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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.
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Suppose that Company A issues 200 new shares effective at the close of market Day 2. We need to account for these new shares in the calculation of the index return Day 3. One way to account for a market capitaliza- tion change is to calculate an adjusted beginning market cap for the return calculation on Day 3. Exhibit 7.11 shows how to adjust the end- ing market cap (32,450) upward for the impact of the new shares issued for Company A.
We do this by adding the market value of the new shares (shares × price) to the ending market capitalization on Day 2. The Day 2 adjusted ending market value is used in the denominator of the Day 3 index return calculation in cell H28. This is analogous to the way we add a purchase to the denominator for the daily security level return calcula- tion for portfolios that are trading. We can use the same adjustment methodology when a security is added to or dropped from an index.
Let’s take a closer look at the calculation of an index level. We are interested in creating a series of numbers (index levels) that represents the change in values of a fixed set of constituents defined on the base date, where each constituent is weighted by its market capitalization within the market as a whole. If there were no share changes from day to day we could calculate the index using this formula:
EXHIBIT 7.10 Index Return with Stock Split
EXHIBIT 7.11 Index Return with Capital Change
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(7.6)
But we also need to take into account periodic additions and dele- tions from the index, including the addition of shares due to a new stock issue by one of the current index constituents. We saw how to do this by adjusting the prior day market capitalization by the new issue when calculating the current day’s return. Another way of handling cap- italization changes is via the introduction of a capitalization adjustment factorthat corrects for the constituent changes and impacts of corporate actions. To use this method we take the following steps:
1. The starting denominator equals the beginning total market capital divided by the starting index level, in our case 1000.
2. Divide each day’s total market cap by the prior day divisor when there are no new or deleted index constituents and no corporate actions requiring adjustment.
3. When there is a capital change, calculate a divisor adjustment factor equal to:
(7.7)
4. Multiply the prior day divisor by the adjustment factor to get today’s divisor.
Exhibit 7.12 shows the calculation of the index levels and returns for each day using the index divisor approach.
EXHIBIT 7.12 Index Level Calculation Using Divisor Adjustment Capitalization weighted index level
Sum Current stock price( ×Current shares) Sum Prior stock price( ×Prior shares)
---×Prior index value
=
Current day adjustment factor
Prior day market cap+New component market cap Prior day market cap
---
=
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EXHIBIT 7.13 Total Return Index
Indices that are calculated in this way are called Laspeyres chain indi- ces. A Laspeyres index is the value of a basket of, in this case, securities where the value of each successive weighted basket of securities is linked together over time. The relative market caps of the individual securities within the index determine the security weights. The number of shares used to determine the weights is fixed at the base period, i.e., only the change in price serves to change the relative weight of the securities within the index over time. The adjustment factor reconciles the prior day index value to the new day’s market capitalization and protects the time series of index levels from changes in the components of the index and corporate actions. We can contrast the Laspeyres index to the Paasche index, where we do adjust the number of shares of each security within the index periodically. Most stock market indices are Laspeyres indices.
One advantage of the divisor approach is that it makes it easy to account for day-to-day market capitalization adjustments. The adjust- ment factor is used to account for many types of corporate actions, including secondary share offerings, stock repurchase programs, rights offerings, and spin-offs. In addition to corporate actions the adjustment factor is changed to account for new additions to or deletions from the index caused by mergers, bankruptcies, and corporate reorganizations.
The index levels calculated so far are price only indices; they do not include income earned on the securities within the index. An index that includes both price appreciation and dividend income is a total return index. If we are calculating a total return index we include the daily div- idend accruals in the calculation. Exhibit 7.13 shows how to use the divisor approach to calculate a total return index.
Fixed Income Indices
Fixed income indices are more difficult to select and construct than equity indices, for several reasons. First, there are many types of bonds, with varying maturities, credit quality, and other differentiating factors.
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The market index appropriate for one investor might have a very differ- ent collection of these bonds than that for another investor, and that index might not be representative of a weighted average of the bond mar- ket as a whole. To facilitate the selection of an index matching the char- acteristics of the portfolio, the fixed income index families offer a variety of subindices, which can be combined in different ways to provide a proper benchmark. Lehman Brothers, JP Morgan, Merrill Lynch, and other providers compile the major fixed income benchmark families.3