Tracking Error

Formula

a_t = R_port,t − R_bench,t
TE  = std(a, ddof=1) · sqrt(P)

P defaults to 252. Reported as a decimal (0.02 means 2 percent annualised), not in basis points.
The information ratio is mean(a) / std(a) · sqrt(P), so IR = annualised mean active return / TE.

Where it shows up

• Index tracking and closet indexing. A mandate that says "stay within 2 percent tracking error" is a constraint on this number, usually estimated on an ex-ante covariance as well as measured ex post. This function is the ex-post measurement.
• Next to the information ratio, which is undefined when tracking error is zero.
• Beta is a different summary of the same pair of series: a slope, not the volatility of the residual gap. Two portfolios can share a beta and have very different tracking errors.

Pitfalls

• Ex post is not ex ante. The standard deviation of realised active returns is not the tracking error a covariance model would predict for the current holdings.
• Annualisation. Multiplying a daily active-return standard deviation by sqrt(252) assumes uncorrelated active returns. A persistent active bet (a permanent sector tilt) has a tracking error that does not scale that way if you change the sampling interval without care. Be consistent about P.
• Levels versus returns. Tracking error is defined on returns. A price spread between an ETF and its basket is a different measurement.

Python reference implementation

Requires numpy. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download tracking-error.py · All snippets

import numpy as np

def tracking_error(port, bench, periods=252) -> float:
    """Annualised sample standard deviation of port minus bench."""
    active = np.asarray(port, dtype=float) - np.asarray(bench, dtype=float)
    if active.ndim != 1 or active.size < 2 or not np.all(np.isfinite(active)):
        raise ValueError("need at least two finite paired returns")
    if periods <= 0:
        raise ValueError("periods must be positive")
    return float(active.std(ddof=1) * np.sqrt(periods))

FAQ

Can tracking error be zero while the portfolio differs from the benchmark?

Yes, if the difference in returns is constant. Standard deviation does not see the mean. A constant lag of one basis point per day is a large cumulative gap and a zero tracking error.

Is tracking error the same as active risk?

In mandate language, active risk usually means this volatility of active returns, sometimes the ex-ante forecast of it. Quote whether the number is realised or predicted. This function is realised.