WHAT IS 'Arithmetic Index'

Arithmetic index refers to a hypothetical portfolio of securities that uses an arithmetic sum to determine changes in its value, without taking the relative value of the securities into account. It is perhaps the simplest way to calculate a hypothetical portfolio, or index.

To calculate an arithmetic index, simply take the daily percent change of each included security, then divide by the total number of those securities. In this way, components with the most price movement in percentage terms exert the most influence on the index return.

There are very few arithmetic indexes, and most focus on equities. One of the best-known is the Value Line Arithmetic Index.

BREAKING DOWN 'Arithmetic Index'

An arithmetic index differs from most hypothetical stock portfolios, which are market-cap weighted. This means stocks backed by companies with the largest market values exert more influence on the index return than lesser-valued companies. Most well-known indexes including the Standard & Poor's 500 Index and the Nasdaq 100 are market-cap weighted.

Arithmetic indexes, however, are equal-weighted, in that each included stock represents the same percentage of the total.

However, not all equal-weighted indexes are arithmetic indexes. Some are geometrically weighted, meaning they take into account the effects of compounding over time. An example is the Value Line Geometric Index.

Neither an arithmetic index nor geometric index should be confused with a price-weighted index, which gives preference to the stocks with the highest prices. In this type of index, a stock moving from $70 to $80 carries more weight than a stock moving from $7 to $17, even though the latter is a much larger move in percentage terms. The Dow Jones Industrial Average is a price-weighted index.

Pros and Cons of Arithmetic Index

An arithmetic index closely mimics the change of a theoretical portfolio if you held each of its components in equal amounts. It also tends not to be dominated by stocks with large market caps or, in the case of a price-weighted index, high prices. Therefore, it gives more representation to mid-and small-cap stocks.

However, investors don't always want this. Market-weighted indexes are more common because they more closely reflect reality: It takes significantly more money to move large-caps, and thus, these stocks seemingly deserve a heavier index weighting. Large-caps also tend to be more widely owned.

In addition, few investors want to hold a large portfolio of stocks in precisely equal amounts in real life, thus making an arithmetic index largely theoretical, and in some ways, impractical. It may be downright impossible for a large investor with tens of billions in the market to accumulate all components of an index in equal proportion, given the small available float of certain holdings, as well as trading costs.

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