The one lesson I truly wish I had learned earlier in life than I did is the Power of Compounding. **This one mathematical equation is the key element to achieving Financial Independence.** Albert Einstein has even been rumored as saying “*Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn’t … pays it*.” Whether this quote is truly attributable to the Father of Relativity or not is unimportant, the content, however, is dead on.

When I teach students about compounding, I pose a riddle which helps to illustrate its true power. Here it is:

“Would you rather have a penny that doubles everyday for 30 days or one million dollars?”

Let’s walk through the math: At the start of Day 1 the penny is worth just that $0.01, by the end of the day it doubles to $0.02, by the end of Day 2 it doubles again to $0.04 and by the end of Day 3 to $0.08 and so on… Here is a table of the first half of the 30-day time period:

At this half way point, I ask the class if they would like to change their selection. Often a number of students who originally picked the penny, thinking it was a trick question, will switch to the $1 million thinking the penny can’t possibly be the right choice if it’s only worth $327.68 after 15 days, with only another 15 days to go. We then proceed with the calculations:

As we move through the calculations, day by day, the students are amazed by how quickly the dollars grow toward the end of the 30-day period. You can see from the table that the penny crosses the million dollar mark on Day 27.

Beyond teaching the mechanics of compounding, this exercise helps to illustrate the need to be patient and the benefits of delayed gratification. Unfortunately, no one has a magic penny like this one, but the same principles apply to investing the dollars we save. Let’s take a look at how.

## The Formula

The formula for compounding a single value at the start of a given time period is 1 plus the rate of return for the period to the power of the number of periods times the investment amount, or:

Investment Amount x (1 + Rate of Return per Period)^Number of Periods

So for the Magic Penny example it would be $0.01 x (1 + 100%)^30

which equals $0.01 x 2^30 the solution for which is $10,737,418.24

(Stick with me here. Math isn’t all that bad, especially when it is working for you!)

## The Variables

There are only three variables in this equation, the *amount* to be compounded, the *rate of return*, and the *number of compounding periods*. In order to comprehend the power of this equation, all three factors should be examined. One observation is that the larger the amount of each variable, the larger the outcome. However, while doubling the amount of the investment, doubles the outcome, doubling the rate of return, and/or doubling the time period, has a far greater effect due to the nature of the exponential calculation.

Let’s look at the table below to see the effects of changes in rate of return and number of periods for a $100 lump sum investment at the beginning of the period:

If you look at the Value and Gain figures in the 5% rate of return columns, you will see that the $100 initial value grew to $128 in year 5 for a gain of $28 during the period. But, as the number of periods double (5 to 10, 10 to 20, and 20 to 40), the gains more than double each time and with increasing force. From period 5 to 10 the factor is 2.25x ($63/$28), from 10 to 20 the factor is 2.62x ($165/$63), and from 20 to 40 the factor is 3.66x ($604/$165).

Doubling the rate of return, in this case from 5% to 10%, also has an increasing impact on the factors: in the 5-year period the factor is 2.18x ($61/$28), in the 10-year period the factors expands to 2.52x ($159/$63), in the 20-year period the factor further expands to 3.47x ($573/$165), and in the 40-year period the factor expands even more to 7.33x ($4,426/$604).

Combining these two effects (doubling the rate of return *and* doubling the time period) you get truly powerful results. Comparing the 10-year 10% gain with the 5-year 5% gain you get a factor of 5.68x ($159/$28). Continuing along this path, the 20-year 10% gain vs. 10-year 5% gain the factor expands to 9.10x ($573/$63). And finally, the 40-year 10% gain vs. 20-year 5% gain the factor expands to **26.82x **($4,426/$165)**!**

Now you have the math behind how someone like Warren Buffett became a multi-billionaire. (He invested capital at a high rate of return for a long period of time!)

Just as in the magic penny example, the effects of compounding are most prominent at the tail end of the time period.

## Historical Rates of Return on Investments

In order to apply this power of compounding, we first need to discuss potential rates of return that one can realistically achieve. An appropriate approach is to look at history for guidance. A great resource for historical returns of stocks and bonds on the internet is from the NYU Stern School of Business. They maintain a chart of historical annual returns for stocks (S&P 500), bills (3-Month Treasury Bill) and bonds (10-year Treasury Bonds) going back to 1928. Open the excel link at the top of the chart to view the arithmetic and geometric averages for various time periods located at the bottom of the spreadsheet. The geometric average is the most appropriate figure to use when measuring performance results of an investment.

For the entire *90-year period* (1928-2017), the average annual return was 9.65% for stocks (S&P 500 Index) and 4.88% for bonds (constant maturity 10-Year Treasury Bond). For the most recent *50-year period* (1968-2017), the average annual return was 10.05% for stocks and 6.76% for bonds. And for the most recent *10-year period* (2008-2017) which started just before the 2008-2009 recession and market decline, the average annual return was 8.42% for stocks and 3.86% for bonds.

I should point out though that these are averages. The returns in any one year vary substantially from the average, especially for stocks. The higher volatility inherent in stock investing is what an investor accepts in exchange for the potential for higher returns. However, volatility of return lessens as holding periods lengthen.

As I wrote in the letter to my daughter a simple approach to replicate investing in these major asset classes is to utilize a low-cost index fund. Major providers, such as Vanguard, Fidelity, Schwab and others offer S&P 500 Index, Total Stock Market Index and Total Bond Market Index funds at extremely low cost, which can be used to easily invest in these broad asset categories.

## A Real-Life Compounding Example

Given our new appreciation of compounding, coupled with our observation of historical returns, let’s run a few scenarios for two hypothetical investors. The first investor, Ms. Early Bird, starts investing as soon as she starts her first job after college or trade school at age 22 investing 20% of her $36,000 starting salary at the end of every month ($600/month). (She is able to put away this percentage and possibly even more since she has yet to settle into a lifestyle that her new salary level might allow.)

Since she is investing for the very distant future (10+ years) she chooses to invest in an S&P 500 Index fund. Using history as a guide, let’s use 9% as her rate of return for this example. Now let’s see what her projected balance is at ages 32, 42, 52 and 62, after 10 years, 20 years, 30 years and 40 years respectively.

Note, this calculation is more involved than our original compounding example as it is a series of multiple investments (monthly) rather than a single amount at the beginning of the period. To perform this calculation it is easiest to use a Time Value of Money (TVM) calculator.

The inputs for the calculation are as follows: Mode = End (she is investing at the end of every month); Present Value = 0 (she is starting out with no money); Payment = -600 (she is investing $600 per month, but since this is an outflow the number is negative); Future Value = leave blank (this is what we will be solving for); Annual Rate (%) = 9 (the rate of return we are using is 9%, be sure to enter “9” and not “0.09”); Periods = 120 (Our first calculation will be for 10 years which is 120 periods, 10 years x 12 months); Compounding = Monthly (she is contributing on a monthly basis).

Once these figures are inputted, just click on the “FV” button to reveal the answer ($116,109 at age 32). Now you can just change the Periods figure to 240 for 20 years and click “FV” for the 20-year result ($400,732 at age 42), 360 for 30-years ($1,098,446 at age 52) and 480 for 40 years ($2,808,792 at age 62).

Now let’s see how a second investor, Mr. Procrastinator, fares. He has too many things going on to be bothered with investing when he first starts working and doesn’t get around to investing until he is 37. While he makes far more money ($60,000 per year) than Ms. Early Bird, he has grown into a lifestyle and has a lot of obligations and can only cut back enough to save 10% of his income ($500) per month. Given his shorter time frame and lower investment amount, even with the same rate of return of 9%, his balances at age 42 (5-years, or 60 months), 52 (15-years, or 180 months) and 62 (25-years, or 300 months) would be $37,712, $189,203 and $560,561 respectively.

Ms. Early Bird clearly has the advantage by starting much earlier and saving a little more per month even though she makes far less. Delaying the start of your investment time line can have serious consequences. And time is the one thing we can never get back.

## The Key to Wealth Creation

** I believe that the most important factor in achieving financial independence is harnessing the power of compounding.** You can cut expenses, live below your means and save as much as possible, but without the added benefit from compounding, reaching financial independence may be a significant challenge. I wish I had learned this lesson earlier in life than I did, but fortunately I wasn’t too late. I hope you find this lesson as valuable as I did when it finally became clear to me – it was truly a life changer.