. We vastly underestimate how much something grows when it compounds over time because of compounding's difficulty. What our brains cannot grasp on the fly tends to be ignored or misunderstood.
But there are ways of doing compounding in your head that are not quite as difficult as you might think.
It helps if you work with a particular growth rate often. 10% per year, for example, is particularly easy because it can be found in Pascal's Triangle. If you know that row 4 of Pascal's Triangle is
1 4 6 4 1
Then you know that every dollar compounded at 10% a year compounds in 4 years to
1×$1 + 4×$0.10 + 6×$0.01 + 4×$0.001 + .1×$0.0001
= $1.4641
$10,000 compounded at 10% a year will grow to $14,641 in 4 years.
Note that if there were no compounding, if someone you lent money to paid you 10% interest on the $10,000 principal each year then paid back the amount borrowed, you would have $10,000 + $1,000×4 = $14,000.
Compounded growth, in this case, gave you $641 beyond what you would have with simple growth, where the growth is not itself growing. This is what makes it so difficult to anticipate. If you know that 4 years of compounding 10% interest gives you 6.41% more - over half a year's additional interest - it doesn't help you anticipate how much more you will have in 10 years of compounding at 10%... $10,000 compounded for 10 years at 10% a year grows to $25,937 and change. This is now $5,937 more than simple interest over that time period would have yielded. Put another way, compounding for 10 years at 10% is the same as growing at 10% simple for 15.937 years. You get an extra 5.937 years of simple 10% return after 10 years versus only 0.641 years additional at 4 years.
Put another way, by increasing your time invested by 150%, you increased your total return by 343%.
And as you continue, the gap itself between a longer and shorter investment period itself compounds and at a rate that is counterintuitively large.
So how do you do compounding in your head if you don't have Pascal's Triangle?
One way is to memorize the doubling time associated with a given rate of return. For 10%, that time is about 7.2 years, something you can calculate in your head using the Rule of 72:
Doubling time in years = 72 ÷ % growth rate per year
Note that for 10%, you don't divide 72 by 0.1 but 10.
7.2% is particularly convenient because it has a doubling time of 72 ÷ 7.2 or 10 years. And, to make life particularly easy, 7.2% is the long-term after-inflation total return of the United States stock market over the very long run. So take whatever you have invested in stocks today and, if you leave it alone and don't tinker, in 10 years, you should have doubled your purchasing power. Note that the dollar amount will be even greater but when you take inflation's growth over that decade, its spending power would have doubled.
Most of us invest over our working lifetimes - regular dollar amounts each year. Every $1 a year invested at 7.2% a year will give you a portfolio worth $14 in a decade.
This is because the first $1 doubles and the last dollar invested - the one invested in year 9 - grows only to $1.072 (7.2%×1 year). Every dollar in years 2 through 9 grows somewhere in between these extremes. The average dollar grows 40%, as it turns out.
This allows you to do the most difficult compounding problem in your head - how much must you invest each year to have a portfolio with $1 million of spending power in today's dollars?
You simply make a table for each decade, starting with the simplest amount, $1 invested each year.
Years 1-10: $1 a year grows to $14.
Years 11-20: $14 at the start of decade doubles to $28.
$1 a year adds another $14 for a total of $42
Years 21-30: $42 doubles to $84.
$1 a year adds another $14 for a total of $96
Years 31-40: $96 doubles to $192.
$1 a year adds another $14 for a total of $206.
Phew! If you don't want to work this out each time, you could just remember the amounts you will have at the end of each decade:
10 years $14
20 years $42
30 years $96
40 years $206
Notice something about this series: like all compounding values, its growth per decade in dollars seems to accelerate from +$14 to +$28 to +$54 to +$110 the final decade. The doubling of the portfolio swamps the effect of the amount added each decade, an effect that you may have experienced later as retirement approaches and you feel less motivated to keep making contributions since they have a shrinking impact on a portfolio much larger than when you started.
Someone who starts such a systematic plan at age 21 would have $206 in today's money in at age 61 for every dollar invested each year.
So, to answer the question of how much is needed to have $1 million, its a matter of simply scaling that $206 to get $1 million:
If $1 per year compounds to $206 in 40 then
$x per year compounds to $1 million...
$1m / x = 206 / 1
x = $1m / 206 = $4,854, rounding off
So, if you invest $4,854 each year in your 401-k or IRA and get 7.2% a year on average for the next 40 years, you will have $1 million of today's purchasing power.
That's just $186.69 per paycheck if you get paid every 2 weeks* or just $13.34 a day.
And, to periodically check if you're on target, you can crunch the numbers in your head!
* Actually, a little less. The $14 per decade assumes that you make the entire contribution in one lump sum each year. You must invest less per year if you spread your contributions out over the year, something especially true if you are investing into a mutual fund or ETF whose value fluctuates frequently.
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