# Which objective function should I choose to minimize tracking error?

Let say I have $n$ assets and their returns over $m$ periods which are represented by a matrix $X \in \mathbb{R}^{m \times n}$, and I have some other asset with return over the same period which is represented by a vector $y \in \mathbb{R}^m$.

My objective is to find a vector of weights $w$ such that

$$w^* = \underset{w}{\arg \min} ~ \text{TE}(w)$$

where $\text{TE}(w)$ is the tracking error defined as follows:

$$\text{TE}(w) = \sqrt{\text{Var}(Xw - y)}$$

and

$$\sum_{i=1}^n w_i =1$$

In short, I want to replicate $y$ using a portfolio of assets $X$.

My idea was to use the exact definition of the tracking error mentioned above through an optimizer.

However, somebody suggested to use the following:

$$w^* = \underset{w}{\arg \min} ~ \sum_{i=1}^m (Xw-y)_i^2$$

I tried both and I get a better tracking error with the first one.

It seems clear to me that both should return exactly the same if indeed there exists some $w$ which perfectly replicates $y$.

What if it's not the case?

Is there another approach?

• Because $Var\left(x\right)=E\left(x^{2}\right)-E\left(x\right)^{2}$ – John Oct 10 '12 at 17:25