Price Return and Total Return of an Index

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:

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

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%

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop Graduate Admission Exam Prep


    Sergio Torrico
    Sergio Torrico
    2021-07-23
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    diana
    diana
    2021-07-17
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    2021-07-16
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    2021-06-28
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Marwan
    Marwan
    2021-06-22
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    2021-05-10
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.