Each index publisher has different methods for defining the universe of securities that are eligible for inclusion in the index and for selecting the actual constituent securities from this universe. Once these decisions have been made the weighting scheme of the securities selected within the index
Measuring Relative Return 109
is the next decision. There are several ways to weight securities within the index. Taking an equity index as an example, we could equal weight each stock in the index. Although some indices are equal weighted, use of an equal-weighted index as a benchmark implies that the investor equal weights stocks within a portfolio. Instead we could weight each index con- stituent according to its proportionate market capitalization. The market capitalization of a stock is equal to the shares outstanding, multiplied by the current market price. The total return for a market cap-weighted index is calculated by weighting each stock return by its proportional market cap. Most of the indices used as manager performance benchmarks are market capitalization weighted, and the examples here use a market cap- weighting methodology. In a market cap-weighted index, larger companies have a greater impact on the total index return than smaller companies. To calculate the market cap we need to determine the number of shares out- standing for each stock. To maintain an index that represents the invest- able marketplace, we can subtract from the total shares outstanding the shares that are closely held or government owned. To reduce double counting we can adjust the share balance by shares that are crossheld by other companies, i.e., two companies that hold stock in each other. Each of the publishers has a different methodology for figuring these adjust- ments. While our examples focus on a basic equity index, indices con- structed to gauge the performance of other asset classes adjust the basic methodology to handle the characteristics of the specific asset class.2
Once we have determined the weighting scheme, we can begin cal- culating index returns. Single period index returns for a basic equity index are calculated by taking the periodic market value of each constit- uent, weighting each constituent return, and then calculating the total index level. Suppose we are starting a new index and our index con- struction methodology filters the universe of stocks down to three stocks with a total beginning market cap of 31,500:
At the start of the measurement period we initialize the index level to an arbitrary base value, or level, say 1000. At the end of the next day we record the change in market value of the three stocks and calculate an index return for the day. Exhibit 7.8 shows the return calculation for Day 1.
2For more information on the construction of indices see Frank Fabozzi (ed.), Pro- fessional Perspectives on Indexing(New Hope, PA: Frank J. Fabozzi Associates, 1997).
110 RETURN MEASUREMENT
EXHIBIT 7.8 Index Return Calculation
The Day 1 index level is equal to 1020. We calculated the level by taking the ratio of the beginning and end market value (32,130/31,500
= 1.02). This gives the growth rate for the index on Day 1. Multiplying the growth rate by the initial index level of 1000 yields the index level at the end of Day 1 (1000 × 1.02 = 1020). We then calculated the index return by taking the ratio of the two daily index levels, 2.00% (1020/
1000 – 1). The index return is also equal to the prior day index level multiplied by (1 + total return for the day). There is an implicit reinvest- ment assumption in the index calculations; the gains in one period are reinvested into the index in the next period.
The change in index levels in a weighted average index represents the weighted average stock price change in the market for the day. It is a summary statistic highly influenced by stocks with large price changes and stocks with large percentage weights within the index.
We calculate index levels in addition to returns in order to build a time series that can be used to determine the performance of the market between any two dates. We can calculate multiperiod cumulative index returns by taking the ratio of the index levels between any two dates.
(7.4)
We can restate the cumulative index return to an annual basis in the same way we do fund returns:
(7.5)
Exhibit 7.9 shows the calculation, using levels, of compound cumu- lative and annualized returns for an index over several periods.
Cumulative index return Index level end of period Index level begin of period ---
1 – ×100
=
Annualized index return 1 Cumulative index return+
( )
365.25
# of days ---
1 – ×100
=
Measuring Relative Return 111
EXHIBIT 7.9 Multiperiod Returns Using Index Levels
It is important to note that multiperiod index returns are inherently time-weighted returns. We link the index growth rates, calculated using index levels, in the same way as we do the subperiod growth rates calcu- lated between cash flow dates for the portfolio. To calculate a dollar weighted index return, we could create a portfolio that purchases shares in the index level in proportion to the contribution and calculate a return on the dollars invested in this portfolio.