📖 指数综合练习

Equity Investments

I. Lesson Focus

This lesson provides comprehensive practice on equity index construction, weighting schemes, return calculations, rebalancing mechanics, and tracking error analysis. Candidates must master the differences between price-weighted, market-capitalization-weighted, and equal-weighted indices, be able to adjust divisors, compute price and total returns, and evaluate the practical implications for index funds.

II. The Problem

A portfolio manager oversees three index funds tracking the SSE 180 (price-weighted), CSI 300 (float-adjusted market-cap weighted), and a custom equal-weighted index. In a volatile 2023 market with frequent dividends, splits, and seasoned equity offerings, the three indices produced materially different performance. The manager must accurately calculate true index returns, understand rebalancing costs, isolate sources of tracking error, and decide which index methodology best suits institutional investors under current conditions. This lesson systematically addresses how to calculate, compare, and apply different equity index methodologies while avoiding common pitfalls.

III. Classification and Construction of Equity Indices

Equity indices are primarily constructed using three weighting methods: price-weighted, market-capitalization-weighted (float-adjusted), and equal-weighted.

Price-weighted indices calculate the index level as the sum of component share prices divided by a divisor. The Dow Jones Industrial Average is the classic example. When a stock split or stock dividend occurs, the divisor must be adjusted to maintain continuity using the formula:
New Divisor = Old Divisor × (Post-adjustment total price / Pre-adjustment total price).

Market-capitalization-weighted indices (most common) use:
Index Level = Σ (Price_i × Shares Outstanding_i × Float Factor) / Divisor.
The weight of each security equals its free-float market capitalization divided by the total free-float market capitalization of the index. These indices automatically become more concentrated in outperforming large-cap stocks and require no routine rebalancing except at reconstitution.

Equal-weighted indices assign an initial weight of 1/N to each constituent. They require periodic rebalancing (typically quarterly), selling recent outperformers and buying recent underperformers. This creates a natural contrarian bias and exposure to smaller stocks.

IV. Index Return Calculations

1. Price Return (does not include dividends):
   $$ R_P = \frac{P_1 - P_0}{P_0} $$ where $P_0$ and $P_1$ are beginning and ending index levels.

2. Total Return (includes reinvested dividends):
   $$ R_{TR} = \frac{P_1 - P_0 + D}{P_0} $$ where $D$ is the total dividend amount from all constituents, weighted by their index weights.

3. Multi-period (chain-linked) return:
   $$ (1 + R_{0,T}) = (1 + R_{0,1}) \times (1 + R_{1,2}) \times \cdots \times (1 + R_{T-1,T}) $$

V. Rebalancing and Reconstitution

• Price-weighted indices adjust only the divisor for splits; no cash rebalancing is needed.
• Market-cap-weighted indices rebalance passively as prices change; active trading occurs mainly during index reconstitution or float-factor updates.
• Equal-weighted indices must be rebalanced regularly, generating higher turnover and transaction costs but potentially capturing size and value premia over time.

Approximate rebalancing cost:
$$ \text{Rebalance Cost} \approx \frac{1}{2} \times \text{Turnover} \times \text{Bid-Ask Spread} $$

VI. Tracking Error

Tracking error is the standard deviation of the difference between the fund’s return and the benchmark index return:
$$ TE = \sqrt{\frac{\sum (R_{fund,t} - R_{index,t})^2}{N-1}} $$ Primary sources include cash drag, rebalancing lag, sampling error (when full replication is impractical), transaction costs, and differences in dividend treatment.

Worked Cases

Case 1: Price-Weighted Index — Divisor Adjustment and Return

Three stocks form a price-weighted index with initial prices A: 40, B: 20, C: 60. Initial divisor = 3, index = 40.
Next day prices: A = 44, B undergoes 1:2 split (post-split quoted price = 10, but economic price for divisor = 20), C = 66.
Post-adjustment total price = 44 + 20 + 66 = 130.
New divisor = 3 × (130 / 120) = 3.25.
New index level = 130 / 3.25 = 40.
Price return for the day = (40 – 40) / 40 = 0%.

Case 2: Market-Cap-Weighted Total Return

Two-stock index:
- Stock X: beginning price 100, 10 million shares, ending price 110, dividend 2 per share.
- Stock Y: beginning price 50, 20 million shares, ending price 48, no dividend.
Beginning total market cap = (100 × 10M) + (50 × 20M) = 2 billion.
Ending price market cap = (110 × 10M) + (48 × 20M) = 2.06 billion.
Total dividends = 2 × 10M = 20 million.
Price return = (2.06 – 2.0) / 2.0 = 3%.
Total return = (2.06 – 2.0 + 0.2) / 2.0 = 4%.

Case 3: Equal-Weighted Index Rebalancing and Tracking Error

A 5-stock equal-weighted index starts with 20% weights each. After one quarter weights drift to 28%, 22%, 18%, 17%, 15%. The manager rebalances back to 20% at quarter-end.
Index quarterly return = 6.8%. The fund earns 6.1% after transaction costs and cash drag. Quarterly return differences over four quarters: +0.7%, –0.4%, +0.9%, +0.5%.
Quarterly tracking error ≈ √[(0.7² – 0.4² + 0.9² + 0.5²) / 3] ≈ 0.78%. Annualized tracking error ≈ 0.78% × √4 ≈ 1.56%.

Traps

Key Formulas

• Price-weighted index = Σ Prices / Divisor
• Market-cap weight = (Price_i × Shares_i × Float Factor) / Total Index Free-Float Market Cap
• Price return: $R_P = (P_1 - P_0)/P_0$
• Total return: $R_{TR} = (P_1 - P_0 + D)/P_0$
• Tracking error: $TE = \sqrt{\frac{\sum(R_{fund}-R_{index})^2}{N-1}}$
• New divisor = Old divisor × (Adjusted price sum / Original price sum)
• Equal-weighted rebalancing: Reset every constituent to exactly 1/N at each rebalance date

Practice Questions

Q1. A price-weighted index consists of three stocks with initial prices 50, 80, and 30. The divisor is 3 and the index level is 53.33. Stock 2 then executes a 1-for-4 split. On the split day the other two stocks trade at 55 and 32, and the post-split price of Stock 2 is 22. The adjusted divisor is closest to:
A. 2.85
B. 3.00
C. 1.85
D. 2.35

Q2. Which statement is correct regarding market-capitalization-weighted indices?
A. They must be actively rebalanced every quarter.
B. They naturally tend to buy stocks that have performed poorly in the past.
C. Weights of large-cap stocks automatically decline over time.
D. Their main advantage is effective control of single-stock risk.

Q3. Compared with market-cap-weighted indices, equal-weighted indices typically exhibit:
A. Lower turnover and transaction costs.
B. Higher tracking error and rebalancing costs.
C. Greater bias toward large-cap stocks.
D. Lower volatility.

Q4. An index fund tracks a total-return index. The index price return is 4.2%, the weighted-average dividend yield of constituents is 1.8%, and the fund suffers 0.3% cash drag. The fund’s most likely realized total return is closest to:
A. 5.7%
B. 4.2%
C. 5.4%
D. 6.0%

Q5. When calculating the return on a price-weighted index without dividends, the return equals:
A. The arithmetic average return of the constituents.
B. The price-weighted average return of the constituents.
C. The market-cap-weighted average return of the constituents.
D. None of the above.

Q6. An equal-weighted index contains 40 stocks and is rebalanced quarterly. Assuming no transaction costs, after each rebalance the weight of each stock should be:
A. Reallocated according to latest market caps.
B. Fixed at 2.5%.
C. Weighted according to prior-period returns.
D. Weighted by free-float market capitalization.

Q7. Tracking error is most likely caused by:
A. The index being price-weighted while the fund uses market-cap weights.
B. The fund fully replicating all constituents and rebalancing daily.
C. The fund holding a small cash balance for subscriptions and redemptions.
D. All index constituents paying no dividends.

Q8. An index begins the period at 1,200 and ends at 1,296. The weighted dividends paid during the period equal 36 index points. The index’s total return is closest to:
A. 8.0%
B. 11.0%
C. 10.0%
D. 5.5%

Answers

Takeaways

• Price-weighted indices revolve around divisor adjustment; stock splits require divisor changes to preserve continuity.
• Market-cap-weighted indices automatically “let winners run” and need no routine rebalancing, but concentration risk can become significant.
• Equal-weighted indices embed a contrarian rebalancing process, higher turnover, and potential exposure to size and value factors.
• Total return indices must incorporate reinvested dividends; price-return indices reflect only level changes.
• Tracking error is the standard deviation of active returns; cash drag and rebalancing lags are frequent culprits.
• The divisor-adjustment formula New Divisor = Old Divisor × (Post-sum / Pre-sum) must be applied mechanically in any price-weighted calculation.