What a stock index is, and how it is weighted

A stock index is two rules and a number. The first rule says which securities are in it. The second says how much of the index each one accounts for. The number is what those rules produce once today's prices are put through them.

Selection can be mechanical or judged. Some indices take the largest companies on a given exchange by market value and stop at a fixed count. Others apply published eligibility tests covering listing venue, share class, trading volume and how much of the company trades freely, then let a committee choose among those that qualify. Both are rules; one has people in the loop. Neither promises a tidy count, which is why an index named for a round number of companies can hold more lines than that when a constituent has two share classes.

Almost every weighting scheme is one formula with one input swapped:

$$\text{Index level} = \frac{\sum_i P_i S_i}{D}$$

$P_i$ is the price of constituent $i$ and $S_i$ is the share count the rule assigns to it. Set $S_i$ to the company's tradeable share count and you have market-value weighting. Set every $S_i$ to 1 and you have price weighting. Choose the $S_i$ so that every product $P_i S_i$ is identical at each review and you have equal weighting.

$D$ is the divisor, and it is what keeps the level continuous. Whenever the total on top changes for a reason that is not a price move, $D$ is reset so the level is identical the instant before and the instant after. A member joining or leaving does that under any rule. A share issue does it under market-value weighting, where the sum tracks share counts. A share split does it under price weighting, where the price falls and the sum falls with it; under market-value weighting a split leaves the sum alone, because three times the shares at a third of the price is the same number. Without the reset, a company joining would register as a market move.

The level carries no information on its own. An index starts at a round number, often 100 or 1,000, on a base date, and every later level is a ratio to that.

Market-capitalisation weighting sets each company's share of the index equal to its share of the constituents' combined value. Market capitalisation is the share price times the number of shares, so weight follows the price the market has already set rather than any single judgement about what a company ought to be worth. That price is itself an aggregate of opinion, which is a different claim from saying it is right.

Most such indices weight by free float rather than by every share issued. Stock held by a founding family, a parent company or a government does not trade, so counting it would hand the index a weight no fund could buy. Float adjustment counts only the shares available to the market.

Three companies are enough to show the whole idea, and they carry the rest of this page. Take all three as fully floated, so that float adjustment does not bite and market value is the whole of each company:

The property that matters is what happens when a price moves. If Alpha doubles, its market value doubles, its weight rises, and a fund tracking the index already owns the shares that did the doubling. No trade is needed, and a falling company shrinks out of the index by itself the same way. That is why market-value weighting is cheap to run, and it is also the honest description of what you hold: automatically more of whatever has already risen and less of whatever has already fallen, measured in weight rather than in shares, since the share counts sit still while the prices move.

That is a consequence of holding a market in proportion, not a forecast about it. It also means the index inherits whatever concentration the market has produced. When a few companies grow into a large share of the total value, they become a large share of the index, and nobody decided that. A rule can be neutral and still leave you concentrated.

A price-weighted index adds up the share prices of its members and divides by the divisor. Weight is proportional to price alone, so Alpha at 300 dollars counts five times as much as Beta at 60 dollars, even though Beta is worth twice as much as Alpha.

Share price is close to meaningless as a measure of size. It is the value of a company divided by however many shares it has chosen to issue, and that share count is a decision rather than a fact about the business. Two identical companies, one of which has split its shares, trade at different prices and therefore carry different weights.

A split makes the point on its own. Suppose Alpha does a three-for-one split before the year starts: the price drops to 100, every holder has three shares for each one held, and nothing at all changes about the company.

Same companies, same percentage moves, two different answers, decided by a share split. Worked examples two and three below run both columns through the arithmetic.

The Dow Jones Industrial Average and the Nikkei 225 are the price-weighted indices most people have heard of. Both were designed to be kept up by hand, and that is the one real argument the rule ever had: a price-weighted index is a column of numbers added up, while a value-weighted one needs a current share count for every member as well. Neither now survives on the merits of price as a weight. They survive on familiarity and on an unbroken run of history that restating them would throw away.

Equal weighting gives every constituent the same share of the index. With 100 members each one is 1 percent, whether it is the largest company in the country or the smallest that qualified.

This changes what the index measures. The smallest members, which a value-weighted version barely notices, now matter as much as the largest, so an equal-weighted index tracks the average constituent rather than the average dollar invested. Its behaviour sits closer to a mid-size company index than its value-weighted twin's does, and the two pull apart most in stretches when a handful of large companies drive the market.

It also has to be maintained. Prices move constantly, so the weights drift away from equal within days and the index is reset on a schedule, commonly quarterly. Resetting means selling what has risen and buying what has fallen. Inside an index that costs nothing, because an index does not trade: the rule simply states new weights. Inside a fund tracking it, that turnover is real trading, with spreads and price impact, and it can realise gains that reach a taxable holder as a distribution. How much of that happens depends on the fund's structure and on the tax rules where it and its holder sit, which differ by country.

Between weighting by value and weighting equally sit the rules a designer reaches for when neither of those works:

• Capped indices limit how large any single constituent can get and redistribute the excess across the rest. Funds in some jurisdictions face diversification limits that an uncapped index would breach.
• Fundamental weighting uses an accounting measure of size, such as revenue, book value or dividends paid, instead of market value.
• Factor indices tilt weights towards a measured characteristic, such as low volatility, small size or high profitability.

Each is still only a weighting rule. Giving it a name does not make it a different kind of object, and the question to ask of any of them is the same one: which companies, and how much of each.

An index is not a fixed set of companies. Membership is reviewed on a published schedule, and the review does two separate jobs worth keeping apart.

Reconstitution changes who is in. Companies are added when they meet the eligibility rules and removed when they stop meeting them, are taken over, go private or fail. Some index families do this once a year on a fixed date. Others review quarterly, and committee-run indices can act between reviews when a constituent is acquired.

Rebalancing changes how much of each. In a value-weighted index this largely happens by itself, because prices move the weights without anyone trading, and the scheduled work is limited to updating share counts and float. In an equal-weighted or capped index, rebalancing is the whole maintenance job.

Two things follow. The first is that a long chart of an index compares different companies at its two ends. An index running for decades has replaced most of its original members, and the ones on the chart today are the ones that kept qualifying. The line is the rule's history, not any single company's, and reading it as though one set of firms produced the whole run is a mistake about what is being plotted.

The second is that the trade is known in advance. Additions and deletions are announced before they take effect, and a fund that tracks the index closely has to hold the new member by then or accept the tracking error of not holding it. That predictability can move a share price ahead of the change, an effect studied under the name the index effect. How large it is has varied a great deal: studies of additions to large indices in recent decades find far less of it than studies of the 1990s did, as the changes became more anticipated and more traded around.

Two gaps get read past constantly.

The first is between an index and the economy it is named after. An index holds listed companies that meet its rules. An economy also contains private firms, partnerships, state-owned enterprises, the self-employed and everything never organised as a listed company. The constituents of a large national index are usually global businesses as well: they are picked on where they list and are domiciled, while their revenue can come from anywhere, so the index answers to conditions the country does not contain. The sector mix drifts the same way, following what has listed and grown rather than what the country produces, and a new industry enters only once it has floated and grown big enough to qualify. An index rises when the price the market puts on its constituents' future profits rises, and that happens either because those profits are expected to be larger or because the return investors demand for holding them has fallen. Neither is the same event as output rising, and the second has nothing to do with the companies at all.

The second gap is between an index level and what a holder earned. The quoted level of most headline indices is a price index: it counts price changes and ignores the dividends paid along the way. The same index almost always has a total return version, which puts each dividend back into the index, normally on the day the shares go ex-dividend rather than on the later day the cash arrives, and therefore rises faster. Germany's DAX is the exception people notice: its headline version is the total return one, with a price version published alongside. The last two worked examples measure that gap.

Then the step from a total return index to your own account. An index charges nothing and trades at no cost; a fund tracking it has expenses and turnover, which is the subject of index funds against active management. Holding every member of an index removes the risk that any single company sinks the result, which is what diversification buys. It does not remove the risk of owning companies at all, and it does not remove the choice buried in the rule: which market, which size of company, which currency. This is educational material about how index rules work, not advice about what to hold.