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# Index Mathematics

By Brian Gonzalez,2014-07-08 16:18
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Index Mathematics

Index Mathematics

a very short course

David M. Blitzer

Managing Director &

Chairman of the Index Committee

Topics

.Index Calculation.Index Analysis.Attribution Analysis.Valuation

Measures

Index Calculation

Divisors, IWF??s and all that

What??s an Index?

.An index is a collection or portfolio of stocks, bonds or other investments selected to represent a particular segment of a market.

?CThe selection defines the investments in the index, ?CIt includes a rule for weighting the constituents in the index, ?CIt includes rules for maintenance when constituents change. .This presentation focuses on stock indices, but similar approaches apply to bonds, commodities and other financial instruments.

Examples?-

Text Box: S&P 500

S&P/TSX 60

S&P/Citigroup Indices

S&P SmallCap 600

S&P Commodities Index

S&P 500 Equal Weight

S&P Hedge Fund Index

.S&P 500

.S&P/TSX 60.S&P/Citigroup Indices .S&P SmallCap 600.S&P Commodities Index.S&P 500 Equal Weight.S&P Hedge

Fund Index

.MSCI EAFE.Lehman Aggregate.FTSE 100But not the Consumer Price Index

or the Index of Leading Economic Indicators

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:

?CCapitalization weighted (Cap weighted)

?CPrice Weighted

?CEqual Weighted

?CAttribute Weighted

.This presentation covers cap weighted stock indices, such as the S&P 500

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&P??s Cap-weighted indices are calculated with divisors using a method developed by the Standard Statistics Company, S&P??s predecessor, when it introduced indices in the 1920??s. This is sometimes called the Base-aggregative approach.

Cap-Weighted Index Defined

.In words, multiply each stock??s 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?

?C??Cap weighted?? means that each stock??s weight in the index is its capitalization

compared to the sum of all the stock??s capitalization. ?CFor 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. ?CInstead the Divisor scales the index to an easier number ?Ce.g. the S&P 500

closed at 1198.22.

DivisorSPIndexjjj/)(?Á=??

The FIVE Index

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

market caps divided by the Divisor.

CompanyIndex PriceShares OutstandingMarket CapWeightsExxon Mobil Corp.60.556,385,358,000 386,633,426,900 26.5%

General Electric35.4710,599,190,000 375,953,269,300 25.8% Microsoft Corp.24.1210,880,222,000 262,430,954,640 18.0% Citigroup Inc.44.625,225,358,000 233,155,473,960 16.0% Johnson & Johnson66.852,973,666,000 198,789,572,100 13.6% Total Market Cap1,456,962,696,900 100.0%

Divisor11,655,701,575.20

Index125.00

Index Shares and

.S&P??s 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.

DivisorIWFSPIndexjjj???Á?Á=

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