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Indices

© Christopher Ting

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Motivation

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Financial and economic indices have evolved into important multi-purpose tools that help investors

track the investment performance

estimate risk

evaluate the performance of investment managers.

Indices also form the basis for new investment products.

Developing and providing indices can be a viable business

Good starting point to learn portfolio management: construction and rebalancing

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Uses of Market Indices

Gauges of market sentiment

Proxies for measuring and modeling returns, systematic risk, and risk-adjusted performance

Proxies for asset classes in asset allocation models

Benchmarks for actively managed portfolios

Model portfolios for such investment products as index funds and exchange-traded funds (ETFs)

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Major Companies in the Indexing Business

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S&P Dow Jones

MSCI

FTSE Russell

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Introduction

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Define a security market index and explain how to calculate the price return and total return of an index for a single period and over multiple periods.

Describe how indices are constructed and managed.

Differentiate between price return and total return

Discuss the use of market indices.

Describe various types of indices.

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What is an Index?

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In general, an index is a quantitative gauge that provides an eclectic, single-value summary of all endogenous and exogenous factors affecting and driving the state of a system.

Example: Temperature

Typically involving many constituent sub-systems

Example: Dow Jones Industrial Average

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Price!

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Price is an index!

Reflects the forces of supply and demand through the market mechanism

Efficient Market Hypothesis – Prices already incorporate and reflect all relevant information about a company stock.

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Index Construction and Management

Which target market should the index represent?

Which securities should be selected from that target market?

How much weight should be allocated to each security in the index?

When should the index be rebalanced?

When should the security selection and weighting decision be re-

examined?

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Target Market Selection

Target market

Defined broadly or narrowly?

Based on an asset class?

Based on geographic

region?

Based on an exchange?

Other characteristics

?

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Different Weighting Methods

Index

Price weighted

Market capitalization

weighted

Equally weighted

Fundamentally weighted

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An Example of Price-Weighted Index

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The first stock index in the world is the Dow Jones Industrial Average, which was unveiled on May 26, 1896

The precursor of DJIA can be traced back to May 4, 1885.

At the start, DJIA had only 12 component stocks.

In 1916, DJIA had 20 stocks.

From 1928, the number of component stocks is fixed at 30.

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Source: http://www.advisorperspectives.com/dshort/guest/Peter-Williams-120108-Dow-Historical-Perspective.php

Dow Jones Industrial Average in Log Scale

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Another Example of Price-Weighted Index

Nikkei 225 index is also price weighted, with some nuances to the “par value.”

The Nikkei 225 index began on September 7, 1950.

Retroactively could be back-calculated to May 16, 1949.

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Examples of Market-Cap Weighted Index

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S&P 500 Index

MSCI Singapore Free Index

Hang Seng Index

Most if not all stock market indexes

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Index Calculation: Price versus Market Cap

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Let Pi be the price of a component stock and ni the number of its shares used for index calculation.

D

Pn

I

N

i

ii 1

N is the total number of components.

D is the divisor needed to scale the sum into a value called index I of an appropriate value.

D

P

I

N

i

i 1

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The number of shares, x, to buy for every stock is found by x ($8 + $12 + $20 + $24) = $40,000.

Hence, x = $40,000/$64 = 625 shares

Index Construction: Price Weighted

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To construct a price-weighted index, buy the same number of shares for every component stock.

If you buy 1 share of every stock, at inception or initially, D = N. The resulting I is the average price of Nstocks

Suppose you have $40,000 and you wan to construct a price-weighted portfolio with the following 4 stocks:

Stock Name W X Y ZPrice per Share $8 $12 $20 $24 Shares Issued 200,000 310,000 602,000 110,000

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Tutorial Question

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For the example in the previous slide, if you want the price-weighted index to be equal to $16, what should be the value of the divisor?

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Index Construction: Price versus Market Cap

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To construct a market-cap-weighted index,

compute the market cap of every component stock

sum up N market caps.

for every stock, divide its market cap by the total market-cap to obtain its fraction of contribution.

Then apportion the fund according to the fraction.

Use the apportioned fund to buy the number of shares at the price with which its market cap is computed.

Stock Name W X Y ZPrice per Share $8 $12 $20 $24 Shares Issued 200,000 310,000 602,000 110,000Market Cap $1,600,000 $3,720,000 $12,040,000 $2,640,000Weight 2/25 1/5 3/5 1/8

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Tutorial Questions

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Given $40,000, how many shares are to be bought in order to construct a market-cap-weighted portfolio?

The amount apportioned to Stock W is (2/25) ×$40000 = $3,200. With this money, you can buy 3200/8 = 400 shares.

Do likewise for Stocks, X, Y, and Z.

If you want the market cap to start at 100 at inception, what should the divisor D be?

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Equal Weight

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Give the same weight to every component stock by having the same dollar amount to buy every stock.

With $40,000 and 4 stocks, we allocate $10,000 for each stock

How many shares do you buy for every component stock?

If you want your index to start at 100, what should the divisor be?

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Factor-Weighted Index

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In a way, the ubiquitous market cap weighted index is based on the “size factor”.

One can use earnings, for example, to decide the weight for constructing the index.

Consider two stocks, A and B

Stock A

Earnings = €20

Market cap = €200

Market cap weight = 20%

Fundamental weight = 40%

Stock B

Earnings = €30

Market cap = €800

Market cap weight = 80%

Fundamental weight = 60%

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Weighting Schemes During Construction

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How much fund out of the total is allocated?

N

i

i

ii

P

Pw

1

P

Nwi

1E

N

j

jj

iii

Pn

Pnw

1

M

N

j

j

ii

F

Fw

1

F

Price weighted

Equally weighted

Market capitalization weighted

Factor weighted

ni is the number of shares of shares of constituent security i.

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Very Important

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After the index is constructed, i.e., the number of shares for each component stock has already been fixed, the index calculation is now based on the shares and not on the weights anymore.

For equally weighted index, the weights will become unequal, and the index becomes unequally weighted. Regular rebalancing is needed to make it equally weighted again.

Likewise, the factor based index needs to be rebalance to better reflect changes in the factor.

But for price- and market-cap-weighted indices, you don’t have to, if the component stocks don’t change.

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Myth of Using Weight for Market Cap Index Calculation

Many people misunderstood that to calculate the market-cap-weighted index even after its inception, you must use the weights of the components.

This is wrong!

The weights are the results, not the starting data for index calculation.

The weight changes as soon as any of the component stock price changes!

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Advantages and Disadvantages

Price weighted

Simple

High price stocks have

greater impact

Stock splits necessitate a

change in divisor

Equally weighted

Simple

Under- and over-representation

Frequent rebalancing

Market capitalization

weighted

Securities held in

proportion to their market

value

Similar to a momentum

strategy

Fundamentally weighted

Ensures a value or

contrarian tilt

Data intensive

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Real World Example: Singapore MSCI Index

The MSCI Singapore Free Index is a free-float adjusted market capitalization weighted index that is designed to track the equity market performance of Singapore securities listed on Singapore Stock Exchange.

The MSCI Singapore Free Index is constructed based on the MSCI Global Investable Market Indexes Methodology targeting a free-float market capitalization coverage of 85%. The index has a base date of December 31, 1987.

https://app2.msci.com/eqb/custom_indexes/sg_performance.html

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Reconstitution

Beginning index

Reconstitution date

Reconstituted index

Change constituent securities?

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Equity Indices

Equity indices

Broad marketWilshire 5000 Total Market

Index

MultimarketMSCI Emerging

Markets

SectorGSTI

Semiconductor Index

StyleDow Jones U.S.

Small-Cap Value Index

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Challenges Facing Fixed Income Index Construction

Lack of pricing

data

Number of

securities

Illiquid securities

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Dimensions of Fixed-Income Indices

Market

Global

Regional

Country or currency zone

Type Corporate

CollateralizedSecuritizedMortgage-

backed

Government agency

Government

Maturity For example, 1–3, 3–5, 5–7, 7–10, 10+ years; short-term, medium-term, or long-term

Credit quality

For example, AAA, AA, A, BBB, etc.; Aaa, Aa, A, Baa, etc.; investment grade, high yield

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Indices for Alternative Investments

Commodities

Real estate

Hedge funds

Indices for

alternativeinvestments

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Commodity Indices

Risk-free interest rate

Changes in futures prices

Roll yield

Commodity index return

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Real Estate Indices

Appraisal indices

Repeat sales indices

Real estate investment trust (REIT) indices

Ownership of properties

Investment in mortgages

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The FTSE EPRA/NAREIT Global REIT Index Family

Source: FTSE International, “FTSE EPRA/NAREIT Global & Global Ex US Indices” (Factsheet 2009).

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Hedge Fund Indices

Hedge funds are private investment vehicles that typically use leverage and long and short investment strategies.

Research organizations maintain databases of hedge fund returns and summarize these returns into indices.

Most indices reflect performance on a broad global level or on a strategy level.

Most hedge fund indices are equal weighted.

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Source: https://www.hedgefundresearch.com/?fuse=indices-str

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Problems Caused by Voluntary Reporting

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Voluntary investment

performance

Survivorship bias

Indices reflect different

performances for the same time

period

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Description of a Security Market Index

Security market index

Price return index Total return index

Constituent securities

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Value of a Price Return Index

D

Pn

V 1PR

N

i

ii

I

VPRI = the value of the price return index

ni = the number of units of constituent securities in the index

N = the number of constituent securities in the index

Pi = the unit price of constituent security i

D = the value of the divisor

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Calculation of Single-Period Price Return

N

i

N

i i

iiiii

I

III

1 1 0

01

0PR

0PR1PR

P

PPwPRw

V

VVPR

PRI = the price return of index portfolio I

PRi = the price return of constituent security i

wi = the weight of security i

Pi1= the price of constituent security i at the end of the period

Pi0= the price of constituent security i at the beginning of the

period

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%29.141429.000.105

00.10500.120PR

00.120100

)18400()24100()12200(V

00.105100

)15400()25100()10200(V

1PR

0PR

I

I

I

Security

Beginning

of Period

Price (€)

Ending of

Period Price

(€)

Dividends

per share

(€)

Shares

Outstanding

LMN 10.00 12.00 0.50 200

OPQ 25.00 24.00 1.00 100

RST 15.00 18.00 0.25 400

Divisor = 100

Example of Single-Period Price Return

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Calculation of Single-Period Total Returns

N

i

N

i i

iiiiiiI

I

IPRIII

1 1 0

01

0PR

01PR

P

Inc PPwTRwTR

V

Inc VVTR

TRI = the total return of the index portfolioIncI = the total income from all securities in the indexTRi = the total return of the constituent security iInci = the total income from security i

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Security

Beginning

of Period

Price (€)

Ending of

Period

Price (€)

Dividends

per share

(€)

Shares

Outstanding

LMN 10.00 12.00 0.50 200

OPQ 25.00 24.00 1.00 100

RST 15.00 18.00 0.25 400

Divisor = 100

00.3100)]25.0400()00.1100()50.0200[(Inc I

%14.171714.000.105

00.300.10500.120TR

I

Calculation of Single-Period Total Return

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Index Values over Multiple Time Periods

T210TRTTR

T210PRTPR

TR1TR1TR1VV

PR1PR1PR1VV

IIIII

IIIII

Calculation of index values over multiple time periods requires geometrically linking the series of index returns.

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Period Return (%) Calculation Ending Value0 1,000(1.00) 1,000.00

1 5.00 1,000(1.05) 1,050.00

2 3.00 1,000(1.05)(1.03) 1,081.50

For an index with an inception value set to 1,000 and price returns of 5 percent and 3 percent for Periods 1 and 2 respectively, the values of the price return index would be calculated as follows:

Calculation Example

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Summary

Price return index

Total return index

Choices in index construction and management

Advantages and disadvantages of different weighting schemes

Rebalancing and reconstitution

Uses of market indices

Equity, fixed income, and alternative investment indices

© Christopher Ting