Index Mathematics
Methodology

March 2024
S&P Dow Jones Indices: Index Methodology

Table of Contents
Introduction

4
Different Varieties of Equity Indices

4

The Index Divisor

5

Supporting Documents

5

Capitalization Weighted Indices6

Definition

6

Adjustments to Share Counts

6

Divisor Adjustments

7

Necessary Divisor Adjustments

8

Capped Market Capitalization Indices10

Definition10

Corporate Actions and Index Adjustments

11

Different Capping Methods

11

Non-Market Capitalization Weighted Indices13

Definition

13

Corporate Actions and Index Adjustments

14

Price Weighted Indices15

Definition

15

Equal Weighted Indices16

Definition

16

Modified Equal Weighted Indices17

Corporate Actions and Index Adjustments

17

Multi-Day Rebalancing

18

Exchange Holidays

18

Freeze Date

19

Total Return Calculations

21

Net Total Return Calculations

22

Post Ex-Dividend Adjustment: Total and Net Total Return Calculation

23

Franking Credit Adjusted Total Return Indices24

Currency and Currency Hedged Indices26

Return Definitions

26

The Hedge Ratio

27

S&P Dow Jones Indices: Index Mathematics Methodology

1Calculating a Currency-Hedged Index

27

Currency Hedging Outcomes29

Index Computation

29

Dynamic Hedged Return Indices32

Currency Hedged Excess Return Indices34

Quanto Currency Adjusted Index

35

Domestic Currency Return Index Calculation

37

Background37

Equivalence of DCR and Divisor Calculations

37

DCR Calculation

38

Essential Adjustments

38

Risk Control Indices

39

Dynamic Rebalancing Risk Control Index

42

Capped Equity Weight Change

43

Excess Return Indices43

Exponentially-Weighted Volatility

44

Exponentially-Weighted Volatility Based on Current Allocations

45

Simple-Weighted Volatility

46

Futures-Based Risk Control Indices47

Exponentially-Weighted Volatility for Futures-Based Risk Control Indices48

Dynamic Volatility Risk Control Indices48

Variance Based Risk Control Indices48

Risk Control 2.0 Indices

49

Constituent Weighting

49

Risk Control 2.0 with Minimum Variance

50

Equity with Futures Leverage Risk Control Indices52

Weighted Return Indices

53

Leveraged and Inverse Indices

55

Leveraged Indices for Equities

55

Leveraged Indices without Borrowing Costs for Equities

56

Inverse Indices for Equities

56

Inverse Indices without Borrowing Costs for Equities

57

Leveraged and Inverse Indices for Futures

57

Daily Rebalanced Leverage or Inverse Futures Indices57

Periodically Rebalanced Leverage or Inverse Futures Indices58

Fee Indices/Decrement and Increment Indices

59

Capped Return Indices64

S&P Dow Jones Indices: Index Mathematics Methodology

2

Dividend Points Indices65

Alternative Pricing

67

Special Opening Quotation (SOQ)

67

Fair Value Indices68

Volume-Weighted Average Price (VWAP)

68

Time-Weighted Average Price (TWAP)

68

Negative/Zero Index Levels

69

Index Turnover70

End-of-Month Global Fundamental Data

71

Monthly Files71

About the Data

71

Output Files

72

Fundamental Data Points

72

Calculations

73

Contact Information
Client Services

Disclaimer

77
77

78
Performance Disclosure/Back-Tested Data

78

Intellectual Property Notices/Disclaimer

79

ESG Indices Disclaimer

81

S&P Dow Jones Indices: Index Mathematics Methodology

3Introduction
This document covers the mathematics of equity index and other quantitative rules-based calculations
and assumes some acquaintance with mathematical notation and simple operations. The calculations arepresented principally as equations, which have largely been excluded from the individual index
methodologies, with examples or tables of results to demonstrate the calculations.
Different Varieties of Equity Indices
S&P Dow Jones Indices’ (S&P DJI) index calculation and corporate action treatments vary according to
the categorization of the indices. At a broad level, indices are defined into two categorizations; Market
Capitalization Weighted and Non-Market Capitalization Weighted Indices.
A majority of S&P DJI’s equity indices are market capitalization weighted and float-adjusted, where eachstock’s weight in the index is proportional to its float-adjusted market value. S&P DJI also offers cappedversions of a market capitalization weighted index where single index constituents or defined groups ofindex constituents, such as sector or geographical groups, are confined to a maximum weight.
Non-market capitalization weighted indices include those that are not weighted by float-adjusted market
capitalization and generally are not affected by notional market capitalization changes resulting from
corporate events. Examples include indices that apply equal weighting, factor weighting such as dividend
yield or volatility, strategic tilts, thematic weighting, or other alternative weighting schemes.
S&P DJI offers a variety of indices and index attribute data calculated according to various methodologies
which are covered in this document:•••

Market Capitalization Indices:oMarket-capitalization indices – where constituent weights are determined either by totalor float-adjusted market capitalization.

oCapped market-capitalization indices − where single index constituents or defined groups
of index constituents, such as sector or geographical groups, are confined to a maximum
index weight.

Non-Market Capitalization Indices:o

Price weighted indices − where constituent weights are determined solely by the prices of
the constituent stocks in the index.

oEqual weighted indices − where each stock is weighted equally in the index.

Derived Indices:oTotal return indices − index level reflect both movements in stock prices and thereinvestment of dividend income.

oLeveraged and inverse indices − which return positive or negative multiples of theirrespective underlying indices.

o

Weighted return indices − commonly known as index of indices, where each underlying
index is a component with an assigned weight to calculate the overall index of indiceslevel.

oIndices that operate on an index as a whole rather than on the individual stocks − theseinclude calculations of various total return methodologies and index fundamentals.

oDividend Point indices − which track the total dividend payments of index constituents.

S&P Dow Jones Indices: Index Mathematics Methodology

4oRisk control, decrement, excess return, currency, currency hedged, domestic currency
return, special opening quotation, turnover and fundamental data calculations.

The Index Divisor
The purpose of the index divisor is to maintain the continuity of an index level following theimplementation of corporate actions, index rebalancing events, or other non-market driven actions.
The simplest capitalization weighted index can be thought of as a portfolio consisting of all available
shares of the stocks in the index. While one might track this portfolio’s value in dollar terms, it would
probably be an unwieldy number – for example, the S&P 500 float-adjusted market value is a figure in thetrillions of dollars. Rather than deal with ten or more digits, the figure is scaled to a more easily handled
number (e.g., 2000). Dividing the portfolio market value by a factor, usually called the divisor, does thescaling.
An index is not exactly the same as a portfolio. For instance, when a stock is added to or deleted from anindex, the index level should not jump up or drop down; while a portfolio’s value would usually change asstocks are swapped in and out. To assure that the index’s value, or level, does not change when stocks
are added or deleted, the divisor is adjusted to offset the change in market value of the index. Thus, thedivisor plays a critical role in the index’s ability to provide a continuous measure of market valuation when
faced with changes to the stocks included in the index. In a similar manner, some corporate actions thatcause changes in the market value of the stocks in an index should not be reflected in the index level.
Adjustments are made to the divisor to eliminate the impact of these corporate actions on the index value.
Supporting Documents
This methodology is meant to be read in conjunction with supporting documents providing greater detailwith respect to the policies, procedures and calculations described herein. References throughout themethodology direct the reader to the relevant supporting document for further information on a specifictopic. The list of the main supplemental documents for this methodology and the hyperlinks to those
documents is as follows:
Supporting Document
S&P Dow Jones Indices’ Equity Indices Policies &Practices Methodology
S&P Dow Jones Indices’ Float Adjustment
Methodology

S&P Dow Jones Indices: Index Mathematics Methodology

URL
Equity Indices Policies & Practices
Float Adjustment Methodology

5Capitalization Weighted Indices
Many of S&P DJI’s equity indices are capitalization-weighted indices. Sometimes these are called valueweighted, or market cap weighted, instead of capitalization weighted. Examples include the S&P 500, theS&P Global 1200 and the S&P BMI indices.
In the discussion below most of the examples refer to the S&P 500 but apply equally to a long list of S&P
DJI’s cap-weighted indices.
Definition
The formula to calculate the S&P 500 is: Pi  Qi

Index Level = i
Divisor(1)

The numerator on the right hand side is the price of each stock in the index multiplied by the number ofshares used in the index calculation. This is summed across all the stocks in the index. The denominatoris the divisor. If the sum in the numerator is US$ 20 trillion and the divisor is US$ 10 billion, the index level
would be 2000.
This index formula is sometimes called a “base-weighted aggregative” method.1 The formula is created bya modification of a LasPeyres index, which uses base period quantities (share counts) to calculate theprice change. A LasPeyres index would be: Pi,1  Qi,oIndex = i Pi,0  Qi,o(2)i

In the modification to (2), the quantity measure in the numerator, Q 0, is replaced by Q1, so the numerator
becomes a measure of the current market value, and the product in the denominator is replaced by thedivisor which both represents the initial market value and sets the base value for the index. The result of
these modifications is equation (1) above.
Adjustments to Share Counts
S&P DJI’s market cap-weighted indices are float-adjusted – the number of shares outstanding is reduced
to exclude closely held shares from the index calculation because such shares are not available toinvestors.
For more information on shares outstanding, please refer to S&P Dow Jones Indices’ Equity Indices
Policies & Practices Methodology.
S&P DJI’s rules for float adjustment are described in more detail in S&P Dow Jones Indices’ Float
Adjustment Methodology or in some of the individual index methodology documents. As discussed there,

1

This term is used in one of the earlier and more complete descriptions of S&P Dow Jones Indices’ index calculations in Alfred
Cowles, Common Stock Indices, Principia Press for the Cowles Commission of Research in Economics, 1939. The book refers to
the “Standard Statistics Company Formula;” S&P was formed by the merger of Standard Statistics Corporation and Poor’s
Publishing in 1941.

S&P Dow Jones Indices: Index Mathematics Methodology

6for each stock S&P DJI calculates an Investable Weight Factor (IWF) which is the percentage of total
shares outstanding that are included in the index calculation.
When the index is calculated using equation (1), the variable Qi is replaced by the product of outstanding
shares and the IWF:
Qi = IWFi  Total Shares i(3)

At times there are other adjustments made to the share count to reflect foreign ownership restrictions or
to adjust the weight of a stock in an index. These are combined into a single multiplier in place of the IWF
in equation (3). In combining restrictions, it is important to avoid unwanted double counting. Let FA
represent the fraction of shares eliminated due to float adjustment, FR represent the fraction of shares
excluded for foreign ownership restrictions and IS represent the fraction of total shares to be excluded
based on the combination of FA and FR.
If FA > FR then IS = 1- FA
If FA < FR then IS = 1-FR
and equation (3) can be written as:
Qi = ISi  Total Shares i

Note that any time the share count or the IWF is changed, it will be necessary to adjust the index divisor
to keep the level of the index unchanged.
Divisor Adjustments
The key to index maintenance is the adjustment of the divisor. Index maintenance – reflecting changes inshares outstanding, corporate actions, addition or deletion of stocks to the index – should not change thelevel of the index. If the S&P 500 closes at 2000 and one stock is replaced by another, after the market
close, the index should open at 2000 the next morning if all of the opening prices are the same as theprevious day’s closing prices. This is accomplished with an adjustment to the divisor.
Any change to the stocks in the index that alters the total market value of the index while holding stock
prices constant will require a divisor adjustment. This section explains how the divisor adjustment is madegiven the change in total market value. The next section discusses what index changes and corporate
actions lead to changes in total market value and the divisor.
Equation (1) is expanded to show the stock being removed, stock r, separately from the stocks that willremain in the index:(
Index Level t −1 = Pi  Qi ) + Pr QriDivisort −1(4)

Note that the index level and the divisor are now labeled for the time period t-1 and, to simplify thisexample, that we are ignoring any possible IWF and adjustments to share counts. After stock r is
replaced with stock s, the equation will read:(Index Level t = Pi  Qi ) + PsQs
i

Divisort(5)

In equations (4) and (5) t-1 is the moment right before company r is removed from and s is added to theindex; t is the moment right after the event. By design, Index Levelt-1 is equal to Index Levelt. Combining(4) and (5) and re-arranging, the adjustment to the Divisor can be determined from the index market value
before and after the change:
S&P Dow Jones Indices: Index Mathematics Methodology

7( Pi  Qi ) + Pr QriDivisort −1(= Index Level = Pi  Qi ) + PsQs
i

Divisort

Let the numerator of the left hand fraction be called MVt-l, for the index market value at (t-1), and thenumerator of the right hand fraction be called MVt, for the index market value at time t. Now, MVt-1, MVt
and Divisort-1 are all known quantities. Given these, it is easy to determine the new divisor that will keep
the index level constant when stock r is replaced by stock s:

Divisort = (Divisort −1) MVt
MVt −1(6)

As discussed below, various index adjustments result in changes to the index market value. When theseadjustments occur, the divisor is adjusted as shown in equation (6).
In some implementations, including the computer programs used in S&P DJI’s index calculations, thedivisor adjustment is calculated in a slightly different, but equivalent, format where the divisor change iscalculated by addition rather than multiplication. This alternative format is defined here. Rearranging
equation (1) and using the term MV (market value) to replace the summation gives:

Divisor =MV
Index Level

When stocks are added to or deleted from an index there is an increase or decrease in the index’s market
value. This increase or decrease is the market value of the stocks being added less the market value of
those stocks deleted; define CMV as the Change in Market Value. Recalling that the index level does notchange, the new divisor is defined as:

DivisorNew =MV + CMV
Index Level

DivisorNew =

MV
CMV
+
IndexLevel IndexLevel

orHowever, the first term on the right hand side is simply the Divisor value before the addition or deletion of
the stocks. This yields the following:

DivisorNew = DivisorOld +CMVIndexLevel(7)

Note that this form is more versatile for computer implementations. With this additive form, the second
term (CMV/Index Level) can be calculated for each stock or other adjustment independently and then all
the adjustments can be combined into one change to the Divisor.
Necessary Divisor Adjustments
Divisor adjustments are made “after the close” meaning that after the close of trading the closing prices
are used to calculate the new divisor based on whatever changes are being made. It is, then, possible toprovide two complete descriptions of the index – one as it existed at the close of trading and one as it will
exist at the next opening of trading. If the same stock prices are used to calculate the index level for thesetwo descriptions, the index levels are the same.

S&P Dow Jones Indices: Index Mathematics Methodology

8With prices constant, any change that changes the total market value included in the index will require adivisor change. For cataloging changes, it is useful to separate changes caused by the management of
the index from those stemming from corporate actions of the constituent companies. Among thosechanges driven by index management are adding or deleting companies, adjusting share counts andchanges to IWFs and other factors affecting share counts.
Index Management Related Changes. When a company is added to or deleted from the index, the netchange in the market value of the index is calculated and this is used to calculate the new divisor. The
market values of stocks being added or deleted are based on the prices, shares outstanding, IWFs andany other share count adjustments. Specifically, if a company being added has a total market cap of US$
1 billion, an IWF of 85% and, therefore, a float-adjusted market cap of US$ 850 million, the market value
for the added company used is US$ 850 million. The calculations would be based on either equation (6)or equation (7) above.
For most S&P DJI equity indices, IWFs and share counts updates are applied throughout the year basedon rules defined in the methodology. Typically, small changes in shares outstanding are reflected inindices once a quarter to avoid excessive changes to an index. The revisions to the divisor resulting fromthese are calculated and a new divisor is determined. Equation (7) shows how the impact of a series ofshare count changes can be combined to determine the new divisor.
Corporate Action Related Changes.
For information on the treatment of corporate actions, please refer to S&P Dow Jones Indices’ Equity
Indices Policies & Practices document. For more information on the specific treatment within an index
family, please refer to that index methodology.

S&P Dow Jones Indices: Index Mathematics Methodology

9Capped Market Capitalization IndicesDefinition
A capped market capitalization weighted index (also referred to as a capped market cap index, cappedindex or capped weighted index) is one where single index constituents or defined groups of index
constituents are confined to a maximum weight and the excess weight is distributed proportionately
among the remaining index constituents. As stock prices move the weights will shift and the modified
weights will change. Therefore, a capped market cap weighted index must be rebalanced from time totime to re-establish the proper weighting. The methodology for capped indices follows an identical
approach to market cap weighted indices except that the indices apply an additional weight factor, or“AWF”, to adjust the float-adjusted market capitalization to a value such that the index weight constraints
are satisfied. For capped indices, no AWF change is made due to corporate actions between
rebalancings except for daily capped indices where the corporate action may trigger a capping.
Therefore, the weights of stocks in the index as well as the index divisor will change due to notional
market capitalization changes resulting from corporate events.
The overall approach to calculate capped market cap weighted indices is the same as in the pure marketcap weighted indices; however, the constituents’ market values are re-defined to be values that will meetthe particular capping rules of the index in question.

Index Level =Index Market Value
Divisor(1)and𝐼𝑛𝑑𝑒𝑥𝑀𝑎𝑟𝑘𝑒𝑡𝑉𝑎𝑙𝑢𝑒 = ∑ 𝑃𝑖 ∗ 𝑆ℎ𝑎𝑟𝑒𝑠𝑖 ∗ 𝐼𝑊𝐹𝑖 ∗ 𝐹𝑥𝑅𝑎𝑡𝑒𝑖𝑖

To calculate a capped market cap index, the market capitalization for each stock used in the calculationof the index is redefined so that each index constituent has the appropriate weight in the index at eachrebalancing date.
In addition to being the product of the stock price, the stock’s shares outstanding, and the stock’s float
factor (IWF), as written above – and the exchange rate when applicable – a new adjustment factor is also
introduced in the market capitalization calculation to establish the appropriate weighting.
AdjustedStock Market Valuei = Pi  Shares i  IWFi  FxRatei  AWFi

where AWFi is the adjustment factor of stock i assigned at each index rebalancing date, t, which adjusts
the market capitalization for all index constituents to achieve the user-defined weight, while maintaining
the total market value of the overall index.
The AWF for each index constituent, i, on rebalancing date, t, is calculated by:
AWFi , t =CWi , tWi , tS&P Dow Jones Indices: Index Mathematics Methodology

10where Wi,t is the uncapped weight of stock i on rebalancing date t based on the float-adjusted market
capitalization of all index constituents; and CWi,t is the capped weight of stock i on rebalancing date t as
determined by the capping rule of the index in question and the process for determining capped weights
as described in Different Capping Methods below.
The index divisor is defined based on the index level and market value from equation (1). The index levelis not altered by index rebalancings. However, since prices and outstanding shares will have changed
since the last rebalancing, the divisor will change at the rebalancing.
So:(Divisor) after rebalancing =(Index Market Value) after rebalancing(Index Value) before rebalancing

where:

Index Market Value = Pi  Sharesi  IWFi  FxRatei  AWFi
i

Corporate Actions and Index Adjustments
All corporate actions for capped indices affect the index in the same manner as in market capitalization
weighted indices.
For more information on the treatment of corporate actions, please refer to S&P Dow Jones Indices’
Equity Indices Policies & Practices document.
Different Capping Methods
Capped indices arise due to the need for benchmarks which facilitate diversification rules. Capping may
apply to single stock concentration limits or concentration limits on a defined group of stocks. At times,
companies may also be represented in an index by multiple share class lines. In these instances,maximum weight capping will be based on company float-adjusted market capitalization, with the weightof multiple class companies allocated proportionally to each share class line based on its float-adjusted
market capitalization as of the rebalancing reference date. Some common, but not an exhaustive list of,examples of the standard S&P DJI methodologies for determining the weights of capped indices using themost popular capping methods are described below.
Single Company Capping. In a single company capping methodology, no company in an index isallowed to breach a certain pre-determined weight as of each rebalancing period. The procedure forassigning capped weights to each company at each rebalancing is as follows:
1. With data reflected on the rebalancing reference date, each company is weighted by floatadjusted market capitalization.
2. If any company has a weight greater than X% (where X% is the maximum weight allowed in theindex), that company has its weight capped at X%.
3. All excess weight is proportionally redistributed to all uncapped companies within the index.
4. After this redistribution, if the weight of any other company(s) then breaches X%, the process isrepeated iteratively until no companies breach the X% weight cap.
Single Company and Concentration Limit Capping. In a single company and concentration limit
capping methodology, no company in an index is allowed to breach a certain pre-determined weight andall companies with a weight greater than a certain amount are not allowed, as a group, to exceed a predetermined total weight. One example of this is 4.5%/22.5%/45% capping (B/A/C in the following
example). No single company is allowed to exceed 22.5% of the index and all companies with a weightgreater than 4.5% of the index cannot exceed, as a group, 45% of the index.
S&P Dow Jones Indices: Index Mathematics Methodology

11

Method 1:
The procedure for assigning capped weights to each company at each rebalancing is as follows:
1. With data reflected on the rebalancing reference date, each company is weighted by floatadjusted market capitalization.
2. If any company has a weight greater than A% (where A% is the maximum weight allowed in theindex), that company has its weight capped at A%.
3. All excess weight is proportionally redistributed to all uncapped companies within the index.
4. After this redistribution, if the weight of any other company(s) then breaches A%, the process isrepeated iteratively until no companies breach the A% weight cap.
5. The sum of the companies with weight greater than B% cannot exceed C% of the total weight.
6. If the rule in step 5 is breached, all the companies are ranked in descending order of their weights
and the company with the lowest weight that causes the C% limit to be breached is identified. The
weight of this company is, then, reduced either until the rule in step 5 is satisfied or it reaches
B%.
7. This excess weight is proportionally redistributed to all companies with weights below B%. Any
stock that receives weight cannot breach the B% cap. This process is repeated iteratively until
step 5 is satisfied or until all stocks are greater than or equal to B%.
8. If the rule in step 5 is still breached and all stocks are greater than or equal to B%, the company
with the lowest weight that causes the C% limit to be breached is identified. The weight of thiscompany is, then, reduced either until the rule in step 5 is satisfied or it reaches B%.
9. This excess weight is proportionally redistributed to all companies with weights greater than B%.
Any stock that receives weight cannot breach the A% stock cap. This process is repeatediteratively until step 5 is satisfied.
For indices that use capping rules across more than one attribute, S&P DJI will utilize an optimization
program to satisfy the capping rules. The stated objective for the optimization will be to minimize thedifference between the pre-capped weights of the stocks in the index and the final capped weights. Thisis done by using an optimization procedure that chooses final weights in such a way to minimize the sumof the squared difference of capped weight and uncapped weight, divided by uncapped weight for eachstock.
Method 2:
A second method of single company and concentration limit capping utilized by S&P DJI for assigning
capped weights to each company at each rebalancing is as follows:
1. With data reflected on the rebalancing reference date, each company is weighted by floatadjusted market capitalization.
2. If either of the defined single-company or concentration-index-weight limits is breached, the floatadjusted market capitalization of all components is raised to a power such as the following:
𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝𝑡 = 𝑊𝑡1−0.01𝑛
where:
Wt = Float-adjusted market capitalization of component t.
n= Number of capping iterations.

3. This process is repeated iteratively until the first iteration where the capping constraints are
satisfied.

S&P Dow Jones Indices: Index Mathematics Methodology

12Non-Market Capitalization Weighted IndicesDefinition
A non-market capitalization weighted index (also referred to as a non-market cap or modified market cap
index) is one where index constituents have a user-defined weight in the index. Between index
rebalancings, most corporate actions generally have no effect on index weights, as they are fixed throughthe processes defined below. As stock prices move, the weights will shift, and the modified weights willchange. Therefore, a non-market cap weighted index must be rebalanced from time to time to reestablish the proper weighting.
The overall approach to calculate non-market cap weighted indices is the same as in the cap-weighted
indices; however, the constituents’ market values are set to a value to achieve a specific weight at eachrebalancing that is divergent from a purely free-float-adjusted market capitalization weighting. Recall two
basic formulae:

Index Level =Index Market Value
Divisor(1)and𝐼𝑛𝑑𝑒𝑥𝑀𝑎𝑟𝑘𝑒𝑡𝑉𝑎𝑙𝑢𝑒 = ∑ 𝑃𝑖 ∗ 𝑆ℎ𝑎𝑟𝑒𝑠𝑖 ∗ 𝐼𝑊𝐹𝑖 ∗ 𝐹𝑥𝑅𝑎𝑡𝑒𝑖𝑖

To calculate a non-market cap weighted index, the market capitalization for each stock used in thecalculation of the index is redefined so that each index constituent has the appropriate user-defined
weight in the index at each rebalancing date.
In addition to being the product of the stock price, the stock’s shares outstanding, and the stock’s float
factor (IWF), as written above – and the exchange rate when applicable – a new adjustment factor is also
introduced in the market capitalization calculation to establish the appropriate weighting.
AdjustedStock Market Valuei = Pi  Shares i  IWFi  FxRatei  AWFi

where AWFi is the adjustment factor of stock i assigned at each index rebalancing date, t, which adjusts
the market capitalization for all index constituents to achieve the user-defined weight, while maintaining
the total market value of the overall index.
The AWF for each index constituent, i, on rebalancing date, t, is calculated by:

AWFi , t =Z* Wi , t
FloatAdjus tedMarketV aluei , t(2)

where Z is an index specific constant set for the purpose of deriving the AWF and, therefore, each stock’s
share count used in the index calculation (often referred to as modified index shares). Wi,t is the userdefined weight of stock i on rebalancing date t.
The index divisor is defined based on the index level and market value from equation (1). The index levelis not altered by index rebalancings. However, since prices and outstanding shares will have changed
since the last rebalancing, the divisor will change at the rebalancing.
S&P Dow Jones Indices: Index Mathematics Methodology

13

So:
(Divisor) after rebalancing =(Index Market Value) after rebalancing(Index Value) before rebalancing

where:

Index Market Value = Pi  Sharesi  IWFi  FxRatei  AWFi
i

Corporate Actions and Index Adjustments
For information on the treatment of corporate actions, please refer to S&P Dow Jones Indices’ Equity
Indices Policies & Practices document. For more information on the specific treatment within an index
family, please refer to that index methodology.

S&P Dow Jones Indices: Index Mathematics Methodology

14

Price Weighted IndicesDefinition
In a price weighted index, such as the Dow Jones Industrial Average, constituent weights are determinedsolely by the prices of the constituent stocks. Shares outstanding are set to a uniform number throughout
the index. Indices using this methodology will adjust the index divisor for any price impacting corporate
action on one of its member stocks; this includes price adjustments, special dividends, stock splits andrights offerings. The index divisor will also adjust in the event of an addition to or deletion from the index.
All other index calculation details follow the standard divisor-based calculation methodology detailed in
the previous Capitalization Weighted Indices section.
For information on the treatment of corporate actions, please refer to S&P Dow Jones Indices’ Equity
Indices Policies & Practices Methodology.

S&P Dow Jones Indices: Index Mathematics Methodology

15

Equal Weighted IndicesDefinition
An equal weighted index is one where every stock, or company, has the same weight in the index, and a
portfolio that tracks the index will invest an equal dollar amount in each applicable instrument. As stock
prices move, the weights will shift, and exact equality will be lost. Therefore, an equal weighted index
must be rebalanced from time to time to re-establish the proper weighting.2
The overall approach to calculate equal weighted indices is the same as in the cap-weighted indices;
however, the constituents’ market values are re-defined to be values that will achieve equal weighting at
each rebalancing. Recall two basic formulae:

Index Level =Index Market Value
Divisor(1)and𝐼𝑛𝑑𝑒𝑥𝑀𝑎𝑟𝑘𝑒𝑡𝑉𝑎𝑙𝑢𝑒 = ∑𝑖 𝑃𝑖 ∗ 𝑆ℎ𝑎𝑟𝑒𝑠𝑖 ∗ 𝐼𝑊𝐹𝑖 ∗ 𝐹𝑥𝑅𝑎𝑡𝑒𝑖
To calculate an equal weighted index, the market capitalization for each stock used in the calculation of
the index is redefined so that each index constituent has an equal weight in the index at each rebalancing
date. In addition to being the product of the stock price, the stock’s shares outstanding, and the stock’s
float factor (IWF), as written above – and the exchange rate when applicable – a new adjustment factor isalso introduced in the market capitalization calculation to establish equal weighting.
𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑𝑆𝑡𝑜𝑐𝑘𝑀𝑎𝑟𝑘𝑒𝑡𝑉𝑎𝑙𝑢𝑒𝑖 = 𝑃𝑖 ∗ 𝑆ℎ𝑎𝑟𝑒𝑠𝑖 ∗ 𝐼𝑊𝐹𝑖 ∗ 𝐹𝑥𝑅𝑎𝑡𝑒𝑖 ∗ 𝐴𝑊𝐹𝑖

(2)

where AWFi (Additional Weight Factor) is the adjustment factor of stock i assigned at each index
rebalancing date, t, which makes all index constituents modified market capitalization equal (and,therefore, equal weight), while maintaining the total market value of the overall index. The AWF for eachindex constituent, i, at rebalancing date, t, is calculated by:

AWFi , t =Z
N * FloatAdjus tedMarketV aluei , t(3)

where N is the number of stocks in the index and Z is an index specific constant set for the purpose ofderiving the AWF and, therefore, each stock’s share count used in the index calculation (often referred toas modified index shares).
The index divisor is defined based on the index level and market value from equation (1). The index levelis not altered by index rebalancings. However, since prices and outstanding shares will have changed
since the last rebalancing, the divisor will change at the rebalancing.

2

In contrast, a cap-weighted index requires no rebalancing as long as there aren’t any changes to share counts, IWFs, returns ofcapital, or stocks added or deleted.

S&P Dow Jones Indices: Index Mathematics Methodology

16

So:

(Index Market Value) after rebalancing(Index Value) before rebalancing(Divisor) after rebalancing =where:

Index Market Value = Pi  Sharesi  IWFi  FxRatei  AWFi
i

Modified Equal Weighted Indices
There are some equal weighted indices that place further restrictions on stocks included in the index. An
example restriction might be a cap on the weight allocated to one sector or a cap on the weight of a single
country or region in the index. The rules could also stipulate a maximum weight for a stock if the index
applies additional liquidity factors (e.g., basket liquidity) when determining the index weights. In any ofthese situations, if a cap is applied to satisfy the restrictions, the excess weight leftover by the cap would
be distributed equally amongst the uncapped companies.
Corporate Actions and Index Adjustments
For more information on the treatment of corporate actions, please refer to S&P Dow Jones Indices’
Equity Indices Policies & Practices document. For more information on the specific treatment within an
index family, please refer to that index methodology.

S&P Dow Jones Indices: Index Mathematics Methodology

17

Multi-Day Rebalancing
A multi-day rebalancing allows indices to transition from the current index constituent weights to a set of target
weights over a pre-determined number of days. The weight increments/decrements from day to day within therebalancing period (i.e., smoothed weights) will be equal in size unless the exchange for a security is closed while
it is still an index business day. Day 1 of the rebalancing period will be the standard effective rebalancing date asstated in the index methodology.
The formula to calculate the smoothed weight for each stock is:(𝑡𝑎𝑟𝑔𝑒𝑡 𝑤𝑒𝑖𝑔ℎ𝑡𝑟,𝑖 − 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑤𝑒𝑖𝑔ℎ𝑡𝑟,𝑖 )𝑠𝑚𝑜𝑜𝑡ℎ𝑒𝑑 𝑤𝑒𝑖𝑔ℎ𝑡𝑡,𝑖 = (× 𝑛𝑢𝑚𝑏𝑒𝑟 𝑟𝑒𝑏𝑎𝑙𝑎𝑛𝑐𝑖𝑛𝑔 𝑑𝑎𝑦𝑡 ) + 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑤𝑒𝑖𝑔ℎ𝑡𝑟,𝑖𝑟𝑒𝑏𝑎𝑙𝑎𝑛𝑐𝑖𝑛𝑔 𝑙𝑒𝑛𝑔𝑡ℎ

where:
smoothed weightt,i= The weight for stock i on day t.

target weighti,r= The weight of stock i that corresponds to the weighting determined byrebalancing r. If stock i is dropping out of the index due to theselection criteria during rebalancing r then target weighti,r is 0.

reference weighti,r= The weight for stock i for the reference date for rebalancing r. If stock i
is not part of the composition of the index on the reference date, thenreference weighti,r is 0.

rebalancing length= The number of days in a multi-day rebalancing. This number isvariable, and is defined by the index methodology.

number rebalancing dayt

= The number of rebalancing days on day t from 1 to rebalancing length.

After the set of smoothed weights for each stock on each rebalancing date is calculated, index shares areset for each stock by utilizing a standard AWF calculation that accounts for forward looking corporate
actions throughout the rebalancing period:
𝐴𝑊𝐹 𝑖,𝑡 =(𝑠𝑚𝑜𝑜𝑡ℎ𝑒𝑑 𝑤𝑒𝑖𝑔ℎ𝑡𝑡,𝑖 ∗𝑧 𝑓𝑎𝑐𝑡𝑜𝑟)(𝑠𝑡𝑜𝑐𝑘 𝑝𝑟𝑖𝑐𝑒𝑟,𝑖 ∗𝑓𝑥 𝑟𝑎𝑡𝑒𝑟,𝑖 ∗𝑠ℎ𝑎𝑟𝑒𝑠 𝑜𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔𝑡,𝑖 ∗𝐼𝑊𝐹𝑟,𝑖 ∗𝑃𝑟𝑖𝑐𝑒 𝐴𝑑𝑗𝑢𝑠𝑡𝑚𝑒𝑛𝑡 𝐹𝑎𝑐𝑡𝑜𝑟)

The 𝑃𝑟𝑖𝑐𝑒 𝐴𝑑𝑗𝑢𝑠𝑡𝑚𝑒𝑛𝑡 𝐹𝑎𝑐𝑡𝑜𝑟𝑡,𝑖 will account for any corporate actions for 𝑠𝑡𝑜𝑐𝑘𝑖 between the reference
date and the rebalance date in question. For example, if there is a 2 for 1 stock split on rebalance day 3of a 5-day rebalancing period, the AWF calculated for the stock on the reference date will use an
adjustment factor of .5. The AWF calculated for days 1 and 2 of the rebalance will use an adjustment
factor of 1.
Day to day calculation of multi-day rebalancings will be conducted using the standard calculation
methodology for weighted indices.
Index shares and AWFs will remain static at their originally announced values throughout the pro-forma
period unless there are corporate actions announced after the pro-forma date and effective prior to theend of the rebalancing period.
Exchange Holidays
Except for the first and penultimate days of the rebalancing period, exchange holidays occurring during
the rebalancing period that do NOT result in an index closure will adjust the individual smoothed weights
S&P Dow Jones Indices: Index Mathematics Methodology

18

of each individual security on holiday. Stocks on holiday on day t will have their smoothed weight frozen
on day t+1. On the first day, stocks will always carry the first smoothed weight of the rebalancing period. If
there is a holiday on the penultimate day of the rebalancing, impacted stocks will smooth to their target
weight a day early and carry that weight over to the final day unless they are being removed from theindex. In the case of a stock removal with an exchange holiday on the penultimate day of the rebalancing
period, the stock weight will be smoothed over the remaining trading days. Similarly, if there are multiple
exchange holidays towards the close of the rebalancing period, the smoothing period will be further
condensed to the remaining trading days within the rebalancing period. If an exchange holiday isannounced after the reference date or there is an unexpected exchange closure, the glide path will be reestablished for the remaining days according to examples 2 and 3 below.
Please see the examples below. All weights in the example are as of the open on the effective date.

Example 1:
Index Weight on Reference Date = 1.2%; Target weight = 1.7%; No. of Rebalancing Days = 5; Weight
Delta = 0.5%; Daily Increment = 0.1%; Day 2 is an exchange holiday.
1. Day 1 weight = 1.2%+0.1%*1 = 1.3%
2. Day 2 weight = 1.2%+0.1%*2 = 1.4%
3. Day 3 weight = Day 2 weight
4. Day 4 weight = 1.2%+0.1%*4 = 1.6%
5. Day 5 weight = 1.2%+0.1%*5 = 1.7%
Example 2:
Index Weight on Reference Date = 1.2%; Target weight = 1.7%; No. of Rebalance Days = 5; Weight Delta= 0.5%; Daily Increment = 0.1%; Day 4 is an exchange holiday.
1. Day 1 weight = 1.2%+0.1%*1 = 1.3%
2. Day 2 weight = 1.2%+0.1%*2 = 1.4%
3. Day 3 weight = 1.2%+0.1%*3 = 1.5%
4. Day 4 weight = 1.2%+0.1%*5 = 1.7%
5. Day 5 weight = Day 4 weight
Example 3:
Index Weight on Reference Date = 1.2%; Target weight = 0.0% (stock removal); No. of Rebalance Days =5; Weight Delta = -1.2%; Daily Increment = -0.3%; Day 4 is an exchange holiday.
1. Day 1 weight = 1.2%-0.3%*1 = 0.9%
2. Day 2 weight = 1.2%-0.3%*2 = 0.6%
3. Day 3 weight = 1.2%-0.3%*3 = 0.3%
4. Day 4 weight = 1.2%-0.3%*4 = 0.0% = Removed from index
5. Day 5 weight = Removed from index
Freeze Date
A multi-day rebalancing process may be put on hold on any given day by utilizing a Freeze Date. On afreeze date, the target weights for a given day in the rebalancing period are carried over from theprevious day. If a freeze date occurs, the rebalancing period is extended by the total number of freeze

S&P Dow Jones Indices: Index Mathematics Methodology

19dates during the rebalancing period. A freeze date will not increase the rebalance length, it will only movethe rebalancing end date.
Multi-day rebalancing capabilities are compatible with standard weighted and equal weighted
methodologies.

S&P Dow Jones Indices: Index Mathematics Methodology

20

Total Return Calculations
The preceding discussions were related to price indices where changes in the index level reflect changes
in stock prices. In a total return index changes in the index level reflect both movements in stock prices
and the reinvestment of dividend income. A total return index represents the total return earned in a
portfolio that tracks the underlying price index and reinvests dividend income in the overall index, not in
the specific stock paying the dividend.
The total return construction differs from the price index and builds the index from the price index anddaily total dividend returns. The first step is to calculate the total dividend paid on a given day and convert
this figure into points of the price index:

TotalDailyDividend = Dividend i  Sharesi(1)i

Where Dividend is the dividend per share paid for stock i and Shares are the index specific shares. Thisis done for each trading day. Dividendi is generally zero except when it goes ex-dividend for a dividend
payment.3 Stocks may commonly issue dividends on a monthly, quarterly, semi-annual, or annual basis.
Some stocks do not pay a dividend and Dividend is always zero. TotalDailyDividend is measured indollars. This is converted to index points by dividing by the divisor for the underlying price index:

IndexDividend =TotalDailyDividend
Divisor(2)

The next step is to apply the usual definition of a total return from a financial instrument to the price index.
Equation (1) gives the definition, and equation (2) applies it to the index:

P + DtTotalRetur n = ( t
) −1
Pt −1
and

DTRt = (IndexLevel t + IndexDividend t− 1)
IndexLevel t −1

where the TotalReturn and the daily total return for the index (DTR) is stated as a decimal. The DTR isused to update the total return index from one day to the next:
Total Return Index t = (Total Return Index t −1)  (1 + DTRt )The Index Value for the Total Return Index can also be calculated as follows:
Total Return Indext = (Total Return Indext-1 )*(Price Return Indext + Index Dividend) / Price Return Indext-1

3

Dividendi can be negative if a dividend correction is applied to a particular stock. In such cases, a total return can have a value
lower than the price return. For more information on dividend corrections please refer to S&P Dow Jones Indices’ Equity Indices
Policies & Practices Methodology.

S&P Dow Jones Indices: Index Mathematics Methodology

21

Net Total Return Calculations
To account for tax withheld from dividends, a net total return calculation is used. The calculation is
identical to the calculations detailed in the previous Total Return section, except each dividend is adjusted
to account for the tax taken out of the payment.
Inserting the withholding rate into the calculation at the first step is all that needs to be done − thecalculation can follow identically from that point forward:

TotalDailyDividend = Dividend i  Shares i  (1 − WithholdingRatei )i

The tax rates used for S&P DJI’s global indices are from the perspective of a Luxembourg investor.
However, in domestic index families, tax rates from the perspective of a domestic investor will be applied.
For more information on the Withholding Rates used for Net Total Return Calculations, please refer to theWithholding Tax Rates by Markets document, available at www.spglobal.com/spdji.

S&P Dow Jones Indices: Index Mathematics Methodology

22

Post Ex-Dividend Adjustment: Total and Net Total
Return Calculation
When S&P DJI recognizes a regular dividend different than the final dividend amount confirmed by thecompany, or for specific markets where companies do not typically confirm cash dividend amounts prior
to the ex-dates, S&P DJI may recognize a post ex-date dividend adjustment.
First, an estimated dividend is reinvested into the index on the dividend’s Ex-Date, following the process
defined in the Total and Net Total Return Calculation section, where is the dividend is converted to index
dividend points by dividing by the divisor for the underlying price index.
Second, a Post-Ex Dividend Adjustment is applied, determined by S&P DJI calculating the difference
between the dividend amount recognized on the original ex-date and the actual dividend amountannounced by the company. Once determined, the adjustment amount is applied, in dividend points, to all
relevant indices using the following formula:
Index Dividend Point Adjustment = (Ddt * Sat) / Divisor Ex-Date
where:
Ddt = the difference between the original and actual dividend amounts. Foreign exchange conversion, if
applicable, is based on the exchange rate on the ex-date, calculated as:
Correction Amount * FX on the Reference Ex-Date
where:
Correction Amount4 = the dividend amounts relative to all the data as of the original dividend
ex-date
Sat = Index shares on the Reference Ex-Date
The adjustment is added to the calculation of the Total Return and Net Total Return indices on thedividend’s effective date. If there are multiple dividend payments with an ex-date of the current day and/or
multiple dividend adjustments, separate index dividend points are calculated for each dividend payment
and aggregated on Index Dividend as follows:
TotalDailyDividend =  IndexDividendPointAdjustment + IndexDividend t
For currency indices, the Index Dividend point adjustment calculated for the headline index is converted
to the corresponding currency in the index calculation, using the foreign exchange rate on the effective
date.

4

For the Index Dividend Point Adjustment, Ddt must be as indicated in the Index Corporate Events – .SDE file for the Index, where
the Reference Date column shows the date of the original dividend ex-date.

S&P Dow Jones Indices: Index Mathematics Methodology

23

Franking Credit Adjusted Total Return Indices
Additional total return indices are available for a number of S&P/ASX Indices that adjust for the tax effectof franking credits attached to cash dividends. The indices utilize tax rates relevant to two segments ofinvestors: one version incorporates a 0% tax rate relevant for tax-exempt investors and a second version
uses a 15% tax rate relevant for superannuation funds. The franking credits attached to both regular andspecial cash dividends are included in the respective calculations.
To calculate the gross dividend points reinvested in the Franking Credit Adjusted Total Return Indices:
Grossed-up Dividend = [As Reported Dividend * (1 – % Franked) + (As Reported Dividend *% Franked / (1 – Company Tax Rate))]The Net Tax Effect of the franking credit is then calculated based on the investor tax rate (i.e. 0% for taxexempt investors and 15% for superannuation funds).
Net Tax Effect = [Grossed-up Dividend * (1 – Investor Tax Rate)] – As Reported Dividend
The Net Tax Effect of each dividend is then multiplied by the index shares of that company to calculate
the gross dividend market capitalization.
Gross Dividend Market Cap = Net Tax Effect * Index Shares
Tax – Exempt= [(Grossed-Up Dividend - Dividend Amount) * Index Shares t ]Superannuation= [(Grossed-Up Dividend Superannuation - Dividend Amount) * Index
Shares t ]Grossed-Up Dividend and Gross Dividend Market Cap must be calculated for all dividend payments
effective on date T.
These are then summed for all dividends going ex on that date and converted to dividend points bydividing by the index divisor.
The only difference between Tax Exempt and Superannuation is the incorporation of tax rates relevant forsuperannuation funds (currently 15%), hence only the steps detailed above differ in each Index
Calculation. Once the Gross Dividend Market Cap is calculated, the following steps are exactly the samefor both Tax-Exempt and Superannuation versions.
Gross Dividend Points = Sum of Gross Dividend Market Caps / Underlying Index Close Divisor
Gross Dividend Points calculate daily, to account for the impact of dividends issued by constituents withan ex-date of date T.
Franking Credit Adjusted Annual Total Return Indices. This index series accrues a pool of grossdividend points on a daily basis and reinvests them across the index annually after the end of the financial
year. Reinvestment occurs at market close on the first trading day after June 30 th. The gross dividend
points are derived by taking the value of the gross dividend market capitalization (less the as reported
dividend market capitalization) and dividing it by the index divisor effective on the ex-date of therespective dividend.
Annual Total Return Index Level is calculated as follows:
1. FCA Index Dividend:
S&P Dow Jones Indices: Index Mathematics Methodology

24On the first trading day after June 30th =(𝐴𝑐𝑐𝑟𝑢𝑒𝑑 𝐺𝑟𝑜𝑠𝑠 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑡−1 + 𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝐼𝑛𝑑𝑒𝑥 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑡 )The Accrued Gross Dividend comes from the S&P Franking Credit Adjusted Annual Total
Return Index on day T-1, for the corresponding version, either Tax-Exempt or
Superannuation.5
On any other trading day = Underlying Index Dividend t
2. FCA Annual Total Return Index t =(𝐹𝐶𝐴 𝐴𝑛𝑛𝑢𝑎𝑙 𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝐼𝑛𝑑𝑒𝑥𝑡−1 ∗ (𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝐼𝑛𝑑𝑒𝑥 𝑡 + 𝐹𝐶𝐴 𝐼𝑛𝑑𝑒𝑥 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑡 ))𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝐼𝑛𝑑𝑒𝑥𝑡−1
The annual series accrues a pool of gross dividend points (adjusted for the net tax effect of franking
credits) on a daily basis, but only reinvests the gross dividend points annually, after the end of thefinancial year.
Therefore, provided it is not the first trading day after June 30th, the Index Dividend for the Franking
Credit Adjusted Daily Total Return Indices is the same as the Index Dividend of the Underlying Index.
Franking Credit Adjusted Daily Total Return Indices. Rather than allowing a separate accrual of grossdividend points, this index series reinvests the gross dividend amount across the index at the close of the
ex-date on a daily basis.
Daily Total Return Index Level is calculated as follows:
1. FCA Daily Total Return Index t =(𝐹𝐶𝐴 𝐷𝑎𝑖𝑙𝑦 𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝐼𝑛𝑑𝑒𝑥𝑡−1 ∗ (𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝐼𝑛𝑑𝑒𝑥𝑡 + 𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝐼𝑛𝑑𝑒𝑥 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑡 + 𝐺𝑟𝑜𝑠𝑠𝑒𝑑 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑃𝑜𝑖𝑛𝑡𝑠𝑡 ))𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝐼𝑛𝑑𝑒𝑥𝑡−1

The net tax effect of franking credits compounds daily over the entire calendar year, therefore, the Index
Dividends for the Franking Credit Adjusted Daily Total Return Indices differ from the Index Dividends of
the Underlying Index on a daily basis.

5

The accrued value of gross dividend points is provided on the corresponding Franking Index Levels File.

S&P Dow Jones Indices: Index Mathematics Methodology

25Currency and Currency Hedged Indices
A currency-hedged index is designed to represent returns for those global index investment strategies
that involve hedging currency risk, but not the underlying constituent risk. 6
Investors employing a currency-hedged strategy seek to eliminate the risk of currency fluctuations andare willing to sacrifice potential currency gains. By selling foreign exchange forward contracts, global
investors are able to lock in current exchange forward rates and manage their currency risk. Profits(losses) from the forward contracts are offset by losses (profits) in the value of the currency, thereby
negating exposure to the currency.
Return Definitions
S&P DJI’s standard currency hedged indices are calculated by hedging beginning-of-period balances
using rolling one-month forward contracts. The amount hedged is adjusted on a monthly basis.
Returns are defined as follows: End Spot Rate
 −1
Beginning Spot Rate Currency Return = Unhedged Return = (1 + Local Total Return ) * (1 + Currency Return ) − 1
Currency Return on Unhedged Local Total Return = (Currency Return ) * (1 + Local Total Return ) Beginning one - month Forward Rate
 −1
Beginning Spot Rate Forward Return = Hedge Return = HedgeRatio  (Forward Re turn − Currency Re turn )Hedged Index Return = Local Total Return + Currency Return on Unhedged Local Total Return +Hedge Return
Hedged Index Level = Beginning Hedged Index Level * (1 + Hedged Index Return)To facilitate index replication, S&P DJI determines the amount of foreign exchange forward contracts soldusing an index rebalance reference date.7 On the index reference date, which occurs one business day
prior to the last business day of prior month, the rebalance forward amounts and currency weights are
determined. As a result of the forward amounts and currency weights determination occurring one
business day prior to the month end rebalance, an adjustment factor is utilized in the calculation of thehedge return to account for the performance of the S&P Dow Jones Indices Currency-Hedged Index on
the last business day of the month. Please refer to the index computation section for further details.

By currency risk, we simply mean the risk attributable to the security trading in a currency different from the investor’s homecurrency. This definition does not incorporate risks that exchange rate changes can have on an underlying security’s priceperformance.
7
Prior to March 1, 2015 S&P Dow Jones Indices’ Currency-Hedged Indices utilized the month-end for both index reference andindex rebalance date.
6

S&P Dow Jones Indices: Index Mathematics Methodology

26

For the standard single-currency monthly currency hedged indices that are on holiday on the lastbusiness day of the month, this holiday treatment is functionally equivalent to months when the last day ofthe month falls on a weekend.
If the hedging currency market is on holiday on the last business day of the month and the S&P DJI
Underlying Index is not, as long as the spot rate in foreign currency per local currency (Sm) and one
month forward spread used to calculate the forward rate (1MF) are not on holiday, the currency andcurrency hedged indices will calculate with no changes to the relevant dates used for Currency Hedged
Calculation.
Currency weights are established based on the composition of the underlying index as of the reference
date. Any changes to the currency weights due to composition changes are not reflected until the
calculation of currency weights on the next reference date. If Reference Date is a holiday, the calculation
uses T-1 values for both underlying index levels and the Spot and One Month Forward Rates.
S&P DJI also offers daily currency hedged indices for clients who require benchmarks with more frequentcurrency hedging. The daily currency hedged indices differ from the standard currency hedged indices byadjusting the amount of the forward contracts that mature at the end of month, on a daily basis, according
to the performance of the underlying index. This further reduces the currency risk from under-hedging orover-hedging resulting from index movement between two monthly rolling periods.
Details of the formulae used in computing S&P DJI’s currency-hedged indices are below.
The Hedge Ratio
The hedge ratio is simply the proportion of the portfolio’s currency exposure that is hedged.
•Standard Currency-Hedged Index. In a standard currency-hedged index, we simply wish toeliminate the currency risk of the portfolio. Therefore, the hedge ratio used is 100%.

•

No Hedging. An investor who expects upside potential for the local currency of the index
portfolio versus the home currency, or does not wish to eliminate the currency risk of the portfolio,
will use an unhedged index. In this case, the hedge ratio is 0, and the index simply becomes thestandard index calculated in the investor’s home currency. Such indices are available in major
currencies as standard indices for many of S&P Dow Jones Indices’ indices.
In contrast to a 100% currency-hedged standard index, which seeks to eliminate currency risk
and has passive equity exposure, over- or under-hedged portfolios seek to take active currency
risks to varying degrees based on the portfolio manager’s view of future currency movements.

•

Over Hedging. An investor who expects significant upside potential for the home currency
versus the local currency of the index portfolio might choose to double the currency exposure. In
this case, the hedge ratio will be 200%.

•

Under Hedging. An investor who expects some upside potential for the local currency of theindex portfolio versus the home currency, but wishes to eliminate some of the currency risk, mightchoose to have half the currency exposure hedged using a 50% hedge ratio.

S&P Dow Jones Indices calculates indices with hedge ratios different from 100% as custom indices.
Calculating a Currency-Hedged Index
Using the returns definitions on prior pages, the Hedged Index Return can be expressed as:
Hedged Index Return = Local Total Return + Currency Return*(1 + Local Total Return) + Hedge
ReturnRearranging yields:

S&P Dow Jones Indices: Index Mathematics Methodology

27

Hedged Index Return = (1 + Local Return) * (1 + Currency Return) -1 + Hedge ReturnS&P Dow Jones Indices: Index Mathematics Methodology

28

Again, using the returns definitions on prior pages with a hedge ratio of 1 (100%), the expression yields:
Hedged Index Return = Unhedged Index Return + Hedge Return
Hedged Index Return = Unhedged Index Return + Forward Return - Currency Return
This equation is more intuitive since when you do a 100% currency hedge of a portfolio, the investor
sacrifices the gains (or losses) on currency in return for gains (or losses) in a forward contract.
From the equation above, we can see that the volatility of the hedged index is a function of the volatility of
the unhedged index return, the forward return, and the currency return, and their pair-wise correlations.
These variables will determine whether the hedged index return series’ volatility is greater than, equal to,or less than the volatility of the unhedged index return series.
Currency Hedging Outcomes
The results of a currency-hedged index strategy versus that of an unhedged strategy vary depending
upon the movement of the exchange rate between the local currency and home currency of the investor.
S&P Dow Jones Indices’ standard currency hedging process involves eliminating currency exposureusing a hedge ratio of 1 (100%).
1. The currency-hedged index does not necessarily give a return exactly equal to the return of theindex available to local market investor. This is because there are two additional returns −currency return on the local total return and hedge return. These two variables usually add to a
non-zero value because the monthly rolling of forward contracts does not result in a perfect
hedge. Further, the local total return between two readjustment periods remains unhedged.
However, hedging does ensure that these two returns remain fairly close.
2. The results of a currency-hedged index strategy versus that of an unhedged strategy varies
depending upon the movement of the exchange rate between the local currency and homecurrency of the investor. For example, a depreciating euro in 1999 resulted in an unhedged S&P
500 return of 40.0% for European investors, while those European investors who hedged their
U.S. dollar exposure experienced a return of 17.3%. Conversely, in 2003 an appreciating euro in2003 resulted in an unhedged S&P 500 return of 5.1% for European investors, while those
European investors who hedged their U.S. dollar exposure experienced a return of 27.3%.
Index Computation
Monthly Return Series (For Monthly Currency Hedged Indices)m= The month in the calculation, represented as 0, 1, 2, etc.

m-1

= The last business day of the prior month.

mr-1= Reference Date. S&P DJI’s standard index reference date for hedged indices is one
business day prior to the last business day of the prior month.

d= Day d for month m.

D= The number of days in month m, counted as calendar days to the last business day.

SPI_EHd= The S&P DJI Currency-Hedged Index level on day t, where t can be defined as d, m-1,
or mr-1.

SPI_MAF= Monthly Index Adjustment Factor to account for the performance of the S&P Dow
Jones Indices Currency-Hedged Index between the index reference and month end
rebalance dates. It is calculated as the ratio of the S&P Dow Jones Indices CurrencyHedged Index level on the reference date and the S&P Dow Jones Indices CurrencyHedged Index level at the end of the month.

S&P Dow Jones Indices: Index Mathematics Methodology29 SPI _ EHmr −1 
SPI _ MAF = 
 SPI _ EHm −1 
SPI_EMd= The S&P Dow Jones Indices Index level, in foreign currency, on day t, where t can be
defined as d or m-1.

HRd

= The hedge return (%) over month m.

Sd= The spot rate in foreign currency per local currency (FC/LC), on day t, where t can be
defined as d or mr-1.

1MFd= The one-month forward spread used to calculate the forward rate, at day t of month d.

Fd= The first front-month forward rate in foreign currency per local currency (FC/LC), on
day t, where t can be defined as d or m-1; calculated as Sd + 1MFd.

Monthly Hedged Calculation for day t of month m:
1. HRd = The hedge return (%) over month m for Monthly Currency Indices
USD to Currency (FX):
𝐻𝑅𝑑 = (

𝑆𝑚𝑟−1
𝑆𝑚𝑟−1
−
) ∗ (𝑆𝑃𝐼_𝑀𝐴𝐹)
𝐷
−
𝑑
𝐹𝑚−1
𝑆𝑑 + () ∗ (𝐹𝑑 − 𝑆𝑑 )𝐷

Currency to USD (FX):

𝐷−𝑑
𝑆𝑑 + ( 𝐷 ) ∗ (𝐹𝑑 − 𝑆𝑑 )𝐹𝑚−1
𝐻𝑅𝑑 = (−) ∗ (𝑆𝑃𝐼_𝑀𝐴𝐹)
𝑆𝑚𝑟−1
𝑆𝑚𝑟−1
2. SPI_EHd = S&P Dow Jones Indices Currency-Hedged Index level
𝑆𝑃𝐼_𝐸𝐻𝑑 = 𝑆𝑃𝐼_𝐸𝐻𝑚−1 ∗ (𝑆𝑃𝐼_𝐸𝑀𝑑+ 𝐻𝑅𝑑 )𝑆𝑃𝐼_𝐸𝑀𝑚−1

Monthly Return Series (For Monthly Multi - Currency Hedged Indices)
S&P DJI indices can be composed of securities listed on multiple exchanges using multiple foreignexchange rates for daily index calculation.
Monthly Multi - Currency Hedged Indices follow a standard Monthly Hedged Calculation, following themain differences listed below:•

The spot rate (Sd ) used for the currency return and the first front-month forward rate (Fd) used tohedge the currency position need to be converted from each of the currencies included in theunderlying index to the hedging currency, i.e., EUR.

The Hedged Return (HRd) is calculated following the same procedure detailed above, but the hedged
return is weighted based on the Adjusted Currency Weights of the underlying index as of the ReferenceDate.
1. HRd = The hedge return (%) over month m for Monthly Currency Indices, for each of thecurrencies on underlying Index.
USD to Currency (FX):
𝐻𝑅𝑑 = (

𝑆𝑚𝑟−1
𝑆𝑚𝑟−1
−
)
𝐹𝑚−1 𝑆 + ( 𝐷 − 𝑑 ) ∗ (𝐹 − 𝑆 )𝑑𝑑𝑑𝐷

S&P Dow Jones Indices: Index Mathematics Methodology

30Currency to USD (FX):
𝐷−𝑑
𝐹𝑚−1 𝑆𝑑 + ( 𝐷 ) ∗ (𝐹𝑑 − 𝑆𝑑 )𝐻𝑅𝑑,𝐶𝑢𝑟𝑟𝑒𝑛𝑐𝑦 = (
−
)
𝑆𝑚𝑟−1
𝑆𝑚𝑟−1
2. Weighted Hedged Return for Monthly Multi - Currency Indices:
𝑊𝐻𝑅𝑚 = ∑(𝑊𝑚𝑟−1,𝐶𝑢𝑟𝑟𝑒𝑛𝑐𝑦 ∗ 𝐻𝑅 𝑑,𝐶𝑢𝑟𝑟𝑒𝑛𝑐𝑦 ) ∗ 𝑆𝑃𝐼_𝑀𝐴𝐹
where:
Wmr-1,Currency = Weight for each of the currencies of the underlying S&P DJI Index as of theReference Date. S&P DJI’s standard index reference date for hedged indices is
one business day prior to the last business day of the prior month.
HRd,Currency= The hedge return (%) over month m for Monthly Currency, for each of thecurrencies on the underlying S&P Dow Jones Index.

WHRm= Weighted Hedged Return for Monthly Multi Currency Indices.

3. SPI_EHd = S&P DJI Currency-Hedged Index level for Multi - Currency Indices:
𝑆𝑃𝐼_𝐸𝑀𝑑
𝑆𝑃𝐼_𝐸𝐻𝑑 = 𝑆𝑃𝐼_𝐸𝐻𝑚−1 ∗ (+ 𝑊𝐻𝑅𝑚 )𝑆𝑃𝐼_𝐸𝑀𝑚−1
For Monthly Multi-Currency Hedged Indices, on any day that is both an underlying index holiday and anindex calculation day, the closing price of the constituents on the holiday day is used for index calculation.
In this case, there is no change to the last business day of the prior month nor the prior month index
reference date.
Daily Return Series (For Monthly Currency Hedged Indices and Daily Currency Hedged Indices)
The daily return series are computed by interpolating between the spot price and the forward price.
For each month m, there are d = 1, 2, 3…D calendar days.
m= The month in the calculation, represented as 0, 1, 2, etc.

m-1

= The last business day of the prior month.

d= Day d for month m.

d-1= Previous day for month m.

D= The number of days in month m, counted as calendar days to the last business day.

SPI_EHd= The S&P Dow Jones Indices Currency-Hedged Index level on day t, where t can be
defined as d or m-1.

SPI_EMd= The S&P DJI Index level, in foreign currency, on day t, where t can be defined as d orm-1.

SPI_ELd= The S&P Dow Jones Indices Index level, in local currency, on day t, where t can be
defined as d-1 or m-1.

HRd

= The hedge return (%) from end of month to day d.

Sd= The spot rate in foreign currency per local currency (FC/LC), on day t, where t can be
defined as d or m-1.

1MFd

= The one month forward spread used to calculate the forward rate, at day d of month m.

S&P Dow Jones Indices: Index Mathematics Methodology

31

Fd

= The first front-month forward rate in foreign currency per local currency (FC/LC), on
day t, where t can be defined as d or d-1, calculated as Sd + 1MFd.

F_Imd= The interpolated forward rate on day t, where t can be defined as d or d-1.

AFd

= The adjustment factor for daily hedged indices as of day d of month m

Daily Hedged Calculation for day T of month m =1. The interpolated forward rate as of day d of month m (F_I md)
When day d is the last business day of month m =𝐹_𝐼𝑚𝑑 = 𝐹𝑀𝑑 = (𝑆𝑑 + 1𝑀𝐹𝑑 )2. When day d is other than the last business day of month m =𝐷−𝑑
𝐹_𝐼𝑚𝑑 = 𝑆𝑑 + () ∗ (𝐹𝑑 − 𝑆𝑑 )𝐷
3. The adjustment factor for daily hedged indices as of day d of month m (AFmd)
𝐴𝐹𝑚𝑑 =𝑆𝑃𝐼_𝐸𝐿𝑑−1
𝑆𝑃𝐼_𝐸𝐿𝑚−1

When day d is the first business day of month m, AFmd = 1
The hedge return for daily currency hedged indices is calculated as follows:
When day d is any other day than the first business day of month m:𝐹𝐼
𝐹𝐼
𝐻𝑅𝑑 = 𝐴𝐹𝑚𝑑 ∗ ( 𝑚𝑑−1 − 𝑚𝑑 ) + 𝐻𝑅𝑑−1
𝑆𝑚−1
𝑆𝑚−1
When day d the first business day of month m:
𝐹𝑑−1
𝐹_𝐼𝑚𝑑
𝐻𝑅𝑑 = 𝐴𝐹𝑚𝑑 ∗ (
−
)
𝑆𝑚−1
𝑆𝑚−1
S&P DJI Daily Currency Hedged Index Level at day d:
𝑆𝑃𝐼_𝐸𝑀𝑑
𝑆𝑃𝐼_𝐸𝐻𝑑 = 𝑆𝑃𝐼_𝐸𝐻𝑚−1 ∗ (+ 𝐻𝑅𝑑 )𝑆𝑃𝐼_𝐸𝑀𝑚−1
Dynamic Hedged Return Indices
Dynamic hedged return indices are rebalanced at a minimum on a monthly basis as per the monthly
series described above, but include a mechanism to ensure that the index does not become over-hedged
or under-hedged beyond a certain percentage threshold. This is measured by taking the percent changeof the current value of the hedged index versus the value of the hedged index on the previous reference
date. If that percentage threshold is crossed during the month an intra-month adjustment is triggered. If
triggered, the hedge is reset to the value of the hedged index on the day the threshold is breached,effective after the close on the following business day, using the current interpolated value of the forward
expiring at the end of the month. Thus, the formulas for dynamic hedged indices become:𝑆𝑃𝐼_𝐸𝐻𝒅= The S&P Dow Jones Indices Currency-Hedged Index level as of day d

𝑆𝑃𝐼_𝐸𝐻𝑟𝑏= The S&P Dow Jones Indices Currency-Hedged Index level at the priorrebalancing date

𝑆𝑃𝐼_𝐸𝐻𝑟𝑓= The S&P Dow Jones Indices Currency-Hedged Index level on the priorreference day. S&P Dow Jones Indices’ standard index reference date forhedged indices is one day prior to the rebalancing date.

S&P Dow Jones Indices: Index Mathematics Methodology

32

𝑆𝑃𝐼_𝐴𝐹= Index Adjustment Factor to account for the performance of the S&P Dow Jones
Indices Currency-Hedged Index between the index reference date andrebalance date. It is calculated as the ratio of the S&P Dow Jones Indices
Currency-Hedged Index level on the reference date and the S&P Dow Jones
Indices Currency-Hedged Index level at the rebalancing date.

𝑆𝑃𝐼_𝐸𝒅= The S&P Dow Jones Indices Index level, in foreign currency, as of date d

𝑆𝑃𝐼_𝐸𝒓𝒃= The S&P Dow Jones Indices Index level, in foreign currency, at the priorrebalancing date

𝐻𝑅𝑑

= The hedge return (%) as of day d since the prior rebalancing date

𝑆𝑑

= The spot rate in foreign currency per local currency (FC/LC) as of date d

𝑆𝑟𝑓= The spot rate in foreign currency per local currency (FC/LC) as of the priorindex reference date

𝐹𝑑

= The forward rate in foreign currency per local currency (FC/LC), as of day d

𝐹_𝐼𝑑= The interpolated forward rate as of day d

𝐹_𝐼𝑟𝑏= The interpolated forward rate as of the prior rebalancing date

The formula for determining if an intra-month rebalancing is triggered is:𝐼𝑓 (𝑎𝑏𝑠((𝑆𝑃𝐼_𝐸𝐻𝒅 /𝑆𝑃𝐼_𝐸𝐻𝑟𝑓 ) − 1)) > 𝑇𝐻
where:TH= Percentage threshold for the index

Then a rebalancing is triggered.8
For the purposes of the Dynamic Rebalancing and determining the hedge absolute deviation fromnotional, when day d is a rebalancing date, the hedged index on reference updates on rebalance dated.9
The interpolated forward rate as of day d is calculated as:
𝐹_𝐼𝑑 = 𝑆𝑑 + (𝐹𝑑 − 𝑆𝑑 ) ∗ (𝐷𝑎𝑦𝑠(𝑑,𝑛𝑟𝑏)
𝐷𝑎𝑦𝑠(𝑑,𝑒𝑥𝑝))

where,
𝐷𝑎𝑦𝑠(𝑑, 𝑛𝑟𝑏)

= Days between date d and next scheduled rebalancing date𝐷𝑎𝑦𝑠(𝑑, 𝑒𝑥𝑝)

= Days between date d and expiry date of the forward rate used

Whenever applicable standard FX market settlement conventions are applied to both the Spot Rate andForward Rate to determine the exact settlement dates to be used in the interpolation.

8

Prior to November 30, 2021, a Dynamic Rebalancing cannot be triggered on two consecutive days. If the prior day signaled adynamic rebalancing, a current day dynamic rebalancing signal was ignored.
9
Prior to November 30, 2021, for the purposes of the Dynamic Rebalancing and determining the hedge absolute deviation fromnotional, when day d is a rebalancing date the hedged index on reference updates on rebalancing date d + 1.

S&P Dow Jones Indices: Index Mathematics Methodology

33

The hedge return for dynamic currency hedged indices is:
𝐻𝑅𝑑 = (

𝐹_𝐼𝑟𝑏 𝐹_𝐼𝑑−
) ∗ 𝑆𝑃𝐼_𝐴𝐹
𝑆𝑟𝑓
𝑆𝑟𝑓

For index value on day d is:
𝑆𝑃𝐼_𝐸𝐻𝒅 = 𝑆𝑃𝐼_𝐸𝐻𝑟𝑏 ∗ (𝑆𝑃𝐼_𝐸𝒅+ 𝑯𝑹𝒅 )𝑆𝑃𝐼_𝐸𝒓𝒃

Currency Hedged Excess Return Indices
Since an excess return index calculates the return on an investment in an index where the investment
was made through the use of borrowed funds, currency risk can be hedged by borrowing funds in thecurrency of the investment. In this scenario the initial value of the index at each hedge period will not beaffected by currency returns, but the amount gained or lost during the period will be affected by returns in
the currency.
When the gain and loss at each hedge period is not hedged, returns are defined as follows:
𝐻𝑒𝑑𝑔𝑒𝑑 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 = 𝐿𝑜𝑐𝑎𝑙 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 + 𝐶𝑢𝑟𝑟𝑒𝑛𝑐𝑦 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑈𝑛ℎ𝑒𝑑𝑔𝑒𝑑 𝐿𝑜𝑐𝑎𝑙 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛

When the gain and loss at each hedge period is hedged, returns are defined as follows:
𝐻𝑒𝑑𝑔𝑒𝑑 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛= 𝐿𝑜𝑐𝑎𝑙 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 + 𝐶𝑢𝑟𝑟𝑒𝑛𝑐𝑦 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑈𝑛ℎ𝑒𝑑𝑔𝑒𝑑 𝐿𝑜𝑐𝑎𝑙 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 + 𝐻𝑒𝑑𝑔𝑒 𝑅𝑒𝑡𝑢𝑟𝑛

For non-convertible currencies, currency return on unhedged local excess return is calculated using thecurrent forward rate based on first front-week forward contract rather than a spot rate for some cases. In
this case, the returns of daily currency hedged excess return indices are calculated as follows (note thatcurrency rates are quoted in local currency per foreign currency in the case of non-convertible
currencies):
𝐹_𝑤𝑒𝑒𝑘𝑁𝐶
𝑚0
𝐻𝑒𝑑𝑔𝑒𝑑 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 = 𝐿𝑜𝑐𝑎𝑙 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 + (𝐿𝑜𝑐𝑎𝑙 𝐸𝑥𝑐𝑒𝑠𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 ∗𝐹_𝑤𝑒𝑒𝑘𝑁𝐶
𝑚𝑑) + 𝐻𝑒𝑑𝑔𝑒 𝑅𝑒𝑡𝑢𝑟𝑛

The hedged return for daily currency hedged excess return indices is calculated as follows:
When day d is the first business day of month m,
𝐻𝑅𝑚𝑑 = 0
When day d is not the first business day of month m,
𝐻𝑅𝑚𝑑 = 𝐴𝐹_𝐸𝑅𝑚𝑑 ∗ (𝑁𝐶𝑁𝐶
𝐹_𝑤𝑒𝑒𝑘𝑚0
𝐹_𝑤𝑒𝑒𝑘𝑚0−) + 𝐻𝑅𝑚𝑑−1
𝑁𝐶
𝑁𝐶
𝐹_𝐼𝑚𝑑−1
𝐹_𝐼𝑚𝑑

where
𝑁𝐶
𝐹_𝑤𝑒𝑒𝑘𝑚𝑑= The first front-week forward rate in local currency per foreign currency (LC/FC) as of
day d of month m𝑁𝐶
𝐹_𝑤𝑒𝑒𝑘𝑚0= The first front-week forward rate in local currency per foreign currency (LC/FC), at theend of prior month, m-1
𝑁𝐶
𝐹_𝐼𝑚𝑑

= The interpolated forward rate in local currency per foreign currency (LC/FC), as of
day d of month m

S&P Dow Jones Indices: Index Mathematics Methodology

34

𝑁𝐶
𝑁𝐶
𝐹_𝐼𝑚𝑑
= 𝑆𝑚𝑑
+(

𝐷−𝑑
𝐷

𝑁𝐶
𝑁𝐶
)
) ∗ (𝐹𝑚𝑑
− 𝑆𝑚𝑑

𝑁𝐶
𝑆𝑚𝑑
= The spot rate in local currency per foreign currency (LC/FC), as of day d of monthm
𝑁𝐶
𝐹𝑚𝑑= The first front-month forward rate in local currency per foreign currency (LC/FC), asof day d of month mD= number of business days in month m

AF_ERmd = The adjustment factor for daily currency hedged excess return indices as of day d ofmonth m𝑆𝑃𝐸𝑅𝐼_𝐸𝐿𝑚𝑑−1
𝐴𝐹_𝐸𝑅𝑚𝑑 =⁄𝑆𝑃𝐸𝑅𝐼_𝐸𝐿 − 1𝑚0where:SPERI_ELmd = The S&P Dow Jones Excess Return Index level, in local currency, as of
day d of month m
SPERI_ELm0 = The S&P Dow Jones Excess Return Index level, in local currency, at theend of the prior month, m-1
Quanto Currency Adjusted Index
A quanto currency adjusted index represents the return of an underlying index from the perspective of aforeign party, and incorporates the respective currency pair return with the underlying index return. It
differs from simply expressing an index in foreign currency because it represents borrowing in the index
currency to fund an investment in assets represented by the index.
For example, suppose a U.S. investor does the following on a daily basis:
1. Borrow 100 GBP in London, secured by the equivalent amount of USD in a U.S. bank
2. Invest 100 GBP in U.K. index stocks in proportion to their index weights
The investor would generate profit or loss equal to the U.K. index return. They would also earn the
combined index return and the currency pair return on the profit/loss. The combined index/currency returnwould not be earned on their principal because the U.K. assets can be sold to satisfy the U.K. loan andclose the position.
Arithmetically, a quanto currency adjusted index can be represented as follows:
SPI_E(t + 1)
SPI_E(t + 1)
S (t + 1)
SPI_QA(t + 1) = SPI_QA(t) × (+(− 1) × (− 1) )SPI_E(t)SPI_E(t − n)S (t)where:𝑆𝑃𝐼_𝑄𝐴(𝑡 + 1) = Quanto Currency-Adjusted Index level, as of day (t+1)

10

𝑆𝑃𝐼_𝑄𝐴(𝑡)= Quanto Currency-Adjusted Index level, as of day (t)𝑆𝑃𝐼_𝐸(𝑡 + 1)

= Underlying Index level, as of day (t+1)

𝑆𝑃𝐼_𝐸(𝑡)= Underlying Index level, as of day (t)𝑆𝑃𝐼_𝐸(𝑡 − 𝑛)

= Underlying Index level, as of day (t-n), where n = (0 or 1), corresponding to thedifference in trading days between the foreign party and the underlying index 10

𝑆(𝑡 + 1)= Spot rate for the currency pair as of date (t+1)

𝑆(𝑡)= Spot rate for the currency pair as of date (t)For example, for foreign parties in an APAC time zone employing such a strategy to acquire U.S. assets, n=1 to account for the
trading day difference between the party and the index.

S&P Dow Jones Indices: Index Mathematics Methodology

35

The index returns can also be expressed as:
Quanto Currency Adjusted Index Returns = Index Returns + (Index Returns′ ) × (𝐶𝑢𝑟𝑟𝑒𝑛𝑐𝑡 𝑅𝑒𝑡𝑢𝑟𝑛𝑠)
Negative/Zero Index Levels. For more information regarding the possibility of negative or zero index
levels, refer to the Negative/Zero Index Levels section.

S&P Dow Jones Indices: Index Mathematics Methodology

36

Domestic Currency Return Index Calculation
Background
Domestic Currency Return (DCR) calculations are used to calculate the return of an index without taking
any exchange rate movements into account. This may be done as a way to perform an attribution on anindex containing constituents which do not all trade in the same currency. By comparing the performanceof the float-adjusted market capitalization weighted index against the performance of the same index
calculated using DCR one can derive the performance due to the exchange rate movements.
In DCR one calculates the period-to-period percentage change of the index from the weighted percentagechange of each security’s local price and then constructs the index levels from the percentage changes.
This is in contrast to a divisor-based index where the process is reversed: the index level is calculated as
total market value divided by the divisor and the period-to-period percentage change is calculated fromthe index levels. Both approaches require an initial base period or divisor value for normalization. For an
index where all of the constituents trade in the same currency both approaches give the same results.
In the DCR calculation, we calculate the percentage change in each security price, weight the percentagechanges by the security’s weight in the index at the start of the period, and then combine the weighted
price changes to calculate the index price change for the time period. The change in the index is, then,applied to the index level in the previous period to determine the current period index level.
Equivalence of DCR and Divisor Calculations
The equivalence of the two approaches – DCR and divisor based – can be understood in two ways. First,
except for the initial base value of an index, it can be defined by either the index levels or the percentagechange from one period to the next. If we defined an index by a time series of index levels (100, 101.2,
103, 105…) we can derive the period to period changes (1.2%, 1.78%, 1.94%...). Given these changes
and assuming the index base is a value of 100 allows us to calculate the index levels. Except for thebase, the two series are equivalent. DCR calculates the changes; the divisor approach calculates thelevels.
This can be shown mathematically:
The divisor calculation approach defines an index as:

Since the initial divisor is defined by the base value and date of the index, we can replace it with the valueof the index market cap at time t=0:

Now we can multiply and divide the term in the summation in the numerator by the price at time t=0
without changing its value.

S&P Dow Jones Indices: Index Mathematics Methodology37

If we look at the term in the numerator for a single stock in the index (i.e., no summation, as there is only
one stock) and rearrange we get:(1)

which is equivalent to the relative price performance for each stock multiplied by its weight in the index.
When this is combined across all constituent stocks, the result is the price performance for the index.
The DCR approach uses the summation of equation (1) across all the stocks in the index to calculate thedaily price performance of the index. Once the daily index performance is calculated, the index level canbe updated from the previous day’s index level.
DCR Calculation

where:
Indext= Index level at date t

Pt= Security price at the close of date t

weightt = Security weight in the index at close of date t
andwhere:
Si,t-1= Shares of stock i

FXi,t-1 = Exchange rate of stock i for currency conversion
Essential Adjustments
The share count (Si,t-1) includes the adjustment for float by multiplying by the investable weight factor(IWF) and for index weight by multiplying by the additional weight factor (AWF) where necessary. Further,when an adjustment to shares is made due to a secondary offering, share buyback or any other corporate
action, this adjustment must be included in Si,t-1 if the adjusted share count takes effect on date t. A priceadjustment due to a corporate action which takes effect on date t should be reflected in Pi,t-1.

S&P Dow Jones Indices: Index Mathematics Methodology

38

Risk Control Indices
S&P Dow Jones Indices’ Risk Control Indices are designed to track the return of a strategy that applies
dynamic exposure to an underlying index in an attempt to control the level of volatility.
The index includes a leverage factor that changes based on realized historical volatility. If realized
volatility exceeds the target level of volatility, the leverage factor will be less than one; if realized volatility
is lower than the target level, the leverage factor may be greater than one, assuming the index allows fora leverage factor of greater than one. A given Risk Control Index may have a maximum leverage factor
that cannot be exceeded. There are no guarantees that the index shall achieve its stated targets.
The return of the index consists of two components: (1) the return on the position in the underlying indexand (2) the interest cost or gain, depending upon whether the position is leveraged or deleveraged.
A leverage factor greater than one represents a leveraged position, a leverage factor equal to one
represents an unleveraged position, and a leverage factor less than one represents a deleveraged
position. The leverage factor may change periodically, on a set schedule, or may change when volatility
exceeds or falls below predetermined volatility thresholds.
For equity indices, the leverage factor will not change at the close of any index calculation day in whichstocks representing 15% or more of the total weight of the underlying index are not trading due to an
exchange holiday. At each underlying index’s rebalancing, and using each stock’s weight at that time, aforward looking calendar of such dates is determined and posted on S&P Dow Jones Indices’ Web site atwww.spglobal.com/spdji.
For digital asset indices, while digital asset markets trade seven days a week (Monday-Sunday), theunderlying index and risk control versions calculate following the standard five business days per weekschedule week (Monday-Friday).
The formula for calculating the Risk Control Index is as follows:

Risk Control Index Return t =
(1)

 t

 Underlying Index tK rb * − 1 + (1 − K rb ) * (1 + InterestRa tei −1 * Di −1,i / 360 ) − 1i = rb +1 Underlying Index rb

The Risk Control Index Value at time t can, then, be calculated as:

RiskControlIndexValuet =(2)(RiskControlIndex Valuerb ) * (1 + RiskControlIndex Return t )Substituting equation (1) into (2) and expanding yields:
Risk Control Index Valuet =S&P Dow Jones Indices: Index Mathematics Methodology

39

Risk Control Index Valuerb *
 
 t

1 + K *  Underlying Index t − 1 + (1 − K ) * 
(1+InterestRate*Di − 1, i / 360 ) − 1 
rb 
rb
i
−
1

 
Underlying Index rb
i = rb +1
  



 



S&P Dow Jones Indices: Index Mathematics Methodology

(3)

40

Excess Return versions of Risk Control Indices are calculated as follow:
Risk Control ER Index Valuet =𝑅𝑖𝑠𝑘𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝐸𝑅 𝐼𝑛𝑑𝑒𝑥 𝑉𝑎𝑙𝑢𝑒𝑟𝑏 ∗𝑡𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔𝐼𝑛𝑑𝑒𝑥𝑡
𝐷𝑖−1,𝑖[1 + [𝐾𝑟𝑏 ∗ (− 1) − 𝐾𝑟𝑏 ∗ [ ∏ (1 + 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑅𝑎𝑡𝑒𝑡−1 ∗) − 1]]]
𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔𝐼𝑛𝑑𝑒𝑥𝑡−1
360
𝑖=𝑟𝑏+1

where:
UnderlyingIndext= The level of the underlying index on day t

UnderlyingIndexrb= The level of the underlying index as of the previous rebalancing daterb

= The last index rebalancing date11

Krb= The leverage factor set at the last rebalancing date, calculated as:
Min(Max K, Target Volatility/Realized Volatilityrb-d)

Max K= The maximum leverage factor allowed in the indexd= The number of days between when volatility is observed and the rebalancing
date (e.g., if d = 2, the historical volatility of the underlying index as of theclose two days prior to the rebalancing date will be used to calculate theleverage factor Krb)

Target Volatility= The target level of volatility set for the index

Realized Volatilityrb-d = The historical realized volatility of the underlying index as of the close of d
trading days prior to the previous rebalancing date, rb, where a trading day isdefined as a day on which the underlying index is calculatedInterest Ratei-1= The interest rate set for the index12

For indices that replicate a rolling investment in a three-month interest rate the above formula is alteredto:
Risk Control Index Valuerb *Risk Control Index Valuet = 


 t

1 + K *  Underlying Index t − 1 + (1 − K ) * (1 + InterestRa tei −1) − 1  
rb 
rb

 
i = rb +1
  
 Underlying Index rb


 



where:
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑅𝑎𝑡𝑒𝑖−1 =1(𝐷𝑖−1,𝑡 ∗ 𝐼𝑅3𝑀𝑖−1 − (𝐼𝑅3𝑀𝑖−1 − 𝐼𝑅3𝑀𝑖−2 − 𝐷𝑖−1,𝑡 ∗ (𝐼𝑅3𝑀𝑖−1 − 𝐼𝑅2𝑀𝑖−1 ) ∗ ( )) ∗ 90)/360
30

where:
Di-1, t= The number of calendar days between day i-1 and day t

IR3Mi-1 = Three-month interest rate on day i-1
IR2M i-1 = Two-month interest rate on day i-113
For indices that are rebalanced daily, the leverage factor is not recalculated at the close of any index
calculation day when stocks representing 15% or more of the total weight of the underlying index are not
trading due to an exchange holiday. If rb is a holiday, then Krb is calculated as follows:11

The inception date of each risk control index is considered the first rebalancing date of that index.
The interest rate may be an overnight rate, such as SOFR or ESTR, or a daily valuation of a rolling investment in a three-monthinterest rate, or zero. A 360-day year is assumed for the interest calculations in accordance with U.S. banking practices.
13
Effective 12/03/2018, the interest rate for EUR-based Risk Control indices is a one-month rate instead of a two-month rate.
Therefore, those indices’ interest rate is depicted as: IR2M i-1 = One-month interest rate on day i-1.
12

S&P Dow Jones Indices: Index Mathematics Methodology

41 Underlying Index rb   RiskContorlIndexValuerb  /
Krb = Krb−1 *   Underlying Index rb−1   RiskcontrolIndexValuerb−1 This shows what the effect will be on rb, given that no adjustment of positions is allowed to occur on suchdays. The leverage factor will adjust solely to account for market movements on that day.
For periodically rebalanced risk control indices, Krb is calculated at each rebalancing and held constantuntil the next rebalancing.
For large position moves, some investors like to rebalance risk control indices intra-period, when the
periodicity is longer than daily. This feature is incorporated in the risk-control framework by introducing abarrier, Kb, on the leverage factor. Intra-period rebalancing is allowed only if the absolute change of theequity leverage factor Kt, at time t, is larger than the barrier Kb from the value at the last rebalancing date.
The equity leverage factor Kt is calculated as:
Kt = Min(Max K, Target Volatility/Realized Volatilityt-d)
If no barrier is provided for the index, then intra-period rebalancing is not allowed.
Dynamic Rebalancing Risk Control Index
The index calculates the theoretical leverage factor on daily basis. If the difference between thetheoretical leverage factor and the leverage factor on the last rebalancing date is less than the Minimum
Daily Allocation Change, the index will not rebalance.
The theoretical leverage factor is determined as:𝑡ℎ𝐾𝑡 = the theoretical leverage factor on day t, calculated daily as:𝑡ℎ𝐾𝑡 = 𝑀𝑖𝑛(𝑀𝑎𝑥 𝐾,

𝑇𝑎𝑟𝑔𝑒𝑡 𝑉𝑜𝑙𝑎𝑖𝑙𝑖𝑡𝑦)
𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑉𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦𝑡−𝑑where:
d = Lag to Rebalancing Date, defined as the number of days between when volatility is observed andthe date which the theoretical leverage factor is calculated for (e.g., if d = 2, the historical volatilityof the underlying index as of the close two days prior to the date which the theoretical leveragefactor is calculated for will be used to calculate the leverage factor thKt)
The trade decision is based on the difference between the theoretical leverage factor and the leveragefactor on the last rebalancing date:
If |𝑡ℎ𝐾𝑡 − 𝐾𝑡−1 | > 𝜃,Thent is a rebalancing day, and𝐾𝑡 = 𝑡ℎ𝐾𝑡
Else
t is not a rebalancing day
𝐾𝑡 = 𝐾𝑡−1
where:
𝜃= Minimum Daily Allocation Change

𝐾𝑡 = the actual leverage factor on day t

S&P Dow Jones Indices: Index Mathematics Methodology

42

Dynamic rebalancing can be combined with monthly rebalancing. In this case, besides intra-monthly
rebalancing triggered by breach of Minimum Daily Allocation Change, the risk control index rebalances
after the close of the last business day of the month.
Capped Equity Weight Change
For daily rebalanced or dynamic rebalanced risk control indices, some investors like to control forexcessive position change. This feature is incorporated in the risk-control framework by introducing a
Maximum Daily Allocation Change , 𝜃̅.
The theoretical leverage factor is determined in the same way as in a Dynamic Rebalanced Risk Control
Index. The trade decision is based on the difference between the theoretical leverage factor and theleverage factor on the last rebalancing date:
If |𝑡ℎ𝐾𝑡 − 𝐾𝑡−1 | > 𝜃,Then:t is a rebalancing day, and𝑀𝑖𝑛(𝐾𝑡−1 + 𝜃̅ , 𝑡ℎ𝐾𝑡 ), 𝑖𝑓 𝑡ℎ𝐾𝑡 − 𝐾𝑡−1 > 0
𝐾𝑡 = {
𝑀𝑎𝑥(𝐾𝑡−1 − 𝜃̅ , 𝑡ℎ𝐾𝑡 ), 𝑖𝑓 𝑡ℎ𝐾𝑡 − 𝐾𝑡−1 ≤ 0
Elset is not a rebalancing day
𝐾𝑡 = 𝐾𝑡−1
where:
𝜃= Minimum Daily Allocation Change (𝜃 >0 for dynamic rebalanced risk control indices, and 𝜃 =0 for daily rebalanced risk control indices).

𝜃̅= Maximum Daily Allocation Change

𝐾𝑡 = the actual leverage factor on day t
Dynamic rebalancing can be combined with monthly rebalancing. In this case, besides intra-monthly
rebalancing triggered by breach of Minimum Daily Allocation Change, the risk control index rebalances
after the close of the last business day of the month.
Excess Return Indices
S&P Dow Jones Indices’ Excess Return Indices are designed to track an unfunded investment in anunderlying index. In other words, an excess return index calculates the return on an investment in an
index where the investment was made through the use of borrowed funds. Thus, the return of an excess
return index will be equal to that of the underlying index less the associated borrowing costs. Most S&P
Dow Jones Indices calculate an excess return index level to mirror an unfunded position.
The formula for calculating the Excess Return Index is as follows: Underlying Index t  Borrowing Rate 
ExcessReturn = 
− 1 −   Dt ,t −1360 Underlying Index t −1  (4)The Excess Return Index Value at time t can be calculated as:
ExcessReturn IndexValue t = (ExcessReturn Index Valuet −1 )  (1 + Excess Return )S&P Dow Jones Indices: Index Mathematics Methodology

(5)

43

Substituting (4) into (5) and expanding the right hand side of (5) yields:

ExcessReturn IndexValue t =  Underlying Index t  Borrowing Rate 
ExcessReturn IndexValue t −1 * 1 + − 1 − Dt ,t −1  360  Underlying Index t −1
 
 
where:

Borrowing Rate = The investment funds borrowing rates, which will differ for each excess
return index14
Dt, t-1= The number of calendar days between date t and t-1

Exponentially-Weighted Volatility
The realized volatility is calculated as the maximum of two exponentially weighted moving averages, one
measuring short-term and one measuring long-term volatility.
RealizedVo latility t = Max (RealizedVo latility S,t , RealizedVo latility L,t )where:
S,t = The short-term volatility measure at time t, calculated as:(6)

RealizedVo latility S,t =252
 VarianceS,tnfor t  T0
  Underlying Index t VarianceS,t = S  VarianceS,t −1 + (1 − S )  ln  Underlying Index t − n 
for t = T0
T0

  Underlying Index i VarianceS,T0 = ln
WeightingF actorS   Underlying Index i − n i = m +1S,i ,m22

L,t = The long-term volatility measure at time t, calculated as:(7)

RealizedVo latility L,t =252
 VarianceL,tnfor t  T0
  Underlying Index t VarianceL,t = L  VarianceL,t −1 + (1 − L )  ln  Underlying Index t − n 
for t = T0
T0

  Underlying Index i VarianceL,T0 = ln
WeightingF actorL   Underlying Index i − n i = m +114

 L,i ,m2

2

Generally, an overnight rate, such as SOFR overnight in the U.S. or ESTR in Europe, will be used. However, in some cases otherinterest rates may be used. A 360-day year is assumed for the interest calculations in accordance with U.S. banking practices.

S&P Dow Jones Indices: Index Mathematics Methodology

44where:T0= The start date for a given risk control indexn= the number of days inherent in the return calculation used for determining volatility15m= the Nth trading date prior to T0N= the number of trading days observed for calculating the initial variance as of the start dateof the index

S= The short-term decay factor used for exponential weighting 16

L= The long-term decay factor used for exponential weighting10

 S,m,i = Weight of date t in the short-term volatility calculation, as calculated based on the following
formula:S,t = (1 − S ) * S

WeightingF actorS =N + m −iT0

S,i ,mi = m +1

L,m,i = Weight of date t in the long-term volatility calculation, as calculated based on the following
formula:L,t = (1 − L ) * L

WeightingF actorL =N + m −iT0L,i,mi = m +1

The interest rate, maximum leverage, target volatility and the lambda decay factors are defined in relation
to each index and are generally held constant throughout the life of the index. The leverage positionchanges at each rebalancing based on changes in realized volatility. There is a two-day lag between thecalculation of the leverage factor, based on the ratio of target volatility to realized volatility, and theimplementation of that leverage factor in the index.
The above formulae can be used for simpler models by the appropriate choice of parameters. For
example, if the short-term and long-term decay factors, S and L are set to the same value (e.g., 5%)than there are no separate considerations for short-term and long-term volatility.
Exponentially-Weighted Volatility Based on Current Allocations
The index calculations are the same as described in the Exponentially Weighted Volatility section above,
except that realized volatility is calculated using the returns derived from the levels of hypotheticalunderlying index based on the current allocations within the underlying index and historical returns of
those constituents, rather than the historical levels of the underlying index.
Underlying Indext = Hypothetical underlying index level on day t, calculated as
𝐾

𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝐼𝑛𝑑𝑒𝑥𝑡 = 𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝐼𝑛𝑑𝑒𝑥𝑡−1 ∗ (1 + ∑ 𝑤𝑖 ∗ 𝑟𝑖,𝑡 )
𝑖=𝑖

15
16

If n = 1 daily returns are used, while if n = 2 two day returns are used, and so forth.
The decay factor is a number greater than zero and less than one that determines the weight of each daily return in thecalculation of historical variance.

S&P Dow Jones Indices: Index Mathematics Methodology

45where:K = number of constituents in current underlying index as of day t
ri,t = return of the i-th constituent in the underlying index on day t
wi = weight of the i-th constituent in current underlying index
Simple-Weighted Volatility
The realized volatility is calculated as the maximum of two simple-weighted moving averages, one
measuring short-term volatility and one measuring long-term volatility.
RealizedVo latility t = Max (RealizedVo latility S,t , RealizedVo latility L,t )where:
S,t = The short-term volatility measure at time t, calculated as:

RealizedVo latility S,t =252
 VarianceS,tnt Underlying Index i VarianceS,t = 1/ NS *lnUnderlying Index i −n 
i =t −Ns +12

L,t = The long-term volatility measure at time t, calculated as:

RealizedVolatility L,t =252
 VarianceL,tntVarianceL,t = 1/ NL *

 UnderlyingIndex i ln
UnderlyingIndex i −n i =t −Nl +1 2where:n= The number of days inherent in the return calculation used for determining volatility 17

NS = The number of trading days observed for calculating variance for the short-term volatility
measure
NL = The number of trading days observed for calculating variance for the long-term volatility
measureUnderlying Indext is defined as in the “Exponentially-Weighted Average Volatility” section.

17

If n = 1 daily returns are used, while if n = 2 two day returns are used, and so forth.

S&P Dow Jones Indices: Index Mathematics Methodology

46

Futures-Based Risk Control Indices
When the underlying index is based on futures contracts, most of the Risk Control methodology follows
the details on the prior six pages. However, there are some differences as detailed below, particularly as
it relates to the cash component of the index.
For such an index, it includes a leverage factor that changes based on realized historical volatility. If
realized volatility exceeds the target level of volatility, the leverage factor will be less than one; if realized
volatility is lower than the target level, the leverage factor may be greater than one. A given risk control
index may have a maximum leverage factor that cannot be exceeded.
For equity risk control indices, the return consists of two components: (1) the return on the position in theunderlying S&P Dow Jones Indices index and (2) the interest cost or gain, depending upon whether the
position is leveraged or deleveraged. For futures-based risk control indices, there is no borrowing or
lending to achieve investment objectives in the underlying index. Therefore, the cash component of theIndex does not exist.
Again, a leverage factor greater than one represents a leveraged position, a leverage factor equal to one
represents an unleveraged position, and a leverage factor less than one represents a deleveraged
position. The leverage factor may change at regular intervals, in response to changes in realized
historical volatility, or when the expected volatility exceeds or falls below predetermined volatility
thresholds, if such thresholds were in place.
The formula for calculating the Risk Control Excess Return Index largely follows that detailed beginningwith equation (1). However, since there is no funding for such indices (as opposed to the case with equity
excess return indices, where the initial investment is presumedly borrowed, and excess cash is, likewise,
assumed to be invested), the interest rate used in the calculation is eliminated:

 Underlying Index tRisk Control Excess Return Index Returnt = Krb * − 1 Underlying Index rb 

(8)

The Risk Control Excess Return Index Value at time t can, then, be calculated as:
RiskControlExcessReturnIndexVa luet =(RiskControlExcessReturnIndex Valuerb ) * (1 + RiskControlExcessReturnIndex Returnt )The formula for calculating the Risk Control Total Return Index, which includes interest earned on
Treasury Bills, is as follows:

Risk Control Total Return Index Return t = Underlying Index t  t
K rb * − 1 + 
(1 + InterestRa tei −1 * Di −1,i / 360 ) − 1
 Underlying Index rb  i =rb+1



(9)

The Risk Control Total Return Index Value at time t can, then, be calculated as:
RiskControlTotalRetu rnIndexVal uet =(RiskControlTotalRetu rnIndex Value rb ) * (1 + RiskControlTotalRetu rnIndex Return t )(10)

Substituting equation (9) into (10) and expanding yields:
Risk Control Total Return Index Valuet =S&P Dow Jones Indices: Index Mathematics Methodology

47

Risk Control Index Valuerb *
 
 t

1 + K *  Underlying Index t − 1 + (1 + InterestRa tei −1 * Di − 1, i / 360 ) − 1  
rb 

 
  
 Underlying Index rb  i = rb +1

 



(11)

where all variables in equations (8)-(11) are the same as those defined for (1)-(3) except:
Interest Ratei-1 = The interest rate set for the index 18
Exponentially-Weighted Volatility for Futures-Based Risk Control Indices
Please refer to the Risk Control 2.0 Indices section of this document for information on ExponentiallyWeighted Volatility. However, for futures-based risk control indices there is a three (3)-day lag between
the calculation of the leverage factor, based on the ratio of target volatility to realized volatility, and theimplementation of that leverage factor in the index.
Dynamic Volatility Risk Control Indices
In dynamic volatility risk control indices, the volatility target is not set as a definition of the index. Rather it
is set at various levels based on the moving average of VIX computed over a predetermined number ofdays (e.g., 30-day moving average).
Variance Based Risk Control Indices
In variance-based risk control indices, a target level of variance is set rather than a target volatility level.
This allows for faster leveraging or deleveraging of allocations based on changes in volatility or variancein the market. For these indices:
Krb = Min(Max K, Target Variance/Realized Variancerb-d)
where variance is defined as per above.
All other index calculations remain the same.

18

In accordance with the S&P GSCI approach, the interest rate for these indices is the 91-day U.S. Treasury rate. A 360-day year isassumed for the interest calculations in accordance with U.S. banking practices.

S&P Dow Jones Indices: Index Mathematics Methodology

48

Risk Control 2.0 Indices
S&P Dow Jones Indices’ Risk Control 2.0 Indices are Risk Control indices, where the cash portion of theinvestment in the standard Risk Control strategy is replaced with a liquid bond index.
The index portfolio consists of two assets, the index for a risky asset A, with weight W, and thecorresponding bond index B, with weight of (1-W). Weight W lies between 0 and 100%. There is no
shorting or leverage allowed in the strategy.
Constituent Weighting
The formula to assign weights to the underlying indices is determined by the following:2
W 2 *  A2 + (1 − W )2 *  B2 + 2 * W * (1 − W ) *  *  A *  B =  Target(1)

where:W= The weight of the risky asset A

A= The volatility of the risky asset A

B= The volatility of the bond index B= The correlation of Index A and B

σ Target= The target volatility

The calculation of volatility and correlation follows the same procedure and conventions as outlined in theprior section for the standard Risk Control strategy.
The quadratic equation above has two solutions to the weight allocated the index A:
W1 = (−b + b 2 − 4a * c ) / 2a
W 2 = (−b − b 2 − 4a * c ) / 2a(2)where:22
a = A+ B− 2 *  *  A * Bb = 2  A  B − 2 B2
2c =  B2 −  Target

The fallback mechanism for the solutions of weight W:
1. If none of the solutions in equation (2) above falls between 0 and 100%, then the strategy fallsback to standard Risk Control, where the maximum leverage is capped at 100%.
2. If both solutions to the equation (2) are valid weights that are greater than 0, then the larger of thetwo, max(W1, W2), becomes the weight of the risky asset A where the maximum leverage iscapped at the level defined by the indices risk control parameters.

S&P Dow Jones Indices: Index Mathematics Methodology

49

The final weights of the underlying assets are determined using the following steps:
Step 1: Determine the weights under the short term parametersa) Determine the short-term variance for assets A and B using the short term exponential parameterwith the same formulae as described in equation (6) under the section Risk Control Indices, withthe returns for assets A and B used in determining the short-term variance for assets A and B.
b) Determine the short-term covariance for assets A and B using similar formulae as described forshort-term covariance calculations in equation (6) under the section Risk Control Indices, but
replacing the squared equity returns with the product of the returns of risky assets A and B.
c) Determine the short-term volatility measure for the risky assets A and B from their respective
variance measures in the same manner as described in equation (6) under the section Risk
Control Indices.
d) Determine the short-term correlation of A and B from the short-term covariance and the shortterm volatility measures.
e) Determine the possible levels for the weights for A and B using equations (1) and (2) above.
Step 2: Determine the weights under the long term parameters
Repeat (a) to (e) in Step 1 above with long-term parameters as described in equation (7) under thesection Risk-Control Indices.
Step 3: Determine the final weight W.
The weight for risky asset A is set equal to the lower of the weight of A as determined in Step 1 and Step
2.
The excess return of the Risk Control 2.0 Indices is calculated as:
RiskControl2.0ExcessReturn t =W * Index AExcessReturn + (1 − W) * Index B ExcessReturnand the Risk Control 2.0 Index value is:
RiskControl2.0IndexV aluet =RiskControl2.0IndexV aluerb * (1 + RiskControl2.0ExcessReturn t )where:
RiskControl2.0IndexV aluerb = The value of the index at the last rebalancing

Risk Control 2.0 total return indices are calculated in a similar way, where the total return is a weighted
sum of total returns of the underlying indices.
Risk Control 2.0 is an extension of standard Risk Control described in detail in the previous section. The
parameters used in Risk Control 2.0 follow exactly the way they are calculated in the standard Risk
Control methodology.
Risk Control 2.0 with Minimum Variance
In Risk Control 2.0 indices with minimum variance, when the quadratic equation (1) has no real solution
for 𝑊𝐴 and 𝑊𝐵 , the fallback mechanism does not switch to standard Risk Control.
2𝑊𝐴2 ∗ 𝜎𝐴2 + 𝑊𝐵2 ∗ 𝜎𝐵2 + 2 ∗ 𝑊𝐴 ∗ 𝑊𝐵 ∗ 𝜌 ∗ 𝜎𝐴 ∗ 𝜎𝐵 = 𝜎𝑇𝑎𝑟𝑔𝑒𝑡(1)

𝑊𝐴 + 𝑊𝐵 = 1(2)where,

S&P Dow Jones Indices: Index Mathematics Methodology

50Instead, the strategy finds the portfolio with minimum variance and then rescales the weight of the risky
asset A and risky asset B to reach the target volatility. The remaining weight is allocated to cash in orderthat total asset weights sum 100%.
If using (1) and (2) for a given asset weight 𝑥 with standard deviation 𝜎𝐴 , the portfolio variance is defined
as a function of 𝑥 as the following:𝑓(𝑥) = 𝑥 2 ∗ 𝜎𝐴2 + (1 − 𝑥)2 ∗ 𝜎𝐵2 + 2 ∗ 𝑥 ∗ (1 − 𝑥) ∗ 𝜌 ∗ 𝜎𝐴 ∗ 𝜎𝐵 Calculating the first the derivative of (3)results in:𝑑𝑓= 2 ∗ 𝑥 ∗ (𝜎𝐴2 + 𝜎𝐵2 − 2 ∗ 𝜌 ∗ 𝜎𝐴 ∗ 𝜎𝐵 ) + 2 ∗ 𝜌 ∗ 𝜎𝐴 ∗ 𝜎𝐵 − 2 ∗ 𝜎𝐵2
𝑑𝑥
Equating the first derivative to zero results in:𝑥∗ =𝜎𝐵2 − 𝜌 ∗ 𝜎𝐴 ∗ 𝜎𝐵
2
𝜎𝐴 + 𝜎𝐵2 − 2 ∗ 𝜌 ∗ 𝜎𝐴 ∗ 𝜎𝐵

Deriving again, the second derivative is always positive and hence, the asset weight 𝑥 ∗ is a local
minimum.
𝑑2𝑓= 2 ∗ (𝜎𝐴2 + 𝜎𝐵2 − 2 ∗ 𝜌 ∗ 𝜎𝐴 ∗ 𝜎𝐵 ) ≥ 2 ∗ (𝜎𝐴 − 𝜎𝐵 )2 ≥ 0𝑑𝑥 2Moreover, given that function (3) is convex over [0,1], 𝑥 ∗ is also a global minimum. Therefore, the asset
weights of the minimum variance portfolio for two risky assets A and B are:
𝑊𝐴𝑀𝑖𝑛 =

2𝜎𝐵
−𝜌∗𝜎𝐴 ∗𝜎𝐵22 −2∗𝜌∗𝜎 ∗𝜎
𝜎𝐴 +𝜎𝐵𝐴 𝐵

𝑊𝐵𝑀𝑖𝑛 = 1 − 𝑊𝐴𝑀𝑖𝑛

(4)
(5)

However, given that equation (1) had no real solution, the portfolio volatility 𝜎𝑀𝑖𝑛𝑖𝑚𝑢𝑚 using weights (4)and (5) is greater that the target volatility. Therefore, (4) and (5) must be rescaled to reach the target
volatility by a scalar 𝜃 as follows:𝜃=𝜎𝑇𝑎𝑟𝑔𝑒𝑡𝜎𝑀𝑖𝑛𝑖𝑚𝑢𝑚(6)

Then the portfolio asset weights are:
𝑊𝐴 = 𝜃 ∗ 𝑊𝐴𝑀𝑖𝑛
𝑊𝐵 = 𝜃 ∗ 𝑊𝐵𝑀𝑖𝑛
Given that 𝜃 < 1, the remaining portfolio weight is allocated to cash to get 100% allocation:
𝑊𝐶 = 1 − 𝑊𝐴 − 𝑊𝐵

S&P Dow Jones Indices: Index Mathematics Methodology

51

Equity with Futures Leverage Risk Control
IndicesS&P Dow Jones Indices’ Equity with Futures Leverage Risk Control Indices measure the performance ofa strategy that combines constant representation of the underlying index with a dynamic weighting to thecorresponding Futures Excess Return Index in order to target a specific level of volatility. When theunderlying index volatility decreases below the target, futures are added to the risk control index toincrease the market exposure and vice versa.
The index includes a leverage factor that represents the target exposure to the underlying index as aresult of both the equity and futures positions. Since representation of the equity position remains
constant at 100%, the resultant dynamic weighting to the futures index equals the leverage factor minus
100%.
The return of the index consists of two components: (1) the return in the underlying index and (2) the
return of a dynamic long or short position in the corresponding Futures Excess Return Index, dependingon whether the index is leveraging or deleveraging in an attempt to achieve the target volatility.

The formula for calculating the Equity with Futures Leverage Risk Control Index Return is as follows:
𝐸𝑞𝑢𝑖𝑡𝑦 𝑤𝑖𝑡ℎ 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 𝑅𝑖𝑠𝑘 𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝐼𝑛𝑑𝑒𝑥 𝑅𝑒𝑡𝑢𝑟𝑛𝑡 =𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔𝐼𝑛𝑑𝑒𝑥𝑡
𝐹𝑢𝑡𝑢𝑟𝑒𝑠𝐸𝑅𝐼𝑛𝑑𝑒𝑥𝑡(− 1) + (𝐾𝑟𝑏 − 100%) ∗ (− 1)
𝑈𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔𝐼𝑛𝑑𝑒𝑥𝑟𝑏
𝐹𝑢𝑡𝑢𝑟𝑒𝑠𝐸𝑅𝐼𝑛𝑑𝑒𝑥𝑟𝑏
where:
𝐹𝑢𝑡𝑢𝑟𝑒𝑠𝐸𝑅𝐼𝑛𝑑𝑒𝑥𝑡= The level of the Futures Excess Return Index on day t

𝐹𝑢𝑡𝑢𝑟𝑒𝑠𝐸𝑅𝐼𝑛𝑑𝑒𝑥𝑟𝑏= The level of the Futures Excess Return Index as of thelast rebalancing date

The leverage factor, Krb, changes based on a 20 trading-day realized historical volatility of the underlying
index. For details on the calculation of the historical volatility please see formulae as described for shortterm, simple-weighted realized volatility under the section Risk Control Indices.
All other parameters are as described in the standard Risk Control Indices section of this document.

S&P Dow Jones Indices: Index Mathematics Methodology

52

Weighted Return Indices
S&P Dow Jones Indices’ Weighted Return Indices combine the returns of two or more underlying indicesusing a specified set of weighting rules to create a new unique index return series. An index that uses theWeighted Return methodology might also be referred to as an “Index of Indices.” Weighted Return indices
may include a cash component which for the purposes of these indices is treated as an underlying index.
S&P Dow Jones Indices offers both daily and periodic rebalance approaches for weighted return indices.
Based on the specification in the individual index methodologies, weighted return indices will becalculated using one of the below formulas:
Daily Rebalancing:𝑁

𝐼𝑛𝑑𝑒𝑥𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑡−1 × (1 + ∑𝑖=1𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑖,𝑡(𝑤𝑒𝑖𝑔ℎ𝑡𝑖,𝑡 × (− 1)) + 𝐶𝑎𝑠ℎ𝑊𝑒𝑖𝑔ℎ𝑡𝑡 × 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑅𝑒𝑡𝑢𝑟𝑛𝑡 )𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑖,𝑡−1

Periodic Rebalancing, accruing interest:𝑁

𝐼𝑛𝑑𝑒𝑥𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑟 × (1 + ∑𝑡𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑖,𝑡(1 + 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑅𝑒𝑡𝑢𝑟𝑛𝑑 ) − 1))(𝑤𝑒𝑖𝑔ℎ𝑡𝑖,𝑟 × (− 1)) + 𝐶𝑎𝑠ℎ𝑊𝑒𝑖𝑔ℎ𝑡𝑟 × (∏𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥
𝑖=1
𝑑=𝑟+1
𝑖,𝑟

Interest Return Options:𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑅𝑎𝑡𝑒𝑡−1
× 𝐴𝐶𝑇(𝑡, 𝑡 − 1), 𝑓𝑜𝑟 𝑠𝑖𝑚𝑝𝑙𝑒 𝑑𝑎𝑖𝑙𝑦 𝑎𝑐𝑐𝑟𝑢𝑎𝑙
𝐴𝑐𝑐𝑜𝑢𝑛𝑡𝑖𝑛𝑔𝐷𝑎𝑦𝑠
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑅𝑎𝑡𝑒𝑡−1 𝐴𝐶𝑇(𝑡,𝑡−1))− 1) , 𝑓𝑜𝑟 𝑎𝑐𝑐𝑟𝑢𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 𝑎𝑛 𝑖𝑛𝑑𝑒𝑥 𝑛𝑜𝑛𝑐𝑎𝑙𝑐 𝑑𝑎𝑦((1 +𝐴𝑐𝑐𝑜𝑢𝑛𝑡𝑖𝑛𝑔𝐷𝑎𝑦𝑠

𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑅𝑒𝑡𝑢𝑟𝑛𝑡 =𝐴𝐶𝑇(𝑡,𝑡−1)
91

{

(
(1 −

1
)
91
× 𝑅𝑖𝑠𝑘𝐹𝑟𝑒𝑒 𝑅𝑎𝑡𝑒𝑡−1 )𝐴𝑐𝑐𝑜𝑢𝑛𝑡𝑖𝑛𝑔𝐷𝑎𝑦𝑠− 1, 𝑓𝑜𝑟 3 𝑚𝑜𝑛𝑡ℎ 𝑅𝑖𝑠𝑘𝐹𝑟𝑒𝑒 𝑅𝑎𝑡𝑒 𝑎𝑐𝑐𝑟𝑢𝑎𝑙

where:
Indext= the value of the top level index on day t

Indexr= the value of the top level index at the previous rebalancing date r19

weighti,t= the weight of component index i on day t

weighti,r= the weight of component index i on the previous rebalancing date r

ComponentIndexi,t = the value of the component index i on day t
ComponentIndexi,r = the value of the component index i on the previous rebalancing date r20

19
20

N

= the number of component indices within the top level index

CashWeightt= the weight of the cash component on day t

CashWeightr= the weight of the cash component on the previous rebalancing date r

InterestReturnt= the return from the interest rate (see Interest Return Options above)

Note that the value is as of the close of the rebalancing date.
Note that the value is as of the close of the previous rebalancing date.

S&P Dow Jones Indices: Index Mathematics Methodology

53InterestReturnt - 1= the interest rate from the previous calculation date t -121

Accounting Days

= the day count convention for InterestRatet-1. Days counts are typically 252, 360,
or 365.

ACT(t, t - 1)

= the calendar day between calculation day t -1 and calculation day t, expressedas the day (t) – (t – 1).

RiskFree Ratet – 1= S&P DJI typically uses the three-month (3M) T-Bill rate published weekly bytreasurydirect.gov.

Component Indices Weightings and Returns:
Daily Rebalancing:
𝑊𝑒𝑖𝑔ℎ𝑡 𝑖,𝑡 =𝐴𝐷𝐽 𝑊𝑒𝑖𝑔ℎ𝑡𝑖,𝑡−1 ∗ ( 1 + 𝐷𝑎𝑖𝑙𝑦 𝑅𝑒𝑡𝑢𝑟𝑛𝑖,𝑡 )(1 + 𝐼𝑛𝑑𝑒𝑥 𝑅𝑒𝑡𝑢𝑟𝑛𝑡 )𝐷𝑎𝑖𝑙𝑦 𝑅𝑒𝑡𝑢𝑟𝑛 𝑖,𝑡 =𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝐼𝑛𝑑𝑒𝑥 𝑖,𝑡−1
𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝐼𝑛𝑑𝑒𝑥 𝑖,𝑡−1

𝐼𝑛𝑑𝑒𝑥 𝑅𝑒𝑡𝑢𝑟𝑛 𝑡 =𝐼𝑛𝑑𝑒𝑥 𝑡−1
𝐼𝑛𝑑𝑒𝑥 𝑡−1

Periodic Rebalancing, accruing interest:
𝑊𝑒𝑖𝑔ℎ𝑡 𝑖,𝑡 =𝐴𝐷𝐽 𝑊𝑒𝑖𝑔ℎ𝑡𝑖,𝑟 ∗ ( 1 + 𝐷𝑎𝑖𝑙𝑦 𝑅𝑒𝑡𝑢𝑟𝑛𝑖,𝑡 )(1 + 𝐼𝑛𝑑𝑒𝑥 𝑅𝑒𝑡𝑢𝑟𝑛𝑡 )𝐷𝑎𝑖𝑙𝑦 𝑅𝑒𝑡𝑢𝑟𝑛 𝑖,𝑡 =𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝐼𝑛𝑑𝑒𝑥 𝑖,𝑡−1
𝐶𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝐼𝑛𝑑𝑒𝑥 𝑖,𝑟𝐼𝑛𝑑𝑒𝑥 𝑅𝑒𝑡𝑢𝑟𝑛 𝑡 =𝐼𝑛𝑑𝑒𝑥 𝑡−1
𝐼𝑛𝑑𝑒𝑥 𝑟where:

21

Weight i,t= the weight of component index i on day t

ADJ Weight i,t-1= the Adjusted close weight of component index i on day t-1

N= the number of component indices within the top-level index

Daily Return i,t= daily return of component index i on day

Index Returnt= the return of the weighted return index on day t.

Note that this can also be a flat rate.

S&P Dow Jones Indices: Index Mathematics Methodology

54

Leveraged and Inverse Indices
Leveraged Indices for Equities
S&P Dow Jones Indices’ Leveraged Indices are designed to generate a multiple of the return of theunderlying index in situations where the investor borrows funds to generate index exposure beyond
his/her cash position. The approach is to first calculate the underlying index, then calculate the daily
returns for the leveraged index and, finally, to calculate the current value of the leveraged index byincrementing the previous value by the daily return. There is no change to the calculation of theunderlying index.
The daily return for the leveraged index consists of two components: (1) the return on the total position in
the underlying index less (2) the borrowing costs for the leverage.
The formula for calculating the Leveraged Index is as follows: Underlying Index t Borrowing Rate Leveraged Index Re turn = K  
− 1 − (K − 1)    Dt ,t −1
Underlying
Index
360


t −1



(1)

In equation (1) the borrowing rate is applied to the leveraged index value because this represents thefunds being borrowed. Given this, the Leveraged Index Value at time t can be calculated as:
Leveraged IndexValuet = (Leveraged Index Valuet −1)  (1 + Leveraged Index Return )(2)

Substituting (1) into (2) and expanding the right hand side of (2) yields:

Leveraged IndexValuet =   UnderlyingIndext − 1 − (K − 1) *  Borrowing Rate   D
Leveraged IndexValuet −1 * 1 + K * t ,t −1 



360


 
   UnderlyingIndex t −1 

(3)where:K (K  1) = Leverage Ratio•

K = 1, no leverage•

K = 2, Exposure = 200%•

K = 3, Exposure = 300%

Borrowing Rate = SOFR Overnight in the U.S. or EURO SHORT TERM RATE (ESTR) in Europe are
two common examples
Dt, t-1 = the number of calendar days between date t and t-1
In the absence of leverage (K=1),
 Underlying Index t 
Leveraged Index Valuet = Leveraged Index Valuet −1 *  Underlying Index t −1 

The leverage position is rebalanced daily. This is consistent with the payoff from futures-based
replication.

S&P Dow Jones Indices: Index Mathematics Methodology

55

Leveraged Indices without Borrowing Costs for Equities
In some cases, leveraged indices that do not account for costs incurred to finance the associatedleverage are calculated. For these indices, the borrowing rate in formulas (1) and (3) is set to zero andthe calculation follows as above.
Inverse Indices for Equities
S&P Dow Jones Indices’ Inverse indices are designed to provide the inverse performance of theunderlying index; this represents a short position in the underlying index. The calculation follows the samegeneral approach as the leveraged index with certain adjustments: First, the return on the underlying
index is reversed. Second, while the costs of borrowing the securities are not included, there is anadjustment to reflect the interest earned on both the initial investment and the proceeds from selling shortthe securities in the underlying index. These assumptions reflect normal industry practice. 22
The general formula for the return to the inverse index is Underlying Index tInverse Index Re turn = −K  
− 1 Underlying Index t −1
Lending
Rate + (K + 1)  
Dt ,t −1
360



(4)

Where the first right hand side term represents the return on the underlying index and the second righthand side term represents the interest earned on the initial investment and the shorting proceeds.
Expanding this as done above for the leveraged index yields:

InverseInd exValuet =  UnderlyingIndex t LendingRat e 
Inverse IndexValuet −1 * 1 − K * 
− 1 − (K + 1) *  Dt ,t −1 

360


 UnderlyingIndex t −1 
 
 

(5)

where:K (K  1) = Leverage Ratio•

K = 1, Exposure = -100%•

K = 2, Exposure = -200%•

K = 3, Exposure = -300%

Lending Rate = SOFR Overnight in the U.S. or ESTR in Europe are two common examples
Dt, t-1 = the number of calendar days between date t and t-1
In the absence of leverage (K =1),

InverseInd exValuet =  UnderlyingIndext − 1 − (2) *  LendingRat e   D
Inverse IndexValuet −1 * 1 − t ,t −1 



360


  UnderlyingIndex t −1  The inverse position is rebalanced daily. This is consistent with the payoff from futures-based replication.

22

Straightforward adjustments can be made to either to include the costs of borrowing securities or to exclude the interest earnedon the shorting proceeds and the initial investment.

S&P Dow Jones Indices: Index Mathematics Methodology

56

Inverse Indices without Borrowing Costs for Equities
In some cases, inverse indices that do not account for any interest earned are calculated. For these
indices, the lending rate in formulas (4) and (5) is set to zero and the calculation follows as above.
Leveraged and Inverse Indices for Futures
S&P Dow Jones Indices’ futures-based Leveraged Indices are designed to generate a multiple of the
return of the underlying futures index in situations where the investor borrows funds to generate index
exposure beyond his/her cash position.
S&P Dow Jones Indices’ futures-based Inverse indices are designed to provide the inverse performanceof the underlying futures index; this represents a short position in the underlying index.
The approach is to first calculate the underlying index, then calculate the daily returns for the leveraged orinverse index. There is no change to the calculation of the underlying futures index.
The leveraged or inverse index may be rebalanced daily or periodically.
Daily Rebalanced Leverage or Inverse Futures Indices
If the S&P Dow Jones Indices futures-based leveraged or inverse index is rebalanced daily, the index
excess return is the multiple of the underlying index’s excess return and calculated as follows:  Underlying IndexER tIndexER t = IndexER t −1  1 +  K * 
− 1   
 Underlying IndexER t −1   

where:K (K ≠ 0) = Leverage/Inverse Ratio•

K = 1, no leverage•

K = 2, leverage exposure = 200%•

K = 3, leverage exposure = 300%•

K = -1, inverse exposure = -100%

A total return version of each of the indices is calculated, which includes interest accrual on the notional
value of the index based on a specified interest rate (e.g., 91-day U.S. Treasury rate), as follows:  IndexERt  + TBRt 
IndexTR t = IndexTR t −1   
 IndexER
t −1 


(6)where:
IndexTRt-1 = The Index Total Return on the preceding business day
TBRt= Treasury Bill Return, as determined by the following formula:




1
TBRt = 1 − 91 * TBARt −1 
360



DeltatDeltat91−1(7)= The number of calendar days between the current and previous business days

S&P Dow Jones Indices: Index Mathematics Methodology

57

TBARt-1= The most recent weekly high discount rate for 91-day U.S. Treasury bills effective on
the preceding business day23

Periodically Rebalanced Leverage or Inverse Futures Indices
If the S&P Dow Jones Indices futures-based leveraged or inverse index is rebalanced periodically (e.g.,
weekly, monthly, or quarterly), the index excess return is the multiple of the underlying index excess
return since last rebalancing business day and shall be calculated as follows: Underlying IndexERt 
IndexERt = IndexERt _ LR  1 +  K * − 1    Underlying IndexERt _ LR
 



where:
IndexERt_LR= The Index Excess Return on the last rebalancing business day, t_LR

UnderlyingIndexERt_LR = The Underlying Index Excess Return value on the last rebalancing
business day, t_LR
t_LR

= The last rebalancing business day

K (K ≠ 0) = Leverage / Inverse Ratio•

K = 1, no leverage•

K = 2, leverage exposure = 200%•

K = 3, leverage exposure = 300%•

K = -1, inverse exposure = -100%

A total return version of each of the indices is calculated, which includes interest accrual on the notional
value of the index based on the 91-day U.S. Treasury rate. The formulae are the same as (6) and (7)above.
Negative Index Levels. For more information regarding the possibility of negative or zero index levels,
refer to Negative/Zero Index Levels section later in this document.

23

Generally, the rates are announced by the U.S. Treasury on each Monday. On Mondays that are bank holidays, Friday’s rates
apply.

S&P Dow Jones Indices: Index Mathematics Methodology

58

Fee Indices/Decrement and Increment Indices
S&P DJI calculates fee indices that are meant to alter the index value of a given underlying index
according to a fixed percentage rate or fixed index points that is applied on a daily basis. This alteration
can be either positive or negative. Fee reduction indices are often also described as Decrement Indices.
Decrement Indices measure the performance of an underlying index with a reduction to the return of theindex representing a fixed, pre-determined amount. Fee addition indices can be described as Increment
Indices. Increment Indices measure the performance of an underlying index with an addition to the returnof the index representing a fixed, pre-determined amount. The pre-determined amount typically
represents a synthetic dividend amount, fund fees, or macroeconomic adjustments, such as interest rates
or inflation targets.
Fee indices can be calculated in a number of ways. The fee can be applied to the index after the return of
the underlying index is calculated, or it can be applied along with the return of the underlying index; thiscan be done with or without a compounding effect on the fee. The different calculations are as follows:
Fixed Percentage Fee Reduction / Addition. A fixed percentage fee reduction or addition multiplies theindex level by a daily portion of an annual fee with no regard for day counts. The formula is as follows:
Decrement Index:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 ×

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒× (1 −)
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1
𝑁

Increment Index:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 ×

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒× (1 +)
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1
𝑁

where:
IndexValuet= The fee reduced / increased index value on day t

IndexValuet-1= The fee reduced / increased index value on day t-1

ParentIndexValuet= The index value of the parent index with(out) fees on day t

ParentIndexValuet-1 = The index value of the parent index with(out) fees on day t-1
Fee= The annual fee percentage

N= The number of days in a year

Standard Fee Reduction / Addition from the Base Date. A standard fee reduction or addition from thebase date multiplies the index level by a pro-rated fee accounting for time since the base date. The
formula is as follows:
Decrement Index:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒0 ×

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒× (1 −
× 𝐴𝐶𝑇(𝑡, 𝑡0 ))𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒0
𝑁

Increment Index:

S&P Dow Jones Indices: Index Mathematics Methodology

59

𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒0 ×

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒× (1 +
× 𝐴𝐶𝑇(𝑡, 𝑡0 ))𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒0
𝑁

where:
IndexValuet= The fee reduced / increased index value on day t

IndexValue0= The fee reduced / increased index value on the base date

ParentIndexValuet = The index value of the parent index with(out) fees on day t
ParentIndexValue0 = The index value of the parent index with(out) fees on the base dateFee

= The annual fee percentage

N= The number of days in a year

ACT(t,t0)= The actual calendar days between day t (exclusive) and the base date(inclusive)

Standard Fee Reduction / Addition. A standard fee reduction or addition multiplies the index level by adaily fee pro-rated to account for non-calculation days (including weekends and holidays). The formula isas follows:
Decrement:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 ×

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1

× (1 −𝐹𝑒𝑒𝑁

× 𝐴𝐶𝑇(𝑡, 𝑡 − 1))

Increment:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 ×

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1

× (1 +𝐹𝑒𝑒𝑁

× 𝐴𝐶𝑇(𝑡, 𝑡 − 1))

where:
IndexValuet= The fee reduced / increased index value on day t

IndexValuet-1= The fee reduced / increased index value on day t-1

ParentIndexValuet= The index value of the parent index with(out) fees on day t

ParentIndexValuet-1 = The index value of the parent index with(out) fees on day t-1
Fee= The annual fee percentage

N= The number of days in a year

ACT(t,t-1)= The actual calendar days between day t (exclusive) and day t-1 (inclusive)

Exponentially Compounding Fee Reduction / Addition. An exponentially compounding fee reduction
multiplies the index level by a daily fee exponentially pro-rated to account for non-calculation days
(including weekends and holidays). The formula is as follows:
Decrement:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 ×

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒 𝐴𝐶𝑇(𝑡,𝑡−1)
× ((1 −))
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1
𝑁

Increment:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 ×

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒 𝐴𝐶𝑇(𝑡,𝑡−1)
× ((1 +))
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1
𝑁

S&P Dow Jones Indices: Index Mathematics Methodology

60where:IndexValuet= The fee reduced / increased index value on day t

IndexValuet-1= The fee reduced / increased index value on day t-1

ParentIndexValuet= The index value of the parent index with(out) fees on day t

ParentIndexValuet-1 = The index value of the parent index with(out) fees on day t-1
Fee= The annual fee percentage

N= The number of days in a year

ACT(t,t-1)= The actual calendar days between day t (exclusive) and day t-1 (inclusive)

Standard Synthetic Dividend. A standard synthetic dividend multiplies the parent index level by an
exponentially pro-rated fee accounting for time since the base date. This fee reduction or addition is afunction of the parent index value and necessarily requires the same base value. The formula is asfollows:
Decrement:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 × ((1 −𝐹𝑒𝑒 𝐴𝐶𝑇(𝑡,𝑡0)))
𝑁

Increment:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 × ((1 +𝐹𝑒𝑒 𝐴𝐶𝑇(𝑡,𝑡0)))
𝑁where:
IndexValuet= The fee reduced / increased index value on day t

ParentIndexValuet = The index value of the parent index with(out) fees on day t
Fee= The annual fee percentage

N= The number of days in a year

ACT(t,t0)= The actual calendar days between day t (exclusive) and the base date(inclusive)

Standard Fee Subtracted from or Added to Return. The standard fee subtracted from return is a fee
reduction that subtracts the fee from the return instead of multiplying the accumulated index level by (1 −Fee). The standard fee added to return is a fee addition that adds the fee to the return instead ofmultiplying the accumulated index level by (1 + Fee). The formula is as follows:
Decrement:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 × (𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒−× 𝐴𝐶𝑇(𝑡, 𝑡 − 1))
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1
𝑁

Increment:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 × (𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒+× 𝐴𝐶𝑇(𝑡, 𝑡 − 1))
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1
𝑁

where:
IndexValuet= The fee reduced / increased index value on day t

S&P Dow Jones Indices: Index Mathematics Methodology

61IndexValuet-1= The fee reduced / increased index value on day t-1

ParentIndexValuet= The index value of the parent index with(out) fees on day t

ParentIndexValuet-1 = The index value of the parent index with(out) fees on day t-1
Fee= The annual fee percentage

N= The number of days in a year

ACT(t,t-1)= The actual calendar days between day t (exclusive) and day t-1 (inclusive)

Fixed Index Point Subtracted from or Added to Return. The fixed index point subtracted from return
subtracts the fee represented as a constant number of index points. The fixed index point added to return
adds the fee represented as a constant number of index points. The formula is as follows:
Decrement:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 ×

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒−× 𝐴𝐶𝑇(𝑡, 𝑡 − 1) × 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒0
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1
𝑁

Increment:
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 ×

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒+× 𝐴𝐶𝑇(𝑡, 𝑡 − 1) × 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒0
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1
𝑁

where:
IndexValuet= The fee reduced / increased index value on day t

IndexValuet-1= The fee reduced / increased index value on day t-1

ParentIndexValuet= The index value of the parent index with(out) fees on day t

ParentIndexValuet-1 = The index value of the parent index with(out) fees on day t-1
Fee= Percentage of fee reduced / increased index base value corresponding to
specified number of index points

N= The number of days in a year

ACT(t,t-1)= The actual calendar days between day t (exclusive) and day t-1 (inclusive)IndexValue0= The fee reduced / increased index value on the base date

Net of Repo Rate Decrement Indices. Net of Repo Rate Decrement Indices apply an additional variable
performance deduction atop a standard fixed percentage or fixed index points decrement index. The
modification to the fixed decrement is based on the quoted spread of a specified Total Return Futures(TRF) contract, which represents the implied equity repo rate.
The additional decrement modification is established once per year at the contract expiry based on the
ASK value of the TRF with the corresponding maturity of the subsequent year. The additional decrement
modification amount is deducted on a daily basis throughout the following 12-month period.
The formula for Standard Fee and Fixed Index Point Subtracted from Returns are as follows:Standard Fee Subtracted from Return:(𝐹𝑒𝑒 + 𝑇𝑅𝐹(𝑟) )𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 × (−× 𝐴𝐶𝑇(𝑡, 𝑡 − 1))
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1
𝑁

Fixed Index Point Subtracted from Return:S&P Dow Jones Indices: Index Mathematics Methodology

62

𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1 × ((𝑇𝑅𝐹(𝑟) )𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐹𝑒𝑒−× 𝐴𝐶𝑇(𝑡, 𝑡 − 1)) −× 𝐴𝐶𝑇(𝑡, 𝑡 − 1) × 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒0
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡−1
𝑁𝑁

Cash Accruals Increment Indices with Compounding. The index is an index of indicesdesigned to measure the performance of the parent index, adding a fixed fee per annum using acash index with a predetermined premium.
𝐶𝑎𝑠ℎ𝐴𝑐𝑐𝑟𝑢𝑎𝑙𝑡 = 𝐶𝑎𝑠ℎ𝐴𝑐𝑐𝑟𝑢𝑎𝑙𝑡−1 ∗ ((1 + 𝐹𝑒𝑒)^ (𝐷𝑡 − 𝐷𝑡−1))
𝑁

𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡
𝐶𝑎𝑠ℎ𝐴𝑐𝑐𝑟𝑢𝑎𝑙𝑡
𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑡 = 𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑟𝑏 ∗ (1 + (− 1) + (− 1))
𝑃𝑎𝑟𝑒𝑛𝑡𝐼𝑛𝑑𝑒𝑥𝑉𝑎𝑙𝑢𝑒𝑟𝑏
𝐶𝑎𝑠ℎ𝐴𝑐𝑐𝑟𝑢𝑎𝑙𝑟𝑏

where:
IndexValuet= The fee increased index value on day t

IndexValuerb= The fee increased index value on day rb

ParentIndexValuet = The index value of the parent index with fees on day t
ParentIndexValuerb = The index value of the parent index with fees on day rb
Fee= The annual fee percentage

N= The number of days in a year

D

= Datet= The current day

rb= The last business day of December

For index specific parameters, please refer to S&P Dow Jones Decrement Indices Parameters document,
available at www.spglobal.com/spdji.
Negative/Zero Index Levels. For more information regarding the possibility of negative or zero index
levels, please refer to the Negative/Zero Index Levels section.

S&P Dow Jones Indices: Index Mathematics Methodology

63Capped Return Indices
In a capped return index, the index return from the prior rebalancing is capped at a pre-defined level. The
overall approach is to first calculate an uncapped index and then compare its return-since-lastrebalancing-day with the return cap. The capped index return takes the smaller value of these two. The
approach can be expressed mathematically as:
𝐼𝑛𝑑𝑒𝑥 𝐿𝑒𝑣𝑒𝑙𝑡 = 𝐼𝑛𝑑𝑒𝑥 𝐿𝑒𝑣𝑒𝑙𝐿𝑅 ∗ (1 + min (𝑅𝑒𝑡𝑢𝑟𝑛𝐶𝑎𝑝,

𝑈𝑛𝑐𝑎𝑝𝑝𝑒𝑑 𝐼𝑛𝑑𝑒𝑥 𝐿𝑒𝑣𝑒𝑙𝑡
𝑈𝑛𝑐𝑎𝑝𝑝𝑒𝑑 𝐼𝑛𝑑𝑒𝑥 𝐿𝑒𝑣𝑒𝑙𝐿𝑅))

where:
index levelt= Index level at date t

index levelLR = Index level at the last rebalancing business day
ReturnCap= Cap on the index return between rebalance dates

S&P Dow Jones Indices: Index Mathematics Methodology

64Dividend Points Indices
S&P Dow Jones Indices’ Dividend Points Indices are designed to track the total dividend payments fromthe constituents of an underlying index. They do not include any changes in the market capitalization ofindex constituents. The index level is based on a running total of dividends of the constituents of theunderlying index. Some indices reset to zero on a periodic basis, generally quarterly or annually, whileothers are cumulative and do not reset periodically. Thus, the index measures the total dividends paid in
the underlying index since the previous rebalancing date, or the base date for indices that do not reset on
a periodic basis. For quarterly indices, the index resets to zero after the close on the third Thursday or
Friday of the last month of the calendar quarter, to coincide with futures and options expiration. For
annual indices, the index resets to zero after the close on the third Thursday or Friday of December, tocoincide with futures and options expiration. Please refer to the table below for index-specific reset dates.
Dividend Points Index
S&P 500 Dividend Points Index
S&P 500 Dividend Points Index (Annual)
S&P/TSX 60 Dividend Points Index
S&P/TSX 60 Dividend Points Index (Annual)
S&P 400 Dividend Points Index (Cumulative)
S&P 600 Dividend Points Index (Cumulative)
Materials Select Sector Dividend Points Index (Cumulative)
Industrial Select Sector Dividend Points Index (Cumulative)
Technology Select Sector Dividend Points Index (Cumulative)
Consumer Staples Select Sector Dividend Points Index(Cumulative)
Consumer Discretionary Select Sector Dividend Points Index(Cumulative)
S&P MLP Dividend Points Index (Cumulative)
Real Estate Select Sector Dividend Points Index (Cumulative)
Utilities Select Sector Dividend Points Index (Cumulative)
Energy Select Sector Dividend Points Index (Cumulative)
Financial Select Sector Dividend Points Index (Cumulative)
Health Care Select Sector Dividend Points Index (Cumulative)
S&P 500 Dividend Points Index (Cumulative)
S&P/ASX 200 Dividend Points Index (Quarterly)
S&P/ASX 200 Dividend Points Index (Annual)

Reset Date (After the Close)
Third Friday of the last month of the quarter
Third Friday of December
Third Friday of the last month of the quarter
Third Friday of December--------------Third Thursday of the last month of the quarter
Third Thursday of December

Except for the S&P/TSX 60 Dividend Points Indices, Dividend Points indices represent regular cash
dividend payments paid by index constituents. The S&P/TSX 60 Dividend Points Indices also include
stock dividends as defined in the S&P/TSX Canadian Indices Methodology. Special dividends are notincluded. Dividends are added on the ex-date, which is consistent with the total return version of theunderlying index. Since stock prices are adjusted for dividends on the ex-date, this method is much morestraightforward than adding dividends on the pay date.
On each trading day, the dividends of all index shares going ex-date are summed up and the total is
divided by the underlying index’s divisor, which transforms the index currency amount (e.g., US$, CAD,
AUD) into index points. Where applicable, dividend payments in another currency are translated into theindex currency at the same exchange rate used for the total return version of the underlying index. Except
for the reset day where applicable, these index points are added to the cumulative total for the respective
period to derive the value of the Dividend Points Index for that day.

S&P Dow Jones Indices: Index Mathematics Methodology

65

The formula for calculating the dividend point index on any date, t, for a given underlying index, x, is:t

DividendIn dex t , x = IDi,xi = r +1

where:
IDi,x = The index dividend of the underlying index x on day i.
t= The current date.

r+1 = The trading date immediately following the reset date of the index (or base date if the index
does not reset periodically).
The index dividend (ID) of the underlying index is calculated on any given day as the total dividend value
for all constituents of the index divided by the index divisor. The total dividend value is calculated as thesum of dividends per share multiplied by index shares outstanding for all constituents of the index whichhave a dividend going ex on the date in question. For more detail concerning the calculation of index
dividends please refer to the Total Return Calculations section of this methodology.

S&P Dow Jones Indices: Index Mathematics Methodology

66

Alternative Pricing
S&P DJI Indices uses alternative pricing for the calculation and publication of certain indices and data
points. Alternative pricing is applied to indices using the approaches outlined below. Details of the pricing
type and application of the pricing for index calculation purposes is indicated in the specific index
methodology.
1. Official Calculation: The daily official index calculation always leverages the alternative pricemethodology.
2. Hybrid Calculation: The alternative price is used in certain instances when calculating the official
index value (e.g., VWAP pricing used for official daily index calculation on the rebalance
implementation while the official close is used for all non-rebalance date calculations)
3. Supplementary Calculation: A supplementary calculation of the index is performed with thealternative price and is published alongside the official closing calculation (e.g., Special Open
Quotation).
Alternative pricing may be captured through vendors or calculated internally by S&P DJI. The formulas
defined in this section are specific to internally calculated alternative pricing. This approach is morecommonly applied to derivative based indices calculated by S&P DJI. S&P DJI leverages exchange
provided prices for official end-of-day index calculations. For each exchange, S&P Dow Jones Indices will
use the relevant price (e.g. last trade, auction, VWAP, official close) as defined in the S&P Dow Jones
Indices' Global Equity Close Prices guide available on https://us.spindices.com/.
Special Opening Quotation (SOQ)
The special opening quotation (“SOQ”) is calculated using the same methodology as the underlying index
except that the price used for each index constituent is the open price at which the security first trades
upon the opening of the exchange on a given trading day. SOQ is calculated using only the opening
prices from the primary exchange, which occur at various times, of all stocks in the index and may occur
at any point during the day. For any stock that has not traded during the regular trading session, theprevious day's closing price is used for the SOQ index calculation. SOQ may be higher than the high,lower than the low and different from the open, as the SOQ is a special calculation with a specific set ofparameters. The open, high, low and close values are continuous calculations, while the SOQ waits until
all stocks in the index are open.
•

U.S. Markets. In the case of a market disruption and if the exchange is unable to provide official
opening prices, the official closing prices utilized are determined based on SEC Rule 123C as
outlined in the Unexpected Exchange Closures chapter of S&P Dow Jones Indices’ Equity Indices
Policies and Practices document.

•

Non-U.S. Markets. In the case of a market disruption and if the exchange is unable to provide
official opening prices, the official closing prices are utilized. If the exchange is unable to provide
official opening or closing prices, the previous closing price adjusted for corporate actions is usedin the calculation of the SOQ.

For M&A target stocks that are suspended or halted from trading on an exchange but are still in indices,
S&P Dow Jones Indices will synthetically derive an SOQ for the suspended security using the deal ratio
terms and the opening price of the acquiring company if the acquirer is issuing stock as part of themerger. If the acquirer is paying cash only, the lower of the previous official close price and the cash
amount are used in the calculation of the SOQ. Similarly, S&P DJI will synthetically derive an SOQ for
spun-off stocks that have not yet begun trading.

S&P Dow Jones Indices: Index Mathematics Methodology

67

Fair Value Indices
Fair Value indices are designed to provide an updated valuation for indices that have ceased calculating
earlier in a given day. The indices are calculated using fair value adjustment factors applied on a stock bystock basis to each stock in the index. The factors are provided by a pricing service which calculates fairvalue adjustments. There may be multiple fair value indices for a given underlying index, due to the use ofdifferent pricing services for each particular index. S&P Dow Jones currently has indices using ICE Data
Services (ICE) and Virtu Financial, Inc. (formerly provided by ITG).
For all stocks in the index, the following are taken from the relevant underlying index: constituents, prices,
and index shares effective as of the next trading date (i.e., the adjusted close data for today). The pricefor each stock is multiplied by the fair value adjustment for that stock to arrive at a fair value price. The
index is then calculated in the same fashion as the underlying index, using the same index shares andindex divisor as the underlying index. Note that the value of a fair value index on a given day, unlike otherindices, is not dependent on the value of that fair value index on the prior day. Rather it is only dependent
on the value of the relevant underlying index and on today’s fair value adjustments.
Volume-Weighted Average Price (VWAP)
Some indices will use VWAP in a specified time window, instead of reported closing values.
Volume Weighted Pricing uses a weighted average price instead of a single closing value. Prices withbigger trading volumes are assigned higher weights. VWAP is calculated by multiplying the price of tradesby their volume, summing that for the applicable time window, and then dividing by the total volume oftrades within that time window, as calculated below:
𝑉𝑊𝐴𝑃𝑖,𝑡 =∑𝑁
𝑗=1 𝑇𝑟𝑎𝑑𝑒𝑉𝑜𝑙𝑢𝑚𝑒𝑖,𝑗 × 𝑇𝑟𝑎𝑑𝑒𝑃𝑟𝑖𝑐𝑒𝑖,𝑗∑𝑁
𝑗=1 𝑇𝑟𝑎𝑑𝑒𝑉𝑜𝑙𝑢𝑚𝑒𝑖,𝑗where:
VWAPi,t= the VWAP for security i on day t over the VWAP observation windowN= the number of trades in the VWAP observation window

TradeVolumei.j= the volume of trade jTradePricei,j= the price of trade jTime-Weighted Average Price (TWAP)
TWAP indicates the Average Price, or Bid Price or Ask Price, that a security is traded at during a
specified time window, rather than its end of day price.
TWAP is calculated by taking a simple average of various snapshots of the price throughout the time
window, written formulaically below:
𝑇𝑊𝐴𝑃𝑖,𝑡 =∑𝑁
𝑗=1 𝑇𝑟𝑎𝑑𝑒𝑃𝑟𝑖𝑐𝑒𝑖,𝑗𝑁

where:
TWAPi,t= the TWAP for security i on day t over the TWAP observation windowN= the number of trades in the TWAP observation window

TradePricei,j= the price of trade j

S&P Dow Jones Indices: Index Mathematics Methodology

68

Negative/Zero Index Levels
A negative index level is possible for certain types of indices including hedged, decrement, leveraged,
and inverse indices, particularly for inverse indices that apply leverage.
•

For indices calculated in real-time, in the event an intraday index calculation results in a zero or
negative value, S&P DJI will publish the zero or negative value as calculated.

•

In the event an end-of-day index calculation results in a zero or negative value, S&P DJI will
publish an official closing index value of zero on that day. Index levels will only be assessed after
the close of trading for purposes of this determination and will not take into consideration intraday
levels for those indices calculated in real-time.

Any index assigned a level of zero will be reviewed by the Index Committee to determine if the index will
be discontinued or the index will be restarted with a new base value. In the event the index is restarted,
S&P DJI will announce such action and will treat these indices as two separate series. Until the Index
Committee has made this determination, the index level will continue to be published with a value of zero.

S&P Dow Jones Indices: Index Mathematics Methodology

69Index TurnoverIndex turnover is a measure of weight changes to an index resulting from corporate events or rebalancing
of an index. Weight changes resulting from market value changes due to market driven price increases ordecreases are not accounted for in an index turnover calculation. All turnover figures provided by S&P
Dow Jones Indices are one-way turnover figures. One-way turnover only views turnover from theperspective of either buying or selling assets. One-way turnover is therefore limited to a maximum amountof 100% which would be equivalent to the deletion of all current index constituents or the addition of all
new constituents. To differentiate between a one-way and two-way turnover approach, a two-way
turnover approach would reflect both the buying and selling of assets. Two-way index turnover would be
200% in the above scenario. A formula of index turnover is provided below. All turnover calculations areprovided by S&P Dow Jones Indices upon request.
𝐼𝑛𝑑𝑒𝑥 𝑇𝑢𝑟𝑛𝑜𝑣𝑒𝑟 =∑𝑖 𝐶𝑜𝑛𝑠𝑡𝑖𝑡𝑢𝑒𝑛𝑡 𝑊𝑒𝑖𝑔ℎ𝑡 𝐶ℎ𝑎𝑛𝑔𝑒
2𝐶𝑜𝑛𝑠𝑡𝑖𝑡𝑢𝑒𝑛𝑡 𝑊𝑒𝑖𝑔ℎ𝑡 𝐶ℎ𝑎𝑛𝑔𝑒 = |𝐶𝑜𝑛𝑠𝑡𝑖𝑡𝑢𝑒𝑛𝑡 𝑊𝑒𝑖𝑔ℎ𝑡 𝐶𝐿𝑆 − 𝐶𝑜𝑛𝑠𝑡𝑖𝑡𝑢𝑒𝑛𝑡 𝑊𝑒𝑖𝑔ℎ𝑡 𝐴𝐷𝐽|
where:
Constituent Weight CLS = Weight of constituent as of the close of business on day T.
Constituent Weight ADJ = Weight of constituent prior to the open on day T+1. This weight will reflect
any adjustments due to corporate events or rebalancing. If the index had
no corporate events or rebalancing, the Constituent Weight CLS will beequal to Constituent Weight ADJ.

S&P Dow Jones Indices: Index Mathematics Methodology

70End-of-Month Global Fundamental Data
The purpose of this section is to give an overview of the End-of-Month (“EOM”) Global Fundamental Datafilings. This section outlines the file types along with their descriptions, general data information, andformulas used to calculate the ratios present in these data files. EOM fundamentals do not include theU.S. Fundamental Data Package.
Global EOM Fundamental Data is disseminated via the following files:
Frequency
Monthly

File TypeIndex Level

File Name
yyyyMMdd_SPTOURUP_EOM.SDL

File Name Extension.SDL

Monthly Files
File Extensions. The following table details the file extensions:File Extension
EOM.SDL

Description
End-of-Month S&P Dow Jones Indices Index Level Files

File Delivery. Monthly files are delivered to clients by the third business day of the following month. For
example, the file 20171031_SPTOURUP_EOM.SDL is delivered to clients no later than November 3,
2017. Files are generated for the last trading day of the month. Therefore, the file name reflects the last
trading day (e.g., October 31, 2017) as shown above.
The EOM.SDL file format details are available in the UFF 2.0 Specifications document available here.
About the Data
For calculation of the Global EOM Fundamental Data values, S&P Dow Jones Indices obtains raw datafrom multiple vendors as of the 25th of every month. The raw data is then validated and used in thecalculation of the ratios listed below.
S&P Dow Jones Indices has 10 Index Level Ratios which are reflected in EOM.SDL files:

24

Ratio24FY0 P/E

Description
Latest reported fiscal year’s price-to-earnings ratio

1 YR FWD P/E

One-year forward (estimated) price-to-earnings ratio

2 YR FWD P/E

Two-year forward (estimated) price-to-earnings ratio

12 MO TRAILING P/E
P/BV
P/CF
P/S
ROE
DIV YLDIND YLD12-month trailing price-to-earnings ratio
Latest reported fiscal year’s price-to-book value ratio
Latest reported fiscal year’s price-to-cash flow ratio
Latest reported fiscal year’s price-to-sales ratio
Latest reported fiscal year’s return on equity
Dividend yield using reported dividend (gross)
Indicated yield using forward looking dividend (gross)

PeriodLatest reported fiscal year
Latest reported fiscal year +one year
Latest reported fiscal year +two years
12-month trailing
Latest reported fiscal year
Latest reported fiscal year
Latest reported fiscal year
Latest reported fiscal year
As per latest reported
As per latest reported

Name as per file.

S&P Dow Jones Indices: Index Mathematics Methodology

71

S&P Dow Jones Indices has five Constituent Level Ratios which are reflected in EOM.SDC files:
Ratio25
PRICE-EARNINGSRATIO (P/E)
PRICE-BOOK VALUERATIO (P/BV)
P/CF
PRICE/SALESIND YLD

Description

Period12-month trailing price-to-earnings ratio

12-month trailing

Latest reported fiscal year’s price-to-book value ratio

Latest reported fiscal year

Latest reported fiscal year’s price-to-cash flow ratio
Latest reported fiscal year’s price-to-sales ratio
Indicated yield using forward looking dividend

Latest reported fiscal year
Latest reported fiscal year
As per latest reported

Output Files
The file naming convention, templates, and field specifications are described below.
There are five EOM file templates included in the Global Fundamental Data Package:•

EOM.SDL – End-of-month index level file•

EOM.SDC – End-of-month constituent level fileoNC_EOM.SDC – End-of-month Constituent file (No Cusip)o

NS_EOM.SDC – End-of-month Constituent file (No Sedol)o

NCS_EOM.SDC – End-of-month Constituent file (No Cusip or Sedol)

Fundamental Data Points
Underlying data point values used for fundamental index level ratio calculations are described below: 26
1. Basic EPS − Continuing Operations (FY0). This is a given company’s basic earnings-pershare excluding extra items for the latest reported fiscal year and is calculated as:
Basic EPS – Continuing Operations (FY0) = (Net Income − Preferred Dividend and Other
Adjustments − Earnings of Discontinued Operations − Extraordinary Item & Accounting
Change) / Weighted Average Basic Shares Outstanding
2. Basic Weighted Average Shares Outstanding (FY0). This is a given company’s basic
weighted average shares outstanding for the latest reported fiscal year.
3. Estimate EPS (FY1). This is a given company’s one year forward estimated earnings-per-shareand represents the aggregated mean of all latest reported fiscal year plus one year estimatesprovided by third-party vendor analysts.
4. Estimate EPS (FY2). This is a given company’s two year forward estimated earnings-per-shareand represents the aggregated mean of all latest reported fiscal year plus two year estimatesprovided by third-party vendor analysts.
5. Basic EPS − Continuing Operations (LTM). This is a given company’s basic earnings-pershare excluding extra items over the last 12 months and is calculated as:
Basic EPS – Continuing Operations (LTM) = (Net Income − Preferred Dividend and Other
Adjustments − Earnings of Discontinued Operations − Extraordinary Item & Accounting
Change) / Weighted Average Basic Shares Outstanding
6. Basic Weighted Average Shares Outstanding (LTM). This is a given company’s basic
weighted average shares outstanding over the last 12 months.

25
26

Name as per file.
All stocks with ADRs are adjusted per the depository receipt ratio except for EPS and Dividend data points.

S&P Dow Jones Indices: Index Mathematics Methodology

72

7. Total Common Equity (FY0). This is a given company’s total common equity for the latest
reported fiscal year and is calculated as:
Total Common Equity (FY0) = Common Stock & APIC + Retained Earnings + Treasury Stock& Other.
8. Cash from Operations (FY0). This is the given company’s cash from operations for the latest
reported fiscal year and is calculated as:
Cash from Operations (FY0) = Net Income + Depreciation and Amortization, Total +Amortization of Deferred Charges, Total − (CF) + Other NonCash Items, Total + Change in Net Operating Assets
9. Total Revenue (FY0). This is the given company’s total revenue for the latest reported fiscal
year and is calculated as:
Total Revenue (FY0) = Revenue + Other Revenue
10. Shares Outstanding. This is the given company’s shares outstanding and provides totalcompany level shares, as reported by stock exchanges, company press releases, and financial
documents. Treasury shares are excluded, and the number is adjusted for corporate actions suchas splits, merger related share issuances, rights offerings, etc.
11. Indicated Annualized Dividend. This is the given company’s latest annualized dividend pershare. It is a forward looking number and is calculated by multiplying the latest dividend paid pershare by the number of dividend payments per year.
Calculations
Monthly calculation of the fundamental data for a given index is done as of the last calendar day of themonth.27
Terminology. Various terms are used in the calculations below and are defined as follows:•

AWF. The Additional Weight Factor (AWF) is the adjustment factor of a stock assigned at eachindex rebalancing date which adjusts the market capitalization for all index constituents toachieve the user-defined weight, while maintaining the total market value of the overall index.

•

IWF. A stock’s Investable Weight Factor (IWF) is based on its free float. Free float can be
defined as the percentage of each company’s shares that are freely available for trading in themarket. For further details, please refer to S&P Dow Jones Indices’ Float Adjustment
Methodology.

•

SO. The shares outstanding of a company.

•

Style. For details, please refer to the S&P U.S. Style Indices Methodology available here.

Index Level Ratios. The formulas below are used to calculate index level ratios: 28
1. FY0 P/E
𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 =𝐵𝑎𝑠𝑖𝑐 𝐸𝑃𝑆 𝐸𝑥𝑐𝑙 (𝐹𝑌0) ∗ 𝐵𝑎𝑠𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 𝐴𝑣𝑔 𝑆𝑂 (𝐹𝑌0) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒 = 𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝐼𝑛𝑑𝑒𝑥 𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 =∑𝑖 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝∑𝑖 𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒

The calculation of fundamental ratios is done based on the index’s current composition as of the date of the fundamental ratio
calculation.
28
With the exception of Dividend Yield and Indicated Dividend Yield, any stock which does not have an underlying value is excluded
from the index level calculation.
27

S&P Dow Jones Indices: Index Mathematics Methodology73

2. 1 YR FWRD P/E𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 =𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝐸𝑃𝑆 𝐹𝑌1 ∗ 𝑆ℎ𝑎𝑟𝑒𝑠𝑜𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒 = 𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝐼𝑛𝑑𝑒𝑥 1𝑦𝑟 𝐹𝑤𝑟𝑑 𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 =∑𝑖 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝∑𝑖 𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒

3. 2 YR FWRD P/E𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 =𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝐸𝑃𝑆 𝐹𝑌2 ∗ 𝑆ℎ𝑎𝑟𝑒𝑠𝑜𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒 = 𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝐼𝑛𝑑𝑒𝑥 2𝑦𝑟 𝐹𝑤𝑟𝑑 𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 =∑𝑖 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝∑𝑖 𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒

4. 12 Month Trailing P/E
𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 =𝐵𝑎𝑠𝑖𝑐 𝐸𝑃𝑆 𝐸𝑥𝑐𝑙 (𝐿𝑇𝑀) ∗ 𝐵𝑎𝑠𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 𝐴𝑣𝑔 𝑆𝑂 (𝐿𝑇𝑀) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒 = 𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝐼𝑛𝑑𝑒𝑥 12 𝑀𝑜𝑛𝑡ℎ 𝑡𝑟𝑎𝑖𝑙𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 =∑𝑖 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝∑𝑖 𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 Value

5. Price-Book Value (FY0)𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 =𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑚𝑚𝑜𝑛 𝐸𝑞𝑢𝑖𝑡𝑦 (𝐹𝑌0) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒 = 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝐼𝑛𝑑𝑒𝑥 𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝐵𝑜𝑜𝑘 𝑉𝑎𝑙𝑢𝑒 =∑𝑖 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝∑𝑖 𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒

6. Price-Cash Flow (FY0)𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 =𝐶𝑎𝑠ℎ 𝑓𝑟𝑜𝑚 𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠 (𝐹𝑌0) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒 = 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝐼𝑛𝑑𝑒𝑥 𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝐶𝑎𝑠ℎ 𝐹𝑙𝑜𝑤 =∑𝑖 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝∑𝑖 𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒

7. Price to Sales (FY0)𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 =𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 (𝐹𝑌0) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒 = 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒

S&P Dow Jones Indices: Index Mathematics Methodology

74

𝐼𝑛𝑑𝑒𝑥 𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝑆𝑎𝑙𝑒𝑠 =∑𝑖 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝∑𝑖 𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝑉𝑎𝑙𝑢𝑒

8. Return on Equity
𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 =𝐵𝑎𝑠𝑖𝑐 𝐸𝑃𝑆 𝐸𝑥𝑐𝑙 (𝐹𝑌0) ∗ 𝐵𝑎𝑠𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 𝐴𝑣𝑔 𝑆𝑂 (𝐹𝑌0) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 = 𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 =𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑚𝑚𝑜𝑛 𝐸𝑞𝑢𝑖𝑡𝑦 (𝐹𝑌0) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐵𝑜𝑜𝑘 𝑉𝑎𝑙𝑢𝑒 = 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝐼𝑛𝑑𝑒𝑥 𝑅𝑂𝐸 =∑𝑖 𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠∑𝑖 𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐵𝑜𝑜𝑘 𝑉𝑎𝑙𝑢𝑒

9. Dividend Yield
𝐼𝑛𝑑𝑒𝑥 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 = ∑(𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑜𝑓 𝑎 𝑠𝑡𝑜𝑐𝑘 ∗ 𝐼𝑛𝑑𝑒𝑥 𝑆ℎ𝑎𝑟𝑒𝑠 𝑜𝑓 𝑎 𝑠𝑡𝑜𝑐𝑘)𝑖

𝑃𝑟𝑖𝑐𝑒 𝐼𝑛𝑑𝑒𝑥 𝑉𝑎𝑙𝑢𝑒 = 𝑇ℎ𝑒 𝑐𝑙𝑜𝑠𝑖𝑛𝑔 𝑖𝑛𝑑𝑒𝑥 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎 𝑔𝑖𝑣𝑒𝑛 𝑠𝑡𝑜𝑐𝑘
𝐷𝐼𝑉 𝑌𝐿𝐷 =𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑑𝑒𝑥 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑
𝑋 100
𝑃𝑟𝑖𝑐𝑒 𝐼𝑛𝑑𝑒𝑥 𝑉𝑎𝑙𝑢𝑒

10. Indicated Yield (IND YLD)
𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 = 𝐼𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑑 𝐴𝑛𝑛𝑢𝑎𝑙 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝐼𝑛𝑑𝑒𝑥 𝐼𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑑 𝑌𝑖𝑒𝑙𝑑 = (∑𝑖 𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 ∗ 𝐷𝑖𝑙𝑢𝑡𝑖𝑜𝑛 𝐹𝑎𝑐𝑡𝑜𝑟) ∗ 100∑𝑖 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝

Constituent Level Ratios. The formulas below are used to calculate constituent level ratios:
1. Price-Earnings Ratio (P/E)
𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 =𝐵𝑎𝑠𝑖𝑐 𝐸𝑃𝑆 𝐸𝑥𝑐𝑙 (𝐿𝑇𝑀) ∗ 𝐵𝑎𝑠𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 𝐴𝑣𝑔 𝑆𝑂 (𝐿𝑇𝑀) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝑃/𝐸 =𝐶𝑙𝑜𝑠𝑒 𝑃𝑟𝑖𝑐𝑒
𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 𝑉𝑎𝑙𝑢𝑒

2. Price-Book Value Ratio (P/BV)𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 𝑉𝑎𝑙𝑢𝑒 =𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑚𝑚𝑜𝑛 𝐸𝑞𝑢𝑖𝑡𝑦(𝐹𝑌0) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 = 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 𝑉𝑎𝑙𝑢𝑒 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝐵𝑜𝑜𝑘 𝑉𝑎𝑙𝑢𝑒 =𝐶𝑜𝑛𝑠𝑡𝑖𝑡𝑢𝑒𝑛𝑡 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝
𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 𝑉𝑎𝑙𝑢𝑒S&P Dow Jones Indices: Index Mathematics Methodology

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3. Price-Cash Flow (P/CF)𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 𝑉𝑎𝑙𝑢𝑒 =𝐶𝑎𝑠ℎ 𝑓𝑟𝑜𝑚 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠 (𝐹𝑌0) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 = 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 𝑉𝑎𝑙𝑢𝑒 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝐶𝑎𝑠ℎ 𝐹𝑙𝑜𝑤 =𝐶𝑜𝑛𝑠𝑡𝑖𝑡𝑢𝑒𝑛𝑡 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝
𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 𝑉𝑎𝑙𝑢𝑒

4. Indicated Yield (IND YLD)
𝐼𝑛𝑑𝑖𝑐𝑎𝑡𝑒𝑑 𝐴𝑛𝑛𝑢𝑎𝑙 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 ∗ 𝐷𝑖𝑙𝑢𝑡𝑖𝑜𝑛 𝐹𝑎𝑐𝑡𝑜𝑟
𝐼𝑛𝑑 𝑌𝑙𝑑 = () ∗ 100
𝐶𝑙𝑜𝑠𝑒 𝑃𝑟𝑖𝑐𝑒
5. Price to Sales
𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 𝑉𝑎𝑙𝑢𝑒 =𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 (𝐹𝑌0) ∗ 𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 1000000
𝑆&𝑃 𝑆ℎ𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 = 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 𝑉𝑎𝑙𝑢𝑒 ∗ 𝑆𝑂 ∗ 𝐼𝑊𝐹 ∗ 𝐹𝑋𝑅𝑎𝑡𝑒 ∗ 𝐴𝑊𝐹 ∗ 𝑆𝑡𝑦𝑙𝑒
𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝑆𝑎𝑙𝑒𝑠 =𝐶𝑜𝑛𝑠𝑡𝑖𝑡𝑢𝑒𝑛𝑡 𝐼𝑛𝑑𝑒𝑥 𝑀𝑎𝑟𝑘𝑒𝑡 𝐶𝑎𝑝
𝐹𝑙𝑜𝑎𝑡 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐷𝑎𝑡𝑎 𝐼𝑡𝑒𝑚 𝑉𝑎𝑙𝑢𝑒

Note: Company level data received from vendors is proportionally assigned to each class of stock. For
example, Altice SA has two classes of stock (Altice SA A and Altice SA B). In order to proportionallyassign company level data to each of these two stock classes, a multiclass factor is used and isdetermined as follows:
𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑆𝑡𝑜𝑐𝑘 𝐴 =𝑆ℎ𝑎𝑟𝑒𝑠 𝑜𝑓 𝑠𝑡𝑜𝑐𝑘 𝐴∑𝑖 𝑆ℎ𝑎𝑟𝑒𝑠 𝑜𝑓 𝑠𝑡𝑜𝑐𝑘𝑠 𝐴 𝑎𝑛𝑑 𝐵

𝑀𝑢𝑙𝑡𝑖𝑐𝑙𝑎𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑆𝑡𝑜𝑐𝑘 𝐵 =𝑆ℎ𝑎𝑟𝑒𝑠 𝑜𝑓 𝑠𝑡𝑜𝑐𝑘 𝐵∑𝑖 𝑆ℎ𝑎𝑟𝑒𝑠 𝑜𝑓 𝑠𝑡𝑜𝑐𝑘𝑠 𝐴 𝑎𝑛𝑑 𝐵

S&P Dow Jones Indices: Index Mathematics Methodology

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80thereof), and all such parties hereby expressly disclaim all warranties of originality, accuracy,completeness, merchantability, or fitness for a particular purpose with respect to any of such standard orclassification. Without limiting any of the foregoing, in no event shall MSCI, S&P DJI, any of their affiliates
or any third party involved in making or compiling any GICS classifications have any liability for any direct,
indirect, special, punitive, consequential or any other damages (including lost profits) even if notified ofthe possibility of such damages.
S&P Dow Jones Indices products are governed by the terms and conditions of the agreements underwhich they may be provided. A license is required from S&P Dow Jones Indices to display, create
derivative works of and/or distribute any product or service that uses, is based upon and/or refers to any
S&P Dow Jones Indices and/or index data.
ESG Indices Disclaimer
S&P DJI provides indices that seek to select, exclude, and/or weight index constituents based on, but notlimited to, certain environmental, social or governance (ESG) indicators, or a combination of thoseindicators, including the following: environmental indicators (including the efficient use of natural
resources, the production of waste, greenhouse gas emissions, or impact on biodiversity); social
indicators (such as, inequality and investment in human capital); governance indictors (such as soundmanagement structures, employee relations, remuneration of staff, tax compliance, respect for humanrights, anti-corruption and anti-bribery matters), specific sustainability or values-related company
involvement indicators (for example, production/distribution of controversial weapons, tobacco products,
or thermal coal), or controversies monitoring (including research of media outlets to identify companies
involved in ESG-related incidents).
S&P DJI ESG indices use ESG metrics and scores in the selection and/or weighting of index constituents.
ESG scores or ratings seek to measure or evaluate a company’s, or an asset’s, performance with respect
to environmental, social and corporate governance issues.
The ESG scores, ratings, and other data used in S&P DJI ESG indices is supplied directly or indirectly by
third parties (note these parties can be independent affiliates of S&P Global or unaffiliated entities) so an
S&P DJI ESG index’s ability to reflect ESG factors depends on these third parties’ data accuracy andavailability.
ESG scores, ratings, and other data may be reported (meaning that the data is provided as disclosed bycompanies, or an asset, or as made publicly available), modelled (meaning that the data is derived using
a proprietary modelling process with only proxies used in the creation of the data), or reported andmodelled (meaning that the data is either a mix of reported and modelled data or is derived from thevendor using reported data /information in a proprietary scoring or determination process).
ESG scores, ratings, and other data, whether from an external and/or internal source, is based on aqualitative and judgmental assessment, especially in the absence of well-defined market standards, anddue to the existence of multiple approaches and methodologies to assess ESG factors andconsiderations. An element of subjectivity and discretion is therefore inherent in any ESG score, rating, orother data and different ESG scoring, rating, and/or data sources may use different ESG assessment or
estimation methodologies. Different persons (including ESG data ratings, or scoring providers, index
administrators or users) may arrive at different conclusions regarding the sustainability or impact of aparticular company, asset, or index.
Where an index uses ESG scores, ratings or other data supplied directly or indirectly by third parties, S&P
DJI does not accept responsibility for the accuracy of completeness of such ESG scores, ratings, or data.
No single clear, definitive test or framework (legal, regulatory, or otherwise) exists to determine ‘ESG’,‘sustainable’, ‘good governance’, ‘no adverse environmental, social and/or other impacts’, or otherequivalently labelled objectives. In the absence of well-defined market standards and due to the existenceof multitude approaches, the exercise of judgment is necessary. Accordingly, different persons mayclassify the same investment, product and/or strategy differently regarding ‘ESG’, ‘sustainable’, ‘good
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81governance’, ‘no adverse environmental, social and/or other impacts’, or other equivalently labelledobjectives. Furthermore, the legal and/or market position on what constitutes an ‘ESG’, ‘sustainable’,‘good governance’, ‘no adverse environmental, social and/or other impacts’, or other equivalently labelledobjectives may change over time, especially as further regulatory or industry rules and guidance areissued and the ESG sustainable finance framework becomes more sophisticated.
Prospective users of an S&P DJI ESG Index are encouraged to read the relevant index methodology andrelated disclosures carefully to determine whether the index is suitable for their potential use case orinvestment objective.

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