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MSc Investment Management

Portfolio Management

Lecture 3Equity Indexing

February, 08, 2012

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Index definition and uses

Indices are designed to provide a concise summary of the pricemovements of their constituents

Uses are:

To provide a record of historical price movements, which facilitatesdetermination of trends

To serve as benchmarks in performance measurement

To act as a basis for index tracking funds, exchange-traded funds(ETFs) and index derivatives

To support portfolio management research and asset allocationdecisions

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Index weighting schemesThere are three weighting schemes:

1. Market value or capitalisation weighting

The largest stock has the largest influence on the index value

Example: S&P 500, FTSE 100

2. Price weighting

The highest price stock has the greatest influence on the indexvalue

Example: Dow Jones Industrial Average Index

3. Equal weighting

All stocks are assigned an equal weight, these are known asunweighted indices

Example: FT 30

Difficult to replicate equal weights, so never used as a benchmarkin index tracking

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Methods of averaging an Index

Arithmetic vs. Geometric average

All indices are calculated as arithmetic averages nowadays exceptunweighted indices (e.g. FT 30)

As the price of a stock increases, the weights adjust automatically forthe consistency with the share amount

Although the weightings change, the share amounts do not,consequently, no rebalancing is necessary

Therefore, arithmetic indices can be tracked exactly by owning allstocks in the index in the proportions suggested by its weights in theindex

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Example

Period Stock X(800 shares

in issue)

Stock Y(100

shares inissue)

Stock Z(100

shares inissue)

Priceweightedarithmetic

index

Unweightedgeometric

index

Valueweightedarithmetic

index

t0 100p 100p 100p 100 100 100

t1 90p 105p 120p 105 104.3 94.5

t2 100p 0p 100p ? ? ?

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Example calculations for period t1

1. Price weighted arithmetic index:

2. Unweighted geometric index:

105100

)100100100(

)12010590(

/01

valueindexpreviousPP tt

3.104100100100100

120105903

valueindexpreviouspricessharebaseofproduct

pricessharecurrentofproductn

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Example calculations for period t1

3. Value weighted arithmetic index:

Task: Calculate the value of the index in period t2 using all threemethods

5.94100100100100100800100

10012010010580090

00

11

x

valueindexpreviousqP

qP

tt

tt

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Comparison of index averaging methods: concluding comments

1. Price weighted arithmetic indices Ignore the number of shares in issue and favour highly priced

shares

Not representative of the real world portfolio limited use asperformance measurement benchmarks

2. Unweighted geometric indices It always understates the price rises and overstates the price

falls of constituents relative to that of a price weighted index

It collapses if the price of an index constituent is zero

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3. Value weighted arithmetic indices More complex calculations and significant data requirements

Replicate the precise effect that changing share values wouldhave on a portfolio comprising the same underlying index

constituents weighted in accordance with their relative marketcapitalisations

They have a broad coverage of the market being represented

Therefore, they are the most suitable indices to asses markettrends, act as performance benchmarks and provide a basisfor index tracking

Comparison of index averaging methods: concluding comments

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Free flotation

The value of market value weighted index as a performancemeasurement benchmark can be compromised if those indexconstituents make significantly less than 100% of their equityavailable to the market are accorded a full market value indexweighting

Restricted supply of the stock prevents a portfolio manager fromholding a full weighting of the stock within their portfolio

Price of the stock will be distorted given the need of index tracking

funds to hold the stock in accordance with their index weighting

Hence, free floating rules were introduced

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Dealing with the free float

FTSE (source: www.ftse.com) Less than or equal to 5% Ineligible Greater than 5% but less or equal to 15% Actual Greater than 15% but less or equal to 20% 20% Greater than 20% but less or equal to 30% 30% Greater than 30% but less or equal to 40% 40% Greater than 40% but less or equal to 50% 50% Greater than 50% but less or equal to 75% 75% Greater than 75% 100%

MSCI

Usually stocks with free-float under 25% are excluded(Lower cut-off for emerging markets)

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The main equity indices

FTSE Index Constituents % of FTSEAll Share

FTSE 100 100 largest companies 80%

FTSE 250 The next 250 largestcompanies

15%

FTSE 350 FTSE100+

FTSE250

95%

FTSE SmallCap FTSE All Share

FTSE 350

5%

FTSE All Share FTSE 350+ FTSESmallCap 100%

FTSE Fledgling Those that do not meetthe size criteria for FTSEAll Share

1.5%

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The main equity indices

Examples of other Indices: FTSE All World Index (2700 stocks, 49 countries, 90-95%

capitalisation of each of the markets, in US$)

MSCI World Index (1700 stocks, 23 countries, 85% capitalisation of

each market, in US$ and local currency)

S&P 500 (500 most widely held NYSE stocks, 80% of NYSE marketcapitalisation)

Nikkei Indices (Nikkei 225 price weighted index of 225 Japanesecompanies representative of the market; more suitable is Nikkei 300which is value weighted)

There are around 3000 indices globally available

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Fundamental Weighted Indices

What are Fundamental Weighted Indices?

Selects, ranks and weights companies, not by marketcapitalisation, but by financial data points, such as sales, cashflow, book value, or dividend yield.

How do they work?

Stocks are reviewed using fundamental factors not the stockprice/mkt cap

The constituents are then weighted in the index according to thefactors themselves notthe mkt cap

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Why the move away from market cap weighted?

Short term share prices are volatile and consequentlymisleadcausing misdirected capital

0%

10%

20%

30%

40%

50%

60%

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

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2000

2001

2002

2003

Telecom & IT Financials

Non-Cyclical Services Other Sectors

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The FTSE GWA (Global Wealth Allocation) Index Series

The FTSE GWA Index Series therefore constructs portfolioswithout referring to share prices stocks should be weightedaccording to their proven ability to create wealth

Wealth is measured by three fundamentals net income, cash

flow & book value

The review process holds every company in the index, but

weights it according to wealth

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FTSE GWA UK Index Top Five

Overweights/Underweights in 2008

Overweight

Company Sector

Wgt in FTSE GWA

UK Index

Wgt in FTSE All-

Share Index Difference

Vodafone Group Mobile Telecommunications 7.02% 5.68% 1.34%

Royal Bank Of Scotland Group Banks 4.36% 3.09% 1.27%

HBOS Banks 3.03% 2.01% 1.03%Barclays Banks 3.13% 2.24% 0.89%

British Energy Group Electricity 0.91% 0.16% 0.75%

Underweight

Company Sector

Wgt in FTSE GWA

UK Index

Wgt in FTSE All-

Share Index Difference

GlaxoSmithKline Pharmaceuticals & Biotechnology 2.97% 5.14% -2.17%

AstraZeneca Pharmaceuticals & Biotechnology 1.59% 2.56% -0.97%HSBC Hldgs Banks 5.72% 6.25% -0.53%

Diageo Beverages 1.00% 1.47% -0.47%

British American Tobacco Tobacco 0.74% 1.15% -0.41%

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FTSE GWA UK Index vs. the FTSE All-Share Index vs. FTSE GWADeveloped markets index; source: www.ftse.com

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Definition of indexing

Index fund: portfolio of securities that replicates the returns of a selectedindex

Process of investing in such a portfolio: indexing

Index funds exist across asset classes but are predominant in equities hence equity indexing

Achievement of optimal diversificationThis strategy is response to both theoretical aspirations and practical

needs

responds to the concept of the efficient market hypothesis andCAPM

maximises control over investment outcomes and minimises costs ofinvestment

no outperformance but assurance of no underperformance

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Development of index funds

In 1970 Wells Fargo Bank introduced the Stagecoach Fund tracking NYSE Composite Index lack of interest, so discontinued

published research helped in understanding how difficult it is tooutperform the market through active strategies

In 1973, Wells Fargo introduced the fund to track more widelyfollowed S&P 500 Index

J. Bogle introduced one of the most popular index tracking fundsVanguard 500 Index Fund in 1976 which also tracks S&P 500

Increased popularity in 1990s

In the US, 20-30% of funds are managed passively nowadays

Three largest pension fund managers in the UK 800bn pensionfund market are index-tracking specialists

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First step in creating an index fund: selecting abenchmark Funds performance may be measured against the market

portfolio or a subset or sector on the market

Pure index fund by definition would tend to perfectly replicate themarket portfolio

Selection of appropriate benchmark is primarily driven by thedesired level of diversification and cost effectiveness

FTSE 100 (large, mature firms)

feasible to buy and hold with no need for particular adjustments

Less diversified and more cost effective

FTSE All Share index (includes younger, smaller-cap firms)

illiquid, more expensive stocks, more frequent revision needed

More diversified and less cost effective

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Customised benchmarks

Choosing the right benchmark is crucial

With an increasing trend in global investing, benchmark error ismagnified as the inappropriate international index is chosen as abenchmark

Customised benchmarks are available used in performancemeasurement of specific, custom-made portfolios

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Approaches to replicating an indexHow perfect should the replication be?

Censusapproach/full replication

perfect replication, buying every stock in the index

stocks in the same weight as in the index

Samplingapproach

select a subset of index optimised to track the benchmark as closelyas possible

needed when: certain securities in the index are very illiquid or wheninvestor is restricted by policy from owning them

Includes: optimisation, stratified sampling and capitalisation

replication approach Consider the availability of derivative products on the index

possibility to create a synthetic index fund as an alternative toholding underlying equities that make up the index

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Sampling approach: Optimization Mathematical algorithm approach

The solution obtained through optimization is the efficient frontier

Investors utility defines where the optimal portfolio lies

Equalise the beta of the replicating portfolio with the beta of thebenchmark

Problem: optimisation relies on the historical estimates of expected

returns, variances, covariances; it assumes normal distribution ofreturns; mathematics that underlies optimisation is difficult to apply toheterogeneous groups of stocks

Not the most widely used approach for tracking a benchmark

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Sampling approach: Capitalisation

Method involves constructing a basket with fewer stocks than inthe index

Purchasing a number of top capitalization stocks in actual

weights and equally weighting the residual stock weightings inthe basket

Example: if top 100 stocks according to size are selected for thebasket, and this accounts for 80% of the total capitalisation of

the index (as FTSE 100 is 80% of capitalisation of FTSE AllShare), the remaining 20% is evenly proportioned among the100 stocks

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Sampling approach: Stratified sampling

The universe of stocks stratified according to certain criteria(industry)

Stock selection within each stratum

matching the weight of the stratum in the basket portfolio as itnaturally exists in the index

Stocks within each stratum are selected using eithercapitalisationranking, valuation methods or optimisation

Problem: no mathematical backing

Strength: not concerned with historical data

Appropriate for high turnover indexes, since historical estimatesare less relevant

Widely used approach forreplication of a benchmark

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Purchasing a basket/replicating portfolio

Program trading or package trading Refers to the purchase or sale of a diversified portfolio of stocks

The cost of executing such a transaction should be smaller thanfor individual stocks as the risk of portfolio is smaller than the riskof individual stock

Informationless trade

Investors transfer execution risk to the broker

broker provides investor with an insurance protection option, the costof which is negotiated with a broker

price protection - bestefforts basis

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Measuring Tracking Error (TE)

Index tracking objective: minimise the cost while retaining the replicating portfolios ability totrack the benchmark index

Holding fewer stocks than in the index generates tracking error

Tracking error represents the risk that the replicating portfoliowill perform differently to benchmark

In statistical terms, there are two commonly used definitions ofex-post tracking error:

Tracking error as Residual risk and Tracking error as Performance volatility

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TE as Residual Risk

Market model suggests that:

where Rp is return of the replicating portfolio, Rb is return of thebenchmark portfolio and is residual risk

TE is then defined as:

TE is model dependant in this definition (model risk!)

Residuals and Variance of residuals can easily be obtained inexcel and more sophisticated statistical packages

bp RR

2var ( ) var ( ) (1 ( , ) )TE iance iance p correlation Rp Rb

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TE as Performance Volatility

TE is defined as the standard deviation of the difference inexpected returns between the replicating/basket portfolio andthe benchmark portfolio:

TE = Standard Deviation (Rp - Rb)

TE depends on benchmark volatility which cannot be controlledby investment manager

For all portfolio betas not equal to 1, the volatility of performanceis larger than residual risk, so TE will be different

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Relationship between TE and size of the replicating/basket portfolio

Hypothetical replication of a benchmark that has 500 stocks willresult in the following tracking errors for different sizes of thebasket portfolio:

Tracking error vs. Size of basket

0

0.5

1

1.5

2

2.5

3

0 100 200 300 400 500 600

Number of stocks in the basket

Trackingerror(%

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Reasons for existence of tracking errors of perfect baskets

Example shows that even if we have perfect replication ofbenchmark, TE is present. The reasons for this are:

1. Odd-lot purchases of stocks vs. round-lots

Index funds are comprised of round lots: the number of shares ofeach stock in the basket is rounded off to the nearest hundred

from the exact number of shares indicated by basket buildingmodel. This might affect the ability of smaller baskets to track theindex

2. Changing composition of the benchmark index

weights of stocks in the index are changing

hold all the stocks as in the index and the weights will be self-adjusting

if fewer stocks held, rebalancing is needed

list of stocks is changing

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Tracking error interpretation

TE statistic has some desirable properties:A fund with a TE of 2% p.a. is expected to have 2/3 of its annualreturns fall within -2% and +2% of the benchmark and 95% of itsreturns within -4% and +4% of the benchmark

Is 2% TE small or large?

This depends on the volatility of the underlying benchmark, the type ofbenchmark and the method used to construct the basket/replicatingportfolio

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Empirical evidence on Tracking Errors

Larsen & Resnick (1998) Data: US, 200 high capitalisation (HC) and 200 low

capitalisation (LC) stocks used to create value weighted andequally weighted indices

Period 1981-1997

Main conclusions: Indexed portfolios from HC indexes have less tracking error and

lower standard deviation of tracking error than indexed portfolios

of LC indexes Value weighted portfolio can be indexed more accurately thanequally weighted ones

The more stocks there is in a tracking basket portfolio, thesmaller the tracking error

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Empirical evidence on Tracking Errors- results of the study

TRACKING ERROR(TE)/STANDARD DEV OF TE FOR ALL INDEXED PORTFOLIOS

No of EW VW

stocks Stratif ied Non-stratif ied Stratif ied Non-stratif ie

HC PORTFOLIOS

10 1.60/1.29 1.89/1.38 1.50/1.19 1.53/1.1320 1.40/1.11 1.47/1.08 1.37/1.01 1.22/0.95

40 1.14/0.79 1.27/0.91 1.02/0.74 1.06/0.80

80 0.78/0.64 0.82/0.65 0.75/0.57 0.71/0.56

LC PORTFOLIOS

10 2.50/2.07 2.59/1.85 2.43/1.90 2.42/1.75

20 1.84/1.38 1.75/1.36 1.79/1.19 1.68/1.27

40 1.51/1.24 1.44/1.04 1.42/1.10 1.43/0.99

80 1.08/0.84 1.08/0.81 1.04/0.89 1.08/0.81

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Difficulties in trading the basket portfolio

Difficult situation arises when stocks are deleted from or added tothe benchmark index

Benchmark is calculated as though the changes were made atclosing prices

However, changes are not publicly announced until the markethas already closed

Therefore, the index fund managers must trade the following dayat prevailing prices which may be less advantageous than theprevious days closing price

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Minimising the costs with indexing

Costs are minimised in three important ways:

1. Minimising of brokerage commission by minimising necessityto transact

almost a buy and hold strategy with a very low turnover

need to transact only to contribute or withdraw funds,

reinvest income or accommodate for the changes in thebenchmark index

2. Market impact is minimised when transaction costs do occur

investment in securities in proportion to their actual weightson the market

largest investment in securities with greatest liquidity3. No asset selection or market timing research involved, so

management fees are low

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Market Impact The price at which a basket of stocks can be bought or sold will

differ substantially from the cash index price which is based onthe prices at which each of the individual stocks comprising theindex last traded

To sell a basket of stocks, the seller would receive the current

bid prices of the individual stocks and vice versa

Market impact is that difference between the cash index and thecost of buying /selling a basket of stocks

major component of transaction costs of index funds

Level of market impact varies over time and depends upon: liquidity (better liquidity implies smaller bid/ask spread and smaller

market impact)

size of the basket (market impact increases with the size of thebasket)

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Return enhancement strategies

There are two strategies by which the indexer can enhance thereturns of his portfolio without compromising the goal oftracking the benchmark index:

1. He can lend securities in basket portfolio to brokerage firms whoneed them for short-sales for example

2. Indexer can engage in index futures arbitrage: they can sell basketportfolio and replace it with a position in the futures (when they areundervalued), investing the cash proceeds in money marketinstruments until the futures are settled use of synthetic indexation

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Synthetic Indexation

By purchasing futures contracts and Treasury bills it is possibleto create a synthetic index fund that will have the same returnsprofile as if one was holding the equities that make up the index

Bare in mind that benchmark with available liquid futures

contract has to be chosen

Advantages of holding long stock position rather than syntheticfund are mainly related to special events such as dividend

payments and special dividends

However, the major drawbacks of holding portfolio of stocks thatreplicates the index are high initial transaction costs, marketimpact, custodial costs and tracking error

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Advantages and disadvantages of a synthetic fund

Advantages are: low transaction costs, no tracking error, noproblems due to dividend reinvestment

Disadvantages are:

price risk related to the fact that futures will be overpriced whenpurchased or that the futures position may have to be rolled out to

the next contract (spread overpriced) variation margin that arises from the fact that futures are marked-

to-the market daily

futures position will outperform the index in the upmarket and viceversa

underhedging technique is used to minimise the risk ofunderperformance due to variation margin:

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The dollar(pound) size of the basket should be stated in terms

of a number of futures contracts

Calculation of the appropriate number of futures contracts:

It HAS to be a round number of futures contracts

If we have a basket portfolio worth 25M replicating FTSE 100

currently trading at 4714 points, the number of futures

contracts to buy to replicate that basket is (note: multiplier forFTSE 100 is 10):

valueindexmultiplier

basketofvaluecontractsfuturesofNo

53033.530471410

25000000

Buying futures for synthetic indexation

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Link between active and passive management

Purely passive plainvanilla index funds vs. purely active styles(market timing and stock picking)

Pure passive funds assume beta of portfolio equal to one, i.e.tracking the stock market index

Pure active fund assumes constant deviation from the market

portfolio in search for positive alphas

Index fund management can be extended into active to a fairlymodest degree being aimed at controlling risk in the following

ways:1. Tilted index funds or Core-satellite approach to indexation

2. Exploiting different sector exposures in indices

3. Asset allocation model applied to the question of rebalancing

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Passive & Active: Core-satellite approach Build a low-risk, low-cost core to a portfolio using index funds, while

pursuing higher returns with more aggressive, satellite active funds orindividual stock proportions

It is based on the belief that only some sectors of the market (e.g. largecap equities) are efficient and they should be indexed and a proportionshould be invested in a less efficient market sector

Exposure of a portfolio to factors that are expected to outperform: P/Eratio, growth etc.

Allows investor to place active bets in the market while retaining thediversification of the index fund

Tilted fund is unlikely to outperform the benchmark index dramatically

Presence of Exchange Traded Funds (ETFs)

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Passive & Active: Exploiting different sector exposure of indices

By combining a long basket/replicating portfolio with one or moreshort futures positions, sector exposure can be realised at alow cost and additional returns could be generated:

Low Cap exposure can be achieved when holding a portfoliowhich is tracking Russell 1000 Index and shorting the S&P 500futures contract

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Passive & Active: Asset allocation applied to rebalancing

Additional cash is not added to the basket unless the expectedreturn of the stock exceeds that of the short-term investmentfund or money market instrument (MMI)

Basically, investor holds an option to exchange one asset (MMI)

for another (equity in the basket portfolio)

The cost of having this choice is the price of the option

Strategy may involve stock index and bonds futures forimplementation of asset allocation decisions to reduce the costsof investing in actual equities and MMIs

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Indexing in emerging markets

Example: Asia & Pacific

Liu (2000)

Passive management relies on the notion of market efficiency are emerging markets efficient?

Index funds have combined expense advantage and increasedturnover advantage of up to 2.26% in the emerging markets

Difficult task of persuading investors that index funds canaccurately track emerging market indices and educating themabout the benefits of indexation

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Reading

Chapter 9 and 10 in Portfolio and Investment Management: State-of-the-art Research, Analysis and Strategies, edited by F.Fabozzi