Stock indices · RiskAtoms Academy

By the end you will be able to

1Explain how a stock index is built: its constituents, rules, base value and divisor.
2Calculate the level and return of a price-weighted and a market-cap-weighted index.
3Explain free-float adjustment, and why the divisor changes after a share split.
4Tell a price index from a total return index.

Why it matters

“The Dow fell 2% today.” “The FTSE 100 hit a record.” Each headline squeezes thousands of trades into one number. How that number is built decides which companies move it, and by how much.

Core explanation

One number for a whole market

A stock index tracks a basket of shares, called its constituents. An index provider publishes the rules: which companies get in (the 500 leading US companies, the 100 largest on the London Stock Exchange), how much each counts, and how the number is kept consistent over time. The index starts at a round base value such as 100 or 1,000, so you can read how far the market has come since then.

The big question is how much each company counts. There are three common answers. Click each to compare.

Why size usually wins. A market-cap-weighted index behaves like owning every company in proportion to its size: it is what the market as a whole earns. That is why index funds, which try to match an index, mostly track cap-weighted ones. A price-weighted index depends on share prices, which companies can change at will through a share split (L1.5.02), so the weights are somewhat arbitrary.

Free float. If a founding family owns half of a company and never sells, those shares are not available to investors. Most providers therefore count only the free float. A company with a 50% free float counts at half its full market value.

Keeping the line smooth. Companies split their shares, issue new ones, or leave the index. None of these should make the index jump, because no investor gained or lost anything. So the provider adjusts the divisor. Companies are added and removed at regular reviews, or rebalances.

Dividends. Most headline indices are price indices: when a company pays a dividend its share price drops, and so does the index, though shareholders lost nothing. A total return index assumes dividends are reinvested. If prices rise 6% in a year and dividends add 3%, a price index starting at 1,000 ends near 1,060, and its total return version near 1,090. Over decades the gap becomes enormous.

The formal version

A price-weighted index, shown in full. With $P_i$ the price of stock $i$ , $n$ stocks and divisor $D$ :

I = \frac{P_1 + P_2 + \cdots + P_n}{D} = \frac{1}{D}\sum_{i=1}^{n} P_i

The cap-weighted index brings in shares, free float and the summation sign together, so let's build it one piece at a time.

Weights and returns, on a few lines. If the divisor does not change between two days, the index return is a weighted average of the stocks' returns $R_i$ , with each weight $w_i$ equal to the stock's share of the index's market value at the start:

\begin{aligned} \frac{I_1}{I_0} - 1 &= \frac{\sum_i P_{i,1} f_i Q_i}{\sum_j P_{j,0} f_j Q_j} - 1 \\[2pt] &= \sum_{i=1}^{n} \underbrace{\frac{P_{i,0} f_i Q_i}{\sum_j P_{j,0} f_j Q_j}}_{w_i} \times \underbrace{\frac{P_{i,1}}{P_{i,0}}}_{1+R_i} \; - 1 \\[2pt] &= \sum_{i=1}^{n} w_i R_i \qquad \text{(because the weights add up to 1)} \end{aligned}

For a price-weighted index the same result holds with $w_i = P_{i,0} / \sum_j P_{j,0}$ . For an equal-weighted index, $w_i = 1/n$ .

Try it

Harbour 3 is a made-up index of three US companies. Push Birch Pharma, the $200 stock, down and watch the price-weighted index fall while the others barely notice. Then switch to free float: Birch's weight halves again, because its founders hold half its shares.

Try itBuild the Harbour 3 index

Atlas Bank+10%

$50 → $55.00 · 400m shares

Birch Pharma−5%

$200 → $190.00 · 40m shares

Cobalt Tech+20%

$10 → $12.00 · 1,200m shares

Shares counted in the cap-weighted index

Price-weighted

988.46

−1.15%

Weights

Atlas Bank19.2%
Birch Pharma76.9%
Cobalt Tech3.8%

Cap-weighted

1,100.00

+10.00%

Weights

Atlas Bank50.0%
Birch Pharma20.0%
Cobalt Tech30.0%

Equal-weighted

1,083.33

+8.33%

Weights

Atlas Bank33.3%
Birch Pharma33.3%
Cobalt Tech33.3%

All three start at 1,000. Weights are at the start of the day. Birch Pharma has the highest price but the smallest company, so it dominates only the price-weighted index.

Worked example

One day in the Harbour 3

Prices in US dollars, shares and market values in millions. All three indices start at 1,000. During the day Atlas Bank rises 10%, Birch Pharma falls 5% and Cobalt Tech rises 20%.

Three weightings, three answers

Step 1 of 6

1
Set the divisors at the start
Stock Price Shares (m) Market value ($m)
Atlas Bank 50 400 20,000
Birch Pharma 200 40 8,000
Cobalt Tech 10 1,200 12,000
Total 260 40,000
Price-weighted divisor: $260 \div 1{,}000 = 0.26$ . Cap-weighted divisor: $40{,}000 \div 1{,}000 = 40$ .

Stock	Price	Shares (m)	Market value ($m)
Atlas Bank	50	400	20,000
Birch Pharma	200	40	8,000
Cobalt Tech	10	1,200	12,000
Total	260		40,000

Regulatory lens

Indices are used to price funds, derivatives and performance fees, so their rules are regulated. The EU Benchmarks Regulation requires index providers to be authorised or registered, to publish their methodology and to manage conflicts of interest. It builds on the IOSCO Principles for Financial Benchmarks of 2013. Lesson L1.6.05 shows why: it covers LIBOR, the interest rate benchmark whose manipulation led to these rules.

Looking ahead

In market risk models, equity exposures are often mapped to indices, and portfolios are stressed by shocking the index (L4.3).

Sparks

SparkAn index that lost half its value by rounding

The Vancouver Stock Exchange launched an index at 1,000 in 1982. It was recalculated after every trade, and each time the result was cut off (not rounded) at three decimal places. The tiny losses added up. By late 1983 the index read about 520, when the correct figure was close to 1,100.

SparkTwelve stocks in 1896

Charles Dow first published his industrial average in 1896, with twelve companies. It has since grown to 30. General Electric was the last of the original twelve to leave, in 2018.

Key takeaways

An index tracks a basket of shares by fixed rules, starting from a base value such as 100 or 1,000.
Price-weighted indices (the Dow, the Nikkei 225) weight by share price; most others weight by free-float market value.
A cap-weighted index return is the weighted sum of its stocks' returns: Σ wᵢRᵢ, with weights from market values.
The divisor is adjusted after splits, new shares and changes of constituents, so the level only moves when prices move.
Price indices ignore dividends; total return indices reinvest them, and the gap grows large over time.

Reaction

Check what stuck

Question 1 · objective 2

An index has two stocks. X is priced at 20 with 100 million shares; Y is priced at 80 with 25 million shares. X rises 10% and Y is unchanged. By what percentage does a price-weighted index rise?

%

Question 2 · objective 2

Same stocks, same moves. By what percentage does a market-cap-weighted index rise?

%

Question 3 · objective 2

A cap-weighted index holds two stocks with weights 60% and 40%. They return +5% and −10%. What is the index return, in per cent?

%

Question 4 · objective 3

A price-weighted index of three stocks priced 30, 60 and 90 has a divisor of 1.8. The 90 stock splits 3-for-1. What must the new divisor be, to one decimal place, so the index does not jump?

Question 5 · objective 4

Over a year, a price index rises 4%. Its companies paid dividends worth about 3% of their value. The total return version rises about:

A1%B4%C7%D12%

Glossary

Stock index: A single number that tracks the value of a defined basket of shares, calculated by fixed rules.
Constituents: The companies included in an index.
Index provider: The firm that designs, calculates and publishes an index, such as S&P Dow Jones Indices, FTSE Russell or MSCI.
Price-weighted: An index in which each stock counts in proportion to its share price.
Cap-weighted: An index in which each stock counts in proportion to its market value (price times shares).
Free float: The fraction of a company's shares freely available to trade, excluding stakes held by founders, governments and other long-term holders.
Divisor: The number an index's total is divided by. Set so the index starts at a round number, and adjusted so corporate actions do not make the level jump.
Rebalance: A scheduled review in which an index provider adds and removes companies and updates shares and float factors.
Price index: An index that tracks share prices only, ignoring dividends.
Total return index: An index that assumes dividends are reinvested, so it tracks the full return.

Deep Dives & Origins

Deep Dives

Bodie, Kane and Marcus, Investments, McGraw Hill, the section on stock market indices in chapter 2. Beginner.
S&P Dow Jones Indices, Index Mathematics Methodology (published online and updated regularly). Intermediate.

Origins

Bodie, Kane and Marcus, Investments, McGraw Hill.
S&P Dow Jones Indices, Index Mathematics Methodology.
Regulation (EU) 2016/1011 on indices used as benchmarks (the Benchmarks Regulation).
IOSCO, Principles for Financial Benchmarks: Final Report, July 2013.

Watch this lesson

Video version

Prefer to watch? The same lesson as a 10-minute video will sit here, made from this page's script.