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Using Regression Coefficients as Benchmark Portfolio Weights

Article Quant Q&A · Author: SRKX
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Summary

A regression of benchmark returns on fund returns can provide portfolio weights that minimize tracking error when the regression is fit without coefficient constraints. The coefficients need not sum to one, so their total has a direct funding implication: if they sum to less than one, the difference can be held in cash; if they sum to more than one, the portfolio requires leverage.

When leverage or other weight limits apply, the unconstrained regression no longer necessarily gives an implementable solution. The response recommends solving a bounded quadratic optimization problem for the coefficients. This is a concise conceptual answer and does not specify estimation choices, transaction costs, rebalancing, or how to handle constraints beyond coefficient bounds.

Key ideas

  • Unconstrained regression coefficients can serve as tracking-error-minimizing weights for the included funds.
  • If the coefficients sum to less than one, the residual allocation can be held in cash.
  • A coefficient sum above one implies leverage is needed to use the regression weights.
  • Weight or leverage constraints call for a bounded quadratic optimization.

Tags

Portfolio constructionStatisticsRisk managementBacktesting
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# How to compute portfolio weights from multivariate regression results?


# How to compute portfolio weights from multivariate regression results?












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.

## Answer by Tal Fishman (score 4, accepted)

https://quant.stackexchange.com/a/3834

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.