Stock market indices

Summary: Stock market indices use sophisticated mathematical formulas to track the performance stock markets and to help inform investors.

Mathematical stock market indices are used for a variety of purposes: as indicators of overall market health and activity, as measures of specific corporate profitability and activity, as performance metrics against which institutional investors (such as mutual fund managers) are measured, and for individual portfolio optimization and risk assessment. Some mathematicians and economists were developing price-based indices as early as the nineteenth century as well as analyzing pricing trends for explanation and prediction of market behavior. The Dow Jones Industrial Average (DJIA), named for journalist Charles Dow and statistician Edward Jones, appeared in 1896. Initially, it was a simple sum or average of the stock prices from 12 large companies. Since then, stock market indices have increased in their variety and mathematical complexity. For example, technical analysts use Fibonacci retracement levels, named after mathematician Leonardo Pisano Fibonacci, in order to model support and resistance levels in the currency market. Mathematicians and statisticians are instrumental in producing these indices. They also conduct theoretical and applied studies of market performance using these indices as data. In 1999, French-American mathematician Benoit Mandelbrot showed that market volatility can be modeled by fractal geometry, which contradicted some aspects of modern portfolio theory. Author and mathematician John Allen Paulos addressed many mathematical stock market issues in his popular book A Mathematician Plays the Stock Market.

Definition and Examples

When describing the performance of the stock market as a whole (or a segment of the market, such as selected large-company stocks, or all small-company stocks, or stocks of all companies belonging to a particular industry), one is usually referring to a stock market index. Such an index is a representation of a hypothetical portfolio that contains a certain quantity of each of the stocks in the market (or market segment). The quantity of each stock in the fictitious portfolio depends upon the “weighting” technique employed.

Some of the more commonly encountered stock indexes include the following:

• • S&P 500, comprised of 500 large-company U.S. stocks that cover about 75% of U.S. equities
• • DJIA, comprised of 30 large-company U.S. stocks
• • Wilshire 5000, comprised of the most common stocks in the United States (although not necessarily exactly 5000 of them)
• • Nikkei 225, an index of Japanese equities
• • FTSE, a collection of indices of British stocks

Building a Stock Market Index

The wide variety of stock indexes fall into several weighting categories, each involving a different mathematical approach to combining stocks within a hypothetical portfolio. One can imagine a potentially unlimited number of ways of creating a portfolio that includes numerous company stocks: for example, a portfolio comprised of one share of each stock, a portfolio comprised of the same dollar amount of each stock, and so on. The most common methods of weighting stocks within an index are price-weighting and market-value-weighting. (To simplify, stock performance is treated as only a function of changes in the stock price over time—as capital gains and losses. In reality, dividends, stock splits, and a variety of other issues must be taken into account, which makes the specific mathematical applications more complex than represented in this entry.)

Price-Weighted Indices

A price-weighted stock index represents a theoretical portfolio that includes one share of each stock comprising the index. The price or value of the index is then equal to the average of individual stock prices. Therefore, the relative impact of a given company stock on the index is a function of the company’s stock price per share: larger prices per share imply greater influence on the index.

Suppose that S_i(t) represents the per-share price of stock i at time t, and let S_i(t) be the value of the index at time t. Then, the price of a price-weighted index could be defined as simply the arithmetic average of the stock prices in the index:

where n is the number of stocks comprising the index.

While the value of a price-weighted index is simple to calculate, typically the measure of most interest to an investor is not the actual price of the index, but rather the percentage change (the rate of return) in the index over a period of time. Let r_i(t,t+1) be the rate of return on stock i during the period from time t to time t+1, and let r_I(t,t+1) be the return on the index between times t and t+1 (assume an annual return period for purposes of this discussion, but returns can also be calculated daily, monthly, quarterly, or over any other period of time).

Then, the return on a price-weighted index is

Multiplying this value by 100 yields the return expressed as a percentage change. The DJIA and other Dow Jones averages are examples of price-weighted indices.

Market-Value-Weighted Indices

A market-value-weighted (also called “value-weighted”) stock index is one that weights the individual per-share stock prices according to the relative market values, or market capitalizations (called “market cap” for short), of the component stocks. A company’s market cap is simply the totally value of its outstanding equity and is calculated as the per share stock price multiplied by the number of stock shares outstanding. Thus, an individual company’s influence on a value-weighted index is a function of the overall equity value, or size, of the company—larger companies have greater influence on the movement of the index.

Using the notation introduced above, and letting Ni represent the number of shares of stock i outstanding, the rate of return on a value-weighted stock index would be

The S&P 500 and other Standard & Poor’s indices are examples of market-value-weighted indices.

Other Types of Index Weightings

While price-weighted and value-weighted indices are common, there are other weighting techniques that can be used. For example, it is possible to create an index that gives equal weight to the return of each stock comprising the index. In such a case, the return on the index would be calculated as

With such an index, the performance of each stock has the same impact on the overall index return as every other stock.

Another possibility in creating an index would be to use geometric, as opposed to arithmetic, averaging. A geometric average is calculated by multiplying n numbers together and taking the n-th root of the product (as opposed to summing the numbers and dividing by n, as with an arithmetic average).

The key in interpreting the various types of stock market indices is to know their underlying construction and to understand and interpret them appropriately. Price-weighting and equal-weighting, for example, can result in very different index performance indications than value-weighting, even relative to the same underlying stock return data. The appropriate index to use in a given situation depends upon the specific purpose in mind. If one wants a measure of market performance that is more influenced by the price movements in the stocks of larger companies, for example, a value-weighted index may be most appropriate. If the sizes of companies are not relevant for analytical purposes, or if the companies that comprise an index are very similar in size and other attributes, a price-weighted or value-weighted index may be appropriate.

Bibliography

Bodie, Zvi, Alex Kane, and Alan Marcus. Investments. New York: McGraw-Hill/Irwin, 2008.

Paulos, John Allen. A Mathematician Plays the Stock Market. New York: Basic Books, 2003.