1.9 Indices: free-float weighting and the divisor

Motivation

"The market fell 2%" means the Nifty 50 fell 2%. An index is a precisely defined portfolio, and its construction determines what the number measures, what index funds must buy, and how index futures and options settle.

1.9.1Free-float capitalisation#

Definitions

For constituent $i$ at time $t$ : price $P_{i,t}$ , shares outstanding $N_i$ , and investible weight factor (IWF) $f_i \in (0,1]$ , the fraction of shares available to ordinary investors (excluding promoter holdings, government strategic stakes and other locked-in holdings). The free-float market capitalisation of the index is

M_t = \sum_i P_{i,t}\, N_i\, f_i. \tag{1.5}

The index level is defined through a divisor $D_t$ :

I_t = \frac{M_t}{D_t}. \tag{1.6}

For the Nifty 50, the base date is 3 Nov 1995 and the base value 1,000, so $D$ at the base was $M_{\text{base}}/1000$ ; the index was launched on 22 Apr 1996 and moved from full to free-float capitalisation on 26 Jun 2009 (NSE Indices factsheet and methodology summaries). It is rebalanced semi-annually.

1.9.2Index returns and the divisor#

Proposition 1.3index return is a cap-weighted average

If the divisor and all $N_i, f_i$ are unchanged between $t-1$ and $t$ , then

\frac{I_t}{I_{t-1}} - 1 = \sum_i w_{i,t-1}\, R_{i,t}, \qquad w_{i,t-1} = \frac{P_{i,t-1}N_i f_i}{M_{t-1}}, \quad R_{i,t} = \frac{P_{i,t}}{P_{i,t-1}} - 1. \tag{1.7}

Proof

$I_t/I_{t-1} = M_t/M_{t-1} = \sum_i P_{i,t}N_i f_i / M_{t-1} = \sum_i \frac{P_{i,t-1}N_i f_i}{M_{t-1}}\cdot\frac{P_{i,t}}{P_{i,t-1}} = \sum_i w_{i,t-1}(1+R_{i,t})$ , and $\sum_i w_{i,t-1} = 1$ .∎

Consequences: weights drift with prices, so a cap-weighted index fund needs no trading except when the basket, share counts or IWFs change. The index is a price index: ordinary dividends are not in $R_{i,t}$ , so the index falls when a constituent goes ex-dividend.

Proposition 1.4divisor adjustment

Suppose that after the close at $t$ the basket changes (a constituent is replaced, shares outstanding change through a rights issue or buyback, or an IWF is revised). Let $M_t$ and $M_t'$ be the free-float capitalisations before and after the change, both at the closing prices of $t$ . The index stays continuous ( $I_t$ unchanged) if and only if the divisor becomes

D_t' = D_t \cdot \frac{M_t'}{M_t}. \tag{1.8}

After the change, Proposition 1.3 holds with the new weights. A split or bonus requires no divisor change.

Proof

Continuity means $M_t/D_t = M_t'/D_t'$ , which is (1.8). For a split of ratio $k$ (one share becomes $k$ ), the theoretical ex-price is $P/k$ and the share count $kN$ (Proposition 1.5 below), so $P N f$ is unchanged, $M_t' = M_t$ and $D_t' = D_t$ . A bonus is the same with $k = (a+b)/b$ .∎

1.9.3Worked example: replacing a constituent#

Worked example

A toy three-stock index (illustrative numbers in the scale of large Indian banks and conglomerates):

Stock	$P$ (₹)	$N$ (shares)	$f$	$PNf$ (₹ crore)
A	2,900	1,353 crore	0.50	19,61,850
B	1,650	760 crore	0.95	11,91,300
C	1,450	696 crore	0.75	7,56,900

$M = \text{₹}39{,}10{,}050$ crore. If the index stands at 25,000, the divisor is $M/25{,}000 = 156.40$ crore. Replace C with D ( $P = 3{,}800$ , $N = 360$ crore, $f = 0.28$ , $PNf = 3{,}83{,}040$ crore). Then $M' = 35{,}36{,}190$ crore and $D' = 156.40 \times 35{,}36{,}190/39{,}10{,}050 = 141.45$ crore. The index is still 25,000. Index funds must sell all of C and buy D at the close of the effective date, which is the source of predictable closing-auction volume on rebalance days. These numbers are the fixtures of test_divisor_keeps_index_continuous_on_replacement in the tests of Exercise m01_mechanics (tests/test_m01_mechanics.py).

1.9.4Dividends and the TRI#

Total return index Let the index dividend in points on day $t$ be $\text{ID}_t = \sum_i d_{i,t}N_i f_i / D_t$ , where $d_{i,t}$ is the dividend per share of constituent $i$ going ex on $t$ . The total return index (for example, Nifty 50 TRI) is built recursively:

\text{TRI}_t = \text{TRI}_{t-1}\cdot \frac{I_t + \text{ID}_t}{I_{t-1}}. \tag{1.9}

It assumes dividends are reinvested in the index on the ex-date. Over long horizons the TRI and the price index diverge by roughly the dividend yield each year. Benchmark mutual funds against the TRI, as SEBI requires.

Practitioner notes

IWF changes and index inclusions move prices, because passive funds must trade a known quantity on a known date. Studying these flows is legitimate; front-running them on non-public information is not.
A cap-weighted index concentrates in the largest stocks. In the toy example A is half the index.
For Nifty futures and options the settlement price is derived from the index, so the index rules are part of the derivative contract.

See it live: current levels of 167 NSE indices, including India VIX, at https://oicharts.com/indices (one page per index, e.g. https://oicharts.com/indices/nifty-50). Compare the Nifty 50 with a sectoral index such as Nifty Bank on a day when one heavyweight moves sharply, and you will see Proposition 1.3 at work.