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Lesson 1 of 7 · 13 min

What an index is: value, price return and total return

An index is a paper portfolio that stands for a market; its price return version tracks only price changes, while its total return version also reinvests every dividend and coupon, so it pulls further ahead over time.

Video lesson · 6 min. Watch it first, then use the text below to go deeper.

In short

A security market index represents a security market, a market segment or an asset class. Most are built as portfolios of marketable securities, the constituent securities.
Index value = sum of (units held × price) ÷ a divisor. The divisor sets a convenient starting value and is adjusted so that events unrelated to price moves do not change the index.
A price return index reflects prices only; a total return index also reflects the reinvestment of all income since inception.
Single-period index return = change in index value (plus income, for total return) ÷ beginning value, or equivalently the weighted average of the constituents' returns.
Over many periods, returns are linked geometrically: $V_T = V_0(1 + R_1)(1 + R_2)\cdots(1 + R_T)$ .
Both versions start at the same value; the total return version then exceeds the price return version by a growing amount.

An index is a portfolio on paper

Watching thousands of prices is impossible, so investors summarise a market with one number. A security market index does that: it represents a given security market, market segment or asset class, such as large Japanese companies, euro corporate bonds or commodities.

Most indexes are defined as a portfolio: so many units of each constituent security. The provider recalculates the portfolio's value regularly, using actual market prices where securities trade, or estimated prices where they do not (common for bonds).

The same index (identical constituents and weights) is usually published in two versions. The price return index (or price index) reflects only the constituents' prices. The total return index reflects prices plus the reinvestment of all income (dividends, interest) received since inception.

Two versions of the same index
	Price return index	Total return index
Reflects	Constituent prices only	Prices plus reinvested dividends and interest
Value at inception	Same as total return version	Same as price return version
Value later	Lower whenever constituents pay income	Higher, by a growing amount
Equal to each other later only if	Constituents pay no income	Constituents pay no income

Index value and the divisor

The value of a price return index is the market value of the index portfolio divided by a divisor: $V_{PRI} = \sum n_iP_i / D$ , where $n_i$ is the number of units of security i held in the index and $P_i$ its price.

The divisor is picked at inception, usually so the index starts at a round number such as 100 or 1,000. After that its job is to keep the index continuous: when something other than a price move would change the sum (a constituent is replaced, a stock splits), the provider resets the divisor so the index value is the same immediately before and after the change.

The divisor is a scaling device, not a source of return. Any index return can be computed from the portfolio values alone, because the divisor cancels out when it does not change during the period.

Think of the divisor as the exchange rate between 'euros in the portfolio' and 'index points'. It only changes when the portfolio itself is restructured, never because prices moved.

Returns over one period

Price return measures price appreciation only: $PR_I = (V_{PRI,1} - V_{PRI,0}) / V_{PRI,0}$ . Total return adds the income paid by all constituents over the period, $Inc_I$ , expressed in index points: $TR_I = (V_{PRI,1} - V_{PRI,0} + Inc_I) / V_{PRI,0}$ .

Note that both use the price index values at the start and end; total return simply adds the income on top. The denominator is always the beginning value.

Because the index is a portfolio, its return also equals the weighted average of the constituents' returns, with each weight being the security's share of the index value at the start of the period: $PR_I = \sum w_iPR_i$ and $TR_I = \sum w_iTR_i$ . A security's own total return is $(P_{1} - P_{0} + Inc) / P_{0}$ .

The second example: the index return is the beginning-weight average of its constituents' returns, so it lies between the worst and the best. Income lifts the total return above the price return.

Index values over many periods

To get an index value after several periods, compound the period returns: $V_{PRI,T} = V_{PRI,0}(1 + PR_{1})(1 + PR_{2})\cdots(1 + PR_{T})$ . The total return index uses the total returns in the same way.

Because each period's income is reinvested and then earns returns itself, the gap between the two versions keeps growing over time: income compounds on top of income, so over long periods the total return index pulls further and further ahead (in any single year the gap can grow by more or less, depending on that year's returns). Over decades the difference between a dividend-paying market's price index and its total return index becomes very large.

Both versions start at 1,000. Price returns of 5%, −3%, 8%, 2%, −6%, 10%, 4% and 6% plus 2% income a year: after one year the gap is 20 points, after eight years it is about 213, because reinvested income compounds.

Formulas

Value of a price return index: $V_{PRI} = \frac{\sum_{i=1}^{N} n_iP_i}{D}$
$n_i$ = units of security i in the index, $P_i$ = its price, N = number of constituents, D = divisor.
Index price return and total return (one period): $PR_I = \frac{V_{PRI,1} - V_{PRI,0}}{V_{PRI,0}}, \qquad TR_I = \frac{V_{PRI,1} - V_{PRI,0} + Inc_I}{V_{PRI,0}}$
$Inc_I$ = income from all constituents over the period, in index points. Both divide by the beginning value of the price index.
Index return as a weighted average: $PR_I = \sum_{i=1}^{N} w_i\frac{P_{i,1} - P_{i,0}}{P_{i,0}}, \qquad TR_I = \sum_{i=1}^{N} w_i\frac{P_{i,1} - P_{i,0} + Inc_i}{P_{i,0}}$
$w_i$ = security i's share of the index value at the start of the period.
Index value over T periods: $V_{PRI,T} = V_{PRI,0}(1 + PR_{I,1})(1 + PR_{I,2})\cdots(1 + PR_{I,T})$
Same form for the total return index with total returns $TR_{I,t}$ .

Worked examples

Example 1 Price and total return from index values

The Nordvest Equity Index stood at 2,400 at the start of the year and 2,460 at the end. Its constituents paid dividends worth 36 index points during the year. What are the index's price return and total return?

Example 2 Index return as a weighted average

An index holds three shares. At the start of the month Kestrel is 50% of the index value, Lumen 30% and Mira 20%. Kestrel moves from EUR 40 to 42 and pays EUR 0.80; Lumen moves from EUR 25 to 24 and pays EUR 0.50; Mira moves from EUR 50 to 55 and pays nothing. What are the index's price and total returns?

Example 3 Linking returns over two periods

A price index and its total return twin both start at 1,000. Price returns are 4% in Year 1 and −2% in Year 2; income adds 2.0% in Year 1 and 2.5% in Year 2. What are both index values after two years?

On the exam

Expect one-period price and total return calculations (from index values or from constituent returns and weights), a two-period index value by compounding, and concept items on when the two index versions are equal.

Dividing by the ending index value instead of the beginning value.
Thinking the price and total return versions are equal on rebalancing dates; they are equal only at inception (or always, if nothing pays income).
Adding period returns instead of compounding them when finding a multi-period index value.
Believing a divisor change creates return; it only keeps the index continuous across non-price events.

Check yourself

Q1A price index moves from 1,250 to 1,300 over a year, and its constituents pay income worth 15 index points. The index's total return is closest to:
Q2An index is 60% stock X and 40% stock Y at the start of a period. X's price rises 10% and Y's falls 5%. The index's price return is closest to:
Q3A total return index starts at 1,000 and earns total returns of 8% and then −5%. Its value after the two periods is closest to:
Q4Three years after inception, the total return version of an index is higher than its price return version. The most likely reason is that:
Q5An index provider changes the divisor when it replaces one constituent with another, most likely to:

0/5Check yourself

Answer the 5 questions: +10 XP for each one you get right first time.

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