Articles

The Mathematics Of Equity Index Pricing
By Mark Labovitz and Jeff Kenyon

Page 1 of 8

The pricing of an index made up of a set of underlying securities appears straightforward. However, there are a variety of factors that make the process nontrivial. These factors include the introduction or removal of instruments when the index is rebalanced, the effects of corporate actions (e.g., dividends, splits, spinoffs, etc.) and the impact of liquidation. This paper describes the process of index pricing, for both equally weighted and market-capitalization-weighted indexes, for price-only and total return variants, with the intent of surfacing assumptions and describing the handling of many special cases.

Assumptions
The pricing described in this paper proceeds from the existence of the set of equities to be used to form the portfolio. The creation of this set of equities, which the authors collectively call the constituents, from the universe of available securities is beyond the scope of this paper. Along with lists of constituents, other data are assumed to be available to the computations. These include equity prices and shares data whose frequency conforms to the interval at which the index price is desired. However, the reader should be aware that beyond the data frequency there are a number of issues affecting the creation of real-time index prices, and these issues are not considered herein. This paper focuses on the use of daily closing values. The following daily closing values are of interest: closing prices, float-adjusted shares, detailed descriptions of corporate actions applicable to and indexed by close of the day, and if multiple currencies are involved, the closing exchange rates between the currencies of interest. Consistent with this time frame, all variable subscripts which refer to time are referring to a series of close-of-the-day points.

Additionally, this paper addresses two index weighting schemes:

Equally weighted (EQ), and
Market-Capitalization-weighted (MC) (also referred to as market-value-weighted [Bodie, et al. 2001])

The price formulations are laid out such that many of the computations performed for these index weighting approaches are identical. For either form of weighting, the index pricing for a given day is formulated as the sum of the closing price × shares × a currency conversion factor, which is multiplied by a quantity describing the previous day’s index price.

An important set of assumptions being made relate to the adjustment of the indexes for applicable corporate actions (CAs), and the impact of those actions upon the mechanical calculation of the index. The methods for performing these adjustments are described below. No discussion of the selection of initial (IPO) index values is given, although there are commonly used conventions in setting such values.

When the index price is not adjusted for the issuance of dividends, the indexes, regardless of whether they are MC or EQ weighted, are denoted as price-only indexes. This is the assumed state of affairs in the material up to the last section. In that final section, the authors provide a discussion for pricing total return indexes, i.e., indexes that are adjusted for the total return of their components (including dividends). After the adjustment on the ex-date that adds back the dividend, the dividend disappears from the computation and the total return is thereafter adjusted by the commonly used computation of price-only, normalized by a previous-day divisor. It is thereby assumed that the dividends are subsequently invested in the aggregate behavior of the index. Total return indexes are computed for both weighting schemes.

Equally Weighted Indexes
After the selection of an index’s constituents, initial pricing of an equally weighted index is as follows. Select a nominal monetary amount¹ for the component value² (in dollars for regional indexes and local currency for country indexes), and call that amount v0. To find the initial total value, we then have

a₀ = v₀ x n (1)