Chapter 8: Indices and Benchmarking — Part 1: Index Fundamentals and Weighting Methodologies
8.1 Understanding Market Indices
It is intuitive that the performance of various assets traded in a financial market is closely tied to the performance of that market as a whole. Market indices serve as a composite measure designed to reflect changes in the value of an entire group of securities traded in a specific market over a period of time. By linking the performance of individual portfolios or holdings to the overall market, investors can obtain a quick, top-down assessment of their investments.
The dictionary definition of an index is "a system of numbers used for comparing values of things that change according to each other or a fixed standard". In securities markets, indices fulfill this exact role by measuring the aggregate change in the value of their underlying constituents from a base period and base value used as a reference point.
8.2 Strategic Uses of Security Market Indices
Originally, the world’s first security market index—the Dow Jones Average—was introduced in 1884 by Charles H. Dow and Edward D. Jones with the single objective of providing a simple, accessible indicator reflecting security market information. However, as financial markets and investment management techniques have evolved, the applications of indices have expanded significantly. Today, security market indices are utilised in four primary ways:
- Providing a Gauge of the Market: Indices act as a barometer for the underlying market segment. They reflect the collective opinion, attitudes, and behaviours of market participants regarding prevailing market dynamics.
- Performance Benchmarking: Because investors find it challenging to evaluate the absolute performance of their portfolios in a vacuum, they use market indices as benchmarks. This allows them to assess how actively managed portfolios and investment managers have performed relative to the market.
- Underlying Portfolios for Passive Investing: In passive investment strategies, indices serve as the "model portfolio" for creating passive investment vehicles, such as index funds and exchange-traded funds (ETFs).
- Proxy for the Market Portfolio in Risk Calculations: As established in Modern Portfolio Theory and Capital Asset Pricing Model (CAPM) frameworks, every security or portfolio must be priced for its systematic risk. Under CAPM, the theoretical "market portfolio" consists of all risky assets in the world. Since constructing such a comprehensive portfolio is practically impossible, broad-based equity market indices are used as the market proxy to calculate asset betas and measure systematic risk.
8.3 Key Factors Differentiating Market Indices
While all indices are designed to reflect the aggregate movements of a group of securities, there are over two million indices available globally (specifically, 2.96 million indices as of a 2019 study by the Index Industry Association). The primary factors that differentiate these indices are their sample size and the scheme of assigning weights to their constituent securities.
1. Sample Size (The Constituent Universe)
The first differentiating factor is the sample size used to construct the index. Although modern supercomputing allows for the inclusion of every stock listed on an exchange, many popular indices continue to rely on a selected sample. Different sample sizes can represent the exact same segment of securities. For example, in the Indian large-cap stock market segment, the BSE SENSEX contains 30 stocks, whereas NIFTY 50 contains 50 stocks. When selecting a sample, index providers must ensure it remains representative of the broader population of securities.
2. Weighting Schemes
The second differentiating factor is the methodology used to allocate weights to each constituent in the sample. The four primary weighting methodologies are:
- Price-Weighting Scheme: Weights are based solely on the market price of the constituent shares.
- Market-Value Weighting Scheme: Weights are based on the total market capitalisation (or free-float market capitalisation) of the companies. This is the most popular method used globally.
- Equal-Weighting Scheme: All constituents carry the same weight, regardless of their stock price or market value.
- Fundamental-Weighting Scheme: Weights are assigned based on fundamental financial variables of the company, such as sales, book value, cash flows, dividends, earnings, profits, or net assets.
8.3.1 Price-Weighted Index Methodology
A price-weighted index is the simplest methodology for assigning weights to constituents. In this type of index, the weight of each security is directly proportional to its current stock price. The Dow Jones Industrial Average is the most prominent global example of a price-weighted index.
Mathematical Formula
The price-weighted index is calculated by summing the closing prices of all constituent securities and dividing the sum by an adjusted divisor:
- Index Value on Day t = (Sum of Closing Prices of all Constituent Stocks on Day t) / Adjusted Divisor on Day t
The Concept of the Index Divisor
The index divisor is a number chosen at the inception of the index. It is initially set so that the index begins with a convenient base value, such as 100 or 1,000.
The index provider must adjust the divisor over time to prevent artificial jumps or drops in the index value due to corporate actions (like stock splits, share issuances, or changes in index constituents) that are unrelated to actual market price movements. For instance, if a constituent stock splits, the divisor is adjusted downward so that the index value immediately after the split remains identical to the value right before the split.
Example 8.1: Divisor Adjustment for a Stock Split
Consider a price-weighted index containing five stocks: A, B, C, D, and E. Assume Stock E undergoes a 2-for-1 stock split, which halves its share price from INR 10 to INR 5.
The table below illustrates how the index divisor must be adjusted to prevent the index value from falling due to the stock split:
| Stock | Share Price Before Split | Share Price After Split (Stock E splits 2-for-1) |
|---|---|---|
| A | INR 4 | INR 4 |
| B | INR 6 | INR 6 |
| C | INR 8 | INR 8 |
| D | INR 9 | INR 9 |
| E | INR 10 | INR 5 |
| Sum of Prices | INR 37 | INR 32 |
| Index Divisor | 3.7 (Inception Divisor) | 3.2 (Adjusted Divisor) |
| Index Value | 10.00 (37 / 3.7) | 10.00 (32 / 3.2) |
Calculation Process:
- Before the split: The sum of prices is INR 37. Dividing by the initial divisor of 3.7 yields an index value of 10.00 (37 / 3.7 = 10).
- After the split: The sum of prices falls to INR 32 due to E's split.
- Divisor Adjustment: To maintain the index value at 10.00, we solve for the new divisor: New Divisor = Sum of New Prices / Pre-Split Index Value. This results in 32 / 10 = 3.2.
- Result: The adjusted divisor of 3.2 ensures that the stock split does not introduce artificial volatility into the index.
Advantages and Limitations of Price-Weighted Indices
Advantages:
- Ease of Calculation: It is mathematically simple, representing the basic arithmetic mean of the constituent stock prices. Before the advent of modern supercomputers, this simplicity was a major advantage.
Limitations:
- High-Price Bias: Because weights are determined by share price, a high-priced stock exerts a disproportionately larger influence on the index than a low-priced stock, regardless of company size. For instance, a 5% price change in an INR 5,000 stock will have a much larger impact on the index value than a 50% price change in an INR 50 stock.
- Downward Bias on High-Growth Companies: Successful, high-growth companies tend to experience rising stock prices and consequently split their shares to remain tradeable for retail investors. Because stock splits automatically lower the share price, these growing companies consistently lose weight within a price-weighted index.
8.3.2 Value-Weighted Index Methodology
A market-value-weighted (or capitalisation-weighted) index allocates weights to constituent securities based on their total market value (market capitalisation). Market capitalisation is determined by multiplying the number of outstanding shares (or free-float shares) by the current market price.
- Market Capitalisation of Stock i = Outstanding Shares * Current Market Price
- Weight of Stock i = Market Capitalisation of Stock i / Sum of Market Capitalisation of All Stocks in the Index
Under this methodology, price changes in larger companies have a more significant impact on the index value than price changes in smaller companies. Furthermore, value-weighted indices automatically adjust for stock splits and other capital changes without requiring divisor adjustments, as a split does not alter a company's total market capitalisation. This makes value-weighted indices the ideal choice for creating index funds and index derivatives.
Example 8.2: Weight and Divisor Calculation at Inception (T0)
Consider a value-weighted index composed of five stocks (A, B, C, D, and E) at inception (T0). If the base value of the index is set at 100, the divisor is calculated by dividing the total market capitalisation by the base value.
| Stock | Current Price | Outstanding Shares | Market Capitalisation | Index Weighting |
|---|---|---|---|---|
| A | INR 3 | 50 | INR 150 | 15.46% (150 / 970) |
| B | INR 1 | 50 | INR 50 | 5.15% (50 / 970) |
| C | INR 7 | 70 | INR 490 | 50.52% (490 / 970) |
| D | INR 9 | 20 | INR 180 | 18.56% (180 / 970) |
| E | INR 10 | 10 | INR 100 | 10.31% (100 / 970) |
| Total | INR 970 | 100.00% |
- Total Market Capitalisation: INR 970
- Index Base Value: 100
- Index Divisor: 970 / 100 = 9.7
Example 8.3: Calculating Index Value and Weights at T1 (After 3 Months)
After three months, the stock prices change, shifting the total market capitalisation and constituent weights:
| Stock | Current Price | Outstanding Shares | Market Capitalisation | Index Weighting |
|---|---|---|---|---|
| A | INR 0.5 | 50 | INR 25 | 2.99% (25 / 835) |
| B | INR 1.0 | 50 | INR 50 | 5.99% (50 / 835) |
| C | INR 7.0 | 70 | INR 490 | 58.68% (490 / 835) |
| D | INR 9.0 | 20 | INR 180 | 21.56% (180 / 835) |
| E | INR 9.0 | 10 | INR 90 | 10.78% (90 / 835) |
| Total | INR 835 | 100.00% |
- New Total Market Capitalisation: INR 835
- Index Value at T1: 835 / 9.7 = 86.08
- Analysis: The index has declined from 100 to 86.08, representing a 13.92% decrease in overall market value.
Example 8.4: Reconstitution and Divisor Adjustment
Suppose the index provider decides to remove Stock A because it no longer meets the index inclusion criteria. In its place, the provider introduces Stock X, which has a current price of INR 6 and 70 outstanding shares, resulting in a market capitalisation of INR 420.
To prevent this change from artificially altering the index value from its current level of 86.08, the index divisor must be recalculated:
| Stock | Current Price | Outstanding Shares | Market Capitalisation | New Index Weighting |
|---|---|---|---|---|
| X | INR 6 | 70 | INR 420 | 34.15% (420 / 1230) |
| B | INR 1 | 50 | INR 50 | 4.07% (50 / 1230) |
| C | INR 7 | 70 | INR 490 | 39.84% (490 / 1230) |
| D | INR 9 | 20 | INR 180 | 14.63% (180 / 1230) |
| E | INR 9 | 10 | INR 90 | 7.32% (90 / 1230) |
| Total | INR 1,230 | 100.00% |
- Post-Reconstitution Total Market Capitalisation: INR 1,230
- Target Index Value: 86.08
- New Index Divisor: 1230 / 86.08 = 14.28
The Concept of Free Float
The free float of a company measures the actual proportion of its outstanding shares that are readily available in the stock market for public investment. Calculating free float is essential for separating strategic, long-term shareholders from liquid public holdings.
Strategic shareholders typically maintain their stakes for corporate control rather than short-term investment returns. Major global and domestic indices, such as the NIFTY 50 and BSE SENSEX, have transitioned from full market capitalisation to free-float market capitalisation methodologies (SENSEX transitioned in September 2003, and NIFTY 50 in June 2009).
Excluded Holdings under Free Float Methodology:
To determine the free-float market capitalisation, the following strategic holdings are excluded from outstanding shares:
- Holdings by founders, promoters, directors, or sponsors.
- Government holdings of a strategic nature.
- Strategic stakes held by private corporate bodies or individuals.
- Cross-holdings (equity held by associate or group companies).
- Equity held by Employee Welfare Trusts.
- Locked-in shares and any shares that cannot be sold in the open market under normal circumstances.
8.3.3 Equal-Weighted Index Methodology
In an equal-weighted index, all constituent securities carry the exact same weight, regardless of their market price or total market capitalisation. Under this methodology, a stock with a share price of INR 2,500 has the same impact on the index as a stock priced at INR 40. Similarly, the largest company by market capitalisation is treated with equal importance to the smallest company in the index.
An equal-weighted index is conceptually equivalent to investing an identical currency amount in each constituent stock.
Example 8.5: Equal-Weighted Index Composition at T0
Suppose an index consists of five stocks: A, B, C, D, and E. At inception (T0), each stock is assigned an equal weight of 20%. The outstanding shares and individual market capitalisations are completely ignored when determining the index's movements.
| Stock | Share Price at T0 | Weight Allocation |
|---|---|---|
| A | INR 6.00 | 20% |
| B | INR 1.00 | 20% |
| C | INR 7.00 | 20% |
| D | INR 9.00 | 20% |
| E | INR 9.00 | 20% |
| Total | 100% |
- Index Base Value at T0: 100
Example 8.6: Calculating Index Value at T1
The following day (T1), the stock prices change, resulting in individual security returns. The return of each constituent is calculated as:
- Constituent Return = (Price on Day t1 - Price on Day t0) / Price on Day t0
The aggregate movement of the equal-weighted index is calculated as the simple arithmetic mean of the percentage changes in the prices of the constituent stocks:
| Stock | Price at T0 | Price at T1 | Stock-Specific Return |
|---|---|---|---|
| A | INR 6.00 | INR 10.00 | 67% (rounded from 66.67%) |
| B | INR 1.00 | INR 1.50 | 50% |
| C | INR 7.00 | INR 8.00 | 14% (rounded from 14.28%) |
| D | INR 9.00 | INR 8.00 | -11% (rounded from -11.11%) |
| E | INR 9.00 | INR 10.00 | 11% (rounded from 11.11%) |
Calculation of Index Return:
- Index Return = (67% + 50% + 14% - 11% + 11%) / 5 = 26.2% (The workbook rounds the average to 26%).
- Index Value at T1: 100 * (1 + 0.26) = 126.
(Note: While this example utilizes an arithmetic average, some equal-weighted indices may use a geometric average calculation instead.)
Advantages and Limitations of Equal-Weighted Indices
Advantages:
- No Price or Value Bias: The index does not suffer from high-price or large-cap concentration biases. Every security has an identical impact on the overall index performance.
Limitations:
- High Transaction and Rebalancing Costs: To maintain equal weights over time, an index fund tracking an equal-weighted index must be rebalanced frequently. Rebalancing requires the fund to sell shares of winning stocks (whose weights have risen above 20%) and buy shares of losing stocks (whose weights have fallen below 20%). This continuous trading activity leads to high portfolio turnover rates and significant transaction costs.
8.3.4 Fundamental-Weighted and Factor-Based Indices
A key criticism of value-weighted indices is that they mathematically overweight overvalued stocks and underweight undervalued stocks. Because weights are driven entirely by market capitalisation, any stock whose price is driven up by market speculation automatically receives a higher weight in a value-weighted index.
To resolve this structural issue, index providers developed fundamental-weighted indices (often referred to as alternative indices or smart-beta indices). These indices weight constituent securities based on fundamental company metrics rather than market prices.
Common Fundamental Weighting Metrics:
- Sales / Revenue
- Book Value
- Cash Flows
- Dividends
- Operating Profits / Net Income
- Net Assets
A fundamental index can be constructed using a single fundamental metric or a combination of multiple weighted factors.
Example: Fundamental Index Weighting and Return Calculation
Suppose a fundamental index contains three companies (A, B, and C) and weights them based on their sales figures from the previous year. The index has a starting base value of 100 at T0:
Table 8.1: Calculating Fundamental Weights at T0
| Company | Previous Year Sales | Fundamental Index Weight | Stock Price at T0 |
|---|---|---|---|
| A | INR 100 Crore | 10% (100 / 1000) | INR 100 |
| B | INR 500 Crore | 50% (500 / 1000) | INR 1,200 |
| C | INR 400 Crore | 40% (400 / 1000) | INR 850 |
| Total | INR 1,000 Crore | 100% |
Table 8.2: Calculating Index Price Movement at T1
After one period, the prices of the three stocks move as follows:
| Company | Stock Price at T0 | Stock Price at T1 | Stock-Specific Return |
|---|---|---|---|
| A | INR 100 | INR 170 | +70.00% |
| B | INR 1,200 | INR 1,500 | +25.00% |
| C | INR 850 | INR 750 | -11.76% |
Calculation of Fundamental Index Return:
- Index Return = (Weight of A * Return of A) + (Weight of B * Return of B) + (Weight of C * Return of C)
- Index Return = (10% * 70%) + (50% * 25%) + (40% * -11.76%)
- Index Return = 7.00% + 12.50% - 4.70% = 14.80%
- Index Value at T1: 100 * (1 + 0.1480) = 114.80
This methodology ensures that the performance of the index is driven by the economic size of the companies (sales) rather than their stock market valuations.
Summary of Index Weighting Methodologies
| Weighting Scheme | Weighting Driver | Major Global Example | Key Advantages | Major Limitations / Biases |
|---|---|---|---|---|
| Price-Weighted | Share Price | Dow Jones Industrial Average | Extremely simple to compute | High-price bias; downward bias on growing companies that split stock |
| Value-Weighted | Market Capitalisation | NIFTY 50, SENSEX, S&P 500 | Self-adjusting for stock splits; ideal for index replication and derivatives | Overweights overvalued companies; underrepresented small-cap segment |
| Equal-Weighted | None (Equal split) | Eurekahedge India Long Short Index | Eliminates price and market-cap biases; higher diversification | High portfolio turnover and transaction costs due to frequent rebalancing |
| Fundamental | Fundamental metrics (Sales, Book Value, etc.) | FTSE RAFI Index Series | Breaks link between market price and stock weight; avoids bubble inflation | Higher data tracking needs; subjective selection of fundamental metrics |
Important Terms & Definitions
- Market Index: A composite measure that reflects changes in the value of an underlying group of securities over time relative to a base standard.
- Index Divisor: A mathematical constant established at an index's inception and adjusted over time to absorb the impact of corporate actions (like stock splits or constituent changes) without changing the index's value.
- Stock Split: A corporate action in which a company increases its outstanding shares while proportionally reducing its share price.
- Free Float: The portion of a company's outstanding shares that are actively traded and available for purchase by public investors, excluding promoter, government, and other strategic holdings.
- smart-beta / Factor-Based Investing: An investment strategy that weights index constituents using factors other than market capitalisation, such as value, growth, momentum, or fundamental size.
Candidates’ Exam-Relevant Takeaways
- 👉 Charles H. Dow and Edward D. Jones introduced the world's first security market index in 1884.
- 👉 Under a price-weighted index, a stock split remains the index value constant, but the divisor will change.
- 👉 Under a value-weighted index, stock splits are automatically adjusted without any changes required in the divisor because total market capitalisation is unaffected by a stock split.
- 👉 A value-weighted index is criticised for having a disproportionate influence from large companies and for overweighting overvalued companies.
- 👉 Free float excludes shares held by Employee Welfare Trusts, strategic promoter holdings, cross-holdings in group companies, and locked-in shares.
- 👉 Equal-weighted index tracking funds experience high portfolio turnover and high transaction costs due to the ongoing need to sell appreciation assets and buy depreciating assets to maintain equal weights.