Price-Weighted, Value-Weighted, and Equal-Weighted Index Returns

A security market index aggregates the prices or values of a set of securities to represent a market or market segment. The three most common weighting methods tested at CFA Level I are: 1. Price-Weighted Index (PWI): Each constituent's weight equals its price divided by the sum of all constituent prices. The index level is the simple arithmetic average of constituent prices, adjusted for a divisor. $\text{PWI} = \frac{\sum_{i=1}^{n} P_i}{D}$ where $D$ is the divisor (initially equal to $n$, adjusted for splits and changes). 2. Value-Weighted (Market-Cap-Weighted) Index: Each constituent's weight equals its market capitalization as a fraction of total market capitalization. $\text{Index Return} = \sum_{i=1}^{n} w_i \cdot R_i, \quad w_i = \frac{P_{i,0} \times Q_i}{\sum_{j} P_{j,0} \times Q_j}$ 3. Equal-Weighted Index (EWI): Each constituent receives an identical weight of $\frac{1}{n}$. $\text{EWI Return} = \frac{1}{n} \sum_{i=1}^{n} R_i$ Index return for any method is calculated over a period as: $\text{Index Return} = \frac{\text{Index}_{t} - \text{Index}_{t-1}}{\text{Index}_{t-1}}$

Scenario: An analyst is constructing three versions of a hypothetical equity index using the same three stocks: Arcturus Corp (ARC), Belmont Industries (BEL), and Cassia Technologies (CAS). The index base value is set at \$1{,}000 on Day 0. The following data are provided:

Stock	Day 0 Price	Day 1 Price	Shares Outstanding
ARC	$80	$88	500
BEL	$50	$47	1,200
CAS	$20	$23	3,000

Additional information: No dividends are paid. No corporate actions (splits, changes) occur between Day 0 and Day 1. The initial divisor for the price-weighted index equals the number of stocks. Calculate each of the following:

1. The Day 0 and Day 1 levels of the price-weighted index, and the one-day percentage return of the price-weighted index.
2. The Day 0 market capitalization weights for each stock in the value-weighted index, the Day 1 value-weighted index level (expressed on a base of \$1{,}000), and the one-day percentage return of the value-weighted index.
3. The one-day percentage return of the equal-weighted index, and the Day 1 equal-weighted index level (expressed on a base of \$1{,}000).
Using the results from parts (1) through (3), identify which index recorded the highest return and provide a one-sentence numerical justification.

Step 1: Individual Stock Returns

First, compute each stock's one-day return, as these are needed in multiple parts: $R_{\text{ARC}} = \frac{88 - 80}{80} = \frac{8}{80} = 10.00\%$ $R_{\text{BEL}} = \frac{47 - 50}{50} = \frac{-3}{50} = -6.00\%$ $R_{\text{CAS}} = \frac{23 - 20}{20} = \frac{3}{20} = 15.00\%$

Part 1: Price-Weighted Index

The initial divisor $D = 3$ (number of stocks). Day 0 PWI level: $\text{PWI}_0 = \frac{80 + 50 + 20}{3} = \frac{150}{3} = 50.00$ Day 1 PWI level: $\text{PWI}_1 = \frac{88 + 47 + 23}{3} = \frac{158}{3} = 52.\overline{6}$ PWI Return: $R_{\text{PWI}} = \frac{52.\overline{6} - 50.00}{50.00} = \frac{2.\overline{6}}{50.00} = 5.33\%$ Expressed on a base of \$1,000: $\text{PWI}_1^{\text{indexed}} = 1{,}000 \times 1.0533 = \$1{,}053.33$ Key insight: In a price-weighted index, higher-priced stocks exert greater influence. ARC ($80) dominates over CAS ($20), even though CAS had the highest return.

Part 2: Value-Weighted Index

Day 0 market capitalizations: $\text{MktCap}_{\text{ARC},0} = 80 \times 500 = \$40{,}000$ $\text{MktCap}_{\text{BEL},0} = 50 \times 1{,}200 = \$60{,}000$ $\text{MktCap}_{\text{CAS},0} = 20 \times 3{,}000 = \$60{,}000$ $\text{Total MktCap}_0 = 40{,}000 + 60{,}000 + 60{,}000 = \$160{,}000$ Day 0 weights: $w_{\text{ARC}} = \frac{40{,}000}{160{,}000} = 0.2500 \quad (25.00\%)$ $w_{\text{BEL}} = \frac{60{,}000}{160{,}000} = 0.3750 \quad (37.50\%)$ $w_{\text{CAS}} = \frac{60{,}000}{160{,}000} = 0.3750 \quad (37.50\%)$ Day 1 market capitalizations: $\text{MktCap}_{\text{ARC},1} = 88 \times 500 = \$44{,}000$ $\text{MktCap}_{\text{BEL},1} = 47 \times 1{,}200 = \$56{,}400$ $\text{MktCap}_{\text{CAS},1} = 23 \times 3{,}000 = \$69{,}000$ $\text{Total MktCap}_1 = 44{,}000 + 56{,}400 + 69{,}000 = \$169{,}400$ Value-weighted index return: $R_{\text{VWI}} = \frac{169{,}400 - 160{,}000}{160{,}000} = \frac{9{,}400}{160{,}000} = 5.875\%$ Alternatively, using weighted returns: $\begin{array}{rcl} R_{\text{VWI}} &=& (0.25)(10.00\%) + (0.375)(-6.00\%) + (0.375)(15.00\%) \\ &=& 2.500\% + (-2.250\%) + 5.625\% \\ &=& 5.875\% \end{array}$ Day 1 VWI indexed level: $\text{VWI}_1 = 1{,}000 \times 1.05875 = \$1{,}058.75$

Part 3: Equal-Weighted Index

Each stock receives a weight of $\frac{1}{3}$. $R_{\text{EWI}} = \frac{1}{3}(10.00\%) + \frac{1}{3}(-6.00\%) + \frac{1}{3}(15.00\%)$ $\begin{array}{rcl} R_{\text{EWI}} &=& \frac{10.00 + (-6.00) + 15.00}{3} \\ &=& \frac{19.00}{3} \\ &=& 6.333\% \end{array}$ Day 1 EWI indexed level: $\text{EWI}_1 = 1{,}000 \times 1.06333 = \$1{,}063.33$

Part 4: Comparison and Conclusion

Index Type	Day 1 Level	One-Day Return
Price-Weighted	$1,053.33	5.333%
Value-Weighted	$1,058.75	5.875%
Equal-Weighted	$1,063.33	6.333%

Conclusion: The equal-weighted index recorded the highest one-day return of $6.333\%$, because it assigns the greatest relative weight to CAS (the best performer at $+15\%$) compared to the other two methods, which underweight CAS due to its low price and moderate market cap relative to BEL.