Construction of optimized portfolios entails complex interaction between three key entities, namely, the risk factors, the alpha factors and the constraints. The problems that arise due to mutual misalignment between these three entities are collectively referred to as Factor Alignment Problems (FAP). Examples of FAP include risk-underestimation of optimized portfolios, undesirable exposures to factors with hidden and unaccounted systematic risk, consistent failure in achieving ex-ante performance targets, and inability to harvest high quality alphas into above-average IR. In this paper, we give a detailed analysis of FAP and discuss solution approaches based on augmenting the user risk model with a single additional factor y. For the case of unconstrained MVO problems, we develop a generic analytical framework to analyze the ex-post utility function of the corresponding optimal portfolios, derive a closed form expression of the optimal factor volatility value and compare the solutions for various choices of y culminating with a closed form expression for the optimal choice of y. Ultimately, we show how the Alpha Alignment Factor (AAF) approach emerges as a natural and effective remedy to FAP. AAF not only corrects for risk underestimation bias of optimal portfolios but also pushes the ex-post efficient frontier upwards thereby empowering a PM to access portfolios that lie above the traditional risk-return frontier.

Axioma Research Paper No. 022

The underestimation of risk of optimized portfolios is a consistent criticism about risk models. Quantitative portfolio managers have historically used a variety of ad hoc techniques to overcome this issue in their investment processes. In this paper, we construct a theory explaining why risk models underestimate the risk of optimized portfolios. We show that the problem is not necessarily with a risk model, but is rather the interaction of expected returns, constraints, and a risk model in an optimizer. We develop an optimization technique that incorporates a dynamic Alpha Alignment Factor (AAF) into the factor risk model during the optimization process. Using actual portfolio manager backtests, we illustrate both how pervasive the underestimation problem can be and the effectiveness of the proposed AAF in correcting the bias of the risk estimates of optimized portfolios.

Axioma Research Paper No. 015

Constraints are now an integral part of the portfolio construction process. With constraints comes the challenge of understanding how they cause the optimal portfolios to deviate from a trade-off dictated by the forecasts of risk and return. We describe the theory and application of a technique able to quantify the impact of individual constraints in several different ways. This includes decomposing the difference between the optimal constrained and unconstrained portfolios and the difference between alphas and implied alphas as described in earlier work by Grinold and others. Furthermore, we introduce a new technique that applies these decompositions on an ex-post basis, providing on understanding of how constraints actually impact realized risk and return.

Axioma Research Paper No. 014

When trades from separately managed accounts are pooled for execution, the realized market-impact cost can be far greater than the sum of the predicted cost over all accounts. Multi-portfolio optimization is a technique for rebalancing multiple portfolios at the same time, considering their joint effects while adhering to account-specific constraints. The interaction of accounts in a multi-portfolio setting can bias particular accounts if fairness is not considered in the solution methodology. With respect to the trading of multiple accounts, fairness is not well-defined. Definitions vary among portfolio managers often based on their particular investment offering. For this reason, we do not prescribe a single best approach for multi-portfolio optimization. Instead, we discuss the pros and cons of two approaches that each has foundations in economic theory, the Cournot-Nash equilibrium and the collusive solution. We present a unified framework capable of solving either problem.

Axioma Research Paper No. 013

Risk model providers commonly report the average R2 value of the asset returns model. Some models, such as statistical models, will consistently have greater R2 values than others. However, strong explanatory power from a returns model does not necessarily translate into an accurate risk model. The ultimate test of a risk model lies in the testing of its risk forecast against realized values.

Research Report No. 012

Daily Risk Changes in September 2008

In this short article, we report some of the dramatic risk changes that occurred during September 2008, and quantify the inaccuracies that occurred with a monthly risk model. The results show that stale risk models seriously misestimated risk during the second half of September. In times like these, there is no justification for using anything other than a risk model that is updated daily.

Author: Anthony Renshaw - PhD

Research Report No. 011

This short article analyzes the market situation on September 15, 2008, using Axioma US fundamental risk model.

In addition to identifying the factors that drove returns that day, the results highlight recent changes in market conditions that may be important to portfolio managers.

Author: Anthony Renshaw - PhD

Research Report No. 010

Typically, the number of factors used in an alpha or risk model is much lower than the number of assets in the portfolio, or in the investable universe. This short note studies the consequence of this “dimensionality gap” and shows how Axioma’s Alpha Factor™ method can limit the potential impact of the difference between the asset space and the factor space to produce improved realized portfolio performance.

Author: Anthony Renshaw - PhD

Research Report No. 009

Using more than one risk model in a portfolio construction strategy allows a portfolio manager to exploit the fact that different risk models measure and capture risk differently. Having both a fundamental and statistical risk model simultaneously in the strategy ensures that the optimized portfolio reflects both points of view. Two risk model strategies can produce just as conservative portfolios and better overall performance than one risk model alone provided that the strategies are calibrated so that both risk models affect the optimal portfolio solution.

Author: Anthony Renshaw - PhD

Research Report No. 008

Robust portfolio optimization has been a core feature of Axioma’s portfolio construction tools for years, and many of our clients use robust optimization in their portfolio construction processes to deliver higher value added. This study reports a series of real world portfolio construction case studies documenting different approaches for implementing robust portfolio optimization and their benefits. The results provide guidance for designing robust portfolio construction strategies.

Author: Anthony Renshaw - PhD

Research Report No. 007