Index divisor

An index is a sum divided by a number. The sum is the total of prices (for a price-weighted index) or market values (for a value-weighted one). The number it is divided by is the index divisor.

The divisor has two jobs.

At inception, it is chosen so the index starts at a tidy figure. The workbook's example: BSE Sensex has an initial value of 100, meaning on day one total market capitalisation divided by the divisor was 100.

Afterwards, it is adjusted to avoid changes in the index value that are unrelated to changes in the prices of the constituents. A stock split halves a share price without anyone losing a rupee. Replacing one constituent with a larger one adds market value without anyone gaining a rupee. Left alone, both would move the index. Changing the divisor cancels those false moves.

The principle is that the index should move only when investors' holdings actually gain or lose value.

Price-weighted index — a stock split (Example 8.1). Five stocks priced Rs 4, 6, 8, 9 and 10 sum to 37; with a divisor of 3.7 the index is 10. Stock E splits 2-for-1 and its price becomes Rs 5. The sum is now 32. The divisor is cut to 3.2, so the index stays at 32 ÷ 3.2 = 10.

Value-weighted index — a constituent change (Examples 8.2 to 8.4).

1. At inception total market capitalisation is 970. For a base of 100, the divisor is 970 ÷ 100 = 9.7.
2. Three months later market cap is 835. Index = 835 ÷ 9.7 = 86.08 — a genuine fall from price moves.
3. Security A is dropped and security X added. Market cap jumps to 1,230 with no price change. To hold the index at 86.08, the new divisor is 1,230 ÷ 86.08.

The workbook prints the new divisor as 14.28; the division gives 14.289, so 14.29 to two decimals. Either way the method is the point: new divisor = new total ÷ index level to be preserved.

A value-weighted index adjusts automatically for splits — the share count doubles as the price halves, so market value is unchanged — but still needs divisor changes for constituent changes. A price-weighted index needs a divisor change for both.

Index value = Σ (prices or market values) ÷ Divisor

Adjusted divisor = Σ after the event ÷ Index value before the event

Prices are illustrative.

A three-stock price-weighted sector index starts at 100:

Divisor at inception = 1,200 ÷ 100 = 12

Stock R splits 2-for-1 and now trades at Rs 300. Nobody's wealth has changed.

New sum      = 200 + 400 + 300 = 900
New divisor  = 900 ÷ 100 = 9
Index        = 900 ÷ 9 = 100    ✓ unchanged

Next day, prices move to Rs 210, Rs 400 and Rs 330:

With the adjusted divisor:  940 ÷ 9  = 104.44   (a real 4.44% rise)
With the old divisor:       940 ÷ 12 = 78.33    (a phantom 21.67% crash)

An investor holding Rs 12 lakh split in the index proportions would have gained money on the day. Without the divisor adjustment, the index would have told them — and every index fund and derivative priced off it — that the market had collapsed.

Chapter 8, section 8.3.1 and Box 8.1, define the index divisor with the BSE Sensex base of 100 and the stock-split Example 8.1; section 8.3.2 carries the value-weighted Examples 8.2 to 8.4 with the 9.7 and reconstitution divisors. Expect a numerical question asking for the adjusted divisor after a split or a constituent change, and a conceptual one on why the divisor is adjusted at all.

• The divisor is adjusted to keep the index level unchanged by non-price events — splits, constituent changes. It is not adjusted for ordinary price moves.
• After a split in a price-weighted index, the divisor falls. The sum of prices falls, so the divisor must fall with it.
• Value-weighted indices adjust automatically for splits, but still need divisor changes when constituents change.
• New divisor = new total ÷ old index level, not old total ÷ new total.
• BSE Sensex's initial value is 100, per the workbook.
• Rounding in the workbook: Example 8.4 gives the new divisor as 14.28; 1,230 ÷ 86.08 is 14.289. If options are 14.28 and 14.29, the workbook answer is 14.28.