Price-weighted and equal-weighted indexes

How does price weighting work?

Lesson 2 showed that every index divides an aggregate value by a divisor. This lesson covers the two simplest weightings; lesson 4 covers market value and fundamentals.

In price weighting each constituent’s weight is its share price divided by the sum of all the prices. The aggregate is the sum of one share of each constituent, and the index value is that sum divided by a divisor . Holding the index is like owning one share of each constituent.

$$V_{PRI} = \frac{\sum_i P_i}{D}$$

$$w_i = \frac{P_i}{\sum_j P_j}, \qquad r = \frac{\sum P_1 - \sum P_0}{\sum P_0}$$

This lesson and the next use one three-stock table. Prices are in USD, shares outstanding in millions and dividends per share are paid during the period.

Stock	Start price	End price	Shares (m)	Dividend
X	80	88	50	2.00
Y	25	24	200	0.50
Z	45	54	40	0.00

What happens when a share splits?

A stock split divides a share’s price, so the sum of prices falls even though nothing economic has changed. To stop the index from dropping, the provider changes the divisor so that the index value is the same just after the split as just before.

$$D_{new} = \frac{\sum P_{after}}{V_{before}}$$

A divisor is also changed when a constituent is replaced.

Price weighting is simple to calculate and understand, but it does not reflect market values, and very high share prices skew it. It is rare today, used mainly by the Dow Jones Industrial Average (30 shares) and the Nikkei 225.

How does equal weighting work?

In equal weighting every constituent gets the same weight, 1 ÷ N, at the start of the period. The index return is then the simple average of the constituents’ returns, measured from the last rebalancing date.

$$w_i = \frac{1}{N}, \qquad r = \frac{1}{N}\sum_i r_i \ \text{(since the last rebalancing)}$$

How is an equal-weighted index built?

Use it

To build an equal-weighted index, hold a number of shares $n_i$ of each constituent so that every holding has the same value at the start. The index value is the value of the holdings divided by a divisor.

$$V_{EQI} = \frac{\sum_i n_i P_i}{D}, \quad n_1 P_1 = n_2 P_2 = \cdots = n_N P_N$$

Only since the last rebalancing

Memorize

The equal-weighted return equals the average of the constituents’ returns only when the weights are equal at the start of the period, that is, measured from the last rebalancing date. After one period here the weights have drifted to 33.7%, 29.4% and 36.8%.

Lesson 5 shows how the provider restores equal weights.

Equal weighting stops the largest companies from dominating and gives smaller companies relatively more weight. Its drawbacks: weights drift and must be rebalanced often (often quarterly), which is costly, and small companies are less liquid and harder to replicate. That is why few managers build portfolios this way.

Two simple weightingsvalue = aggregate ÷ divisor

• Price-weighted Memorize aggregate = sum of one share of each
  • Weight price ÷ sum of prices
  • Return change in the sum of prices
  • Upkeep new divisor after splits and swaps
• Equal-weighted Memorize aggregate = equal holdings n_i × P_i
  • Weight 1 ÷ N at each rebalancing
  • Return average of returns since the last rebalancing
  • Upkeep frequent rebalancing

How do the two methods compare on the same table?

They give different returns because the weighting rule is different. Price weighting follows the change in the sum of prices (10.67%). Equal weighting averages the returns (8.67%). A third method, market-cap weighting, gives 5.19% on the same table, and the next lesson shows how.

Go deeper Why price-weighted indexes survive, and what splits do over decades

Price weighting is the oldest method. It needs only prices and a divisor, so it could be calculated by hand a century ago. Some of the most quoted indexes still use it. Because companies tend to split their shares as prices rise, the shares that have done best get the lowest prices per share, and so lose weight. Over decades a price-weighted index drifts away from the winners. That is a quirk of the method, not a view about the companies.

Go deeper The hidden tilts of equal weighting

Giving the same weight to a very small company and a giant means equal weighting holds far more small companies than the market does. It also sells winners and buys losers each time it rebalances, which is a mild contrarian pattern. Both tilts mean its returns can differ from a market-cap index even when the target market is the same.