Consider the following scenario:

- Stock share \(A\) has an initial price \(S_{A}(0)=\$ 25\), at time \(t=0\), which is increased by \(20 \%\) to \(S_{A}(T)=\$ 30\) at time \(t=T\). The total market capitalization is \(\$ 500\) million (hence, 20 million shares are outstanding).

- Stock share \(B\) has an initial price \(S_{B}(0)=\$ 100\), which drops by \(10 \%\) to \(S_{B}(T)=\$ 90\) at time \(t=T\). The total market capitalization is \(\$ 100\) million (hence, one million shares are outstanding).

A price-based index would initially be

\[\frac{25+100}{2}=625\]

where we assume a divisor \(D=2\), which is really inconsequential when considering percentage changes in the index. At the end of the time horizon, the new index value would be

\[\frac{30+90}{2}=60\]

with a drop of \(4 \%\). Note that the price drop of the more expensive stock share dominates here, but this does not reflect the true market weights. Let us consider a market-value-weighted index, with initial value

\[\frac{25 \quad 20 \quad 10^{6}+100 \quad 1 \quad 10^{6}}{10^{6}}=600\]

where we set \(D=10^{6}\). The new index value would be

\[\frac{30 \quad 20 \quad 10^{6}+90 \quad 1 \quad 10^{6}}{10^{6}}=690\]

with an increase of \(15 \%\).