equity 2: Security Market Indexes

What is a security market index and what is it made up of?

A portfolio of marketable securities representing a market, market segment, or asset class.

• constituent securities: individual securities included in an index.

What does a price return index vs total return index measure?

price return: Only changes in prices of constituent securities.

total return: Price changes plus reinvested income (dividends and/or interest).

price return index formula

$$V_{PRI}=\frac{\sum_{i}^{N}n_{i}p_{i}}{D}$$

• $V_{PRI}$ = the value of the price return index

• $n_i$ = the number of units of constituent security i held in the index portfolio

• $N$ = the number of constituent securities in the index

• $P_i$ = the unit price of constituent security i

• $D$ = the value of the divisor: scaling factor used to maintain continuity in index values.

Formula for index price return using index values?

$$PR_{I}=\frac{V_{PRI1}-V_{PRI0}}{V_{PRI0}}$$

• $PR_I$ = the price return of the index portfolio (as a decimal number, i.e., 12 percent is 0.12)

• $V_{PRI1}$ = the value of the price return index at the end of the period

• $V_{PRI0}$ = the value of the price return index at the beginning of the period

Formula for price return of a constituent security?

$$PR_{i}=\frac{P_{i1}-P_{i0}}{P_{i0}}$$

• $PR_i$ = the price return of constituent security i (as a decimal number)

• $P_{i1}$ = the price of constituent security i at the end of the period

• $P_{i0}$ = the price of constituent security i at the beginning of the period

Formula for weighted average index price return?

$$\sum_{i}^{N}w_{i}\cdot\frac{P_{i1}-P_{i0}}{P_{i0}}$$

$$=w_1PR_1+w_2PR_2+\ldots+w_{N}PR_{N}$$

Formula for index total return?

$$TR_{I}=\frac{V_{PRI1}-V_{PRI0}+INC_{I}}{V_{PRI0}}$$

• $TR_{I}$ = total return of constituent security i (as a decimal number)

• $V_{PRI1}$ = the value of the price return index at the end of the period

• $V_{PRI0}$ = the value of the price return index at the beginning of the period

• $INC_{I}$ = total income (dividends and/or interest) from all securities in the index held over the period

Formula for total return of a constituent security?

$$PR_{i}=\frac{P_{i1}-P_{i0}+Inc_{i}}{P_{i0}}$$

• $TR_{i}$ = the total return of constituent security i (as a decimal number)

• $P_{i1}$ = the price of constituent security i at the end of the period

• $P_{i0}$ = the price of constituent security i at the beginning of the period

• $INC_{I}$ = total income (dividends and/or interest) from all securities in the index held over the period

Formula for weighted average index total return?

$$\sum_{i}^{N}w_{i}\cdot\frac{P_{i1}-P_{i0}+Inc_{i}}{P_{i0}}$$

$$=w_1TR_1+w_2TR_2+\ldots+w_{N}TR_{N}$$

Formula for multi-period price return index value?

$$V_{PRIT}=V_{PRI0}\left(1+PR_{I1}\right)\left(1+PR_{I2}\right)\cdot\cdot\cdot\left(1+PR_{IT}\right)$$

• $V_{PRI0}$ = the value of the price return index at inception

• _{$V_{PRIT}$} = the value of the price return index at time t

• $PR_{IT}$ = the price return (as a decimal number) on the index over period t, t = 1, 2, …, T

Formula for multi-period total return index value

$$V_{TRIT}=V_{TRI0}\left(1+TR_{I1}\right)\left(1+TR_{I2}\right)\cdot\cdot\cdot\left(1+TR_{IT}\right)$$

• $V_{TRI0}$ = the value of the index at inception

• _{$V_{TRIT}$} = the value of the total return index at time t

• $TR_{IT}$ = the total return (as a decimal number) on the index over period t, t = 1, 2, …, T

What are the main decisions in index construction?

1. Define target market

2. Select securities

3. Determine weights

4. Decide rebalancing frequency

5. Re-examine selection/weighting rules

What can define a target market for an index?

• Asset class

• Geography

• Exchange

• Sector

• Company size

• Investment style

• Credit quality

• Duration

What is the investment universe?

The set of securities eligible for inclusion in the index.

What are two approaches to security selection?

• Include nearly all securities

• Use a representative sample

Which indexes use subjective selection committees?

Examples include the S&P 500 and S&P BSE SENSEX.

what are the four main index weighting methods?

1. Price weighted

2. Equal weighted

3. Market-cap weighted

4. Fundamental weighted

in a price-weighted index, what determines a stock’s weight?

Its price relative to the sum of all constituent prices.

Higher-priced stocks receive larger weights.

• Used in Dow Jones Industrial Average

Formula for price-weighted index weight?

$$w_{i}^{P}=\frac{P_{i}}{\sum_{}^{}P_{i}}$$

Main advantage/disadvantage of price weighting?

Advantages:

• Simple to calculate.

• Easy to understand.

disadvantages:

• Weights are arbitrary.

• Higher stock price ≠ larger company.

• Stock splits distort weights.

What happens after a stock split in a price-weighted index?

• Stock splits change weights of all securities.

• Divisor must be adjusted to maintain index continuity.

• Without adjustment, index value changes artificially.

Formula for equal-weighted index weights?

$$w_{i}^{E}=\frac{1}{N}$$

advantages and disadvantages of equal weighting?

Advantages

• Simple method.

• Greater diversification across securities.

Disadvantages

• Overweights small companies.

• Underweights large companies.

• Requires frequent rebalancing as price changes cause weights to drift

• Higher transaction costs.

Formula for market-cap weight?

$$w_{i}^{M}=\frac{Q_{i}P_{i}}{\sum_{j}^{N}Q_{j}P_{j}}$$

• $w_i$ = fraction of the portfolio that is allocated to security i or weight of security i

• $Q_i$ = number of shares outstanding of security i

• $P_i$ = share price of security i

• $N$ = number of securities in the inde

advantages and disadvantages of market cap method?

Advantages

• Represents market portfolio well.

• Low turnover.

• Requires little rebalancing.

• Easy for passive investing.

Disadvantages

• Stocks that rise get larger weights.

• May overweight overvalued stocks.

• May underweight undervalued stocks.

• Has a momentum bias.

What is Float-Adjusted Market-Cap Weighting?

• Uses only publicly tradable shares.

• Excludes restricted or closely held shares.

• Most modern indexes use float adjustment

Formula for float-adjusted market-cap weight?

$$w_{i}^{M}=\frac{f_{i}Q_{i}P_{i}}{\sum_{}^{}f_{j}Q_{j}P_{j}}$$

$f_i$ = fraction of shares outstanding in the market float

$Q_i$ = number of shares outstanding of security i

$P_i$ = share price of security i

$w_i$ = fraction of the portfolio that is allocated to security i or weight of security i

advantages / disadvantages of float0adjusted method?

Advantages

• More realistic for investors.

• Improves investability.

• Better reflects investable market opportunity.

• Closely held firms receive lower weights.

Disadvantages

• Still inherits momentum bias from market-cap weighting.

What is the goal of fundamental weighting?

• Weights are based on company fundamentals.

• Not based on stock price.

• avoids market-cap weighting’s tendency to overweight overpriced stocks.

Possible measures:

• Earnings

• Book value

• Revenues

• Dividends

• Cash flow

• Employees

Formula for fundamental weighting?

$$w_{i}^{F}=\frac{F_{i}}{\sum_{}^{}F_{j}}$$

$F_i$ = fundamental size measure of company i

characteristics of fundamental method?

• Creates a value tilt.

• Overweights high fundamental-to-price stocks.

• Underweights expensive stocks

rebalancing a fundamental method?

As prices rise:

• Weights may decrease

As prices fall:

• Weights may increase

Creates a contrarian effect.

market cap eighting vs fundamental cap weighting?

Market-Cap Weighting

• Momentum-oriented.

• Winners receive larger weights.

Fundamental Weighting

• Value-oriented.

• Rebalancing shifts toward cheaper stocks

comparison of all methods

Price-weighted indexes:

• Weight by stock price

• Sensitive to stock splits

Equal-weighted indexes:

• Equal starting weights

• Require frequent rebalancing

Market-cap-weighted indexes:

• Weight by company size

• Most common method

• Momentum bias

Float-adjusted indexes:

• Use only publicly tradable shares

Fundamental indexes:

• Weight by accounting measures

• Have value tilt

• contrarian rebalancing

what is rebalancing and how often is it done?

• Adjusting constituent weights to maintain consistency with the index weighting method.

• Usually occurs quarterly.

Why is rebalancing necessary?

• Security prices change over time.

• This changes portfolio weights automatically.

• Indexes drift away from target weights.

• Rebalancing restores desired weights.

what is turnover?

• buying and selling securities.

• Higher turnover increases transaction costs.

• Rebalancing creates turnover.

Equal-Weighted Index Rebalancing

Equal weights do not stay equal after price changes.

Stocks with strong performance gain larger weights.

Stocks with weak performance lose weight.

Rebalancing requires:

• Selling outperformers

• Buying underperformers

Price-Weighted Indexes rebalancing

none required

• weights automatically depend on price

Market-Cap-Weighted Indexes rebalancing

Mostly self-rebalancing.

• stock prices rise Weights increase automatically

• stock prices fall: Weights decrease automatically

• Only limited adjustments are needed.

Adjustments mainly occur due to:

• Mergers

• Acquisitions

• Liquidations

• Corporate actions

What is index reconstitution?

Reconstitution changes the securities in the index.

Securities may be:

• Added

• Removed

• Retained

Why are indexes reconstituted?

Ensures the index still represents the target market.

• Re-applies original inclusion criteria.

Examples of events leading to reconstitution?

• Bankruptcies

• Delistings

• Mergers

• Acquisitions

steps for the reconstitution process?

• Review constituent securities

• Apply inclusion rules again

• Remove ineligible securities

• Add qualifying securities

• Reapply weighting method

rebalancing vs reconsitution?

Rebalancing:

• Adjusts weights

• Maintains target weighting scheme

• Creates turnover

Reconstitution:

• Changes constituent securities

• Reapplies inclusion criteria

• Reflects market changes

List the major uses of market indexes.

1. Gauges of market sentiment

2. Proxies for returns and risk measurement

3. Asset allocation proxies

4. Benchmarks for active managers

5. Bases for index funds and ETFs

How do indexes act as gauges of market sentiment?

• They reflect the collective opinions and behavior of investors

• Dow Jones Industrial Average - only has 30 stocks

How do indexes act as Benchmark of manager performance?

• Indexes evaluate portfolio manager performance.

• Benchmark should match manager strategy.

how are indexes used in CAPM and Market Portfolio

Indexes are used to calculate:

• Beta

• Systematic risk

• Expected returns

how are indexes used as proxies for asset classes

Indexes represent asset classes.

Used in portfolio allocation models.

Provide historical:

• Returns

• Risk data

• Correlations

What is a broad market index?

An index that represents an entire equity market

• typically covering over 90% of the market.

• Wilshire 5000 → us equity market

What is a multi-market index?

An index that combines securities from multiple countries or regions.

• used to measure the equity returns of a geographic region, development stage, or whole world

• eg. MSCI

  • Developed markets

  • Emerging markets

  • Frontier markets

what are Fundamental Multi-Market Indexes

Some indexes weight countries by GDP.

• prevents a country with previously high stock returns from being overweighted in a multi-market index.

What is a sector index?

An index that tracks a specific economic sector.

What are style indexes?

Indexes that classify stocks by size, value, or growth characteristics.

• Large cap

• Mid cap

• Small cap

High turnover

What defines value vs growth stocks in style indexes?

value stocks: Low valuation ratios (e.g., low P/E, low P/B, high dividend yield).

growth stocks: High expected earnings growth and higher valuation ratios.

List the six style index categories.

1. Large-cap value

2. Large-cap growth

3. Mid-cap value

4. Mid-cap growth

5. Small-cap value

6. Small-cap growth

Why are fixed-income indexes harder to construct than equity indexes?

• more securities

• many are illiquid

• pricing is less transparent.

  • dealers market

  • infrequent trading

main categories of fixed income securities

1. Aggregate (broad market) indexes

2. Market sector indexes

3. Style indexes

4. Economic sector indexes

5. Specialized indexes

What are key dimensions used to classify fixed-income securities?

1. Issuer type

2. Financing type (collateralized vs general obligation)

3. Currency

4. Maturity

5. Credit quality

6. Inflation protection

What do commodity indexes consist of?

Futures contracts on commodities (not the physical commodities).

Why do commodity indexes with the same commodities still differ?

• commodities have no market-cap equivalent so use different weighting methods

  • equal weighting

  • global production values,

  • fixed weights that the index provider determines

• different indexes have significantly different commodity exposures and risk and return characteristics.

Why do commodity index returns differ from spot commodity returns?

Commodity indexes are based on the prices of commodity futures contracts not the spot prices of commodities.

Returns are affected by:

• Spot price changes

• Roll yield (futures rollover)

• Collateral interest (risk-free rate)

What are the main types of real estate indexes?

• Appraisal-based indexes

• Repeat-sales indexes

• REIT indexes

characteristics of REIT?

• Based on market prices

• Continuously updated

• More liquid than physical real estate

what are hedge fund indexes?

Measure performance of hedge fund strategies.

Hedge funds use:

• Leverage

• Long/short positions

• Derivatives

how are hedge funds constructed?

• Equal-weighted averages

• Based on reporting hedge funds (issue)

index comparison table

Index	Reflects	Number of Constituent Securities	Weighting Method	Notes
Dow Jones Industrial Average	Large U.S. stocks	30	Price	Stocks are chosen by Wall Street Journal editors
Nikkei Stock Average	Large Japanese stocks	225	Modified price	Price weighted and adjusted for high-priced shares
TOPIX	All stocks on the Tokyo Stock Exchange First Section	Variable	Market capitalization, adjusted for float	Has a large number of small illiquid stocks making it hard to replicate. Contains 93% of the market cap of Japanese equities
MSCI All Country World Index	Stocks in 23 developed and 24 emerging markets	Variable	Market capitalization, adjusted for float	Available in both U.S. dollars and local currency
S&P Developed Ex-U.S. BMI Energy Sector Index	Global energy stocks outside the United States	Variable	Market capitalization, adjusted for float	Is the model portfolio for an ETF
Barclays Capital Global Aggregate Bond Index	Global investment-grade bonds	Variable	Market capitalization	Formerly compiled by Lehman Brothers
Markit iBoxx Euro High-Yield Bond Indexes	Below investment-grade bonds	Variable	Market capitalization	Represents liquid portion of market and rebalanced monthly
FTSE EPRA/NAREIT Global Real Estate Index	Global real estate	Variable	Market capitalization, adjusted for float	Represents publicly traded REITs
HFRX Global Hedge Fund Index	Global hedge funds	Variable	Asset weighted	Contains a variety of hedge fund strategies and is weighted based on the amount invested in each hedge fund
HFRX Equal Weighted Strategies EUR Index	Global hedge funds	Variable	Equal weighted	Contains same strategy funds as HFRX Global Hedge Fund Index and is equal weighted
Morningstar Style Indexes	U.S. stocks grouped by value/growth and market cap	Variable	Market capitalization, adjusted for float	Nine categories classified by combinations of three cap categories and three value/growth categories