# Relationship between Tracking Error and Beta to benchmark

Analyzing an indexed portfolio, can we say there is any relationship between ex-ante TE and Beta to benchmark?

Tracking error is the volatility of the difference in returns between the portfolio and the benchmark.

Beta can be calculated as correl(portfolio, bmk) * ( vol portfolio / vol bmk).

I am trying to assess if a change in Beta would be systematically matched by a change in TE.

## 1 Answer

If you define tracking error as the volatility of the difference in returns between the portfolio and the benchmark, then the Beta of a portfolio needs to be 1 to have the best TE and deviations from 1 will cause an increase in TE over this optimal value.

If you define TE, as a few people do, as the SEE of a regression of portfolio returns on the benchmark, then this kind of TE is not affected by the beta of the portfolio. (The regression automatically adjusts for a different beta).

• Just to make it more explicit for the OP, if a model of returns is $R^{(p)}_t = \alpha + \beta R^{(b)}_t + \epsilon_t$ then the difference $R^{(p)}_t - R^{(b)}_t = \alpha + (\beta - 1) R^{(b)}_t + \epsilon_t$. If $\beta \neq 1$, non-zero benchmark returns $R^{(b)}_t$ will show up in the difference $R^{(p)}_t - R^{(b)}_t$. Furthermore $\operatorname{Var}(R^{(p)}_t - R^{(b)}_t) = (\beta-1)^2 \operatorname{Var}(R^{(b)}_t) + \operatorname{Var}(\epsilon_t)$. – Matthew Gunn May 22 '19 at 13:41