Index Mathematics 12-05

8/7/2019 Index Mathematics 12-05

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Index Mathematicsa very short course

David M. Blitzer

Managing Director &Chairman of the Index Committee


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Topics

Index Calculation

Index Analysis

Attribution Analysis

Valuation Measures


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Index CalculationDivisors, IWFs and all that


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Whats an Index?

An index is a collection or portfolio of stocks, bonds or other

investments selected to represent a particular segment of a

market.

The selection defines the investments in the index,

It includes a rule for weighting the constituents in the index,

It includes rules for maintenance when constituents change.

This presentation focuses on stock indices, but similar

approaches apply to bonds, commodities and other financial

instruments.


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Kinds of Indices

Indices can be defined by the kinds of investments included

such as stock indices, bond indices, commodity indices, etc.

Indices can also be defined by the weight schemes used,

including:

Capitalization weighted (Cap weighted)

Price Weighted

Equal Weighted

Attribute Weighted

This presentation covers cap weighted stock indices, such as

the S&P 500


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Divisors and Calculation

To understand how and when index funds trade, one must

understand how the index is put together.

The key to index calculation and maintenance is the Divisor.

S&Ps Cap-weighted indices are calculated with divisors using amethod developed by the Standard Statistics Company, S&Ps

predecessor, when it introduced indices in the 1920s. This is

sometimes called the Base-aggregative approach.


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Cap-Weighted Index Defined

In words, multiply each stocks Price (P) by its Shares (S) to find its

market cap and then divide the sum of market caps by the divisor.

Why is it done this way? Cap weighted means that each stocks weight in the index is its capitalization

compared to the sum of all the stocks capitalization.

For the S&P 500, this sum is about $11 trillion. We could quote the index each

day as the total capitalization, say $10,978,453,672.22. That would be

cumbersome. Further, if we changed a stock, the index would change.

Instead the Divisor scales the index to an easier number e.g. the S&P 500

closed at 1198.22.

DivisorSPIndex jj

j /)( =


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The FIVE Index

In this five stock index, the index is the sum of the

market caps divided by the Divisor.

Company Index Price Shares Outstanding Market Cap Weights

Exxon Mobil Corp. 60.55 6,385,358,000 386,633,426,900 26.5%

General Electric 35.47 10,599,190,000 375,953,269,300 25.8%

Microsoft Corp. 24.12 10,880,222,000 262,430,954,640 18.0%

Citigroup Inc. 44.62 5,225,358,000 233,155,473,960 16.0%

Johnson & Johnson 66.85 2,973,666,000 198,789,572,100 13.6%

Total Market Cap 1,456,962,696,900 100.0%

Divisor 11,655,701,575.20Index 125.00


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1012/14/2005

Index Shares andFloat Adjustment

S&Ps US indices are moving to float adjustment.

This means that when the indices are calculated they only include

shares that are available to investors, shares that are part of the float.Unavailable shares include blocks owned by executives or directors if

they total more than 10%.

S&P reviews each company in the index to determine what percentage

of its shares are available to the public. This percentage is called the

investable weight factor or IWF.

This means that the calculation of the index is now a bit different.

Everything about the Divisor is the same as before.

Divisor

IWFSP

Index

jjj =

h d


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1112/14/2005

The FIVE IndexFloat Adjusted

Company

Index

Price

Shares

Outstanding IWF Market Cap Weights

Exxon Mobil Corp. 60.55 6,385,358,000 1.00 386,633,426,900 26.9%

General Electric 35.47 10,599,190,000 1.00 375,953,269,300 26.1%Microsoft Corp. 24.12 10,880,222,000 0.93 244,060,787,815 17.0%

Citigroup Inc. 44.62 5,225,358,000 1.00 233,155,473,960 16.2%

Johnson & Johnson 66.85 2,973,666,000 1.00 198,789,572,100 13.8%

Total Market Cap 1,438,592,530,075 100.0%

Divisor 11,508,740,240.60

Index 125.00

We have added IWFs and recalculated the index to reflect the newinformation. Note that the divisor is smaller because we have removed

some market cap and Microsofts weight is lower. Other stocks have

higher weights. This is essentially what was done on March 18th.


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1212/14/2005

Counting Shares

A companys shares can be counted in different ways and sometimes

have different names. FASB and GAAP define basic and diluted share

counts. The number of shares used in index calculations start withbasic shares.

However, a companys shares change often as employee options are

exercised or shares are bought back or other things happen. Because applying all these small changes would require almost daily

index and divisor adjustments, S&P only makes immediate adjustments

when a change is over 5%. Otherwise, share counts are updated once

a quarter. This is the 5% rule.

Index Shares refers to the share count S&P uses in its index

calculations. Index Shares are multiplied by the IWF in calculating the

index.


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Index Shares and 5%

Index Shares is the share count S&P uses in calculating the index. It

is multiplied by the IWF.

If an index fund manager wants to track the index exactly he needs to

know this number, exactly. Data on index shares are valuable.

Once a quarter on the 3rd Friday of the last month of the quarter when

futures expire index shares are updated to reflect all those little, lessthan 5%, changes that were ignored.

None of this changes with float adjustment except that now an index

fund manager needs both Index Shares and the IWF for each stock inthe index.


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Indices Change The stocks in an index change from time to time because of mergers,

acquisitions or bankruptcies and sometimes companies just dont fit

anymore.

Polaroid was once a hot growth Nifty Fifty company. Eventually it slipped and

was moved to the small cap index. Later it was removed before it filed for

bankruptcy.

Microsoft is less than 20 years old as a public company today it is # 3 in the

S&P 500.

ExxonMobil, #1 recently, used to be two companies. In fact, it is descended from

the Standard Oil Trust and was one company, then several and now one again.

Some indices change quarterly or annually on schedules; most of S&Psinvestable indices change only when necessary.

When indices change, the index level should not change otherwise

market moves would be confused by index maintenance. The Divisor is the

key to maintenance.

Adj ti th Di i


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Adjusting the Divisorwhen a Stock Changes

In words, calculate the market capitalization of the index before and after the

change. Before the change we know the market cap, the index level and thedivisor. After the change, we know the market cap (with the new stock instead of

the old stock), we know the index level (which will not change) and we can solve

for the new divisor.

S&P does the calculations after the market closes. The index opens the next daywith the new stocks and the new divisor, but the index level only changes if the

prices of the stocks change.

Thats why our press releases sometimes say, After the close on May 1st, XYZ

company will replace ABC company in the S&P 500.

newjj

j

jnewnewnew

oldjj

j

joutoutout

DIWFSPIWFSPIndex

IndexDIWFSPIWFSP

/)(

/)(

+=

==+


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Solving for the New Divisor

Using the right hand side of the equation from the last slide we

have the definition of the index.

Solve this for the Divisor to calculate the new divisor.

DivisorIWFSPIWFSPIndex jjj

jnnn /)][( +=

IndexIWFSPIWFSPDivisor jjj

jnnn /)][( +=

B k f A i C


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Bank of America CorpReplaces Citigroup Inc.

Company

Index

Price

Shares


Exxon Mobil Corp. 60.55 6,385,358,000 1.00 386,633,426,900 26.9%

General Electric 35.47 10,599,190,000 1.00 375,953,269,300 26.1%

Microsoft Corp. 24.12 10,880,222,000 0.93 244,060,787,815 17.0%Citigroup Inc. 44.62 5,225,358,000 1.00 233,155,473,960 16.2%

Johnson & Johnson 66.85 2,973,666,000 1.00 198,789,572,100 13.8%

Total Market Cap 1,438,592,530,075 100.0%

Divisor 11,508,740,240.60

Index 125.00

Company

Index

Price

Shares


Exxon Mobil Corp. 60.55 6,385,358,000 1.00 386,633,426,900 27.9%


Bank of America Corp 44.01 4,053,638,000 1.00 178,400,608,380 12.9%

Johnson & Johnson 66.85 2,973,666,000 1.00 198,789,572,100 14.4%

Total Market Cap 1,383,837,664,495 100.0%

Divisor 11,070,701,315.96

Index 125.00

The Divisor can also


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The Divisor can alsoAdjust for Shares Changes

In words, if the shares of stock k change by s, the Divisor is adjusted so

the index level doesnt change. This way, an index can be adjusted when

companies issue or buy back shares. The same basic approach is used for

stock price adjustments, such as when a company spins off a unit.

As with stock changes, these are done after the close.

There are many kinds of corporate actions that require divisor adjustments.

One common action that does not require a divisor adjustment is a stock

split shares increase and prices decrease in proportion.

nj

j

jkkk

ojj

j

jkkk

DSPIWFsSPIndex

IndexDIWFSPIWFSP

/)]))([(

/)(

+=

=+

10% Share Increase in


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10% Share Increase inJohnson & Johnson

Company

Index

Price

Shares


Exxon Mobil Corp. 60.55 6,385,358,000 1.00 386,633,426,900 26.5%


Citigroup Inc. 44.62 5,225,358,000 1.00 233,155,473,960 16.0%

Johnson & Johnson 66.85 3,271,032,600 1.00 218,668,529,310 15.0%

Total Market Cap 1,458,471,487,285 100.0%

Divisor 11,667,771,898.28

Index 125.00

Compared to the slide on page 11, the share count for JNJ rises10%, the divisor is adjusted, and JNJs weight rises also.

The other stocks weights decline slightly.


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The Divisor - Summary

By using a Divisor, we create an index with an easy-to-use level.

Adjustments to the Divisor assure that changes in the indexs

level reflect changes in the market, not adjustments for

corporate actions.

The index has meaning over long spans of time and we can use

it to measure long-term investment returns over years, decades

and even centuries.


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Index AnalysisTotal returns, shares to buy and more


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Total Return

Some or maybe all of the stocks in an index will pay dividends.

In the S&P 500, about 380 of the stocks pay dividends.

Dividends make a big difference and ignoring dividends means

ignoring money. Therefore, two kinds of index returns are

calculated:

Price return -- the gain or loss from changes in the price of stocks in

the index, ignoring dividends, and

Total Return -- the gain or loss from owning stocks in the index and

re-investing their dividends in the index. Mutual fund returns are

compared to total return indices.

To track total return, a Total Return Index (TRI) is calculated.


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Total Return Index

)/$( DivisorDividendDt =

First the total dollars paid in dividends each day (or week or month)across the entire index is calculated,

This is divided by the Divisor to calculate dollars of dividends per index

point (above) The result is combined with the price change in the index to calculate

the daily (weekly or monthly) total return and applied to the total return

index (below)

][1

1

+=

t

tttt

Index

DIndexTRITRI


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Total Return

S&P calculates total return based on the Ex-Date. Ex-Datestands for excluding dividend date the first day when

someone who buys the stock does not receive the dividend. The

Ex-Date is determined by the record date and settlement rules. S&P uses the change in the index level between the day before

the ex-date and the ex-date. In the equation, t equals the ex-

date.

][ 11

+

= t

tt

tt Index

DIndex

TRITRI


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Calculating Weights

tCapTotalMarke

IWFSP

IWFSP

IWFSP

Weight

jjj

jjj

jjj

j

)(

)(

)( =

=

The weight of each stock in the index is the proportion of its

market cap to the total market cap of the index.

This is a key variable in a lot of index analysis.

How Much of Each Stock do


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How Much of Each Stock doIndex Funds Own?

tCapTotalMarkeweightMarketCapStock jj=

For the market overall

SizeFundweightapPortfolioCStock jj =

For the fund, the amount in stock j is the weight multiplied by the

portfolios size:

So, the proportion of each stocks total cap held in the fund is:

tCapTotalMarke

SizeFund

tCapTotalMarkeweight

FundweightOwnsFund

j

j=

=

How Much of Each Stock do


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If index funds as a group

own 10% of the S&P 500,then the funds own 10% of

the float adjusted market cap

of each stock in the index.

Therefore, if S&P adds a $10

billion float-adjusted marketcap stock to the index, the

index funds together will buy

up $1 billion of the stock.

How Much of Each Stock doIndex Funds Own?


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Build a Portfolio

To determine how much to buy of each stock to build a

$1,000,000,000 (one billion dollar) fund based on the FIVEindex we can use the weights calculated on slide 11.

We can check this by calculating what proportion of each stocks

market cap the FIVE Fund will hold.


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Building a Fund

Company Weights Fund Size

Total For

Each Stock SharesExxon Mobil Corp. 26.9% 1,000,000,000$ 268,758,122$ 4,438,615

General Electric 26.1% 1,000,000,000$ 261,334,090$ 7,367,750

Microsoft Corp. 17.0% 1,000,000,000$ 169,652,478$ 7,033,685

Citigroup Inc. 16.2% 1,000,000,000$ 162,071,934$ 3,632,271Johnson & Johnson 13.8% 1,000,000,000$ 138,183,376$ 2,067,066

Multiply the fund size ($1 billion) by the weights to find thedollars invested in each stock. Divide this by the price of the

stock to find how many shares the fund must buy.

Checking the Ownership


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Checking the Ownershipin Each Stock

Company Market Cap Weights Fund Size

Total for

Each Stock

How Much of

Each Stock is

Owned

Exxon Mobil Corp. 386,633,426,900$ 26.9% 1,000,000,000$ 268,758,122$ 0.070%

General Electric 375,953,269,300$ 26.1% 1,000,000,000$ 261,334,090$ 0.070%Microsoft Corp. 244,060,787,815$ 17.0% 1,000,000,000$ 169,652,478$ 0.070%

Citigroup Inc. 233,155,473,960$ 16.2% 1,000,000,000$ 162,071,934$ 0.070%

Johnson & Johnson 198,789,572,100$ 13.8% 1,000,000,000$ 138,183,376$ 0.070%

Index MC 1,438,592,530,075$

Divisor 11,508,740,240.60 Check: 1,000,000,000$ 0.070%Index 125.00

For each stock, calculate the percentage of its market cap that isOwned by the FIVE Fund, compare this to the percentage the

Five Fund is of the index market cap. They should match.


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Attribution

AnalysisWhat makes the index rise or fall?


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Measuring Why the Index Rose

Prices change, stocks change and the index moves.

What stock drives the index up or down?

The answer depends on individual stock weights as well as the

price changes.

A stocks contribution depends on both its weight and the price

change, as shown on the following slide.

Attribution Analysis


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ySpreadsheet

Company

Index

Price

Shares


Contri-

bution

Exxon Mobil Corp. 60.55 6,385,358,000 1.00 386,633,426,900 26.9% 0.20%

General Electric 35.47 10,599,190,000 1.00 375,953,269,300 26.1% -0.35%

Microsoft Corp. 24.12 10,880,222,000 0.93 244,060,787,815 17.0% -0.79%

Citigroup Inc. 44.62 5,225,358,000 1.00 233,155,473,960 16.2% 0.86%

Johnson & Johnson 66.85 2,973,666,000 1.00 198,789,572,100 13.8% -0.38%

Total Market Cap 1,438,592,530,075 Sum -0.45%

Divisor 11,508,740,241

Index 125.00

Company

Index

Price

Shares


Price

Change

Exxon Mobil Corp. 61 6,385,358,000 1.00 389,506,838,000 27.1% 0.7%

General Electric 35 10,599,190,000 1.00 370,971,650,000 25.8% -1.3%Microsoft Corp. 23 10,880,222,000 0.93 232,727,948,580 16.2% -4.6%

Citigroup Inc. 47 5,225,358,000 1.00 245,591,826,000 17.1% 5.3%

Johnson & Johnson 65 2,973,666,000 1.00 193,288,290,000 13.4% -2.8%

Total Market Cap 1,432,086,552,580

Divisor 11,508,740,241Index 124.43 -0.45%


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Valuation

MeasuresIs the market over valued?

EPS


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EPS

On Wall Street, corporate profits mean the EPS on the S&P 500.

How is it measured?

For a company, EPS is earnings (properly defined) divided by

shares outstanding.

For the index, the Divisor plays the role of shares outstanding.

When S&P moved to float adjustment we evaluated the impact

of the change in EPS and found only a small shift less then

some index changes in the recent past.

C l l ti I d EPS


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Calculating Index EPS

Divisior

IWFSharesEPSEPS jjj

=

Start with each companys EPS, multiply by its shares and IWF

to find its earnings, in dollars, for the shares in the index. The

sum of each companys dollar earnings is then divided by the

divisor to get the index EPS.

P/E R ti


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P/E Ratios

EPS

LevelIndexEPIndex =/

One of the most widely used valuation measures is the P/E

ratio. The markets P/E is the P/E ratio on the S&P 500. What

does that mean?

EPCosallofEarnings

CapMarketIndexTotal

Divisor

Divisor

EPS

LevelIndex/==

Di id d


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Dividends

Recently dividends have been an important factor in valuing

stocks. The markets dividend yield the dividend yield of theS&P 500 is closely watched.

For dividend yield, the measure used is Indicated Annual

Dividends covering only ordinary dividends and assumingany increases are not reversed.

Di id d Yi ld


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Dividend Yield

Its calculation is similar to the P/E ratio.

levelIndex

sharedividends

Yield

/

=

Divisor

IWFSdividendsshareDividends

jjj =

/

Understanding Indices


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Understanding Indices

This presentation has barely scratched the surface in exploring

how indices work and what can be done with them. S&Ps indices are a key part of many products and services.

The S&P 500 is one of our most valuable brands.

Clients depend on indices and index data. As a result, clients

can be demanding, fussy and occasionally difficult to deal with.

On top of that, there is often more than one way to use indices

or solve an analytical index problem.

Hopefully the background here will make it easier to answer

questions about our indices.

In Answering Questions


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In Answering Questions

S&P Index Analysis and Index Services doesnt give investment

advice or advise people on how to trade. We deal only in public information.

Critical index announcements are made available to everyone on

an equal basis and are posted on our web site.

Other data is posted on Index Alert.

If in doubt about answering a specific inquiry, ask the client to

send an e-mail to Index_Services it will be referred to the Index

Committee if necessary.


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David M. Blitzer

Managing Director &Chairman of the Index Committee

(212) 438-3907

[email protected]

Thank you

Index Mathematics 12-05

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