# How to calculate annualised tracking error?

I have 36 months of relative returns and I need to calculate the annualised tracking error.

So, using 36 months of returns is it simply like below:

stdev(36 months of returns) * sqrt(12)


Why the sqrt(12)?

## 1 Answer

$\sqrt{12}$ annualizes monthly deviations.

But I don't understand why you measure tracking error with stdev. It should be $$ATE = \sqrt{\frac{12}{36}\sum_{i=1}^{36}(r_{b,i}-r_{t,i})^2}$$ where $r_{b,i}$ is benchmark return for month $i$ and $r_{t,i}$ is tracking portfolio return for same period. So you shouldn't substract average error inside square.

• This is correct, in particular, for ETFs. The scaling is needed for annualization. The same treatment is also employed for historical volatility estimation based on daily asset prices. – Gordon Nov 2 '15 at 14:42
• I think his "returns" are as indicated in the question "relative" returns so they correspond to $\bar{r}_i = r_{t,i} - r_{b,i}$, then he uses the approach from wiki $TE=\sqrt{\text{Var}(\bar{r}_i)}$ is that wrong? – SRKX Nov 3 '15 at 3:37
• I have also encountered tracking error formulas without the negative average excess return term. Which is it? – user1627466 Aug 9 '17 at 6:52
• SRKX, it's Ok but correct to get annualised value. – hvedrung Aug 10 '17 at 14:00