📖 指数构建方法：等权重

Equity Investments

I. Lesson Focus

This lesson explains the construction, return calculation, rebalancing mechanics, advantages, and disadvantages of equal-weighted indexes. Candidates must be able to compute equal-weighted index returns, understand the impact of periodic rebalancing, and compare equal-weighted indexes with price-weighted and market-capitalization-weighted indexes.

II. The Problem

Suppose you are managing a fund that tracks the CSI 800. If an equal-weighted methodology is used, each constituent starts with an identical weight of 1/N. When one stock rallies sharply and its weight drifts away from 1/N, must the index be rebalanced quarterly? How much turnover and transaction cost does rebalancing create? In a prolonged bull market, will an equal-weighted index systematically underperform its market-cap-weighted counterpart? These are the practical and exam-critical questions this lesson resolves.

III. Core Logic of Index Construction and the Three Primary Methods

Index construction centers on assigning weights to constituents. The three dominant approaches are price-weighted, market-capitalization-weighted, and equal-weighted.

An equal-weighted index assigns exactly the same weight to every stock: $w_i = \frac{1}{N}$, where $N$ is the number of constituents. The method ignores share price and company size, aiming for pure “one stock, one vote.”

IV. Steps to Construct an Equal-Weighted Index

1. Select the constituent universe (e.g., from the CSI 500).
2. Assign each stock an initial target weight $w_i = \frac{1}{N}$.
3. Determine the initial divisor so the index equals a convenient starting value (commonly 1,000).
4. Calculate the index level each day:
   $$Index_t = \frac{\sum_{i=1}^N (P_{i,t} \times \frac{1}{N})}{Divisor}$$
5. Rebalance periodically (typically quarterly) to reset every weight back to $\frac{1}{N}$.

V. Calculating Returns on an Equal-Weighted Index

The single-period return of an equal-weighted index equals the arithmetic average of constituent returns:
$$R_{EW,t} = \frac{1}{N}\sum_{i=1}^N R_{i,t}$$

This arithmetic-mean construction naturally produces a small-cap bias and a value bias relative to market-cap-weighted benchmarks.

VI. Rebalancing Mechanics and Turnover Costs

Because prices change at different rates, weights drift. Large gainers automatically increase in weight; laggards shrink. Quarterly rebalancing is therefore required: sell winners and buy losers.

This “sell high, buy low” discipline creates a contrarian (reversal) effect. The index tends to perform well in range-bound or volatile markets but systematically lags capitalization-weighted indexes during strong, uninterrupted bull markets.

VII. Advantages and Disadvantages of Equal-Weighted Indexes

Advantages
- Prevents “winner-take-all” concentration found in cap-weighted indexes.
- Provides natural exposure to smaller and value-oriented stocks.
- Rebalancing can add a positive return contribution from the reversal effect.

Disadvantages
- High turnover and transaction costs.
- Liquidity strain (must buy many small-cap names).
- Persistent underperformance versus cap-weighted indexes in long bull markets.
- Limited capacity; unsuitable for very large asset pools.

Worked Cases

Case 1: Single-Period Return Calculation

An equal-weighted index contains four stocks with period returns of +5%, –2%, +8%, and +1%.
The index return is:
$$R_{EW} = \frac{5\% - 2\% + 8\% + 1\%}{4} = 3\%$$

Case 2: Weight Drift and Rebalancing

Three stocks A, B, and C start at ¥100 each and equal weights of 33.33%. Ending prices are A = ¥130, B = ¥90, C = ¥110.
Ending natural weights before rebalancing:
A: $\frac{130}{330} = 39.39\%$
B: $\frac{90}{330} = 27.27\%$
C: $\frac{110}{330} = 33.33\%$

After rebalancing, all weights return to exactly 33.33%. The manager sells A and buys B.

Case 3: Multi-Period Cumulative Effect of Rebalancing

Three stocks experience the following successive returns:
Period 1: A +30%, B –10%, C +5% → equal-weighted return = (30 – 10 + 5)/3 = 8.33%
Period 2: A –15%, B +25%, C +10% → equal-weighted return = (–15 + 25 + 10)/3 = 6.67%

Without rebalancing, the weights at the start of Period 2 deviate from 1/3, altering the realized return. Quarterly rebalancing restores target weights, captures the “sell high, buy low” gain, and keeps cumulative performance close to the compounded arithmetic-average path.

Traps

Key Formulas

• Target weight per constituent: $w_i = \frac{1}{N}$
• Single-period index return: $R_{EW,t} = \frac{1}{N}\sum_{i=1}^N R_{i,t}$
• Index level: $Index_t = \frac{\sum (P_{i,t} \times \frac{1}{N})}{Divisor}$
• Rebalancing frequency: typically quarterly
• Conceptual relation: Equal-weighted ≈ Cap-weighted + small-cap premium – momentum effect

Practice Questions

Q1. The single-period return of an equal-weighted index is closest to:
A. The weighted-average return of constituents
B. The arithmetic average return of constituents
C. The geometric average return of constituents
D. The return of the largest constituent

Q2. Which of the following is NOT a typical disadvantage of an equal-weighted index?
A. Higher transaction costs
B. Overexposure to small-cap stocks
C. Underperformance versus cap-weighted indexes in long bull markets
D. Complete elimination of style bias

Q3. An equal-weighted index has 200 constituents. If one stock rises 50% while all others are unchanged, its weight immediately before rebalancing is closest to:
A. 0.5%
B. 0.75%
C. 1.0%
D. 1.5%

Q4. The rebalancing rule of an equal-weighted index is essentially:
A. A momentum strategy
B. A contrarian (reversal) strategy
C. A growth strategy
D. A quality strategy

Q5. Relative to a market-cap-weighted index, an equal-weighted index typically exhibits:
A. Higher concentration risk
B. Lower small-cap exposure
C. Greater potential rebalancing return
D. Lower turnover costs

Q6. The primary role of the divisor in an equal-weighted index is to:
A. Set the index level to a convenient target value on the base date
B. Eliminate the effect of stock splits
C. Adjust for changes in the number of constituents
D. All of the above

Q7. During a strong, one-directional bull market, an equal-weighted index will most likely:
A. Significantly outperform a cap-weighted index
B. Significantly underperform a cap-weighted index
C. Perform about the same
D. Cannot be determined

Q8. Four stocks in an equal-weighted index start at ¥100 each. Ending prices are ¥105, ¥98, ¥112, and ¥90. Without rebalancing, which stock will have the largest ending weight?
A. First stock
B. Second stock
C. Third stock
D. Fourth stock

Answers

Takeaways

• Equal-weighted indexes assign every constituent an identical weight of $1/N$.
• Index return equals the arithmetic average of stock returns and embeds small-cap and value tilts.
• Periodic rebalancing is mandatory; weights drift with price changes.
• The rebalancing rule is inherently contrarian and performs best in volatile or mean-reverting markets.
• Primary drawbacks are high turnover, liquidity demands, and underperformance in strong bull markets.
• Equal-weighted, price-weighted, and cap-weighted methods form the three foundational index-construction techniques tested at Level I.