Price Return, Total Return, and Weighting Methods for Security Market Indexes

A security market index measures the performance of a defined group of securities. Three core calculations appear on the CFA Level I exam: 1. Price Return (PR) captures only capital appreciation: $PR_{index} = \frac{V_1 - V_0}{V_0}$ where $V_0$ and $V_1$ are the index values at the beginning and end of the period. 2. Total Return (TR) adds income (dividends) to the price return: $TR_{index} = \frac{V_1 - V_0 + D}{V_0}$ where $D$ is the total dividends paid by index constituents over the period, expressed in index-value terms. 3. Index Value depends on the weighting scheme. For a price-weighted index: $V_{PW} = \frac{\sum_{i=1}^{n} P_i}{D_{adj}}$ where $D_{adj}$ is the divisor, adjusted for corporate actions. For a market-capitalization-weighted (value-weighted) index: $V_{MCW,t} = V_{MCW,0} \times \frac{\sum_{i=1}^{n} P_{i,t} \times Q_{i,t}}{\sum_{i=1}^{n} P_{i,0} \times Q_{i,0}}$ 4. Equal-weighted index return is the simple arithmetic average of each constituent's individual return: $TR_{EW} = \frac{1}{n}\sum_{i=1}^{n} TR_i$ Understanding how to compute each measure—and how corporate actions (e.g., stock splits) alter the price-weighted divisor—is essential for the exam.

Scenario: Regional Equity Index — Beginning of Year to End of Year An analyst tracks a regional equity index composed of four stocks: Alpha (A), Beta (B), Gamma (G), and Delta (D). The index is maintained in three parallel versions: (1) a price-weighted index, (2) a market-capitalization-weighted index (base value = 1,000 at inception), and (3) an equal-weighted index. The following data apply at the start of the year (January 1):

Stock	Price (Jan 1)	Shares Outstanding (millions)	Price-Weighted Divisor (Jan 1)
Alpha	$80	50	4.00 (initial)
Beta	$60	80
Gamma	$40	120
Delta	$20	200

Mid-year corporate action (July 1): Alpha executes a 2-for-1 stock split. The price-weighted index divisor must be adjusted so the index value is unaffected on the split date. At July 1 (just before the split), the prices of Beta, Gamma, and Delta are unchanged at $60, $40, and $20 respectively. Alpha's pre-split price is $88. End-of-year data (December 31):

Stock	Price (Dec 31)	Shares Outstanding (millions)	Annual Dividend Per Share
Alpha	$50	100 (post-split)	$1.00
Beta	$72	80	$2.00
Gamma	$35	120	$0.50
Delta	$27	200	$0.75

Additional assumption: For total return calculations on the price-weighted and market-cap-weighted indexes, express dividends as an index-level contribution using the same divisor/base methodology applied to prices. Calculate each of the following:

1. The price-weighted index value on January 1 (using the initial divisor of 4.00).
2. The adjusted price-weighted divisor after Alpha's 2-for-1 split on July 1.
3. The price-weighted index value on December 31 using the adjusted divisor.
4. The price return and total return of the price-weighted index for the year.
5. The market-cap-weighted index value on December 31 (base value = 1,000 on January 1).
6. The price return and total return of the market-cap-weighted index for the year.
7. The equal-weighted index total return for the year.
Rank the three indexes from highest to lowest total return and explain in one sentence which index benefited most from dividend income relative to price return.

Step 1: Price-Weighted Index Value on January 1

$V_{PW,0} = \frac{P_A + P_B + P_G + P_D}{D_{adj,0}} = \frac{\$80 + \$60 + \$40 + \$20}{4.00} = \frac{\$200}{4.00} = \$50.00$ The price-weighted index opens the year at 50.00.

Step 2: Adjusted Price-Weighted Divisor After the Split

On July 1, just before the split, the index value must be preserved. Pre-split prices: Alpha = $88, Beta = $60, Gamma = $40, Delta = $20. $V_{PW,\text{pre-split}} = \frac{88 + 60 + 40 + 20}{4.00} = \frac{208}{4.00} = 52.00$ After the split, Alpha's price halves to $44. The new divisor $D_{new}$ must keep the index at 52.00: $52.00 = \frac{44 + 60 + 40 + 20}{D_{new}} = \frac{164}{D_{new}}$ $D_{new} = \frac{164}{52.00} = 3.15385 \approx \mathbf{3.1538}$

Step 3: Price-Weighted Index Value on December 31

Using the adjusted divisor and December 31 prices (Alpha post-split = $50): $V_{PW,1} = \frac{50 + 72 + 35 + 27}{3.1538} = \frac{184}{3.1538} = \mathbf{58.3369} \approx 58.34$

Step 4: Price Return and Total Return of the Price-Weighted Index

Price Return: $PR_{PW} = \frac{V_{PW,1} - V_{PW,0}}{V_{PW,0}} = \frac{58.34 - 50.00}{50.00} = \frac{8.34}{50.00} = 0.1668 = \mathbf{16.68\%}$ Dividend contribution to the price-weighted index: Dividends per share (annual): Alpha = $1.00 (post-split, 100M shares but per-share basis for index), Beta = $2.00, Gamma = $0.50, Delta = $0.75. $D_{PW} = \frac{1.00 + 2.00 + 0.50 + 0.75}{D_{new}} = \frac{4.25}{3.1538} = 1.3475$ Total Return: $TR_{PW} = \frac{V_{PW,1} - V_{PW,0} + D_{PW}}{V_{PW,0}} = \frac{58.34 - 50.00 + 1.3475}{50.00} = \frac{9.6875}{50.00} = 0.19375 = \mathbf{19.38\%}$

Step 5: Market-Cap-Weighted Index Value on December 31

Market caps on January 1 (in millions): $\begin{array}{rcl} \text{Alpha} &=& 80 \times 50 = \$4{,}000M \\ \text{Beta} &=& 60 \times 80 = \$4{,}800M \\ \text{Gamma} &=& 40 \times 120 = \$4{,}800M \\ \text{Delta} &=& 20 \times 200 = \$4{,}000M \\ \text{Total}_{0} &=& \$17{,}600M \end{array}$ Market caps on December 31 (in millions): $\begin{array}{rcl} \text{Alpha} &=& 50 \times 100 = \$5{,}000M \\ \text{Beta} &=& 72 \times 80 = \$5{,}760M \\ \text{Gamma} &=& 35 \times 120 = \$4{,}200M \\ \text{Delta} &=& 27 \times 200 = \$5{,}400M \\ \text{Total}_{1} &=& \$20{,}360M \end{array}$ $V_{MCW,1} = 1{,}000 \times \frac{20{,}360}{17{,}600} = 1{,}000 \times 1.15682 = \mathbf{1{,}156.82}$

Step 6: Price Return and Total Return of the Market-Cap-Weighted Index

Price Return: $PR_{MCW} = \frac{1{,}156.82 - 1{,}000}{1{,}000} = 0.15682 = \mathbf{15.68\%}$ Dividend contribution in index-value terms: Weight each stock's dividend yield by its beginning market-cap weight: $w_A = \frac{4{,}000}{17{,}600} = 0.22727, \quad w_B = \frac{4{,}800}{17{,}600} = 0.27273, \quad w_G = \frac{4{,}800}{17{,}600} = 0.27273, \quad w_D = \frac{4{,}000}{17{,}600} = 0.22727$ Dividend yield per stock (annual dividend / beginning price): $\begin{array}{rcl} dy_A &=& \frac{1.00}{80} = 0.01250 \\ dy_B &=& \frac{2.00}{60} = 0.03333 \\ dy_G &=& \frac{0.50}{40} = 0.01250 \\ dy_D &=& \frac{0.75}{20} = 0.03750 \end{array}$ Weighted average dividend yield: $dy_{MCW} = (0.22727)(0.01250) + (0.27273)(0.03333) + (0.27273)(0.01250) + (0.22727)(0.03750)$ $= 0.002841 + 0.009091 + 0.003409 + 0.008523 = 0.023864 = 2.3864\%$ Dividend index contribution = $1{,}000 \times 0.023864 = 23.864$ index points. Total Return: $TR_{MCW} = PR_{MCW} + dy_{MCW} = 15.68\% + 2.39\% = \mathbf{18.07\%}$ (Alternatively: $\frac{1{,}156.82 - 1{,}000 + 23.86}{1{,}000} = \frac{180.68}{1{,}000} = 18.07\%$)

Step 7: Equal-Weighted Index Total Return

Each stock's total return (price appreciation + dividend yield, using January 1 prices): $\begin{array}{rcl} TR_A &=& \frac{50 - 80 + 1.00}{80} = \frac{-29.00}{80} = -0.36250 = -36.25\% \\[6pt] TR_B &=& \frac{72 - 60 + 2.00}{60} = \frac{14.00}{60} = 0.23333 = 23.33\% \\[6pt] TR_G &=& \frac{35 - 40 + 0.50}{40} = \frac{-4.50}{40} = -0.11250 = -11.25\% \\[6pt] TR_D &=& \frac{27 - 20 + 0.75}{20} = \frac{7.75}{20} = 0.38750 = 38.75\% \end{array}$ $TR_{EW} = \frac{-36.25\% + 23.33\% + (-11.25\%) + 38.75\%}{4} = \frac{14.58\%}{4} = \mathbf{3.645\%}$

Step 8: Ranking and Dividend Observation

Summary table:

Index Type	Price Return	Total Return	Dividend Contribution
Price-Weighted	16.68%	19.38%	2.70%
Market-Cap-Weighted	15.68%	18.07%	2.39%
Equal-Weighted	~2.38%*	3.65%	~1.27%

*Equal-weighted price return = $\frac{-37.5\% + 20\% + (-12.5\%) + 35\%}{4} = \frac{5\%}{4} = 1.25\%$... corrected below. Note on equal-weighted price return (verification): $PR_A = \frac{50-80}{80} = -37.5\%, \quad PR_B = \frac{72-60}{60} = 20\%, \quad PR_G = \frac{35-40}{40} = -12.5\%, \quad PR_D = \frac{27-20}{20} = 35\%$ $PR_{EW} = \frac{-37.5 + 20 - 12.5 + 35}{4} = \frac{5}{4} = 1.25\%$ Dividend-only contribution (EW) = $3.645\% - 1.25\% = 2.395\%$, consistent. Final ranking by total return (highest to lowest): $\text{Price-Weighted } (19.38\%) > \text{Market-Cap-Weighted } (18.07\%) > \text{Equal-Weighted } (3.65\%)$ The price-weighted index benefited most from dividend income relative to its price return because high-priced stocks (Beta at $60) carry disproportionately large weights, and Beta paid the highest absolute dividend ($2.00), amplifying the dividend contribution per index unit relative to the other weighting schemes.