Research Papers

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.

Authors: Anureet Saxena, Robert A. Stubbs

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.

Authors: Robert A. Stubbs, Dieter Vandenbussche

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.

Authors: Robert A. Stubbs, Dieter Vandenbussche

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

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

Research demonstrates significant risk model errors are likely when risk models are not up-to-date.

Author: Anthony Renshaw - PhD

Research Report No. 006

Axioma's Applied Research team offers a practical method for predicting when leverage is likely to improve the performance of a long-only portfolio.

Author: Anthony Renshaw - PhD

Research Report No. 005

Understanding the impact of constraints imposed during portfolio construction and being able to quantify that impact to the asset owners is critical.

This research paper demonstrates how to measure the effects that individual constraints have on a portfolio, compared with a portfolio dictated exclusively by a trade-off of forecast risks and returns.

Author: Robert A. Stubbs - PhD, Dieter Vandenbussche - PhD

Research Report No. 004

Portfolio managers who rely on mean-variance efficiency often find that their portfolios are unintuitive or do not behave as expected. In this paper we discuss how estimation error can affect the quality of your portfolio and what you can do about it.

Authors: Sebastian Ceria - PhD, Robert A. Stubbs - PhD

Research Report No. 003

Axioma's patent-pending Alpha Factor™ method can be used to improve your risk estimation by reducing risk model portfolio selection bias.

Authors: Anthony Renshaw - PhD, Robert A. Stubbs - PhD, Stefan Schmieta - PhD, Sebastian Ceria - PhD

Research Report No. 002

A practical look at methods for computing effective error estimates to be used as inputs for robust optimization.

Authors: Robert A. Stubbs - PhD, Pamela Vance - PhD

Research Report No. 001