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.

Related Section

Don't miss... Money Math, which has sections on interest, home mortgage, Fibonacci and gold.

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.

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.

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 - Dec 2006

dow 2006

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 has recovered all of the lost ground since then and is now above the 2000 peak. It is clear that predictions based on past performance are very dangerous since our original model predicted that we would be at about 20,000 by the end of 2006.

The dramatic surge from 1995 may have been an anomoly and maybe we are back on the long-term trend line.

Where is the Dow Jones Index Now?

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

Updated Model - to 2005

Let's see what the model becomes, taking into account the volatility since 2000. Here's how it looks now that I have added in one more data point, the closing value of the Dow at the end of 2005 (10,717):

dow 2005

We see that the market had "got ahead of itself" in 2000, but was below the trend line at the end of 2005.

These models are interesting to consider but should not be relied on for predicting the market. Markets can go up and down for thousands of reasons. See these models as an interesting application of mathematics.

Also, don't forget Japan...

In the late 1980s, Japan had explosive growth in sharemarket prices, similar to the DJIA in the late 1990s. The Japan market started to plunge in the early 1990s and in 2005 it was about one quarter of its peak value.

Here's the graph:

nikkei

Where is the Nikkei now?

The Nikkei has recovered somewhat, but still has a long way to go. Don't ever believe professional financial experts who tell you markets always go up. There are a lot of people in Japan who lost a lot of money on the markets.

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.

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