Which objective function should I choose to minimize tracking error?

Question

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?

Answer 1

When performing a tracking error optimization, you will obtain the same result by using the tracking error squared, which is just the variance of the relative portfolio weights. This would be just finding the minimum variance portfolio, but with conditions on the weights. For instance, it would be equivalent to instead set up the variance minimization assuming you have a fixed -100% weight on the benchmark and optimize with the overall sum of weights equal to zero.

For a short time horizon, if you can assume that the expected return is approximately zero, then your second formula is equivalent to minimizing the variance. When the time horizon is longer and it is no longer approximately true that the expected return on each asset is zero, then the second formula will not produce the same weights.

Answer 2

It is better to use a factor model, if one is available. Are you asking this question because you don't have access to one?

Also, what is the nature of the asset you want to track? Is it an index or a single security? What asset class? What risk factors is it exposed to (e.g. interest rate and credit risk vs. stock market volatility and other equity factors)? The answer to your question depends on what you are tracking.