### The Value and Return of an Index

Every index weighting method has a formula that calculates the weighting of a given constituent security within an index. For the following examples, the same portfolio of three securities will be used to help illustrate the weighting methods. Note that while income from constituent securities could hypothetically be reinvested in the index, the index level adjusts only for price changes.

$$\begin{array}{l|r|r|r} \textbf{Security} & \textbf{Beg. Price/Share} & \textbf{Income} & \textbf{End Price/Share} \\ \hline \text{Security A} & 500 & 0 & 750 \\ \text{Security B} & 20 & 1 & 21 \\ \text{Security C} & 45 & 1 & 25 \\ \hline \textbf{Total} & & \mathbf{2} & \\ \end{array}$$

## Price Weighting

Here, the weight of each security i is given by:

$$w_{i}^P = \frac{ P_i } { \sum_{i=1}^{N}{P_i} }$$

Where:

wi = fraction of the portfolio that is allocated to the security or weight of the security

N = number of securities in the index

Pi = price of the security

#### Example of a price-weighted index

$$\begin{array}{l|rrrrrrr} \textbf{Security} & \textbf{Beg. Value} & \textbf{Income} & \textbf{End Value} & \textbf{Shares} & \textbf{Beg. Weight} & \textbf{End Weight} \\ \hline \text{Security A} & 500 & 0 & 750 & 1 & 88.5\% & 94.2\% \\ \text{Security B} & 20 & 1 & 21 & 1 & 3.5\% & 2.6\% \\ \text{Security C} & 45 & 1 & 25 & 1 & 8.0\% & 3.1\% \\ \hline \textbf{Total} & \mathbf{188.33} & & \mathbf{265.33} & \mathbf{3} & \mathbf{1} &\mathbf{1} \\ \textbf{Price Return} & \mathbf{40.9\%} \\ \textbf{Total Return} & \mathbf{41.2\%} \\ \end{array}$$

One share of each security is held in the index and the divisor is set to 3 – the total number of shares. The high weighting of Security A in both periods is perhaps the most interesting part of this index. While this is an extremely concentrated index, price/share is nevertheless very important in how securities are weighted in price indices even though it remains a mostly arbitrary figure.

## Equal Weighting

Here, the weight of each security i is given by:

$$w_{i}^E = \frac{ 1 } { N }$$

Where:

N = number of securities in the index

#### Example of an equal-weighted index

$$\begin{array}{l|rrrrrr} \textbf{Security} & \textbf{Beg. Value} & \textbf{Income} & \textbf{End Value} & \textbf{Shares} & \textbf{Beg. Weight} & \textbf{End Weight} \\ \hline \text{Security A} & 500 & 0 & 750 & 1 & 33.3\% & 48.3\% \\ \text{Security B} & 500 & 25 & 525 & 25 & 33.3\% & 33.8\% \\ \text{Security C} & 500 & 11 & 278 & 11.11 & 33.3\% & 17.9\% \\ \text{Total} & 1,500.00 & 36.11 & 1,552.78 & 37.11 & 100\% & 100\% \\ \hline \textbf{Price Return} & \mathbf{3.5\%} \\ \textbf{Total Return} & \mathbf{5.9\%} \\ \end{array}$$

As shown in the table, each of the three securities is equally-weighted within the index at the beginning of the period, but largely strayed from the initial weights. While the price index posted returns above 40%, returns of the three-security portfolio were much more modest when beginning with the same values.

## Market-capitalization

Here, the weight of each security i is given by:

$$w_{i}^M = \frac{ Q_i P_i } { \sum_{i=1}^{N}{Q_i P_i } }$$

Where:

Qi = number of shares outstanding of security

#### Example of a market-capitalization index

$$\begin{array}{l|rrrrrrr} \textbf{Security} & \textbf{Beg. Value} & \textbf{Income} & \textbf{End Value} & \textbf{Out. Shares(mm)} & \textbf{Beg. Weight} & \textbf{End Weight} \\ \hline \text{Security A} & 1,500 & 0 & 2,250 & 3 & 25.6\% & 46.7\% \\ \text{Security B} & 300 & 15 & 315 & 15 & 5.1\% & 6.5\% \\ \text{Security C} & 4,050 & 90 & 2,250 & 90 & 69.2\% & 46.7\% \\ \text{Total} & 5,850.00 & 105.00 & 4,815.00 & 108.00 & 100\% & 100\% \\ \hline \textbf{Price Return} & \mathbf{-17.7\%} \\ \textbf{Total Return} & \mathbf{-15.9\%} \\ \end{array}$$

Note that the column that recorded constituent shares as part of the index portfolio is now set equal to millions of outstanding shares. The beginning and ending values now reflect the market capitalizations of the companies in the index portfolio. Since Security C has by far the largest amount of shares outstanding, it accounts for more than two-thirds of market capitalization when multiplied out with the share price at the beginning of the period. As a result of Security C’s heavy weighting, the market-capitalization index performed very poorly during the period.

Here, the weight of each security i is given by:

$$w_{i}^{fM} = \frac{ f_i Q_i P_i } { \sum_{i=1}^{N}{f_iQ_i P_i } }$$

Where:

fi = fraction of shares outstanding in the market float

#### Example of a float-adjusted market-capitalization index

$$\begin{array}{l|rrrrrrr} \textbf{Security} & \textbf{Beg. Value} & \textbf{Income} & \textbf{End Value} & \textbf{Out. Shares(mm)} & \textbf{Beg. Weight} & \textbf{End Weight} &\textbf{Market Float %} \\ \hline \text{Security A} & 750 & 0 & 1,125 & 3 & 15.4\% & 31.7\% & 50.0\% \\ \text{Security B} & 270 & 14 & 284 & 15 & 5.5\% & 8.0\% & 90.0\% \\ \text{Security C} & 3,848 & 86 & 2,138 & 90 & 79.0\% & 60.3\% & 95.0\% \\ \text{Total} & 4,867 & 99.00 & 3,546.00 & 108.00 & 100\% & 100\% \\ \hline \textbf{Price Return} & \mathbf{-27.1\%} \\ \textbf{Total Return} & \mathbf{-25.1\%} \\ \end{array}$$

To calculate float-adjusted market capitalization, we just multiply each market capitalization by its respective fraction of shares outstanding in the market float. Since Security A is more closely held with just 50% market float, its weight is significantly reduced in the float-adjusted index in comparison to the unadjusted market capitalization index. In this scenario, the performance of the index becomes highly skewed by Security C’s losses during the period.

Calculate and analyze the value and return of an index given its weighting method

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