Why the Dow Will Hit a Million, Eventually

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At an event earlier this month, Warren Buffett, one of the most successful investors of all time, revealed his prediction that the Dow Jones industrial average (DJIA) will be "over 1 million" in 100 years.

With the DJIA currently sitting at about 22,400, is it even reasonable to think that the stock market could grow that much? Growth from the current value of the Dow to 1 million would represent an increase of about 45 times over. My first impression was that a value of 1 million for the Dow is very high, and Mr. Buffett must be either confused or overly optimistic to put forth such a prediction.

But since this prediction came from someone who clearly has a good understanding of investments and the stock market, I decided to check out the math behind this prediction to see if it makes sense.

An important part of Warren Buffett's prediction is the "in 100 years" part. One hundred years is a long time, and although it may be surprising, Warren Buffett's prediction of the Dow topping 1 million is actually quite reasonable given the historical performance of the market. In fact, the prediction of the Dow reaching 1 million in 100 years may even be conservative.

Here's the math

Let's look at what kind of growth rate would be required for the Dow to reach 1 million in 100 years. As I mentioned, the Dow would need to grow by 45 times its current value. When thinking about investment growth, it is informative to look at the growth in terms of the number of doublings that would be required.

2ⁿ = 45

n ln(2) = ln (45)

n = ln(45) / ln(2)

n = 3.81 / 0.693

n = 5.5

So the market value would need to double 5.5 times from its current value to reach 1 million. Let's look at this in the form of a table to make sure it makes sense:

From the table above, you can see that doubling the current Dow five times yields 716,800, and doubling six times yields over 1 million, so the number of doublings for the Dow to reach 1 million must be somewhere in between. Our estimate of 5.5 doublings makes sense.

So the Dow would need to double 5.5 times in 100 years — or in other words, it would need to double every 18.2 years: 100 years / 5.5 doublings = 18.2 years to double.

The next step to checking out Mr. Buffett's prediction is to figure out what rate of growth would be required for the value of the Dow to double every 18.2 years.

For a quick estimate, I turned to the "Rule of 72." The Rule of 72 is a handy approximation to find how many years it will take an investment to double — simply divide 72 by the annual rate of growth. I flipped the Rule of 72 formula around to check the rate of growth required:

72 / growth rate = years to double

72 / growth rate = 18.2 years

Solve for growth rate:

72 = 18.2 x growth rate

growth rate = 72 / 18.2 = 3.96 percent annual growth

So the "Rule of 72" approximation tells us that an annual growth rate of 3.96 percent would be required to double the Dow every 18.2 years, which is the rate of growth needed for the Dow to hit 1 million in 100 years.

If you don't want to settle for an approximation, or if you are just geeky in a cool sort of way, you can do a more exact calculation:

2P = Pe^Yr

2P = Pe^(18.2)r

ln(2) = 18.2r

r = ln(2) / 18.2

r = 0.038 or 3.8 percent

The approximation from the Rule of 72 matches pretty closely with the exact calculation, so it seems we have nailed down the rate of growth that is required for the Dow to reach 1 million.

It turns out that that an annualized growth rate of 3.8 percent is well within the historical growth rate of the stock market over the past 100 years. The average rate of return from the stock market is typically considered to be as high as 7 percent.

Of course the stock market does not march steadily along at an average rate of growth year after year. The market swings up and down from day to day and follows longer upward and downward trends during bull and bear markets. But over the long haul, the average trend for the stock market has been upward at a rate of well over 3.8 percent average growth over the past 100 years.

In addition to the mathematical consideration of the rate of growth required for the Dow to reach 1 million in 100 years, another consideration is whether the world's people and natural resources will continue to sustain economic growth over the next 100 years. With development of exciting new technologies and emerging global markets to drive growth, it seems reasonable that the stock market could keep going up.

So it looks like Mr. Buffett's thinking makes good sense as usual, and the prediction of the Dow 1 million makes perfect sense.

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