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 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.
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?
Let’s first calculate the sum of the values of both constituent securities.
Stock A value = 50 x 10 = 500
Stock B Value = 30 x 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 = 3,500/D
D = 3,500/1,000
D = 3.5
Price Return and Total Return
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
IncI = the total income from all securities in the index over the period
The price return calculation is simply the difference in value between the two periods divided by the beginning value. Calculating total return involves adding income into the numerator. Another way to calculate these returns would be to sum up the weighted returns of each constituent security in the index portfolio.
RI = w1R1 + w2R2 + … + WNRN
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
This formula works for both price and total return calculations.
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.
|Constituent Security||Units (billions)||Beginning Value||Dividend||Ending Value|
|Enout Stationary Corp||4||75||2.70||91|
|Smith & Smith||2.75||100||3.00||115|
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.
IncI = (5 * 2.15) + (7.75 * 1.20) + (4 * 2.70) + (5 * 0) + (2.75 * 3)
IncI = 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%
Reading 45 LOS 45b:
Calculate and interpret the value, price return, and total return of an index