About Me

I am a Quantitative Researcher at Millennium Management, where I engage in risk modeling and portfolio optimization.

Previously, I was a Postdoctoral Research Scientist in the Department of Industrial Engineering and Operations Research at Columbia University working under the supervision of Garud N. Iyengar. My research sits at the intersection of machine learning and robust optimization theory, and their application to solve problems in the financial industry. I also used to work at RBC Global Asset Management as a Quantitative Investments Researcher. During my time at RBC I developed data-driven portfolio construction models, implemented statistical tools, and evaluated the performance of existing investment strategies.

I completed my PhD degree at the Department of Mechanical and Industrial Engineering at the University of Toronto under the supervision of Roy H. Kwon. I completed my undergraduate degree (B. Eng. Hons.) in Mechanical Engineering at McGill University.

Research Interests

Convex optimization
Supervised learning and predictive modelling
End-to-end integrated learning and optimization
Robust and distributionally robust optimization
Optimal asset allocation and portfolio construction
Factor models, risk attribution, and risk diversification

Education

Ph.D. in Industrial Engineering (Operations Research), University of Toronto
Advisor: Roy H. Kwon
Thesis: Advances in risk parity portfolio optimization

B.Eng. (Hons) in Mechanical Engineering, McGill University
Advisor: Michael P. Paidoussis
Thesis: Stability of pipes subjected to fluid flow induced vibration with flexible boundary conditions in discharge

Publications

Journal publications

Costa, G. and Iyengar, G. N. (2023). Distributionally Robust End-to-End Portfolio Construction. Arxiv preprint, arXiv:2206.05134.

We propose an end-to-end distributionally robust system for portfolio construction that integrates the asset return prediction model with a distributionally robust portfolio optimization model. We also show how to learn the risktolerance parameter and the degree of robustness directly from data. End-to-end systems have an advantage in that information can be communicated between the prediction and decision layers during training, allowing the parameters to be trained for the final task rather than solely for predictive performance. However, existing end-toend systems are not able to quantify and correct for the impact of model risk on the decision layer. Our proposed distributionally robust end-to-end portfolio selection system explicitly accounts for the impact of model risk. The decision layer chooses portfolios by solving a minimax problem where the distribution of the asset returns is assumed to belong to an ambiguity set centered around a nominal distribution. Using convex duality, we recast the minimax problem in a form that allows for efficient training of the end-to-end system.

Costa, G. and Kwon, R. H. (2022). Data-driven distributionally robust risk parity portfolio optimization. Optimization Methods and Software, 37(5), 1876-1911.

We propose a distributionally robust formulation of the traditional risk parity portfolio optimization problem. Distributional robustness is introduced by targeting the discrete probabilities attached to each observation used during parameter estimation. Instead of assuming that all observations are equally likely, we consider an ambiguity set that provides us with the flexibility to find the most adversarial probability distribution based on the investor's confidence level. This allows us to derive robust estimates to parametrize the distribution of asset returns without having to impose any particular structure on the data. The resulting distributionally robust optimization problem is a constrained convex--concave minimax problem. Our approach is financially meaningful and attempts to attain full risk diversification with respect to the worst-case instance of the portfolio risk measure. We propose a novel algorithmic approach to solve this minimax problem, which blends projected gradient ascent with sequential convex programming. By design, this algorithm is highly flexible and allows the user to choose among alternative statistical distance measures to define the ambiguity set. Moreover, the algorithm is highly tractable and scalable. Our numerical experiments suggest that a distributionally robust risk parity portfolio can yield a higher risk-adjusted rate of return when compared against the nominal portfolio.

Costa, G. and Kwon, R. H. (2020). A robust framework for risk parity portfolios. Journal of Asset Management, 21 (5), 447-466.

We propose a robust formulation of the traditional risk parity problem by introducing an uncertainty structure specifically tailored to capture the intricacies of risk parity. Typical minimum variance portfolios attempt to introduce robustness by computing the worst-case estimate of the risk measure, which is not intuitive for risk parity. Instead, our motivation is to shield the risk parity portfolio against noise in the estimated asset risk contributions. Thus, we present a novel robust risk parity model that introduces robustness around both the overall portfolio risk and the assets’ marginal risk contributions. The proposed robust model is highly tractable and is able to retain the same level of complexity as the original problem. We provide a general procedure by which to create an uncertainty structure around the asset covariance matrix. We quantify this uncertainty as a perturbation on the nominal covariance estimate, which allows us to intuitively embed robustness during optimization. We then propose a specific procedure to construct a robust risk parity portfolio through a factor model of asset returns. Computational experiments show that the robust formulation yields a higher risk-adjusted rate of return than the nominal model while maintaining a sufficiently risk-diverse portfolio.

Costa, G. and Kwon, R. H. (2020). Generalized risk parity portfolio optimization: An ADMM approach. Journal of Global Optimization, 78, 207-238.

The risk parity solution to the asset allocation problem yields portfolios where the risk contribution from each asset is made equal. We consider a generalized approach to this problem. First, we set an objective that seeks to maximize the portfolio expected return while minimizing portfolio risk. Second, we relax the risk parity condition and instead bound the risk dispersion of the constituents within a predefined limit. This allows an investor to prescribe a desired risk dispersion range, yielding a portfolio with an optimal risk–return profile that is still well-diversified from a risk-based standpoint. We add robustness to our framework by introducing an ellipsoidal uncertainty structure around our estimated asset expected returns to mitigate estimation error. Our proposed framework does not impose any restrictions on short selling. A limitation of risk parity is that allowing of short sales leads to a non-convex problem. However, we propose an approach that relaxes our generalized risk parity model into a convex semidefinite program. We proceed to tighten this relaxation sequentially through the alternating direction method of multipliers. This procedure iterates between the convex optimization problem and the non-convex problem with a rank constraint. In addition, we can exploit this structure to solve the non-convex problem analytically and efficiently during every iteration. Numerical results suggest that this algorithm converges to a higher quality optimal solution when compared to the competing non-convex problem, and can also yield a higher ex post risk-adjusted rate of return.

Costa, G. and Kwon, R. H. (2020). A regime-switching factor model for mean–variance optimization. Journal of Risk, 22(4), 31-59.

We formulate a novel Markov regime-switching factor model to describe the cyclical nature of asset returns in modern financial markets. Maintaining a factor model structure allows us to easily derive the asset expected returns and their corresponding covariance matrix. By design, these two parameters are calibrated to better describe the properties of the different market regimes. In turn, these regime-dependent parameters serve as the inputs during mean-variance optimization, thereby constructing portfolios adapted to the current market environment. Through this formulation, the proposed model allows for the construction of large, realistic portfolios at no additional computational cost during optimization. Moreover, the viability of this model can be significantly improved by periodically rebalancing the portfolio, ensuring proper alignment between the estimated parameters and the transient market regimes. An out-of-sample computational experiment over a long investment horizon shows that the proposed regime-dependent portfolios are better aligned with the market environment, yielding a higher ex-post rate of return and lower volatility, even when compared against competing portfolios.

Costa, G. and Kwon, R. H. (2019). Risk parity portfolio optimization under a Markov regime-switching framework. Quantitative Finance, 19(3), 453-471.

We formulate and solve a risk parity optimization problem under a Markov regime-switching framework to improve parameter estimation and to systematically mitigate the sensitivity of optimal portfolios to estimation error. A regime-switching factor model of returns is introduced to account for the abrupt changes in the behaviour of economic time series associated with financial cycles. This model incorporates market dynamics in an effort to improve parameter estimation. We proceed to use this model for risk parity optimization and also consider the construction of a robust version of the risk parity optimization by introducing uncertainty structures to the estimated market parameters. We test our model by constructing a regime-switching risk parity portfolio based on the Fama--French three-factor model. The out-of-sample computational results show that a regime-switching risk parity portfolio can consistently outperform its nominal counterpart, maintaining a similar ex post level of risk while delivering higher-than-nominal returns over a long-term investment horizon. Moreover, we present a dynamic portfolio rebalancing policy that further magnifies the benefits of a regime-switching portfolio.

Wu, D., Kwon, R. H., and Costa, G. (2017). A constrained cluster-based approach for tracking the S&P 500 index. International Journal of Production Economics, 193, 222-243.

We consider the problem of tracking a benchmark target portfolio of financial securities in particular the S&P 500. Linear integer programming models are developed that seeks to track a target portfolio using a strict subset of securities from the benchmark portfolio. The models represent a clustering approach to select securities and also include additional constraints that aim to control risk and transactions costs. Lagrangian and semi-Lagrangian methods are developed to compute solutions to the tracking models. The computational results show the effectiveness of the linear tracking models and the computational methods in tracking the S&P 500. Overall, the models and methods presented can serve as the basis of the optimization module in an optimization-based decision support for creating tracking portfolios.

Kheiri, M., Paidoussis, M. P., Costa, G., and Amabili, M. (2014). Dynamics of a pipe conveying fluid flexibly restrained at the ends. Journal of Fluids and Structures, 49, 360-385.

The subject of this paper is the study of dynamics and stability of a pipe flexibly supported at its ends and conveying fluid. First, the equation of motion of the system is derived via the extended form of Hamilton׳s principle for open systems. In the derivation, the effect of flexible supports, modelled as linear translational and rotational springs, is appropriately considered in the equation of motion rather than in the boundary conditions. The resulting equation of motion is then discretized via the Galerkin method in which the eigenfunctions of a free-free Euler–Bernoulli beam are utilized. Thus, a general set of second-order ordinary differential equations emerges, in which, by setting the stiffness of the end-springs suitably, one can readily investigate the dynamics of various systems with either classical or non-classical boundary conditions. Several numerical analyses are initially performed, in which the eigenvalues of a simplified system (a beam) with flexible end-supports are obtained and then compared with numerical results, so as to verify the equation of motion, in its simplified form. Then, the dynamics of a pinned-free pipe conveying fluid is systematically investigated, in which it is found that a pinned-free pipe conveying fluid is generally neutrally stable until it becomes unstable via a Hopf bifurcation leading to flutter. The next part of the paper is devoted to studying the dynamics of a pinned-free pipe additionally constrained at the pinned end by a rotational spring. A wide range of dynamical behaviour is seen as the mass ratio of the system (mass of the fluid to the fluid+pipe mass) varies. It is surprising to see that for a range of rotational spring stiffness, an increase in the stiffness makes the pipe less stable. Finally, a pipe conveying fluid supported only by a translational and a rotational spring at the upstream end is considered. For this system, the critical flow velocity is determined for various values of spring constants, and several Argand diagrams along with modal shapes of the unstable modes are presented. The dynamics of this system is found to be very complex and often unpredictable (unexpected).

Conference Proceedings

Kheiri, M., Paidoussis, M. P., and Costa, G. (2014). Dynamics of a Pipe Conveying Fluid Flexibly Supported at the Ends. Proceedings of the ASME 2014 Pressure Vessels and Piping Conference. Volume 4: Fluid-Structure Interaction. Anaheim, California, USA. July 20–24, 2014.

The subject of this paper is the study of dynamics and stability of a pipe flexibly supported at its ends and conveying fluid. First, the equation of motion of the system is derived via the extended form of Hamilton’s principle for open systems. In the derivation, the effect of flexible supports, modelled as linear translational and rotational springs, is appropriately considered in the equation of motion rather than in the boundary conditions. The resulting equation of motion is then discretized via the Galerkin method in which the eigenfunctions of a free-free Euler-Bernoulli beam are utilized. Thus, a general set of second-order ordinary differential equations emerge, in which, by setting the stiffness of the end-springs suitably, one can readily investigate the dynamics of various systems with either classical or non-classical boundary conditions.

Teaching

Lecturer

MIE377H1 - Financial Optimization Models
Division of Engineering Science, University of Toronto
Winter 2018 - 2021
Instructor rating: 4.7 / 5.0
MIE236 - Probability
Department of Mechanical and Industrial Engineering, University of Toronto
Fall 2020
Instructor rating: 4.2 / 5.0
ECE302H1 - Probability and Applications
Department of Electrical and Computer Engineering, University of Toronto
Fall 2019
Instructor rating: 4.0 / 5.0
ECE Instructor award recipient (Fall 2019)
MIE375H1 - Financial Engineering
Division of Engineering Science, University of Toronto
Fall 2018
Instructor rating: 4.5 / 5.0

Teaching Assistant

MMF2000H - Risk Management
Master of Mathematical Finance Program
University of Toronto
Fall 2018 - 2020, Summer 2017 - 2018
MIE479H1 - Capstone Design
Mechanical and Industrial Engineering
University of Toronto
Fall 2019 - 2020
MMF1921H - Operations Research
Master of Mathematical Finance Program
University of Toronto
Summer 2017 - 2020
MIE1621H - Non-Linear Optimization
Mechanical and Industrial Engineering
University of Toronto
Fall 2016

Presentations

Data-driven distributionally robust risk parity portfolio optimization

INFORMS Annual Meeting
November 2020, Virtual.

Generalized Risk Parity Portfolio Optimization: An ADMM Approach

CASCON x EVOKE, IBM Annual Academic and Research Conference
November 2019, Markham, ON.
CORS Annual Conference
May 2019, Saskatoon, SK.

A Regime-Switching Framework for Portfolio Optimization

4th Industrial-Academic Workshop on Optimization and Artificial Intelligence in Finance at The Fields Institute
November 2018, Toronto, ON.
Student Seminar Series - University of Toronto Operations Research Group
November 2018, Toronto, ON.

Hidden Markov Model for Risk Parity Optimization

Master of Mathematical Finance Symposium
January 2018, Blue Mountain, ON.

Contact

gcosta151@icloud.com