📖 指数构建方法：价格加权

Equity Investments

I. Lesson Focus

This lesson explains the construction, calculation, maintenance, and biases of price-weighted indexes. Candidates must master the divisor adjustment mechanism for stock splits, index reconstitutions, and be able to compute index values and returns under various scenarios. The material directly supports later readings on equity index types and benchmark selection.

II. The Problem

An investor wants to track the performance of a portfolio of three blue-chip stocks trading at CNY 200, CNY 150, and CNY 50. Simply averaging the prices gives equal importance to a CNY 1 move in any stock regardless of economic size. This creates distortions: high-priced stocks dominate the index, and corporate actions such as splits distort the level unless the divisor is adjusted. CFA exams frequently test the correct divisor adjustment, the resulting weights, and systematic performance differences versus market-cap-weighted or equal-weighted indexes.

III. Price-Weighted Index – Concept and Construction

A price-weighted index sums the prices of constituent stocks and divides by a divisor. Each stock’s influence is proportional to its absolute share price, independent of shares outstanding or market capitalization.

Construction steps: 1. Select constituents and record initial prices. 2. Sum the initial prices. 3. Choose an initial divisor (commonly equal to the number of stocks so the index equals the arithmetic mean price). 4. Index level = Sum of prices / Divisor.

Formula: $$ PWI_t = \frac{\sum_{i=1}^{n} P_{i,t}}{D_t} $$

where $P_{i,t}$ is the price of security i at time t and $D_t$ is the divisor at time t.

The Dow Jones Industrial Average (DJIA) is the most famous price-weighted index.

IV. Divisor Adjustment Mechanism

The divisor must be changed whenever an event alters the sum of prices without reflecting genuine economic performance. Primary events are stock splits, stock dividends, and index reconstitutions.

Adjustment formula: $$ D_{new} = D_{old} \times \frac{\sum P_{new}}{\sum P_{old}} $$

or equivalently, $$ D_{new} = \frac{\text{New sum of prices}}{\text{Target index level}} $$

After adjustment the index level remains unchanged at the instant of the corporate action.

V. Return Calculation and Weighting

The return on a price-weighted index equals the arithmetic average of the percentage price changes of its constituents, weighted by their prices.

Weight of stock i: $$ w_i = \frac{P_i}{\sum P_j} $$

Consequently, a 1% move in a high-priced stock has a larger impact on the index than the same percentage move in a low-priced stock. This creates a systematic “price bias.”

VI. Comparison with Market-Cap-Weighted and Equal-Weighted Indexes

• Price-weighted: High-priced stocks dominate; splits automatically reduce a stock’s influence unless the divisor is adjusted.
• Market-cap-weighted: Weights proportional to total market value; reflects economic importance.
• Equal-weighted: Each stock has identical weight; requires periodic rebalancing.

Price-weighted indexes tend to overweight high-priced (often mature) companies and underweight lower-priced (often growth) companies irrespective of their true economic size.

Worked Cases

Case 1: Basic Construction

Three stocks have initial prices of 100, 60, and 40. The initial divisor is set at 3, giving an index level of (100 + 60 + 40) / 3 = 66.67.

At the end of period 1 the prices are 110, 55, and 48.
New index = (110 + 55 + 48) / 3 = 71.
Index return = (71 – 66.67) / 66.67 ≈ 6.49%.

Case 2: Stock Split and Divisor Adjustment

Continuing from Case 1, stock A executes a 2-for-1 split; its price falls to 55 while B and C remain at 55 and 48. The new price sum is 158.

To keep the index at 71, the new divisor must be 158 / 71 ≈ 2.225.
Post-adjustment index = 158 / 2.225 = 71 (continuous).

Case 3: Index Reconstitution

Current index: prices 120, 80, 30; divisor = 2.3; index level = 100.
Stock C (30) is replaced by stock D priced at 90. New price sum = 120 + 80 + 90 = 290.
To keep the index at 100, the new divisor = 290 / 100 = 2.9.
Thereafter the index is calculated as sum of prices / 2.9.

Traps

Key Formulas

• Price-weighted index: $ PWI_t = \frac{\sum P_{i,t}}{D_t} $
• Divisor adjustment: $ D_{new} = D_{old} \times \frac{\sum P_{new}}{\sum P_{old}} $
• Stock weight: $ w_i = \frac{P_i}{\sum P_j} $
• Index return: $\frac{PWI_t - PWI_{t-1}}{PWI_{t-1}}$
• Price bias: higher-priced stocks receive disproportionately large weights

Practice Questions

Q1. A price-weighted index consists of three stocks priced at 40, 60, and 100 with a divisor of 2. The current index level is closest to:
A. 66.67
B. 100
C. 200
D. 50

Q2. In a price-weighted index, the stock with the largest weight is the one with:
A. The largest market capitalization
B. The highest share price
C. The largest number of shares outstanding
D. The highest beta

Q3. If a constituent stock undergoes a 2-for-1 split and the divisor is not adjusted, the price-weighted index will:
A. Increase
B. Decrease
C. Remain unchanged
D. Depend on the prices of the other stocks

Q4. The primary bias of a price-weighted index is:
A. Size bias
B. Price bias
C. Liquidity bias
D. Sector bias

Q5. Which index construction method requires divisor adjustment when a stock split occurs?
A. Market-cap-weighted
B. Equal-weighted
C. Price-weighted
D. Fundamental-weighted

Q6. A price-weighted index has a divisor of 3.2 and constituent prices of 50, 80, and 110. If the 110 stock rises to 121 while the others are unchanged, the index increases by approximately:
A. 3.125%
B. 10%
C. 6.25%
D. 9.09%

Q7. When a price-weighted index replaces one constituent, the divisor is adjusted primarily to:
A. Increase the index return
B. Maintain index continuity at the moment of replacement
C. Force the new stock to have exactly one-third weight
D. Reflect changes in company fundamentals

Q8. Compared with a market-cap-weighted index, a price-weighted index typically:
A. Better represents the overall economy
B. Overweights high-priced stocks
C. Automatically rebalances
D. Requires no ongoing maintenance

Answers

Takeaways

• A price-weighted index equals the sum of share prices divided by a divisor; higher-priced stocks receive larger weights.
• Stock splits and index reconstitutions require divisor adjustment to keep the index continuous.
• The index exhibits clear price bias, overweighting high-priced stocks and underweighting lower-priced stocks.
• Its return equals the price-weighted arithmetic average return of constituents.
• The DJIA is the classic example of a price-weighted index.
• Compared with market-cap weighting, price weighting is insensitive to company size measured by total capitalization.