Tracking Error Formula
Tracking error is the standard deviation of the difference between a portfolio's returns and its benchmark's returns. A passive index fund aims for near-zero tracking error; an active manager accepts higher tracking error in pursuit of excess return. It is the denominator of the information ratio.
Formula
Copy the exact expression or work through it step by step below.
TE = stdev(R_p - R_b) = sqrt( (1/(n-1)) x sum_i ( (R_p,i - R_b,i) - mean_active )^2 )
Annualized: TE_annual = TE_period x sqrt(N) Variables
R_p,i - R_b,i
Active return in period i
Portfolio return minus benchmark return for each period. The series of these differences is what tracking error summarizes.
mean_active
Mean active return
Average of the active returns over the sample. Tracking error measures the dispersion of active returns around this mean, not around zero.
n
Number of observations
Count of return periods. The n-1 divisor gives the sample standard deviation, the standard convention for an estimate from data.
N
Periods per year
Annualization factor (252 daily, 52 weekly, 12 monthly). Tracking error scales by the square root of N like any volatility.
Step By Step
- 1
Compute the active return for each period as portfolio minus benchmark return.
Active returns over four months: +0.5%, -0.2%, +0.8%, -0.1%.
- 2
Find the mean of the active returns.
Mean = (0.5 - 0.2 + 0.8 - 0.1)/4 = 1.0/4 = 0.25%.
- 3
Compute each active return's deviation from that mean, square it, and sum the squares.
Deviations: +0.25, -0.45, +0.55, -0.35 (in percent); squares 0.0625, 0.2025, 0.3025, 0.1225; sum = 0.69 (percent-squared).
- 4
Divide by n-1 and take the square root for the per-period tracking error, then annualize.
0.69 / 3 = 0.23; sqrt(0.23) = 0.4796% per month; annualized 0.4796% x sqrt(12) = 1.66%.
Worked Example
Enhanced index fund, four monthly active returns
Active returns
+0.5%, -0.2%, +0.8%, -0.1%
Periods per year
12
Mean active = 0.25%. Deviations from mean (in %): 0.25, -0.45, 0.55, -0.35. Squared: 0.0625, 0.2025, 0.3025, 0.1225, summing to 0.69. Sample variance = 0.69 / (4-1) = 0.23 (percent-squared). Monthly TE = sqrt(0.23) = 0.4796%. Annualized = 0.4796% x sqrt(12) = 0.4796% x 3.464 = 1.66%.
Annualized tracking error of about 1.66%. This sits in the typical range for an enhanced-index strategy: tight enough to stay close to the benchmark, but loose enough to express modest active views. Pure passive funds usually report tracking error well under 0.5%.
Common Variations
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Sources & References
- Active Portfolio Management — Richard C. Grinold and Ronald N. Kahn, McGraw-Hill (1999)
- Practical Portfolio Performance Measurement and Attribution — Carl R. Bacon, Wiley (2008)
Related Content
Keep the topic connected
Information Ratio Formula
The information ratio formula: active return over a benchmark divided by tracking error. Measures the consistency of excess return, with an example.
Annualized Volatility Formula
The annualized volatility formula scales per-period return standard deviation by the square root of periods per year. Root-time scaling explained.
R-Squared Formula
The R-squared formula: the share of a portfolio's return variance explained by its benchmark. The squared correlation behind beta and alpha.
Beta Formula
The beta formula: covariance of asset and market returns over market variance. The CAPM regression slope measuring systematic risk, with an example.