A security market index represents a given security market, market segment, or asset class. Most indexes are constructed as portfolios of marketable securities.
The value of an index is calculated on a regular basis using either the actual or estimated market prices of the individual securities, known as constituent securities, with the index.
For each security market index, investors may encounter two versions of the same index (i.e., an index with identical constituent securities and weights):
One version based on price return and one version based on total return. As the name suggests, a price return index, also known as a price index, reflects only the prices of the constituent securities within the index.
A total return index, in contrast, reflects not only the prices of the constituent securities but also the reinvestment of all income received since inception.
At inception, the values of the price and total return versions of an index are equal. As time passes, however, the value of the total return index, which includes the reinvestment of all dividends and/or interest received, will exceed the value of the price return index by an increasing amount.
The value of a price return index is calculated as:
where
- VPRI = the value of the price return index
- ni = the number of units of constituent security i held in the index portfolio
- N = the number of constituent securities in the index
- Pi = the unit price of constituent security i
- D = the value of the divisor
The divisor is a number initially chosen at inception. It is frequently chosen so that the price index has a convenient initial value.
The index provider then adjusts the value of the divisor as necessary to avoid changes in the index value that are unrelated to changes in the prices of its constituent securities.
Index return calculations, like calculations of investment portfolio returns, may measure price return or total return.
Price measures only price appreciation or percentage change in price.
Total return measures price appreciation plus interest, dividends, and other distributions.
Calculation of Single-Period Returns
For a security market index, price return can be calculated in two ways: either the percentage in value of the price return index, or the weighted average of price returns ofthe constituent securities.
The price return of an index can be expressed as:
where
- PRI = the price return of the index portfolio (as a decimal number, i.e., 12 percent is 0.12)
- 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
The price return of each constituent security can be expressed as:
where
- PRi = the price return of constituent security i (as a decimal number)
- Pi1 = the price of constituent security i at the end of the period
- Pi0 = the price of constituent security i at the beginning of the period
The price return of the index equals the weighted average of price returns of the individual securities,
where:
- PRI = the price return of index portfolio (as a decimal number)
- PRi = the price return of constituent security i (as a decimal number)
- N = the number of individual securities in the index
- wi = the weight of security i (the fraction of the index portfolio allocated to security i)
- Pi1 = the price of constituent security i at the end of the period
- Pi0 = the price of constituent security i at the beginning of the period
can be rewritten simply as:
where
- PRI = the price return of index portfolio (as a decimal number)
- PRi = the price return of constituent security i (as a decimal number)
- wi = the weight of security i (the fraction of the index portfolio allocated to security i)
- N = the number of securities in the index
Total return measures price appreciation plus interest, dividends, and other distributions. Thus, the total return of an index is the price appreciation, or change in the value of the price return index, plus income (dividends and/or interest) over the period, expressed as a percentage of the beginning value of the price return index. The total return of an index can be expressed as:
where
- TRI = the total return of the index portfolio (as a decimal number)
- 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
- IncI = the total income (dividends and/or interest) from all securities in the index held over the period
The total return of an index can also be calculated as the weighted average of total returns of the constituent securities. The total return of each constituent security in the index is calculated as:
where
- TRi = the total return of constituent security i (as a decimal number)
- P1i = the price of constituent security i at the end of the period
- P0i = the price of constituent security i at the beginning of the period
- Inci = the total income (dividends and/or interest) from security i over the period
The total return of an index can be calculated as the weighted average of total returns of the constituent securities,
can be rewritten simply as
where
- TRI = the total return of the index portfolio (as a decimal number)
- TRi = the total return of constituent security i (as a decimal number)
- wi = the weight of security i (the fraction of the index portfolio allocated to security i)
- N = the number of securities in the index
Calculation of Index Values over Multiple Time Periods
The calculation of index values over multiple time periods requires geometrically linking the series of index returns. With a series of price returns for an index, we can calculate the value of the price return index with the following equation:
where
- VPRI 0 = the value of the price return index at inception
- VPRIT = the value of the price return index at time t
- PRIT = the price return (as a decimal number) on the index over period t, t = 1, 2, …, T
Similarly, the series of total returns for an index is used to calculate the value of the total return index with the following equation:
where
- VTRI 0 = the value of the index at inception
- VTRIT = the value of the total return index at time t
- TRIT = the total return (as a decimal number) on the index over period t, t = 1, 2, …, T
