Risk parity refers to an asset allocation method which consists in budgeting the risk. It is today the main alternative method to the traditional mean-variance portfolio optimization. Contrary to this last one, risk parity portfolios appear robust. It is largely used by institutional investors such as pension funds, sovereign funds or insurance company to manage their equity exposure or their stock/bond asset mix.

- Introduction to Risk Parity and Budgeting (Roncalli)
Although portfolio management didn't change much during the 40 years after the seminal works of Markowitz and Sharpe, the development of risk budgeting techniques marked an important milestone in the deepening of the relationship between risk and asset management. Risk parity then became a popular financial model of investment after the global financial crisis in 2008. Today, pension funds and institutional investors are using this approach in the development of smart indexing and the redefinition of long-term investment policies. Introduction to Risk Parity and Budgeting provides an up-to-date treatment of this alternative method to Markowitz optimization. It builds financial exposure to equities and commodities, considers credit risk in the management of bond portfolios, and designs long-term investment policy. The first part of the book gives a theoretical account of portfolio optimization and risk parity. The author discusses modern portfolio theory and offers a comprehensive guide to risk budgeting. Each chapter in the second part presents an application of risk parity to a specific asset class. The text covers risk-based equity indexation (also called smart beta) and shows how to use risk budgeting techniques to manage bond portfolios. It also explores alternative investments, such as commodities and hedge funds, and applies risk parity techniques to multi-asset classes. The book's first appendix provides technical materials on optimization problems, copula functions, and dynamic asset allocation. The second appendix contains 30 tutorial exercises. Solutions to the exercises, slides for instructors, and Gauss computer programs to reproduce the book's examples, tables, and figures are available on the book's website.

Go to the book's website

- Constrained Risk Budgeting Portfolios: Theory, Algorithms, Applications & Puzzles (Richard, Roncalli)
In this paper we consider the problem of risk budgeting when we face constraints. We show how to formulate the portfolio optimization problem. The numerical solution is obtained by mixing the method of cyclical coordinate descent (CCD), alternating direction method of multipliers (ADMM), proximal operators and Dykstra's algorithm. Moreover, we discuss the scaling puzzle, which is related to the homogeneity property of risk measures. The working paper is available at SSRN.

- Risk Parity Portfolios with Skewness Risk: An Application to Factor Investing and Alternative Risk Premia (Bruder, Kostyuchyk, Roncalli)
In this paper we extend the volatility-based risk parity approach in order to take into account the jump risk. For that, we develop a skewness-based risk measure by considering a Gaussian mixture model. We apply this portfolio construction to the traditional asset mix policy between equities and bonds when we introduce a short volatility exposure. Moreover, we show interesting results concerning some asset management problems: factor investing, alternative risk premia and CTA strategies. The working paper is available at SSRN.

- A Fast Algorithm for Computing High-dimensional Risk Parity Portfolios (Griveau-Billion, Richard, Roncalli)
In this paper we propose a cyclical coordinate descent (CCD) algorithm for solving high dimensional risk parity problems. We show that this algorithm converges and is very fast even with large covariance matrices (n > 500). Comparison with existing algorithms also shows that it is one of the most efficient algorithms. The working paper is available at SSRN.

- Introducing Expected Returns into Risk Parity Portfolios: A New Framework for Asset Allocation (Roncalli)
This paper is a generalization of the risk parity approach of Bruder and Roncalli (2012) when we consider a generalized risk measure that takes into account expected returns. After deriving the theoretical properties of such risk budgeting portfolios, we apply this new model to asset allocation. First, we consider long-term investment policy and the determination of strategic asset allocation. We then consider dynamic allocation and show how to build risk parity funds that depend on expected returns. The working paper is available at SSRN.

- Risk Parity Portfolios with Risk Factors (Roncalli, Weisang)
This paper is a generalization of the risk parity approach of Bruder and Roncalli (2012) if we consider risk factors instead of assets. It appears that the problem is trickier. For instance, we could obtain multiple solutions. We propose then to formulate the diversification problem in terms of risk factors as an optimization program. We illustrate our methodology with some real life examples and backtests, which are: budgeting the risk of Fama-French equity factors, maximizing the diversification of an hedge fund portfolio and building a strategic asset allocation based on economic factors. The working paper is available at SSRN.

- Managing Risk Exposures using the Risk Budgeting Approach (Bruder, Roncalli)
This paper can be viewed as a generalization of the results obtained in the article published in the

*Journal of Portfolio Management*. We derive theoretical properties of risk budgeting portfolios, when the risk budgets are not the same. We also study the existence, the uniqueness and the optimality of such portfolios, and compare them to weight budgeting portfolios. Applications to strategic asset allocation, bonds and equities management and multi-asset diversified funds are provided. The working paper is available at SSRN. - The Properties of Equally Weighted Risk Contribution Portfolios (Maillard, Roncalli, Teiletche)
In this paper, we derive theoretical properties of risk parity portfolios. In particular, we show that the ERC portfolio always exists and is unique. We also show that the ERC portfolio is located between the minimum variance portfolio and the equally-weighted portfolio. An empirical application to commodities is provided. This paper has been published in the Journal of Portfolio Management (Summer 2010). The working paper is available at SSRN (with two other applications on US equities and multi-asset classes).

- The Smart Beta Indexing Puzzle (Cazalet, Grison, Roncalli)
In this article, we consider low volatility indexing. We show that there is a trade-off between the volatility reduction and the risks of such solutions. Thus, we measure the relationships between volatility on the one hand, and diversification, tracking error, liquidity and performance on the other. This paper is available at SSRN.

- Beyond Risk Parity: Using Non-Gaussian Risk Measures and Risk Factors (Roncalli, Weisang)
Application of the risk factor parity with non-gaussian risk measures. Examples concern the Fama-French model, the yield curve, portfolios of hedge funds, strategic asset allocation and smart indexing (slides available here).

- From Portfolio Optimization to Risk Parity (Roncalli)
A survey on risk parity applications and some results on robustness of optimized and risk parity portfolios (slides available here).

- Managing Sovereign Credit Risk in Bond Portfolios (Bruder, Hereil, Roncalli)
In this research, we show how risk budgeting techniques could be useful to manage credit risk in bond portfolios and we compare risk-based indexation with debt-weighted and gdp-weighted indexations. It is certainly the first application of risk parity to bonds. This article has been published in Journal of Indexes Europe (November 2011). An extended version of this article is available at SSRN. A summary of this research appears in the French journal La Revue Banque in October 2011.

- Measuring the Risk Concentration of Investment Portfolios (Hereil, Roncalli)
This short paper illustrates the use of the Gini coefficient to measure the risk concentration of equity indexes. A comparison between the ERC Eurostoxx index and the Eurostoxx 50 index is provided. This article has been published in the Bloomberg Risk newsletter (July 2011). A copy is available here.

- Risk-Based Indexation (Demey, Maillard, Roncalli)
Market-cap indexation presents some limits (trend-following strategy, lack of diversification and bubble bias). Alternative-weighted indexation correspond to passive indexes where the weights are not based on market capitalization. We generally distinguish two forms: fundamental indexation and risk-based indexation. Different forms of risk-based indexation exist: minimum variance, equally-weighted, mdp/msr and ERC. In this paper, we compare these four alternative indexation. This paper is available at SSRN.

- Interviews and Newspapers Articles
- Portfolio Institutional Smart Beta: Exploring the New Age of Index Investing (report available here)
- L'Agefi Suisse L'ère de la sélection active du béta est arrivée (The Time to Actively Pick Beta Has Arrived) (article reproduit dans L'Agefi Luxembourg)
- Life & Pension Risk, Risk Parity Investment Strategies Present Opportunities for Insurers by Charlotte Moore
- Investment Europe A 'better place' for sovereign risk by Caroline Allen (copy available here)

- Professional Conferences
- Frontiers in Finance, Petits Déjeuners de la Finance (October 2, 2013) -- Risk Parity: A (New) Tool for Asset Management (slides available here)
- EDHEC-Risk Seminar, Analyzing Sovereign Risk for Portfolio Management Decisions (June 13, 2012) -- Managing Sovereign Risk Through a Risk Budgeting Approach (slides available here)
- Forum GI (March 13, 2012) -- Vers une gestion indicielle efficiente pour les obligations (slides available here)
- Association Française des Gestionnaires Actif-Passif, (November 17, 2011) -- Managing Sovereign Credit Risk in Bond Portfolios (slides available here)
- CFA Institute (October 13, 2011) -- Managing Sovereign Credit Risk in Bond Portfolios (slides available here)
- CFA Institute (March 10, 2011) -- How Quantitative Methods Can Help To Understand Some Asset Management Problems? (slides available here)
- Frontiers in Finance, Petits Déjeuners de la Finance (May 12, 2010) -- Risk-Based Indexation (slides available >here)

- Academic Conferences
- Multifractals, Non-stationarity, and Risk, ENGREF AgroParisTech (July 4, 2012) -- From Portfolio Optimization to Risk Parity (slides available here)
- 6th R/Rmetrics Meielisalp Workshop & Summer School on Computational Finance and Financial Engineering (June 24, 2012) -- From Portfolio Optimization to Risk Parity (slides available here)
- EPEE Seminar, University of Evry (June 14, 2012) -- Managing Sovereign Risk Through a Risk Budgeting Approach (slides available here)
- Séminaire de Mathématiques Appliquées, Laboratoire de Mathématiques Jean Leray & Atelier Finance & Risque, Laboratoire LEMNA, Nantes University (March 29, 2012) -- Managing Risk Exposures using the Risk Budgeting Approach (slides available here)
- Financial Risks International Forum (March 22, 2012) -- Managing Sovereign Credit Risk in Bond Portfolios using the Risk Budgeting Approach (slides available here)
- ESSEC WG RISK (January 18, 2012) -- Portfolio Optimization versus Risk-Budgeting Allocation (slides available here)
- Séminaire CEPN - LAGA, University of Paris 13 (Mai 10, 2011) -- How Quantitative Methods Can Help To Understand Some Asset Management Problems? (slides available here)

Back to the main page.