## Calculation of stock performance

Now that we have a basic understanding of the principles that underline their calculations, we review in this section how these indexes are calculated. Obviously, constructing an aggregate measure of stock market performance is not as easy as it may initially appear. It is clear by now that coverage is not the only feature setting these indexes apart from each other; their construction must address several other concerns, including the weight given to each security included in the index.

For illustrative purposes we will use hypothetical data to demonstrate how a price-weighted average is calculated. Assume an index is constituted of two stocks, X and Y. Company X sells for \$50 and has 10 million shares outstanding. Company Y has 1 million shares outstanding and sells for \$10.

Under the price-weighted arithmetic average methodology, the price of the index is

In concept, this type of index is supposed to reflect the average price of stocks in the index. However, a problem arises whenever there is a stock split. Dow Jones indexes account for such splits by explicitly adjusting the denominator. An adjusted divisor is divided into the post-split sum of stock prices so that the index value remains the same before and after the split.

This methodology has long been criticized for allowing a bias associated with adjusting the denominator. The argument is that this adjustment scheme has the net effect of reducing the investment in the split shares and increasing the investment in the non-split shares. Since high growth shares are more likely to split than low growth shares, they will tend to lose weight within the index, which causes it to experience a downward bias. Box 2.1 represents a relatively recent discussion of this bias.

Continuing with our example, suppose now that the larger market capitalization company underperforms in the next year. The stock price of company X drops to \$46 whereas company Y finishes the year at \$12.00 per share. The new price-weighted arithmetic average is

In both calculations, the price-weighted index does not recognize the difference in the number of shares outstanding. This difference, on the other hand, is very important in the calculation of a value-weighted index. If the same numbers are used, the average price of the index would be calculated as follows:

Total value = Price X number of shares

= (\$50 X 10,000,000) + (\$10 X 1,000,000) = \$510,000,000.

Total value of all shares

Average index price =

Total number of shares \$510,000,000

11,000,000

If the prices of the stocks constituting the index change to \$46 and \$12, respectively, the new total value of the shares is

= (\$46X10,000,000) + (\$12X1,000,000) = \$472,000,000.

And the average value of a share of stock becomes = \$472,000,000/11,000,000 = \$42.91

When the two prices are treated equally, the simple price-weighted index drops by 3.45% (from \$30 to \$29), but when the value weighted average is used, the decrease is a more significant 8.04% (from \$46.36 to \$42.91) because it gives more weight to the underperformance of X, the larger market capitalization company.

Furthermore, indexes based on arithmetic averages are simple to duplicate regardless of the weighting scheme. No rebalancing is needed, as the weights respond to price changes by adjusting automatically to be consistent with the share amounts. This certainly has serious implications for the management of the replication portfolio. Also, one does not have to explicitly account for stock splits in calculating value-weighted indexes. This is not the case for indexes weighted by per-share prices, as already explained.

The alternative to the price-weighted and the value-weighted averages is the equal weighted average price, which assumes an equal dollar invested in each stock. As previously noted, VLG assumes equally weighted positions in every stock regardless of its market value or price. Instead of adding the prices of the various stocks and dividing by the number of stocks included in the index, it uses a geometric average that takes the nth root, with n equal to the number of stocks, after multiplying the various prices.

For each stock in the index, a ratio of its closing price today to the close on the previous trading day is calculated first. Next, a single number is obtained by multiplying together the ratios of all of the stocks included in the index. The final step consists of raising this number to the power defined by the reciprocal of the 1675 stocks currently included in the index. The result of these calculations is the ratio of today's VLG price to the previous trading day's close. The percentage price change is derived by subtracting 1 from this value and multiplying the result by 100.

Using our previous example, if company X's stock drops to \$46 from \$50, its ratio is 0.920. Conversely, ifY's stock goes from \$10 to \$12, its ratio is 1.20. The geometric average return is calculated as follows

Under the same circumstances, a price-weighted average dropped by 3.45%; a value weighted average fell by a more significant 8.04%, due to the underperformance of X; whereas the equally weighted geometric average showed a solid increase of 5.07%. In this hypothetical example, the equally weighted geometric average had clearly understated the impact of the price drop of the larger market capitalization company X.

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