# Subject: Stocks - Types of Indexes

Last-Revised: 10 Jul 1998
Contributed-By: Susan Thomas, Chris Lott (contact me)

There are three major classes of indexes in use today in the US:

Equally weighted price index
An example is the Dow Jones Industrial Average.
Market capitalization weighted index
An example is the S&P500 Industrial Average.
Equally weighted returns index
The only one of its kind is the Value-Line index.

The first two are widely used. All my profs in the business school claim that the equally weighted return indexs is weird and don't emphasize it too much.

Now for the details on each type.

Equally Weighted Price Index
As the name suggests, the index is calculated by taking the
average of the prices of a set of companies:
```Index =  Sum (Prices of N companies) / divisor
```
In this calculation, two questions crop up:

1. What is "N"? The DJIA takes the 30 large "blue-chip" companies. Why 30? Well, you want a fairly large number so the index will (at least to some extent) represent the entire market's performance. Of course, many would argue (and rightly so) that 30 is a ridiculously small number in today's markets, so a case can be made that it's more of a historical hangover than anything else.

Does the set of N companies change across time? If so, how often is the list updated (with respect to the companies that are included)? In the case of the DJIA, yes, the set of companies is updated periodically. But these decisions are quite judgemental and hence not readily replicable.

If the DJIA only has 30 companies, how do we select these 30? Why should they have equal weights? These are real criticisms of the DJIA-type index.

2. The divisor is not always equal to N for N companies. What happens to the index when there is a stock split by one of the companies in the set? Of course the stock price of that company drops, but the number of shares have increased to leave the market capitalization of the shares the same. Since the index does not take the market cap into account, it has to compensate for the drop in price by tweaking the divisor. For examples on this, look at pg. 61 of Bodie, Kane, and Marcus, Investments. The DJIA actually started with a divisor of 30, but currently uses a number around 0.3 (yes, zero point 3).

Historically, this index format was computationally convenient. It just doesn't have a very sound economic basis to justify it's existence today. The DJIA is widely cited on the evening news, but not used by real finance folks. I have an intuition that the DJIA type index will actually be BAD if the number of companies is very large. If it's to make any sense at all, it should be very few "brilliantly" chosen companies. Because the DJIA is the most widely reported index about the U.S. equity markets, it's important to understand it and its flaws.

Market capitalization weighted index
In this index, each of the N companies' price is weighted by the market capitalization of the company.
```        Sum (Company market capitalization * Price) over N companies
Index = ------------------------------------------------------------
Market capitalization for these N companies
```
Here you do not take into account the dividend data, so effectively you're tracking the short-run capital gains of the market.

Practical questions regarding this index:

1. What is "N"? I would use the largest N possible to get as close to the "full" market as possible. By the way, in the U.S. there are companies that make a living on only calculating extremely complete value-weighted indexes for the NYSE and foreign markets. CMIE should sell a very complete value-weighted index to some such folks.

Why does S&P use 500? Once again, a large number of companies captures the broad market, but the specific number 500 is probably due to historical reasons when computating over 20,000 companies every day was difficult. Today, computing over 20k companies for a Sun workstation is no problem, so the S&P idea is obsolete.

2. How to deal with companies entering and exiting the index? If we're doing an index containing "every single company possible" then the answer to this question is easy -- each time a company enters or exits we recalculate all weights. But if we're a value-weighted index like the S&P500 (where there are only 500 companies) it's a problem. For example, when Wang went bankrupt, S&P decided to replace them by Sun -- how do you justify such choices?

The value-weighted index is superior to the DJIA type index for deep reasons. Anyone doing modern finance will not use the DJIA type index. A glimmer of the reasoning for this is as follows: If I held a portfolio with equal number of shares of each of the 30 DJIA companies then the DJIA index would accurately reflect my capital gains. But we know that it is possible to find a portfolio which has the same returns as the DJIA portfolio but at a smaller risk. (This is a mathematical fact).

Thus, by definition, nobody is ever going to own a DJIA portfolio. In contrast, there is an extremely good interpretation for the value weighted portfolio -- it yields the highest returns you can get for its level of risk. Thus you would have good reason for owning a value-weighted market portfolio, thus justifying it's index.

Yet another intuition about the value-weighted index -- a smart investor is not going to ever buy equal number of shares of a given set of companies, which is what the equally weighted price index tracks. If you take into consideration that the price movements of companies are correlated with others, you are going to hedge your returns by buying different proportions of company shares. This is in effect what the market capitalization weighted index does, and this is why it is a smart index to follow.

One very neat property of this kind of index is that it is readily applied to industry indexes. Thus you can simply apply the above formula to all machine tool companies, and you get a machine tool index. This industry-index idea is conceptually sound, with excellent interpretations. Thus on a day when the market index goes up 6%, if machine tools goes up 10%, you know the market found some good news on machine tools.

Equally weighted returns index
Here the index is the average of the returns of a certain set of companies. Value Line publishes two versions of it:

• The arithmetic index:
```( VLAI / N ) =  Sum (N returns)
```
• The geometric index:
```VLGI  = { Product (1 + return) over N } ^ { 1 / n },
```
which is just the geometric mean of the N returns.

Notice that these indexes imply that the dollar value on each company has to be the same. Discussed further in Bodie, Kane, and Marcus, Investments, pg 66.

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