Why Price-weighted Indices do not require "rebalancing" ?

Ref : CFAI text 2013 V5, P85, Exhibit 2 and 1st paragraph & P.91, last paragraph, statements :

Price-weighted indices are not rebalanced because the weight of each constituent security is determined by its price.

The weights of each constituent security changed after stock-split or even without stock-split. Why rebalancing is not required ?

Suppose that an index has 3 stocks:

  • A, at $90/share
  • B, at $45/share
  • C, at $15/share

For simplicity, assume that the value of the index is 150, so that the divisor is 1. Share A represents 90/150 or 60% of the index, B represents 45/150 or 30% of the index, and C represents 15/150 or 10% of the index. The representation is proportional to the share price. If you own one share each of A, B, and C, you have replicated the index.

If stock A’s price rises to $100/share (and B’s and C’s remain unchanged), then the index value is 160, A represents 100/160 of the index, B represents 45/160 of the index, and C represents 15/160 of the index. Again, the representation is proportional to the share price, so if you own 1 share each of A, B, and C, you have replicated the index: no change in the number of shares, so no rebalancing.

If A then has a 2:1 stock split, so that its share price is now $50, then the index value doesn’t change (it’s still at 160), so the divisor changes: (50 + 45 + 15) / 160 = 0.6875. Stock A represents (50/0.6875)/160 (= 50/110) of the index, stock B represents (45/0.6875)/160 (= 45/110) of the index, and stock C represents (15/0.6875)/160 (= 15/110) of the index; the representations are still proportional to the stock prices. In this case, however, you would now own 2 shares of A, so you would have to sell 1 to replicate the index. In this case, their claim of no rebalancing is incorrect.

(You would also have to sell and buy shares – obviously – if one stock in the index were replaced with another.)

In cap-weighted indices, there is no rebalancing necessary, even with stock splits, reverse stock splits, or stock dividends.

So, in case of stock split, rebalancing is required, whereas in case of price change, rebalancing is not required ?

If so, is the statements quoted by me wrong ? Thanks !