The value of the price return index can be calculated as follows:
Where,
VPRI = Value of Price-Return Index
ni = Number of units of security i
Pi = Price of security i
N = Number of equal securities in an index
D = Value of the Divisor
In the above equation,
a. The number of units of security depends upon the type of weightings used; and
b. The value of divisor is a number initially chosen at inception so that the index has a convenient initial value. The divisor changes over time so that changes in the index reflect the price changes only.
1.1. Single Period Return
There are two types of returns on the indices, i.e. the price return and the total return. They can be calculated as follows:
a. The Price Return. It is nothing but the percentage change in the value of the index. It can be calculated directly, as follows:
Where,
VPRI1 = Value of Index at Time 1
VPRI0 = Value of Index at Time 0
The indirect way of calculating the price return on the index is by calculating the weighted average of each of the constituent components or constituent security.
The price return on the individual security can be calculated as follows:
Where,
Pi1 = Price of Security at Time 1
Pi0 = Price of Security at Time 0
Now in order to calculate the return on the index, the sum of the return on each of the constituent security multiplied by the weight in the index is calculated.
Thus, the return on the index can be calculated as follows:
b. Total Returns. The total return on the index includes the return due to the price change plus any other income received from the security. Thus, the total return on the index can be calculated directly, as follows:
Where,
VPRI1 = Value of Index at Time 1
VPRI0 = Value of Index at Time 0
I1 = Income
The indirect way of calculating the price return on the index is by calculating the weighted average of total return on each of the constituent components or constituent security.
The total price return on the individual security can be calculated as follows:
Where,
Pi1 = Price of Security at Time 1
Pi0 = Price of Security at Time 0
I1 = Income on security i
Now in order to calculate the return on the index, the sum of the total return on each of the constituent security multiplied by its weight in the index is calculated.
Thus, the return on the index can be calculated as follows:
1.2. Multi-Period Returns
The multi-period return is the geometric progression of the returns in each period since inception.
a. Thus, we can calculate the multi-period price return on the index as follows:
VPRIT = VPRI0 (1 + PRI1) (1 + PRI2) … … … (1 + PRIt)
b. And, we can calculate the multi-period total return on the security as follows:
VTRIT = VTRI0 (1 + VTRI1) (1 + VTRI2) … … … (1 + VTRIt)