Standard portfolio constraints can lead to a 1.3–1.4 times underestimation of ex post active risk in optimized equity portfolios. The bias arises from systematic risk exposures caused by the constraints that are not included in the risk model. The authors successfully demonstrate a methodology for eliminating such exposures and the resulting biases, and in doing so, they are able to achieve a higher realized efficient frontier.
Factor alignment problems (FAPs) between alpha model factors and those captured in a risk model are well documented in the quantitative equity investing literature. FAPs can lead to unrecognized systematic risk exposures and a subsequent underestimation of ex post active risk. The authors extend the literature to consider how constraints applied within the optimization may do the same. They test across both small- and large-cap stock universes and use different risk models. They present two theorems that underlie the empirical results and propose a method of suppressing misalignment between constraints and the risk model.
How Is This Research Useful to Practitioners?
In considering the impact of constraints, the authors simulate a small-cap strategy with growth and short-term momentum as the only alpha factors, both of which exist in a risk model and thus contribute nothing to factor misalignment. The optimization includes constraints on sector and industry exposure as well as on active position bounds. The realized active risk is a highly significant 1.2–1.3 times the ex ante risk, although the ex ante total risk is unbiased. The industry and sector constraints are included in the risk model. Therefore, the source of the bias is solely from systematic risk arising from the long-only and active bet constraints. Such constraints curtail the extreme decile positioning. Furthermore, such positioning is shown to have quite different characteristics compared with the broader alpha itself, and it is not captured in the risk model.
The authors augment the constraints from their standard setup with bounds on portfolio turnover, transaction costs, market impact, and average daily volume as well as by halving the active deviation limits for the least liquid quartile of stocks. The ex ante active risk prediction is further biased upward by as much as 1.3–1.4 times, with the bias again peaking at around a 1% active risk target. The results are found to be robust when the authors switch to a large-cap stock universe and medium-term alpha factors and risk model.
Finally, they present the alpha alignment factor (AAF) approach to portfolio optimization. It adds a penalty to the optimization to suppress systematic risk that arises from constraint-driven factor misalignment. Using a broad market universe of stocks with complete alpha and risk model alignment, the authors show that their approach generates superior realized efficient frontiers compared with standard mean–variance optimization (MVO) regardless of the risk target. It also reduces the ex ante risk bias to insignificance at the 95% level.
How Did the Authors Conduct This Research?
The authors run monthly backtests for the period of 1998–2010 using the small-cap S&P 600 Index universe. The optimization relies on a short-horizon risk model developed by Axioma, where two of the authors are employed, with risk targets ranging between 0.5% and 3%. In robustness testing, the large-cap S&P 500 Index is used together with Axioma’s medium-horizon risk model. In the AAF testing, the Russell 3000 Index universe is used for monthly back tests for the period of 1999–2010 with an augmented version of Axioma’s medium-horizon risk model.
The authors present two important theoretical results. First, they show the solution to the constrained MVO is the solution to the unconstrained MVO in alpha space tilted toward the solution to an unconstrained MVO in constraint space. The degree of tilt is proportional to the degree of both constraint violation and risk aversion, consistent with the observation of the greatest bias being for ordinary risk targets. High-risk targets have a low-risk aversion parameter, and low-risk targets have little constraint violation. The second theorem demonstrates the interchangeability of alphas and constraints within the constrained MVO. The concept highlights how constraint-driven FAPs have as much potential to bias risk predictions as do their alpha-driven counterparts.
The authors make a convincing case for the importance of the problem of misalignment between constraints and the risk model in MVO, with significant downward biases observed in the ex ante risk. The bias is likely to be widespread among quantitative equity managers because most common constraints can induce it, so raising its profile is worthy in itself. The authors’ solution is a powerful way to alleviate the problem, and although it may be overly technical for widespread manager-level implementation, it would certainly have the potential to improve portfolio outcomes if included in standard portfolio optimization packages.