# Index Divisor

The key to index maintenance is the adjustment of the index divisor. Index maintenance (e.g., addition or deletion of constituents) should not change the index level. This is achieved by adjusting the index divisor. This section describes how the adjustment of the index divisor is made given the change in the total market value of the constituents.

The equation of index level is expanded to show the constituent being removed, constituent r, separately from the constituents that will remain in the index:

$$
{Index Level}{ }*{t0}=\frac{\left(\sum*{i} P\_{i} \* Q\_{i}\right)+P\_{r} Q\_{r}}{{Divisor}\_{t0}}
$$

Note that the index level and index divisor are now labeled for the time *t0*. To simplify, we are ignoring any possible events except the replacement of a constituent. After constituent *r* is replaced with constituent *s*, the equation is:

$$
{Index Level}{ }*{t1}=\frac{\left(\sum*{i} P\_{i} \* Q\_{i}\right)+P\_{s} Q\_{s}}{{Divisor}\_{t1}}
$$

In the equations, *t0* is the moment right before constituent *r* is deleted from and constituent *s* is added to the index, and *t1* is the moment right after the event. By design, *IndexLevelt0* is equal to *IndexLevelt1*. Combining and rearranging these two equations, the adjustment to the index divisor can be determined from the index market value before and after the change:

$$
\frac{\left(\sum\_{i} P\_{i} \* Q\_{i}\right)+P\_{r} Q\_{r}}{{Divisor}*{t0}} = \frac{\left(\sum*{i} P\_{i} \* Q\_{i}\right)+P\_{s} Q\_{s}}{{Divisor}\_{t1}}
$$

Therefore, the new index divisor after the maintenance is:

$$
{Divisor}*{t1} = {Divisor}*{t0}\*\frac{\left(\sum\_{i} P\_{i} \* Q\_{i}\right)+P\_{s} Q\_{s}}{\left(\sum\_{i} P\_{i} \* Q\_{i}\right)+P\_{r} Q\_{r}}
$$


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