Power of Compounding

Options are necessary tools for serious investors. The reason is not due, as many suspect, to the leverage they provide. It is not due to the small, limited potential losses to which option buyers are exposed. While these are certainly benefits of options, the real benefit is that options allow you to strategically manage risk and access the greatest financial power of all – compound interest. Because options hedge risk, you can eliminate potential unwanted risks by purchasing options or by collecting money for accepting risks you’re willing to take. Both of these actions allow investors to tap into the most powerful of all forces – compounding. 

It is reported that Albert Einstein, when asked to give his opinion on the greatest invention of all time simply stated, "compound interest." Why would a brilliant physicist pick such a seemingly simple answer? It's because he saw time as the fourth dimension of the universe and recognized the power it had to create wealth if we allowed it to work on our money. Time normally has harmful effects on assets. Cars depreciate, bridges crack, factories rust, and plumbing leaks. But this is not so for investments. Investments earn money and then they earn money on that money. It is the “interest on interest” that creates the magnificent compounding force that you will soon agree should be considered the greatest invention of all time. If not, then it is certainly the greatest invention for investors.

This reference guide explains the science behind the answer. If you understand compounding, it will change the levels of risk you are willing to take and the way you pick your investments.

Understanding Simple Interest

Most of us understand the idea behind interest. It is money paid to another for the use of money. However, there are many ways to calculate how that interest is paid and, depending on the method, the differences can be dramatic.

To understand compound interest, let's first look at the most basic of all calculations, which is called “simple interest.” With simple interest, you only earn interest on the principal, which is the amount initially invested. For instance, if you deposit $1,000 in an account that earns 5% simple interest per year then you will earn $1,000 * .05 = $50 per year. That’s all you’ll ever get each and every year. It does not matter that the total account value is growing because interest is only paid on the original $1,000 principal.

The formula for simple interest is:

Interest = Principal * Rate * Time

Using the formula, $1,000 invested at 5% for two years would yield $1,000 * .05 * 2 = $100 interest for a total account value of $1,100. For a five-year period, the interest would equal $1,000 * .05 * 5 = $250 for a total account value of $1,250. Again, $50 per year (5% of $1,000) is deposited to the account each and every year regardless of the account size or duration of the investment.

Understanding Compound Interest

Rather than calculate interest on the initial investment, we can also calculate it based on the total account value. When interest is paid on the total account value, you earn interest on the principal but you also earn interest on the interest.

To demonstrate, let’s use the previous example and assume you deposit $1,000 that earns 5% per year. But this time we’re going to compound the interest.

After the first year, your account will be worth $1,000 * 1.05 = $1,050 just as it would be with simple interest. However, in the second year, you will earn the 5% interest on the $1,050 balance for a total of $1,050 * 1.05 = $1,102.50. Let’s take a closer look at what’s happening with that calculation.

At the end of the first year, we said your account is worth $1,050, which can be broken down into the two component parts of $1,000 principal plus $50 interest. In the second year, you are paid interest on the initial $1,000 investment but also on the $50 that was deposited to your account as interest. The grand total paid to the account in the second year is then $52.50 as follows:

$1,000 * .05 = $50

$50 * .05 = $2.50

Because your account balance was initially $1,050 then it must be $1,050 + $52.50 = $1,102.50 after the second year, which is exactly what we calculated by taking 5% of the total account value. The important point to understand with compounding is that you earned $50 on your principal but you also earned an additional $2.50 that would not have been realized under simple compounding. Okay, so the difference is nothing to get too excited about now. But wait until you see what happens years later.

To understand the effects of compounding, it will help to have a formula rather than going through the individual steps for each year. Fortunately, the formula for compounding is simple. If you deposit $1,000 and earn 5%, you will have $1,000 * 1.05 = $1,050 after the first year. Because interest is paid on the entire balance, you will have $1,050 * 1.05 = $1,102.50 after the second year. All we have to do is multiply the total account balance by one plus the interest rate (where the interest rate is expressed as a decimal) for each year the investment is held.

If the investment is held for two years, you would have $1,000 * 1.05 * 1.05 = $1,102.50 which is what we calculated previously. Because we are multiplying by a constant 1.05 for each year then that is mathematically equivalent to 1.05 raised to the amount of time the investment is held. For instance $1,000 * 1.052 = $1,102.50.

So the general formula for compound interest is:

Principal * (1+interest rate) time

How much money would you have after three years? The answer is easy once you know the formula. You would have $1,000 * 1.053 = $1,157.63

After five years, the account would be worth $1,000 * (1.05)5 = $1,276.28 as compared to $1000 * .05 * 5 = $250, for a total value of $1,250 with simple interest for a $26.28 difference. So while you only gained an additional $2.50 worth of interest in the first year by compounding, that has increased to $26.28 after five years.

You can see the effects of compounding in Figure 1:

Figure 1


Figure 1 shows time on the horizontal axis and total account value on the vertical axis. The height of each bar represents the total account value for that year. The blue portion of the bar shows how much of that value is due to simple interest while the red portion shows how much is due to compounding – interest on interest.

For example, in year one, the account grows from $1,000 to $1,050 and all of that $50 increase is due to simple interest. In year two, the account grows to $1,102.50 and $100 of that value is due to simple interest and the remaining $2.50 is due to compound interest. Notice how the blue bars increase by a constant $50 every year while the red bars are expanding.

The important thing to notice is that the amount of money due to compound interest (red bars) grows as time goes on!

Look at the power of compounding 30 years forward as shown in Figure 2:

Figure 2


The portion of value due to compounding eventually surpasses that of simple interest – if you allow enough time. It's like a snowball rolling down a hill that gets bigger as it collects more snow.  As it gets bigger, it rolls faster and collects even more snow at a faster rate. It becomes an accelerating cycle, which is why Figure 2 is curved along the red portions rather than climbing in a straight line as with the blue. If we trace a line across the tops of the blue and red bars in Figure 2, it is easier to see this power of compounding as in Figure 3:

Figure 3 


Notice in Figure 3 how the compound interest (red line) pulls away from the simple interest (blue line) as time progresses. Not only does the red line pull away, but it also pulls away at an accelerating rate. The reason for this can be understood by the formulas. The simple interest formula is principal * rate * time. Because principal and rate are constants, the only factor that changes the amount of interest earned is time. And that means that the interest you earn is directly proportional to the amount of time the principal is invested. In other words, it is a constant growth formula and that’s why the blue line is straight.

However, the formula for compounded interest formula is principal * (1+interest rate) time and uses time as an exponent. In other words, it is an exponential formula and that’s why the red line in Figure 3 accelerates away from the blue line. It is the interest on interest that causes that effect. The longer you allow compounding to work, the faster that red line accelerates. Compounding is truly a powerful force.

Just How Powerful Is Compounding?

Question 1: Start with a stack of pennies (a very, very large stack). Place one penny on a checkerboard square and double it as you move to each square. You would have two pennies on the second square and four on the third square and so forth. How much money will you have on the 64th square?

Question 2: Here is a great example from Malcolm Gladwell’s excellent book The Tipping Point. The first step begins with a single sheet of paper. Step two: Put two pieces of paper on top. Step three: Put four pieces of paper on top. Step four: Place eight pieces of paper on top. How high is the stack after continuing for a total of 50 steps?

Think about these questions. Guess big! Guess bigger! And you’ll still probably be far below the correct answer. 

If you start with one penny on the first square of the checkerboard, two on the second, four on the third and so on, how much money will you have sitting on the 64th square? You would beat Bill Gates' highest level of net worth at $90 billion – one million times over! That’s more than 70,000 times the U.S. GDP for 2006.

If you start with a sheet of paper and double the stack 50 times, it would reach to the sun; that’s 93 million miles away. To further demonstrate the power of compounding, if you double the stack just one more time, you reach all the way back to earth. So while it takes 50 steps to reach to the sun, just one more step has the power to bring you all the way back to your starting point.


Does Compounding Apply To The Stock Market?

We've been talking about the power of compounding and have demonstrated its accelerating power with a hypothetical account that pays a fixed rate of interest. So what does that have to do with stock investing?

Even though stocks do not pay annual interest as in our examples, they do have returns in the sense that their values tend to rise over time and some even pay dividends. The point is that whether you refer to the increases in your account as interest or as capital gains, the effects are identical.

For example, if you buy $1,000 worth of stock and it later rises 5% then it is valued at $1,050. If it rises 5% again the next year, your investment is then worth $1,050 * 1.05 = $1,102.50 just as it was with the compound interest example. So while stocks do not actually pay interest, their gains affect your total account value and not just the initial investment. In this way, the stock market compounds returns just as if interest was paid annually so the power of compounding definitely applies to stock market investing. It’s the art of managing the growth so that the science of compounding works best for you. That’s where options come in.

Here's the mysterious part: After the first row of the checkerboard is completed, your total is only a measly $1.28. You'll break the $327 mark after the second row is completed and $83,886 after the third. There are only eight rows so it hardly looks like you'll reach that astronomical level, but you will. It all happens in the last few squares, when the compounding force really takes effect. A similar effect can be seen in Figure 3. Notice that there’s not too much difference between the red and blue lines after 10 years but a substantial difference after 30. That’s the same effect that’s causing the checkerboard question to be so mystifying.

That’s a big number to comprehend so here’s another comparison that might make it easier. The stack of pennies sitting on the last square could stretch to the sun and back more than 98,000 times. Pretty amazing when you consider the sun is 93 million miles from Earth. Actually, when viewed this way it doesn’t seem to make the number any more comprehensible. Einstein was right – compounding is the greatest invention ever.

A Mathematical Illusion

How can we use the power of compounding to help you meet your investment goals? The key is to manage the losses and not to spend your time looking for the next big winner in the market. And for that, you need to use options.

If you are a true investor (rather than a speculator) you have the most important ingredient of success on your side – time. Time allows compounding to work and it is the final few years that really accelerate the account value. Recall the checkerboard example if you're doubtful!

However, it's easy to lose sight of this power, which can be demonstrated with a simple example.  Say you have the following information of market returns over a ten-year period:


Let's assume the market is expected to perform in a similar way over the next ten years. Just by looking at the returns in the above table, what rate of return would you strive for your annual returns in order to replicate the returns of the market? Think about it for a moment and then continue reading.

Have an answer? If you’re like most people, you would want to pick stocks or mutual funds expected to yield somewhere between 18% to 20% since that appears to be about the average for the positive return years. Further, the negative return years are relatively small and only occurred three out of the ten years. They don't appear to have much of an impact.

Regardless of the answer you chose, it's probably safe to say that you'd have no desire to consider a risk-free investment at 10% per year. After all, how could a 10% investment compete with those large returns in seven of the ten years? 

The truth is that a steady 10% investment actually beats the above returns. The reason has to do with compounding. The risk-free investment keeps the steady pace of growth and gets a great compounding effect. There are no losses of value during the holding period. Remember that we compared compounding to a snowball rolling down a hill. A steady 10% allows the snowball to roll faster and faster and accelerate all the way to the end. The result is that it collects far more snow at the bottom of the hill.

However, any negative return, no matter how small, has a two-fold negative effect. First, a negative return reduces the size of the snowball in the middle of the hill. Second, it is similar to tapping the brakes on the snowball. The result is that this snowball takes longer to get to the bottom (a longer investment horizon) and it will be smaller when it reaches the bottom (fewer dollars).

If you're not convinced the 10% investment would win, let's consider a $1,000 deposit and look at the ending account values for each year:



After ten years, the stock portfolio would be worth $2,578 while the risk-free portfolio would be worth $2,593. Granted, there’s not a lot of difference between these two examples but that was done intentionally. It was designed to show that the effects of compounding are greatly reduced by unexpected losses that occur from time to time. In fact, if we take out the negative returns for the stocks and replace them with zeros, the final result is $3,666, which is 41% larger than the risk-free investment. It’s the small, infrequent losses that really damage the compounding effect.

The problem is worsened if the losses occur near the end of your expected holding period (or retirement for IRA accounts) when the account value has grown to a sizeable amount and you have less time to recoup those losses. 

We never know when the bear markets will occur, which emphasizes the importance of hedging and the use of options. In order for compounding to work, it is necessary to not lose value along the way. While it is also important to receive adequate returns in order for compounding to work, don't think they need to be enormous. As we've demonstrated, even a "meager" 10% can be a large amount when steadily compounded for many years. By using options, we can reduce the risk and the damage that can occur during down years and actually achieve incremental returns above the market averages. 

Just as the tortoise proved to the hare in Aesop's fable, slow and steady wins the race. It is this strategy that serious investors follow and that’s why you need to understand options. Options allow you to hedge certain risks and lock in gains while staying in the market. Don't think you need to step up to the plate with the biggest bat you can find and swing for the fences if you are serious about making money. Steady compounding is a far more powerful force.  

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