What Is Tracking Error? Formula and Example

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Tracking error is the standard deviation of the difference between a portfolio’s returns and its benchmark’s returns over a set period. It measures how consistently a portfolio moves with its benchmark, not how far ahead or behind it finished. An index fund that trails its index by the same small amount every period has very low tracking error. A fund that beats the index one year and lags it the next has high tracking error, even if its average return matches the index exactly.

The measure is used in two settings. For passive products such as index mutual funds and exchange-traded funds, it shows how faithfully the fund replicates its index. For active managers, it measures how much risk the manager takes by departing from the benchmark. In the active setting it is often called active risk.

How Tracking Error Works

The Tracking Error Formula

Start with a series of portfolio returns (P) and benchmark returns (B) measured over the same periods: daily, weekly, monthly or annually. For each period, compute the active return:

Active return (period t) = P(t) − B(t)

Tracking error is the standard deviation of that series:

Tracking error = √[ Σ (Dₜ − D̄)² / (n − 1) ]

where Dₜ is the active return in period t, D̄ is the average active return, and n is the number of periods. Dividing by n − 1 gives the sample standard deviation, the usual choice. Some sources divide by n (the population version), which gives a slightly lower figure for short histories.

A second definition uses the root mean square of active returns, without subtracting the average:

Tracking error (RMS) = √[ Σ Dₜ² / n ]

The standard-deviation version captures only fluctuation around a steady gap. The RMS version counts both the fluctuation and any persistent gap, so a fund that lags by a constant amount every period has a nonzero RMS figure but a standard deviation of zero. Check which definition a published figure uses.

Annualizing Tracking Error

Tracking error is normally quoted on an annual basis, even when it is calculated from monthly or daily returns. The standard conversion multiplies by the square root of the number of periods in a year:

• Monthly data: annual tracking error = monthly tracking error × √12
• Weekly data: × √52
• Daily data: × √252 (trading days)

Hypothetical example: a monthly tracking error of 0.30% annualizes to about 0.30% × 3.464 = 1.04%. This scaling assumes active returns are independent from one period to the next. If they are autocorrelated, as can happen when a fund’s holdings are priced with a lag, the square-root rule misstates the annual figure.

Interpreting the Number

If active returns are roughly normally distributed, about two-thirds of periods should fall within one tracking error of the average active return, and about 95% within two.

Hypothetical example: an active fund has an average active return of +1% per year and an annual tracking error of 3%. Its annual return would land between 2% below and 4% above the benchmark in roughly two out of three years, and between 5% below and 7% above in roughly 19 out of 20.

Tracking error is never negative. The sign of the average active return, not tracking error, shows whether the portfolio beat or lagged its benchmark.

Example of Calculating Tracking Error

Suppose a large-cap fund benchmarked to the S&P 500 posted these hypothetical annual returns over five years:

Step 1: average active return. (−0.6 − 0.6 + 0.7 − 0.8 − 0.5) / 5 = −1.8 / 5 = −0.36%. On average, the fund trailed its benchmark by 0.36 percentage points a year.

Step 2: deviations from the average. −0.24, −0.24, +1.06, −0.44, −0.14.

Step 3: square and sum. 0.0576 + 0.0576 + 1.1236 + 0.1936 + 0.0196 = 1.452.

Step 4: divide and take the square root.

• Sample version: √(1.452 / 4) = √0.363 ≈ 0.60%
• Population version: √(1.452 / 5) = √0.2904 ≈ 0.54%

Step 5 (optional): RMS version. The squared active returns are 0.36, 0.36, 0.49, 0.64 and 0.25, which sum to 2.10. √(2.10 / 5) = √0.42 ≈ 0.65%.

The fund’s annual tracking error is about 0.60% on the sample basis. Year 3, the only year the fund outperformed, drives most of it. The average gap of −0.36% alone would hide that the fund’s relationship with its index was less stable than four of the five years suggest.

In a spreadsheet, apply STDEV.S (or STDEV.P for the population version) to a column of active returns. Five annual observations are a thin sample. Analysts usually use at least three years of monthly returns, which gives 36 data points.

Tracking Error vs. Tracking Difference, Active Risk and Beta

Tracking Error vs. Tracking Difference

Tracking difference is the gap between a fund’s return and its index’s return over a period, such as −0.20% for a year. It shows how much the investor actually gained or lost relative to the index. Tracking error shows how much that gap fluctuated from period to period.

Hypothetical example: Fund A lags its index by exactly 0.20% every year, roughly its expense ratio. Its tracking difference is −0.20% a year and its standard-deviation tracking error is zero. Fund B lags by 0.05% one year and beats the index by 0.05% the next. Its average tracking difference is near zero, but its tracking error is higher than Fund A’s. A long-term holder pays Fund A’s lag every year, while Fund B’s swings mostly cancel out, so comparing index funds on tracking error alone can point to the more expensive product.

Tracking Error vs. Active Risk

In most usage, active risk and tracking error mean the same thing: the standard deviation of active returns. Some writers reserve “active risk” for the forward-looking, model-based estimate and “realized tracking error” for the historical figure. The math is the same; only the inputs differ.

Tracking Error vs. Beta

Beta measures how sensitive a portfolio is to benchmark moves; tracking error measures how much its return differs from the benchmark’s. A beta near 1.0 with holdings close to the index usually means low tracking error, but it does not guarantee it: a concentrated portfolio can have a beta of 1.0 and still deviate sharply because of stock-specific moves. A portfolio with a beta of 0.5 shows substantial tracking error simply because it moves half as much as the benchmark.

Tracking Error vs. Standard Deviation of Returns

A fund’s standard deviation of returns measures its total volatility. Tracking error measures its relative volatility against a chosen benchmark. An emerging-markets index fund can be very volatile yet have tiny tracking error, because it is volatile in step with its index.

Ex-Post vs. Ex-Ante Tracking Error

Ex-post tracking error, also called realized or historical tracking error, is the backward-looking figure calculated above from actual returns. It is used for performance evaluation.

Ex-ante tracking error, also called predicted or forecast tracking error, estimates how much the current portfolio is likely to deviate from the benchmark. It is built from active weights, the portfolio weight minus the benchmark weight for each security or risk factor, combined with a covariance matrix of returns:

Ex-ante tracking error = √[ (wₚ − w_b)ᵀ Σ (wₚ − w_b) ]

where wₚ and w_b are the portfolio and benchmark weight vectors and Σ is the covariance matrix of the securities or factors. If every active weight were zero, predicted tracking error would be zero. Each overweight or underweight adds to it, and correlations determine whether those bets reinforce or offset each other.

The two serve different jobs. Ex-post shows how the manager actually behaved over the whole window, including positions no longer held. Ex-ante shows how much relative risk the current portfolio carries and reacts immediately to new positions, but its covariance estimates come from historical data and can understate risk when correlations shift in a market shock.

Quantitative managers often build portfolios around a target ex-ante tracking error, choosing exposures to style, size, momentum or leverage factors so predicted deviation stays within a set limit. Institutional mandates commonly set such a limit, often called a tracking error budget.

What Causes Tracking Error in ETFs and Mutual Funds

An index is a calculation, not a portfolio. It pays no fees, holds no cash and trades without cost, so every real fund that replicates it runs into frictions. Sector, international and dividend funds face more of these frictions and tend to show higher tracking error than broad equity and bond funds.

Fees and Expenses

Management fees are the most direct cause of a gap. The expense ratio is deducted continuously, so a fund’s net asset value tends to run below the index by roughly that amount. Because the fee is steady, it mainly widens tracking difference rather than tracking error. Securities lending income or efficient trading can partly offset it.

Sampling and Optimization

Buying every thinly traded security in an index at index weights could push prices up and add trading costs. Many funds instead hold a representative sample of more liquid securities chosen to mimic the index, a technique called sampling or optimization. Returns from the missing securities show up as tracking error, and illiquid holdings add more through wide bid-ask spreads and stale prices.

Diversification Constraints

Diversified funds under the U.S. Investment Company Act of 1940 must keep, for at least 75% of total assets, no more than 5% in any single issuer. Tax rules for regulated investment companies add separate tests. An index dominated by a few very large companies, common in some sector indexes, can require weights a fund cannot legally hold, forcing it to underweight those names.

Cash Drag

Indexes are always fully invested; funds are not. Dividends, interest, inflows and trading proceeds sit as cash until reinvested, earning a different return from the index. High-dividend funds are especially exposed to this cash drag. Open-end mutual funds also hold cash for redemptions, while ETFs typically meet large redemptions through in-kind exchanges with authorized participants.

Index Reconstitution and Rebalancing

When an index adds, removes or reweights constituents, the fund must trade to follow, paying costs and possibly missing the prices the index uses. Indexes with heavy turnover from mergers, spin-offs and rule-based reshuffles produce more of this error.

Capital-Gains Distributions

Funds that sell appreciated securities to follow index changes or meet redemptions may distribute taxable capital gains. ETFs do so less often than mutual funds because of in-kind redemptions, but they are not immune. Pre-tax tracking figures miss this, but after tax it makes the investor’s experience diverge from the index’s.

Securities Lending

Some funds lend securities to short sellers for a fee. The income can offset part of the expense ratio, but it adds a return stream the index lacks, which can raise or lower measured tracking error.

Premiums and Discounts to NAV

An ETF’s market price can trade at a premium or discount to the value of its underlying basket. Authorized participants usually arbitrage the gap away by creating or redeeming shares, but it can be large for thinly traded ETFs. Tracking error calculated on market price can therefore differ from tracking error on NAV, and an investor’s experience depends on the prices paid and received.

Currency Hedging

Currency-hedged international funds use forward contracts to neutralize exchange-rate moves. The cost of rolling those forwards depends largely on interest-rate differentials between the two currencies and on market volatility. That cost is not always captured in the expense ratio, and it can push the fund away from its hedged index.

Futures Roll

Many commodity funds hold futures rather than the physical commodity and, typically each month, sell expiring contracts and buy later-dated ones. This is the roll. In contango, when later contracts cost more, each roll buys exposure at a higher price, and the fund can lose value even when the spot price is flat or rising slightly. In backwardation, the roll adds return. Either way, performance can drift far from the spot price many investors treat as the benchmark.

Daily Leverage Resets

Leveraged and inverse ETFs use swaps, futures and forwards to deliver a multiple, such as 2x or −1x, of an index’s daily return, rebalancing every day. Over longer periods, compounding and volatility make their returns differ from that multiple of the index’s return, so they can show large tracking error by design.

How Tracking Error Is Used

• Choosing index funds. Among funds tracking the same index, low tracking error together with a small tracking difference over the same window signals tight replication.
• Evaluating active managers. For an active fund, tracking error is a budget the manager spends in pursuit of excess return. High tracking error with weak average active return means a lot of risk for little reward. Very low tracking error with active fees may signal a “closet indexer.” Short periods of high tracking error with strong returns can reverse, so a longer track record is more reliable.
• Measuring skill per unit of risk. The information ratio divides average active return by tracking error. It is the relative-return counterpart to the Sharpe ratio, showing how much excess return, or alpha, a manager generated per unit of active risk.

Limitations of Tracking Error

• It is symmetric. Outperformance and underperformance raise it equally, so the number alone cannot say whether deviation helped or hurt.
• It depends on choices. Window length, return frequency, sample or population formula and annualization method all change the result, so published figures are comparable only when these match.
• It is only as good as the benchmark. A poorly matched benchmark can inflate the figure, and a well-matched one can still hide concentrated risk the benchmark shares.
• History may not repeat. Ex-post figures describe one period’s conditions. Ex-ante covariance estimates can break down when correlations change suddenly.
• Normality is an assumption. Fat tails or one extreme period, like Year 3 in the example above, distort the probability-band reading.

FAQs

The Bottom Line

Index fund investors should judge funds on tracking error and tracking difference together, since a steady fee drag can hide behind low tracking error. For active managers, tracking error works as a risk budget, best weighed against average active return over a long record. Published figures are comparable only when window, return frequency, formula and annualization method match.