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Penny Doubled: A Compounding Story

What if I told you I would give you $1,000,000 OR the amount of a penny doubled every day for a month?

If you’re like most people, you would probably think that one million dollars sounds like a lot more than a handful of pennies, and you would therefore choose the million dollars.

But would you be right…?

The Grain of Rice Fable

I’m not the first person to pose a question like this. This question is based on a famous fable, often referred to as the grain of rice fable, or the rice and the chessboard.

Grain of Rice Fable

The story goes something like this…

In India, a long time ago, a king ruled who loved playing chess and enjoyed challenging others to play against him. So, when a visiting sage came through, the king challenged the sage to a game of chess. The king told the sage he would wager any reward the sage could imagine.

The sage took the king up on his offer, but he requested a surprising reward – rice.

Making the request even stranger, the sage asked for the rice in a specific way. If the sage won, the king would put one grain of rice on the chessboard’s first square, and then he would double the amount for every subsequent square.

The king considered this a humble request and happily obliged. The two played, and the sage won.

True to his word, the king ordered a servant to bring out a bag of rice. The king laid one grain of rice on the first chessboard, two on the second, four on the third, and so on. But before long, the king realized he had a problem.

The king not only didn’t have enough rice on hand to fulfill the request for all 64 squares; the king didn’t have enough rice in his entire kingdom to satisfy the need.

By the 15th square, the king would have had to place 16,384 grains of rice; by the 25th square, 16,777,216; and by the time the king reached the final square, he would have had to give the sage 18,000,000,000,000,000,000 grains of rice, which is more than 200 billion tons of rice, or enough to cover the entire country of India with a meter of rice.

As the king began to realize what this reward entailed, the sage told the king that there was no need for the king to pay out his reward right away; the king could pay him over time.

This is what the king did. And the sage lived out the rest of his days as the richest man in the world.

Compounding Rice Fable

The Math Behind the Fable

Now we return to the original question.

Would you still take the million dollars instead of one penny doubled every day for 30 days?

If you’re re-thinking your answer, you’re right to do so. If you took the penny doubled every day for thirty days, by the 30th day you would have $5,368,709.

This is often hard to believe and doesn’t quite feel right. So, let’s check the math. The formula for compounding is:

\[FutureValue = PresentValue * (1 + r)^t\]

In our example, we’re doubling a penny, a 100% growth rate, for 29 days since we do not double it on the first day. Let’s do this mathematically and check it using Excel:

\[FutureValue = PresentValue * (1 + r)^t \\ FutureValue = 0.01 * (1 + 1.0)^{29} \\ FutureValue = 0.01 * 2^{29} \\ FutureValue = 0.01 * 2^{29} = $536,870,912\]

A Penny Doubled Every Day For 29 Days

A Penny Doubled For A Month

Day Amount
1 $0.01
2 $0.02
3 $0.04
4 $0.08
5 $0.16
6 $0.32
7 $0.64
8 $1.28
9 $2.56
10 $5.12
11 $10.24
12 $20.48
13 $40.96
14 $81.92
15 $163.84
16 $327.68
17 $655.36
18 $1,310.72
19 $2,621.44
20 $5,242.88
21 $10,485.76
22 $20,971.52
23 $41,943.04
24 $83,886.08
25 $167,772.16
26 $335,544.32
27 $671,088.64
28 $1,342,177.28
29 $2,684,354.56
30 $5,368,709.12

Here we have the numbers laid out day by day. As you can see, it takes a while to get going. On day 15, the halfway point, we’ve only reached $163.84. But three days later, we’ve exceeded a thousand dollars. Three days after that, we’ve exceeded ten thousand dollars. By the 28th day, we’re already over a million.

Clearly, the penny doubled every day for thirty days is a much better deal than the million dollars.

Compound Interest

This is good to know if the offer ever arises, but as it’s highly doubtful you’ll ever be in this fortuitous situation, why is this worth knowing?

The answer may surprise you.

If you plan to retire, invest in the stock market, pay off credit card debt, take out a mortgage, or any number of other financial activities that make up daily life, you’ll need to understand the concept behind both of our two doubling examples - compound interest.

Compound Interest

Compound interest is the math behind the magic of investing.

Or, in more concrete terms, compound interest is the interest earned on the initial amount you put in, and even more importantly, on the interest earned on interest.

Let’s look at a super simple example to see what we mean by “interest earned on interest”.

Your friend borrows $10,000 from you and agrees to pay you back plus 5% interest, accruing annually. Five percent of $10,000 is $500. After two years, how much does the friend owe you?

It’s easy (and common) to think that since the friend owes 5% per year (and 5% is $500), the friend owes $500 times two. But that’s incorrect.

After the first year, the friend owes the principal (what they originally borrowed) plus one year of interest. This comes out to $10,000 (the principal) plus $500 (one year of interest) or $10,500.

The interest in the second year is now based on this new amount - $10,500. This means the second year of interest is calculated by multiplying 5% by $10,500, coming out to $525.

At the end of the second year, the friend now owes the principal ($10,000) plus the first year of interest ($500) plus the second year of interest ($525). This brings their total to $11,025.

This may not seem like a huge difference. If you multiply$500 by two, you will get $1,000. So, compounding only interest creates a difference of $25 in the total payment.

But this is where the examples we mentioned above come into play. Doubling rice or a penny doesn’t lead to much right away. It’s only after time that we start to see the impact. While 5% is not the same as doubling, the power of compound interest over time is still quite impactful.

To see just how impactful, let’s look at a very similar example to the one above, but in the context of investing over a more extended period.

Compound Interest and the Stock Market

We just saw how compounding interest works if you let a friend borrow $10,000 at 5% interest for two years (you earn an extra $25 thanks to compounding interest).

Let’s now look at some more complex compound interest calculations.

Let’s say that instead of lending that $10,000 to a friend, you invest it in the stock market. The annual rate you earn on the money (also known as the annual rate of return) remains 5%, BUT the timeline changes. You invest the $10,000 when you’re 25 years old and leave it untouched until you’re 65 years old. Over those 40 years, you neither add any money nor take any out.

How much do you end up with after 40 years? $70,400

With an annual interest rate of only 5%, you ended up with an over 700% return on your money.

And this is a rather conservative annual rate of return. The historical average annual return for the S&P 500 (a benchmark that aims to mirror the stock market’s overall returns as a whole) is around 10%.

If you use this 10% annual rate of return in place of the 5%, the growth becomes even more dramatic. How dramatic?

If you invested $10,000 and had an annual interest rate of 10%, after 40 years, you would have $452,593.

That’s right: an interest rate twice as high would mean more than six times higher returns over 40 years.

Before we move on, I do want to clarify two points about basing calculations on historical averages.

First of all, it’s a _ historical_ average, meaning it’s based on past data. There is no way to predict the future rate of return. It may end up close to this number, or it may not.

The second point is that it’s a historical average, meaning that not every year saw a 10% return rate. Some years saw higher returns, and some saw lower.

Neither of these points impacts the takeaway from our example, though - compound interest plays a massive role in returns on stock market investments.

Compound Interest and Time Horizon

By now, you’re likely starting to see a pattern. Whether it’s doubling or an interest rate of 5%, compound interest gains its power over time. Now you know why they say time is money.

Time Is Money

When applied to investing in the stock market, this means that the longer your money is invested, the more time compound interest has to work its magic. To see just how powerful a factor time is, let’s look at a few scenarios.

In each of the scenarios, you make an initial investment of $10,000, and for ten years, you invest $500 a month, but in each scenario, your time horizon is different.

In each scenario, you invest the same amount of money - a $10,000 initial investment plus $60,000 in additional contributions ($500 x 12 months x 10 years). This means your total invested amount in each scenario is $70,000.

Let’s see how much that $70,000 becomes in each of our three scenarios.

Scenario #1 - 10 Year Time Horizon

Let’s say you make the initial $10,000 investment when you’re 50 years old. For the next ten years, you invest $500 each month. After ten years with an annual rate of return of 10%, you’re 60 years old and ready to retire. How much money do you have?

In this first scenario, your $70,000 investment is now worth $121,562.

Your total return is 74%.

Scenario #2 - 20 Year Time Horizon

For this second scenario, let’s say you invest $10,000 when you’re 40 years old. You still retire at 60, which means you have a 20-year time horizon. You also invest $500 each month for ten years, but you do it when you’re between the ages of 40 and 50. From 50 to 60, you make no additional contributions. How much money do you have when you’re 60?

In this second scenario, your $70,000 investment is worth $315,301 when you’re 60.

Your total rate of return in this scenario is 450%.

Scenario #3 - 30 Year Time Horizon

For our final scenario, let’s say you make an initial investment of $10,000 when you’re 30 years old. For the next ten years, you make a monthly deposit of $500. Between the ages of 40 and 60, you make no additional deposits. How much money do you have when you’re 60?

In this second scenario, your $70,000 investment is worth $817,808 when you’re 60.

Your total rate of return in this scenario is 1,168%.

Breaking Down the Three Scenarios

Below is a chart breaking down each of the three scenarios.

  Initial Investment Activity from age 30-40 Activity from age 40-50 Activity from age 50-60 Total Invested Value at Age 60 with 10% Annual Rate of Return
Scenario #1 $10,000 N/A N/A Initial Investment + $500/month $70,000 $121,562
Scenario #2 $10,000 N/A Initial Investment + $500/month N/A $70,000 $315,301
Scenario #3 $10,000 Initial Investment + $500/month N/A N/A $70,000 $817,808

Again, these examples assume an annual return rate that’s the same every year, which is practically impossible. These rates also fail to account for fees, dividends, and many other investing aspects that impact a real-world portfolio. But even keeping all this in mind, the impact time has on compound interest is quite clear.

I would also like to add that just because you’re not 30 years old doesn’t mean that investing may not prove worthwhile. The takeaway is not that investing is only for the young, but that the sooner you invest, the longer compound interest has to work its magic.

The Rule of 72

We’ve included many numbers in this post, so before we wrap up, I’ll provide you with an incredibly simple and straightforward tool to help you think about compound interest - The Rule of 72.

Rule of 72

The Rule of 72 is a way to figure out how long it would take for your money to double. According to the Rule of 72, you divide 72 by your annual rate of return, and this gives you the amount of time it would take for your money to double.

Written out as a formula, it looks like this:

\[YearsToDouble = \frac{72}{AnnualRateOfReturn}\]

For example, if we stick with our example of $10,000 and a 10% annual rate of return, we divide 72 by 10, which gives us 7.2. This means that it would take 7.2 years for our $10,000 investment to reach a value of $20,000.

\[YearsToDouble = \frac{72}{10\%} \\ YearsToDouble = 7.2\]

When Compound Interest Works Against You

So far, we’ve only looked at how compound interest can benefit you, but compound interest doesn’t always work in your favor.

To conclude this post, we’ll look at the three main ways compound interest can negatively impact you and what you can do to limit the negative impact.

Fees

As briefly mentioned above, the investment scenarios’ returns did not include fees or other expenses that come with investing. These may have a significant impact on your total returns.

Suppose you’re comparing two investment options, such as two different mutual funds. In that case, you should make sure to consider the expense ratio (the amount you pay in fees and other expenses) and the impact it has on your portfolio.

The simplest way to do this is by using a fee calculator.

The differences in investment fees may sound minimal and often differ by mere fractions of a percent. But as we’ve already seen, small numbers can add up quickly once compounding begins. Let’s look at an example.

Let’s say you invested $10,000 and contributed $500 per month ($6,000 per year) with an annual rate of return of 10%. Your time horizon is 30 years.

If you had an expense ratio of 0.25%, the fees would have cost you $63,967. With an expense ratio of 0.5% in the same scenario, you would end up paying $124,505. This means that a quarter of a percent ultimately cost you over $60,000.

Withdrawals

Another example of when compound interest may work against you is when you make withdrawals.

An excellent example of this is when you withdraw from a 401(k) or other retirement savings account. While withdrawals may prove unavoidable in certain circumstances, you should avoid them whenever possible. That’s because when you make a withdrawal, you don’t just lose the money you withdrew; you also lose all the compound interest you would have earned on that money.

This means that even if you deposit the same amount you withdrew, you’ll still end up with less than you would have since you missed out on the compound interest from that sum not being invested.

Debt

The final way compound interest may negatively impact you is also, arguably, the most detrimental - debt.

Drowning in Debt

With debt, compound interest now benefits the lender and costs you more. This is not to say that all debt is necessarily bad, but it is essential to understand how much debt is genuinely costing you.

Compound interest is why high-interest debt, such as credit card debt, is so insidious.

For example, let’s look at the true cost of a credit card purchase if you only make the minimum payment.

If you put a $5,000 purchase on a credit card with a 20% annual interest rate and make the minimum payment every month, you’ll end up paying $2,884 in interest. This makes the actual cost of your $5,000 purchase around $7,884.

So, while compound interest can undoubtedly benefit you, it’s also critical to keep in mind how it may negatively impact you.

leo

Leo Smigel

Based in Pittsburgh, Analyzing Alpha is a blog by Leo Smigel exploring what works in the markets.