While classical regression provides a valuable picture of the “average” outcome, averages are dangerously misleading in a world defined by asymmetric risks and opportunities. A portfolio’s true character is revealed not in calm markets, but in crises and bubbles. Therefore, our methodology moves beyond the mean to model the entire distribution of outcomes, with a particular focus on the tails, where risk is most acute.

To achieve this deeper level of insight, we employ quantile regression. Unlike classical regression, which minimizes the sum of squared errors to find the single best-fit line for the mean, quantile regression minimizes a weighted sum of absolute deviations. This allows us to model any specific quantile of the distribution, from the 10th percentile (representing poor outcomes) to the 90th percentile (representing strong outcomes).

4.1 Application: Analyzing Investment Style Across Market Regimes

We use quantile regression to stress-test a manager’s strategy across market regimes. By analyzing performance at the 10th, 50th, and 90th percentiles, we uncover whether their stated style holds firm during periods of market stress or if it opportunistically deviates, exposing clients to unintended risks.

By regressing a fund’s returns on various style indexes at different quantiles, we can answer a critical question: Does the manager’s strategy change during periods of market stress or exuberance? If the estimated coefficients on the style indexes are statistically different across the various quantiles—a hypothesis that can be formally tested using a Wald test—it implies that the manager’s style is not static. This analysis provides a more nuanced view of a manager’s true behavior than a simple mean-based regression ever could.

4.2 Application: Value-at-Risk (VaR) Estimation

This makes quantile regression the natural and superior tool for Value-at-Risk estimation. VaR is fundamentally a quantile problem: it seeks to identify the maximum potential loss a portfolio might experience at a given confidence level. Rather than relying on flawed assumptions of normality, we model the 5th or 1st percentile of the return distribution directly, providing a more robust and realistic assessment of potential downside risk. As noted by researchers Engle and Manganelli, this approach can produce more accurate VaR estimates, particularly when returns exhibit non-normal characteristics like skewness and fat tails.

By incorporating these advanced techniques, we build more robust and nuanced portfolios. This rigorous analytical process is protected by a strict framework for model validation and implementation.