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Assuming that I performed a multivariate regression and I found a set of $k$ coefficients $\alpha_1, ..., \alpha_k$ for each of the factors $F_1, ... F_k$. I have then computed the following relationship:

$$y_t = \sum_{i=1}^k \alpha_i F_{i,t} + \epsilon_t$$

So far, I only looked at this equation to understand where the risk is coming from.

I was wondering if we could use this approach to create a portfolio made of funds $F_i$ which aims to match a benchmark (and hence $y$ are the returns of the benchmark).

We cannot use the correlation coefficient directly, as we do not have $\sum_{i=1}^k \alpha_i=1$.

I believe there is no real way to convert this result into a portfolio, am I right?

Note: I would normally use an optimizer that minimize the tracking error, I am just wondering if this approach is possible.

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I am a bit confused by your question. What would be the goal of this portfolio? What sort of application do you have in mind? – Tal Fishman Jul 19 '12 at 13:49
Right sorry it's unclear. $y$ are the historical returns of a benchmark, $F$ are historical returns of funds. I was thinking about using this approach to minimize tracking error. I'll edit the question. – SRKX Jul 19 '12 at 14:01
You'll need to estimate the joint distribution of $F$ and $\epsilon$, project them to the horizon, convert to linear returns (if you're dealing with logs) and then perform a benchmark relative optimization. – John Jul 19 '12 at 14:32
Isn't a lot of the explanatory power for y going to live in the error term from the multivariate regression and therefore effectively do the same as just solving the numerical relationship by minimizing the tracking error? What were you hoping to gain with this? – Hansi Jul 24 '12 at 22:18
@Hansi I was just experimenting this approach. – SRKX Jul 25 '12 at 7:03
up vote 3 down vote accepted

If $\sum_{i=1}^k \alpha_i<1$, then you could just leave the remainder of the portfolio in cash. If $\sum_{i=1}^k \alpha_i>1$, that means you will have to take on some leverage in order to minimize tracking error. If you have a leverage constraint, then you can run this as a quadratic program with bounds on your coefficients. A regression should give the same result as an optimizer with no bounds.

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