📖 股票指数导论

Equity Investments

I. Lesson Focus

This lesson defines equity indices, explains their construction methods (price-weighted, market-cap weighted, and equal-weighted), distinguishes between price return and total return indices, and demonstrates how to calculate index returns under each weighting scheme. Candidates must be able to compute index values and periodic returns, understand divisor adjustments, and recognize the investment implications of different index methodologies.

II. The Problem

Suppose you manage a passive index fund that must closely track a well-known benchmark such as the CSI 300 or the S&P 500. Different index weighting methodologies applied to the exact same group of stocks can produce materially different index returns on the same day. How do you determine whether an index is fairly valued? How do you calculate its true economic return to investors? How do you explain performance differences across indices to clients? This lesson systematically solves these core questions of index construction, calculation, and practical application.

III. Basic Concepts of Equity Indices

An equity index is a single numerical statistic that aggregates the price performance of a basket of stocks. It serves as a market benchmark, a reference for passive investing, performance evaluation, and derivative pricing.

Two core elements define every index: - Constituents: The specific stocks included, selected by index providers (S&P Dow Jones, MSCI, SSE, SZSE) according to rules on market capitalization, liquidity, and sector representation. - Weighting Scheme: How much influence each stock exerts on the index. This is the single most important determinant of index behavior.

IV. The Three Primary Weighting Methods

1. Price-Weighted Indices

Index level = (Sum of constituent share prices) / Divisor
Weights are proportional to the absolute share price; higher-priced stocks have greater influence regardless of company size.
Classic example: Dow Jones Industrial Average (DJIA).
Advantage: simple arithmetic.
Disadvantage: economically irrational (price level ≠ economic importance); stock splits require divisor adjustments to maintain continuity.

Price-weighted return: $$ R_{PW} = \frac{\sum P_t - \sum P_{t-1}}{\sum P_{t-1}} $$ (after divisor adjustment).

2. Market-Capitalization Weighted Indices

Also called value-weighted.
Index level = (Current total market cap / Base period total market cap) × Base index value.
Weight of stock i = (Price_i × Shares outstanding_i) / Total index market cap.

Classic examples: S&P 500, CSI 300, MSCI Emerging Markets.
Advantage: automatically reflects economic size; larger companies have greater impact.
Disadvantage: can create “growth stock bubbles” as the largest firms dominate.

3. Equal-Weighted Indices

Each stock is assigned an identical weight of 1/N (N = number of constituents) and the index is rebalanced periodically.
Advantage: prevents mega-cap dominance and gives small stocks equal voice.
Disadvantage: high turnover and transaction costs; increased liquidity risk in smaller names.

Comparison of Weighting Schemes

V. Price Return vs. Total Return Indices

• Price Return Index: Reflects only changes in constituent share prices; dividends are ignored. Many headline indices (early versions of the Shanghai Composite) are price-return only.
• Total Return Index: Assumes dividends are reinvested back into the index, measuring the actual return an investor would earn. $$ \text{Total Return Index}_t = \text{Price Return Index}_t \times \left(1 + \frac{\text{Dividends}}{\text{Price}}\right) $$

CFA exams frequently test the distinction: when dividend yields are high, total return indices significantly outperform price return indices over time.

VI. Index Maintenance and Adjustments

Index providers regularly perform: - Constituent additions and deletions - Free-float adjustment: only publicly tradable shares are included in capitalization - Divisor adjustment: preserves index continuity after splits, dividends, or constituent changes - Rebalancing: especially critical for equal-weighted indices to restore 1/N weights

Worked Cases

Case 1: Price-Weighted Index Calculation

Three stocks: A ($40), B ($20), C ($10). Initial divisor = 3, index = (40+20+10)/3 = 23.33.
Next day prices: A $44, B $19, C $12. Calculate the new index level and return.

Solution:
New price sum = 44 + 19 + 12 = 75
New index = 75 / 3 = 25
Return = (25 − 23.33) / 23.33 ≈ 7.16%

Case 2: Market-Cap Weighted vs. Equal-Weighted Comparison

Stock data (market caps in millions):

Next day prices: A $55, B $27, C $25. Compute both index returns.

Market-cap weighted return:
New total cap = 550 + 216 + 150 = 916
Return = (916 − 860) / 860 ≈ 6.51%

Equal-weighted return:
A return = (55−50)/50 = 10%
B return = (27−30)/30 = −10%
C return = (25−20)/20 = 25%
Equal-weighted index return = (10% − 10% + 25%) / 3 = 8.33%

The equal-weighted index outperformed because the smallest stock (C) had the largest price gain.

Case 3: Price Return vs. Total Return

An index rises from 1,000 to 1,050 points while dividends contribute the equivalent of 1.8% of the index level. Calculate both returns.

Price return = (1,050 − 1,000) / 1,000 = 5.0%
Total return = 5.0% + 1.8% = 6.8% (assuming frictionless reinvestment)

Traps

Key Formulas

• Price-weighted index = $\frac{\sum P_i}{D}$ (D = divisor)
• Market-cap weight = $\frac{P_i \times Q_i}{\sum (P_j \times Q_j)}$
• Index price return = $\frac{I_t - I_{t-1}}{I_{t-1}}$
• Total return index ≈ Price index × (1 + dividend yield)
• Equal-weighted single-period return = $\frac{1}{N} \sum r_i$
• Divisor adjustment: $D_{new} = D_{old} \times \frac{\text{New price sum}}{\text{Old index value}}$

Practice Questions

Q1. Which index type is most likely to be disproportionately influenced by a high-priced but relatively small-capitalization stock?
A. Market-cap weighted
B. Price-weighted
C. Equal-weighted
D. Free-float adjusted

Q2. A price-weighted index contains three stocks priced at $60, $30, and $10 with a divisor of 2 (index = 50). If the first stock undergoes a 1-for-2 split, what should the new divisor be to keep the index continuous?
A. 1.0
B. 1.5
C. 2.0
D. 3.0

Q3. Compared with a price return index, a total return index normally:
A. Exhibits lower volatility
B. Delivers superior long-term performance
C. Excludes dividends
D. Is simpler to calculate

Q4. Which statement about equal-weighted indices is correct?
A. Large-cap stocks always receive higher weights
B. They require frequent rebalancing
C. Transaction costs are usually lower than cap-weighted indices
D. They naturally avoid value factor exposure

Q5. An index closed yesterday at 1,200 and today at 1,260. Dividends during the period contributed 1.5%. Today’s total return is closest to:
A. 5.0%
B. 6.5%
C. 6.0%
D. 4.5%

Q6. The main purpose of applying a free-float adjustment when constructing an index is to:
A. Increase index volatility
B. Reflect only shares that are actually available to trade
C. Increase the weight of high-priced stocks
D. Reduce rebalancing frequency

Q7. If the three largest stocks in a market-cap weighted index collectively exceed 40% weight, this most likely indicates:
A. The index uses equal weighting
B. High market concentration
C. The index is price-weighted
D. The index has not been rebalanced

Q8. Which of the following is NOT a common index maintenance adjustment?
A. Periodic constituent review and replacement
B. Free-float factor updates
C. Daily forced rebalancing of all weights
D. Divisor adjustment to maintain continuity

Answers

Takeaways

• Price-weighted indices weight by absolute share price, market-cap indices by total capitalization, and equal-weighted by 1/N with periodic rebalancing.
• The DJIA is the classic price-weighted index; the S&P 500 and CSI 300 are classic market-cap weighted.
• Total return indices include reinvested dividends and outperform price return indices over long horizons when dividends are material.
• Index maintenance relies on divisor adjustments and free-float factors to keep the index continuous and representative of actually tradable shares.
• Equal-weighted indices offer greater small-cap exposure but incur higher turnover and transaction costs.
• CFA Level I frequently tests calculation differences among the three weighting schemes, the impact of splits on divisors, and the distinction between price and total return.