📖 指数周测（10 题）

Equity Investments

I. Lesson Focus

II. The Problem

A portfolio manager runs an enhanced index fund benchmarked to the CSI 800. In 2023, dramatic style rotation caused the fund’s return to deviate noticeably from the benchmark. The client asks, “Why did the ‘index-tracking’ fund underperform?” This lesson uses 10 high-quality questions to test mastery of price-weighted, market-cap-weighted, equal-weighted, and fundamental-weighted index calculations, rebalancing frequency, sources of tracking error, replication methods (full, stratified sampling, optimization), and common pitfalls. The goal is to build the ability to identify key information quickly in exam scenarios and avoid calculation traps.

III. Classification and Weighting Methods of Equity Indexes

Equity indexes are primarily classified by construction methodology into price-weighted, market-capitalization-weighted (value-weighted), equal-weighted, and fundamental-weighted indexes.

• Price-weighted indexes: Index level = (∑ stock prices) / divisor. Classic example: Dow Jones Industrial Average (DJIA). High-priced stocks receive higher weights; stock splits require divisor adjustment.
• Market-cap-weighted indexes: Weight = (individual free-float market cap / total index market cap). Most common (CSI 300, CSI 800, S&P 500). Automatically “buy high, sell higher,” but creates concentration risk during growth-stock bubbles.
• Equal-weighted indexes: Each constituent has identical weight and must be rebalanced periodically. Advantages include natural small-cap tilt; disadvantages are high turnover and transaction costs.
• Fundamental-weighted indexes: Weights are based on fundamentals such as sales, cash flow, dividends, or book value, avoiding the valuation bias of cap weighting.

Key Formulas: - Price-weighted index = $\frac{\sum P_i}{D}$ - Market-cap-weighted return ≈ $\sum w_i \times R_i$, where $w_i = \frac{P_i \times Q_i}{\sum (P_j \times Q_j)}$

IV. Index Replication Strategies and Tracking Error

Three main replication approaches exist:

1. Full replication: Hold all constituents at target weights. Lowest tracking error but expensive for indexes with many illiquid small-cap names.
2. Sampling replication (stratified or optimized sampling): Select a representative subset. Preferred for indexes with >1,000 constituents.
3. Optimized/synthetic replication: Use index futures, swaps, or other derivatives. Lowest cost but introduces counterparty risk and roll risk.

Tracking error (TE) is the standard deviation of the difference between portfolio and benchmark returns: $$ TE = \sqrt{\frac{1}{T-1}\sum_{t=1}^{T}(R_{p,t} - R_{b,t})^2} $$ Major sources of tracking error: constituent deviation, rebalancing frequency, cash drag, dividend treatment, and expense ratio.

V. Rebalancing and Index Maintenance

• Market-cap-weighted indexes generally use passive rebalancing, adjusting only when constituents change.
• Equal-weighted indexes require regular rebalancing (quarterly or semi-annually); otherwise weights drift with price changes.
• Higher rebalancing frequency reduces tracking error but increases transaction costs; an optimal trade-off exists.

Worked Cases

Case 1: Price-Weighted Index and Divisor Adjustment

Three stocks have initial prices of ¥40, ¥80, and ¥120. The initial divisor is 3, so the index level is 80. Stock B undergoes a 1:2 split, reducing its price to ¥40. Calculate the new divisor and the index level immediately after the split.

Solution: Pre-split total price = 40 + 80 + 120 = 240, divisor = 3, index = 80.
Post-split prices = 40 + 40 + 120 = 200.
To keep the index continuous, new divisor $D_{new} = 200 / 80 = 2.5$.
If next-day prices are ¥42, ¥41, and ¥118, the new index level = $(42+41+118)/2.5 = 80.4$.

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

An index contains stocks A, B, and C with the following data:

Total initial market value = ¥4,600 (10k).

Calculate both the market-cap-weighted and quarterly-rebalanced equal-weighted index returns.

Solution: Market-cap weights: A = 21.74%, B = 34.78%, C = 43.48%.
Market-cap-weighted return = 0.2174×20% + 0.3478×(-10%) + 0.4348×20% = 8.70%.

Equal weights: 1/3 each.
Equal-weighted return = (20% – 10% + 20%)/3 = 10.00%.

Equal weighting outperforms during this style-rotation period due to its contrarian rebalancing effect.

Case 3: Tracking-Error Calculation

An index fund produces excess returns over four quarters of 0.8%, –0.4%, 1.2%, –0.9%, and 0.3%. Calculate the monthly tracking error and its annualized value.

Solution: Mean excess return = (0.8 – 0.4 + 1.2 – 0.9 + 0.3)/5 = 0.2%.
Variance = $\frac{1}{4}[(0.6)^2 + (-0.6)^2 + (1.0)^2 + (-1.1)^2 + (0.1)^2]$ = 0.725.
Monthly TE = $\sqrt{0.725} \approx 0.851\%$.
Annualized TE ≈ 0.851% × $\sqrt{12}$ ≈ 2.95%.

Traps

Key Formulas

• Price-weighted index = $\frac{\sum P_i}{D}$
• Market-cap weight $w_i = \frac{P_i Q_i}{\sum P_j Q_j}$
• Tracking error $TE = \sqrt{\frac{\sum (R_p - R_b)^2}{T-1}}$
• Annualized TE ≈ monthly TE × $\sqrt{12}$
• After rebalancing, equal-weighted index restores all weights to $1/N$
• Fundamental weights proportional to fundamentals, independent of market cap
• Index total return = price return + reinvested dividend return

Practice Questions

Q1. Which index construction method naturally embeds a “sell winners, buy losers” rebalancing mechanism?
A. Price-weighted B. Market-cap-weighted C. Equal-weighted D. Fundamental-weighted

Q2. A price-weighted index contains three stocks priced at 50, 80, and 120 with a divisor of 2.5. If the second stock undergoes a 1:4 split, the new divisor is closest to:
A. 0.625 B. 1.25 C. 2.0 D. 2.5

Q3. Which statement about tracking error is correct?
A. Full replication always produces zero tracking error
B. Tracking error is affected only by management fees
C. Optimized replication usually has lower tracking error than stratified sampling
D. Cash drag is an important source of tracking error

Q4. Compared with a market-cap-weighted index, an equal-weighted index typically exhibits:
A. Higher concentration risk B. Lower turnover C. Greater exposure to small-cap stocks D. Lower rebalancing costs

Q5. An index fund produces quarterly excess returns of 1.2%, –0.8%, 2.1%, and –1.5%. Its quarterly tracking error is closest to:
A. 1.48% B. 1.65% C. 2.19% D. 3.12%

Q6. When constructing a broad-based emerging-market equity index, the most commonly used replication strategy is:
A. Full replication B. Optimized replication C. Stratified sampling D. Synthetic replication

Q7. Which of the following is not an advantage of fundamental-weighted indexes over market-cap-weighted indexes?
A. Avoidance of growth-stock valuation bubbles B. Natural value tilt C. Lower turnover D. Improved diversification

Q8. An index has total market capitalization of ¥5,000 billion. One constituent rises 15% while all other stocks and share counts remain unchanged. The change in that stock’s weight is best described as:
A. An increase of exactly 15% B. An increase of 0.15 percentage points C. Dependent on its initial weight D. Unchanged

Answers

Takeaways

• Price-weighted indexes are sensitive to high-priced stocks and require continuous divisor adjustment.
• Market-cap weighting automatically tilts toward outperforming stocks and can create concentration risk.
• Equal-weighted indexes need regular rebalancing and exhibit small-cap/value rebalancing effects.
• Tracking error is the standard deviation of excess returns; cash drag, fees, and rebalancing are primary drivers.
• The larger the number of constituents, the more likely sampling or optimized replication is used.
• “Passive” indexing does not mean zero trading—corporate actions and constituent changes still generate turnover.