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Dow Jones Industrial Average (DJIA) Model

The Dow Jones Industrial Average (DJIA) is a weighted average of 30 of the largest stocks on the New York Stock Exchange. It is a very important barometer of the health of the US economy and has a large influence on stock markets around the world.

When we look at a graph of the DJIA from 1945 to 2000, we notice it is close to an exponential growth curve that we met in the last section.

 

It can be useful to model the shape of the curve so we can make a prediction for the future direction of the curve. (As we see later, this is a dangerous thing to do, but interesting, especially if we want to be rich.) A "model" is just a fancy word for a function (or formula) that closely matches some observed data.

Exponential Model

Let's just consider one portion of the Dow Jones index, from 1980 to 2000, where dramatic growth occurred. We can model this curve using an exponential function.

The first step is to obtain some data values. The following are the DJIA closing values on the last trading day of the years indicated:

1980	963
1985	1546
1990	2633
1995	5117
2000	10787

The plot of this data is as follows:

We observe that it this is a similar curve to our exponential growth curve that we met before. As time goes on, the exponential growth curve gets steeper and steeper.

Using Excel, we can add a trend line to our data and in the process, find a model. If you right-click on the graph (not the background) you will see the option "Add Trendline...".

Choose "Exponential" for the Type and then click OK.

In "Options" check "Display equation on chart" (this is the model, or equation, that we want).

We now get the following chart, showing the exponential graph and the model:

We see that the exponential curve (orange) passes closely through our data points.

That model may need some explaining.

2E-101 means 2 × 10^-101. This is a very small number indeed. "E" here means base 10.

e^0.1206x is the exponential function with base e, which we met in the section Natural Logarithms.

So the graph is y = (2 × 10^-101)e^0.1206x, where x is the year.

The model assumes that in "Year 0", the value of the DJIA was 1. This is a reasonable assumption for the sake of this model, even though the DJIA did not exist before 1896.

Dow Jones Prediction

In Excel, you can use the model to make a prediction (see the "Options" screenshot above). I have predicted what will happen, given the same growth rate, out to 2010:

This is called extrapolation, where you continue trend lines outside of the given data.

Wow - this indicates that if we had invested $10,000 in 2000, we can expect to have about $33,000 in 2010. Good deal! But wait, will it always go up like this? Let's look at what really happened.

Update - Jan 2009

The DJIA has been volatile since the peak of early 2000. There was a plunge after the September 11 terrorist attacks on the US in 2001 and it plunged even further before the Iraq war started in 2003. However, the Dow recovered all of the lost ground and peaked on 9 Oct 2007 at 14,163. It is clear that predictions based on past performance are very dangerous since our original model predicted that we would be at about 23,000 by the end of 2006 and almost 30,000 by the end of 2008.

Where is the Dow Jones Index Today?

Here is the most recent DJIA chart (updated every 5 minutes when the market is open).

Updated Models

Model 1: Assuming Growth Will Continue

The financial planners are an optimitstic crowd and try to tell us that the market will eventually always go up. If we assume they are right, here's an exponential model of the DJIA that uses all data points from 1st Jan 1970 to 31 Dec 2008.

The market excesses of the late 1990s and the difficulties of Sep to Dec 2008 appear as anomolies in this model. If we use the model to extrapolate forward to 2020, this is how it looks:

By 2020, the model predicts the Dow will be at around 40,000.

Of course, we will only see such growth in the market if the US economy is in good health and grows as it has done in the past 40 years. However, there are many factors involved (including the unravelling of the credit crunch, and the large numbers of people moving into retirement from now to 2020), so don't be too hopeful.

Model 2: Assumes Growth to Oct 2007 then Decline

Let's assume now that the end of the exponential rise of the Dow occurred in Oct 2007. Here's how the model looks now that I have added in 3 more data points (the low of 7286 in Oct 2002, the close of 2005 and the highest point of the Dow in Oct 2007):

Excel has revised the model and we note that after 1990, the actual Dow data is quite volatile compared to the model. We see that the market had "got ahead of itself" in 2000 (during the Dot.com boom); was below the trend line in 2003; and was close to the model in Oct 2007.

Let's Turn to Japan...

In the late 1980s, Japan had explosive growth in sharemarket prices, similar to the DJIA in the late 1990s, and again up to Oct 2007. The euphoria in Japan was driven by healthy export growth, but especially by a housing and construction boom. The real estate bubble burst in the early 1990s and the Japan market started to plunge. Japan has been in and out of recession ever since, and the latest stock meltdown in Sep through to Dec 2008 has seen the value of the Nikkei 225 return to values last seen in early 2003, and before that, in 1983. Investors who were in the market during the 1980s did very well, but since then, many people have lost a lot of money.

The Nikkei 225 is an index of the top 225 companies in Japan, similar to the DJIA in the USA. The graph of the Nikkei 225 since 1967 is as follows:

The early part of this chart is quite similar to the exponential rise of the DJIA and it is interesting that both stock bubbles were in part fuelled by real estate bubbles. If the DJIA unwinds over the next 20 years in a similar fashion to the Nikkei, we might see something like the following.

In this next graph, I have superimposed the DJIA (in orange) and the Nikkei (in dark blue). The period from 2003 to 2007 for the DJIA has a remarkably similar shape to the runup for the Nikkei from Dec 86 to Dec 1989. The wipeout that followed is also very similar. The Dow's low of near 7500 in Oct 08 corresponds to the Nikkei's low of around 20000 in late 1990.

Since its peak, the Nikkei has been in a downward spiral. Here's an exponential decay model for the Nikkei, to the end of 2008 (note the negative in the exponential term):

So will the Dow follow the Nikkei's pattern over the next 20 years? We hope not, but if so, we can expect the DJIA to be somewhere around 3000 to 4000 at that time, or about 50% of its current value. That will make a lot of retirees seriously unhappy.

Many commentators are saying that the Japanese did not address the issues regarding bank disclosures early enough in the 1990s and that's why their economy has never really recovered. However, the US Federal Reserve has already reached 0% interest rates (like the Japanese did) and have nowhere else to go now except for stimulus packages (like the Japanese have been trying, with little success, for 20 years.)

Where is the Nikkei right now?

This is an up to date reading for the Nikkei:

Don't ever believe professional financial experts who tell you markets always go up.

Past performance is not a guide to future returns, as can be seen from the above updates! Think about your investments carefully. But don't give up on investing - it is arguably the most important application of mathematics that you will ever do...

 

What Did Excel Do for Us Before?

The model that Excel produced for us is based on the following process. Since it is obviously an exponential growth curve, we are looking for a curve in the form

y = me^x + b,

where y is the value of the DJIA, x is the year (starting at 0 for 1980) and we need to find the constants m and b.

We use the formulae (as used in statistics) and subsititute in the values from our data table.

The calculations are pretty horrible - it is best to use a spreadsheet program to do them. But then, use Excel as I did above and it finds the model for you in one step.

Using Excel's built in trendline, you can specify what the y-intercept is for the exponential curve (this will affect the value of b). I used the default setting where b = 0.

 

Need a break? Play a math game. Well, they all involve math... No, really!

Chapter Contents

• Exponential & Logarithmic Functions
• 1. Definitions: Exponential and Logarithmic Functions
• 2. Graphs of Exponential and Logarithmic Equations
• 3. Logarithm Laws
• 4. Logarithms to Base 10
• 5. Natural Logarithms (base e)
• Dow Jones Industrial Average
• 6. Exponential and Logarithmic Equations
• World Population Live
• 7. Graphs on Logarithmic and Semilogarithmic Paper

• Comments, Questions?

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