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028A fund's annual volatility is 18%, its benchmark's is 16%, and the correlation between their returns is 0.95. What is the fund's tracking error?Correlation and diversificationCoreMSCIMonterrey · 2013
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Quick instinct: roughly how big is the tracking error?
Show the worked solution
About 5.73%. Tracking error is the volatility of the fund's return minus the benchmark's. Its variance is 18 squared plus 16 squared minus 2 x 0.95 x 18 x 16, which is 324 plus 256 minus 547.2, or 32.8. The square root is 5.73%, nearly three times the 2-point gap in volatilities.
What exactly is tracking error measuring?
Two friends walk to the same office. How far apart they are at any moment depends less on how fast each walks than on whether they take the same streets. Tracking error is the volatility of the return difference, fund minus benchmark, so it depends on how much the two move apart, not on how much each moves. The fund and index can both swing wildly and still track closely if they swing together. That is why the formula needs the correlationA number from minus 1 to 1 describing how closely two returns move together; 1 means perfect lockstep., not just the two volatilities.
The relationship
$TE = \sqrt{\sigma_F^2 + \sigma_B^2 - 2\rho\,\sigma_F\sigma_B} = \sqrt{324 + 256 - 547.2} = \sqrt{32.8} = 5.73\%$
\sigma_F, \sigma_B fund and benchmark volatility, 18% and 16%
\rho correlation of their returns, 0.95
What it says in wordsThe variance of a difference is the two variances added, less twice the part they share.
Drawing the two volatilities as sides 18 and 16 at the angle whose cosine is 0.95 makes tracking error the short third side, 5.73%; nudging correlation from 0.95 to 0.99 cuts it to 3.12%, and dropping it to 0.90 raises it to 7.85%.
Why does the correlation matter more than the volatilities?
Look at the table in the figure. Keeping 18 and 16 fixed, moving correlation from 0.99 to 0.90 takes tracking error from 3.12% to 7.85%, more than doubling it. At high correlations each hundredth of correlation moves tracking error a lot, because the large shared term 2 x rho x 18 x 16 almost cancels the two variances. Now hold correlation at 0.95 and give both sides 16% volatility: tracking error is still 5.06%. The volatility gap contributes a little; the imperfect correlation contributes most.
Close with the limit. The formula uses a correlation estimated from history, and correlations drift, often falling in stressed markets. A fund reporting 5.7% tracking error in calm years can run well above it in a sell-off, so a risk team watches realised tracking error alongside the model figure.
Where candidates lose it
The fast wrong answer is 2%, subtracting the volatilities. It silently assumes correlation of exactly 1, which the question has just told you is false. Candidates who say it have treated volatility as if it were a return.
The second trap is fumbling the formula under pressure. Anchor it to one line you already know: the variance of A minus B is var A plus var B minus twice the covariance. Everything else follows.
What the interviewer asks next
What correlation would give a tracking error of exactly 2%?
The fund's beta to the benchmark is 1.07. Split the tracking error into a beta part and a residual part.
Why might a fund with low tracking error still underperform its benchmark every year?
Asked at MSCI, Financial Tools, Monterrey, 2013 (Wall Street Oasis): What's the tracking error formula?

\sigma_F, \sigma_B	fund and benchmark volatility, 18% and 16%
\rho	correlation of their returns, 0.95

Try it first

What exactly is tracking error measuring?

Why does the correlation matter more than the volatilities?

Where candidates lose it

What the interviewer asks next