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# Price Return and Total Return of an Index

## Index Value

The formula for calculating the value of a price return index is as follow:

$$V_{PRI} = \frac{ \sum_{i=1}^{N}{n_iP_i} } { D }$$

Where:

VPRI = the value of the price return index

ni = the number of units of constituent security  held in the index portfolio

N = the number of constituent securities in the index

Pi = the unit price of constituent security

Di = the value of the divisor

While the formula for calculating the value of an index may seem somewhat complicated at first glance, it is similar to calculating the value of any other normal portfolio of securities as it involves adding up the values of constituent securities. Index value calculation has just one additional step of dividing the sum of constituent securities’ values by a divisor, which is usually chosen at the inception of the index to set a convenient beginning value and then adjusted to offset index value changes unrelated to changes in the prices of constituent securities.

#### 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:

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

PRI = the price return of the index portfolio

VPRI1 = the value of the price return index at the end of the period

VPRI0 = the value of the price return index at the beginning of the period

TRI =    the total return of the index portfolio

IncomeI = 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.

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

RI = the return of the index portfolio number (as a decimal number)

Ri = the return of constituent security i (as a decimal number)

wi = 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.

VPRI0 = (5 × 107) + (7.75 × 55) + (4 × 75) + (5 × 660) + (2.75 × 100)

VPRI0 = 535 + 426.25 + 300 + 330 + 275 = 1,866.25

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

VPRI1 = (5 × 116) + (7.75 × 62) + (4 × 91) + (5 × 750) + (2.75 × 115)

VPRI1 = 580 + 480 + 364 + 375 + 316.25 = 2,115.75

And one more time to calculate portfolio income.

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

Income= 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%

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