6B.8 Index funds and ETFs: tracking

Motivation

An index fund promises the index return minus its cost. How close it gets is measured by two numbers that sound alike and are not: tracking difference, which is what you lose on average, and tracking error, which is how variable the loss is.

Tracking difference and tracking error#

Definitions

For daily fund returns $r_{F,t}$ (from NAV) and index returns $r_{I,t}$ (from the TRI):

TD = R_I - R_F, \qquad TE = \sqrt{252}\;\sigma\big(r_{F,t} - r_{I,t}\big), \tag{6B.24}

where $R_I, R_F$ are compound returns over the period (annualised for periods longer than a year). This module defines $TD$ as index minus fund, so a positive number is a shortfall; AMC disclosures often report fund minus index, a negative number. Read the sign convention before comparing.

Indian rules (checked 6 Nov 2026) [S]: SEBI's circular on the development of passive funds of 23 May 2022 (SEBI/HO/IMD/DOF2/P/CIR/2022/69) caps the tracking error of equity ETFs and index funds at 2%, measured on one year of rolling daily data, and caps the tracking difference of debt ETFs and index funds at 1.25% a year; AMCs disclose tracking error daily and tracking difference monthly on their own and AMFI's websites; ETFs disclose iNAV during the day (6B.9)¹.

Decomposing tracking difference#

Proposition 6B.8.1decomposition of tracking difference

Over a short period, let the fund hold a fraction $w_c$ in cash earning $r_c$ , and the rest in securities with weights $(1 - w_c)w_{I,i} + d_i$ , where $w_{I,i}$ are the index weights and $\sum_i d_i = 0$ (sampling and timing deviations). Let the fund pay expenses at rate $c$ , incur trading costs $\kappa$ (from flows, rebalancing and corporate actions), earn securities-lending income $\ell$ , and lose $\delta$ from dividend timing (the TRI reinvests a dividend on the ex-date; the fund receives cash later and earns $r_c$ on it in the meantime). Then, to first order,

TD \approx c + w_c\,(R_I - r_c) + \kappa - \ell + \delta - \textstyle\sum_i d_i R_i. \tag{6B.25}

Its expectation is $c + w_c\,\mathbb E[R_I - r_c] + \kappa - \ell + \delta$ , and tracking error comes from the random terms: approximately $TE^2 \approx w_c^2\sigma_I^2 + \sigma^2\big(\sum_i d_iR_i\big)$ plus the variance of the timing terms.

Proof

The gross return of the portfolio is $\sum_i [(1 - w_c)w_{I,i} + d_i]R_i + w_cr_c = (1 - w_c)R_I + \sum_i d_iR_i + w_cr_c$ , using $R_I = \sum_i w_{I,i}R_i$ . Subtract expenses and trading costs, add lending income, and subtract the dividend-timing loss: $R_F = R_I - w_c(R_I - r_c) + \sum_i d_iR_i - c - \kappa + \ell - \delta$ . Rearrange for $R_I - R_F$ . "First order" means ignoring products of small terms (compounding within the period).∎

Worked example 6B.11

A Nifty 50 index fund with $c = 0.20\%$ , an average cash balance of 1% while the index earns 6 points more than cash a year, trading costs of 0.05%, lending income of 0.02% and a dividend-timing loss of 0.02%: expected $TD \approx 0.20 + 0.06 + 0.05 - 0.02 + 0.02 = 0.31\%$ a year. With index volatility of 15%, the cash drag alone contributes $0.01 \times 15\% = 0.15\%$ of tracking error. The fund's expense ratio is two-thirds of its tracking difference; the rest is operations.

The cost of index changes#

Index changes and front-running When an index adds a stock, every fund tracking it must buy at the close of the effective date. The change is announced weeks ahead, so other traders buy first and sell to the index funds at the close: the price rises before inclusion and partly reverses after. Petajisto (2011) estimated that this "index premium" costs US index funds a material amount a year for the S&P 500 and more for the Russell 2000, a cost that appears in no expense ratio. The cost is larger for indices with mechanical, predictable rules and high turnover (small-cap and factor indices) and smaller for the Nifty 50. A fund can reduce it by trading some of the change before or after the effective date, at the price of tracking error.

Practitioner notes

Compare index funds on one-year and three-year tracking difference, not on the expense ratio alone. AMFI and AMC websites publish both TD and TE.
A tracking difference much below the expense ratio usually means lending income or luck; one much above it means cash drag, high flows or poor execution.
Factor and smart-beta index funds have higher turnover and therefore higher hidden costs than market-cap index funds; their TD understates the drag relative to a buy-and-hold portfolio because the index itself bears the rebalancing cost.