Price Return and Total Return of an Index

Index Value

The formula for calculating the value of a price return index is as follows:
$$ V_{PRI} = \frac{ \sum_{i=1}^{N}{n_iP_i} } { D } $$

Where:

V_PRI = The value of the price return index.

n_i = The number of units of constituent security held in the index portfolio.

N = The number of constituent securities in the index

P_i = The unit price of constituent security.

D_i = The value of the divisor.

The formula for calculating an index’s value might seem complex, but it’s akin to calculating a regular securities portfolio. You add up the values of its constituent securities. There’s an extra step: dividing this total by a divisor. This divisor is usually selected when the index begins to set a convenient starting value. It’s adjusted over time to account for changes in the index value unrelated to the securities’ prices.

Example 1: Index Value

An index is made up of two constituent securities, Stock A and Stock B. What beginning divisor must be used to achieve a beginning value of 1,000?

$$
\begin{array}{l|r|r}
\textbf{Security} & \textbf{Units} & \textbf{Price/Unit} \\
\hline
\text{Stock A} & 50 & 10 \\
\text{Stock B} & 30 & 100 \\
\end{array}
$$

Let’s first calculate the sum of the values of both constituent securities.

Stock A value = 50 × 10 = 500

Stock B value = 30 × 100 = 3,000

Stock A value + Stock B value = 3,500

The divisor must be set such that this figure is adjusted down to 1,000.

$$ 1,000 = \frac{ 3,500 } { D } $$

$$ D = \frac{ 3,500 } { 1,000 } $$

$$ D = 3.5 $$

Price Return and Total Return

The price return calculation – the return from the index in percentage terms – is simply the difference in value between the two periods divided by the beginning value.

$$ PR_I = \frac{ V_{ PRI1 } – V_{ PRI0 } } { V_{ PRI0 } } $$

The formula for total return is the same, except we need to add the income generated from the securities, usually in the form of dividends:

$$ PR_I = \frac{ V_{ PRI1 } – V_{ PRI0 } + \text{Income}_I } { V_{ PRI0 } } $$

PR_I = The price return of the index portfolio.

V_PRI1 = The value of the price return index at the end of the period.

V_PRI0 = The value of the price return index at the beginning of the period.

TR_I =    The total return of the index portfolio.

Income_I = The total income from all securities in the index over the period.

Another way to calculate these returns would be, to sum up the weighted returns of each constituent security in the index portfolio.

$$ R_I = w_1R_1 + w_2R_2 + … + w_NR_N $$

R_I = The return of the index portfolio number (as a decimal number).

R_i = The return of constituent security i (as a decimal number).

w_i = The weight of security i (the fraction of the index portfolio allocated to security.

Note that this formula works for both price and total return calculations.

Example 2: Price Return and Total Return

Calculate the one-year price return and total return for the Uncommon & Riches 5, a fictional index made up of five constituent securities. The divisor’s value begins and ends the year at 1.

$$
\begin{array}{l|r|r|r}
\textbf{Constituent Security} & \textbf{Units (billions)} & \textbf{Beginning Value} & \textbf{Dividend} & \textbf{Ending Value} \\
\hline
\text{Orange} & 5 & 107 & 2.15 & 116 \\
\text{Macrotough} & 7.75 & 55 & 1.20 & 62 \\
\text{Enout Stationary Corp} & 4 & 75 & 2.70 & 91 \\
\text{Draintree} & 0.5 & 660 & 0.00 & 750 \\
\text{Smith & Smith} & 2.75 & 100 & 3.00 & 115 \\
\end{array}
$$

Let’s first calculate the beginning index price by multiplying the number of units and price of each constituent security and totaling the values.

V_PRI0 = (5 × 107) + (7.75 × 55) + (4 × 75) + (5 × 660) + (2.75 × 100)

V_PRI0 = 535 + 426.25 + 300 + 330 + 275 = 1,866.25

We’ll do the same calculation again, except replace the beginning values with ending values.

V_PRI1 = (5 × 116) + (7.75 × 62) + (4 × 91) + (5 × 750) + (2.75 × 115)

V_PRI1 = 580 + 480 + 364 + 375 + 316.25 = 2,115.75

And one more time to calculate portfolio income.

Income_I = (5 × 2.15) + (7.75 × 1.20) + (4 × 2.70) + (5 × 0) + (2.75 × 3)

Income_I = 10.75 + 9.30 + 10.80 + 8.25 = 39.10

The one-year price return for the Uncommon & Riches 5 comes out to: (2,115.75 – 1,866.25)/1,866.25 = 13.37%

To calculate the total return, we’ll add in the portfolio income: (2,115.75 + 39.10 – 1,866.25)/1,866.25 = 15.46%