Tracking Error

August 24, 2024 Portfolio Management

Standard deviation of periodic portfolio-minus-benchmark returns, used to measure benchmark-relative or active risk.

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Tracking error is the standard deviation of periodic differences between portfolio returns and benchmark returns. It measures how variable benchmark-relative performance has been or is expected to be; it is not the same as the portfolio’s average underperformance or outperformance.

Key Takeaways

  • Tracking error measures the volatility of active returns, not a one-period return gap.
  • Tracking difference measures average or cumulative underperformance or outperformance over a stated period.
  • Portfolio and benchmark observations must use aligned dates, frequency, currency, valuation, and total-return conventions.
  • Low tracking error indicates consistency relative to the benchmark, not low absolute investment risk.

Formula

First calculate active return for each period:

$$ AR_t=R_{p,t}-R_{b,t} $$

Historical, or ex post, tracking error is commonly estimated as the sample standard deviation of those active returns:

$$ TE=s(AR_t)=\sqrt{\frac{\sum_{t=1}^{n}(AR_t-\overline{AR})^2}{n-1}} $$

Where (\overline{AR}) is mean active return and (n) is the number of aligned observations.

If periodic active returns satisfy the assumptions behind square-root-of-time scaling, tracking error may be annualized as:

$$ TE_{annual}\approx TE_{period}\sqrt{m} $$

Here, (m) is the number of observations per year, such as 12 for monthly data. The approximation can mislead when active returns are autocorrelated, exposures change, or valuations are smoothed.

Worked Example

Suppose a fund has six monthly active returns, stated in percentage points:

0.10%, -0.10%, 0.20%, -0.20%, 0.10%, -0.10%

Their mean is 0.00%, so the sample tracking error is:

$$ TE_{monthly}=\sqrt{\frac{0.10^2+(-0.10)^2+0.20^2+(-0.20)^2+0.10^2+(-0.10)^2}{6-1}}\%=0.155\% $$

Approximate annualized tracking error is:

$$ 0.155\%\sqrt{12}\approx0.54\% $$

This small sample is only an arithmetic illustration. A production estimate requires more observations and a disclosed methodology.

Tracking Error vs. Tracking Difference

If a fund returns 9% for a year while its benchmark returns 10%, the -1 percentage-point gap is a tracking difference or active return for that year. It is not enough information to calculate tracking error because tracking error requires a series of periodic return differences.

MeasureCalculationWhat it shows
Tracking differencePortfolio return minus benchmark return over a periodDirection and amount of relative performance
Tracking errorStandard deviation of periodic active returnsVariability or consistency of relative performance
Information RatioMean active return divided by tracking errorActive return earned per unit of active risk

A fund can have a persistent negative tracking difference but low tracking error if it trails its index by a similar amount each period, perhaps because of stable expenses. It can also have near-zero average tracking difference but high tracking error if positive and negative deviations offset over time.

Ex Post and Ex Ante Tracking Error

  • Ex post tracking error is calculated from realized active returns.
  • Ex ante tracking error is forecast using portfolio holdings, active weights, factor exposures, covariances, and a risk model.

The two figures answer different questions and need not match. Ex ante estimates depend on model assumptions and current holdings; ex post estimates depend on the historical sample.

What Causes Tracking Error?

For an index-tracking fund, sources can include:

  • fees and operating expenses not reflected in the index
  • sampling instead of full replication
  • transaction costs and index-rebalancing trades
  • cash balances, subscriptions, and redemptions
  • taxes or withholding differences
  • valuation timing, stale prices, and fair-value adjustments
  • currency hedging and different market closing times
  • securities lending income or implementation differences

For an active portfolio, intended security, sector, factor, duration, credit, or currency positions also create tracking error. Higher active risk is neither inherently good nor bad; it should be consistent with the mandate and risk budget.

How to Evaluate Tracking Error

There is no universal appropriate level. A passive fund generally seeks a narrow, stable relationship with its index, while an active strategy intentionally accepts benchmark-relative risk. In both cases, verify:

  1. benchmark suitability
  2. observation frequency and sample length
  3. arithmetic versus log returns
  4. gross-versus-net return treatment
  5. total-return, dividend, tax, currency, and valuation alignment
  6. sample-standard-deviation and annualization conventions
  7. whether the figure is ex post or ex ante

Tracking error must also be read with tracking difference. Consistent underperformance is economically important even if its variability is low.

  • Information Ratio: Uses tracking error as the denominator of benchmark-relative return efficiency.
  • Benchmark Index: Supplies the comparison return for each active-return observation.
  • Standard Deviation: Provides the statistical operation used to calculate historical tracking error.
  • Alpha: Measures benchmark- or model-relative value added rather than its periodic variability.
  • Beta: Measures sensitivity to benchmark movements, not dispersion around benchmark returns.

Sources

FAQs

Is lower tracking error always better?

No. Low tracking error is usually consistent with an index-tracking mandate, but an active strategy may deliberately accept more active risk. The appropriate level depends on the objective and benchmark.

Can tracking error be zero while a fund underperforms?

In theory, yes. If the fund trails the benchmark by exactly the same amount every period, active-return variability is zero even though tracking difference is negative.

Does tracking error measure absolute portfolio risk?

No. A portfolio and benchmark can both be highly volatile while staying close to each other. Tracking error measures only their relative-return variability.

This page is for financial education and does not recommend a fund, benchmark, manager, or active-risk target.

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