Let us consider how to manage an index for a stock market on which two stocks are traded. Company \(A\) has 50 shares outstanding, with current price \(\$ 2\), and company \(B\) has 10 shares outstanding, with current price \(\$ 10\). The current value of a price-based index is 6 , whereas the value of a market-value-weighted index is 100 . Let us consider the following scenario: The price of Company \(A\) 's stock increases to

\(\$ 4\) per share, and Company \(B\) 's stock splits 2 for 1 and is priced at \(\$ 5\). How will the values of the price-based and market-value-weighted indexes change?

To begin with, we have to find the divisors. The current divisor for the price-based index is clearly \(D=2\), since

\[\frac{2+10}{2}=6\]

Then, it is important to notice that, actually, the second stock price did not change. The drop from \(\$ 10\) to \(\$ 5\) merely reflects the split. After the change in price of the first share, without considering the stock split, the new index would be

\[\frac{4+10}{2}=7\]

The new divisor is changed in order to reflect the split without introducing a discontinuity in the index:

\[\frac{4+5}{D}=7 \quad D=\frac{9}{7}\]

The divisor for the market-value-weighted index is found as follows:

\[\frac{2 \quad 50+10 \quad 10}{D}=100 \quad D=2\]

However, the stock split is inconsequential for this index and does not require any adjustment in the divisor. Hence, the new index value is

\[\frac{50 \quad 4+20 \quad 5}{2}=150\]