Introduction to Risk Parity and Budgeting

Transcription

1 Chapman & Hall/CRC FINANCIAL MATHEMATICS SERIES Introduction to Risk Parity and Budgeting Thierry Roncalli CRC Press Taylor &. Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business A CHAPMAN & HALL BOOK

2 Contents Introduction List of Figures List of Tables List of Symbols and Notations xiii xvii xxi I From Portfolio Optimization to Risk Parity 1 1 Modern Portfolio Theory From optimized portfolios to the market portfolio The efficient frontier Introducing the quadratic utility function Adding some constraints Analytical solution The tangency portfolio Market equilibrium and CAPM Portfolio optimization in the presence of a benchmark The Black-Litterman model Computing the implied risk premia The optimization problem Numerical implementation of the model Practice of portfolio optimization Estimation of the covariance matrix Empirical covariance matrix estimator Hayashi-Yoshida estimator GARCH approach Factor models Designing expected returns Regularization of optimized portfolios Stability issues Resampling techniques Denoising the covariance matrix Shrinkage methods Introducing constraints 53 vii

3 Why regularization techniques are not sufficient How to specify the constraints Shrinkage interpretation of the constrained solution 65 2 Risk Budgeting Approach Risk allocation principle Properties of a risk measure Coherency and convexity of risk measures Euler allocation principle Risk contribution of portfolio assets Computing the risk contributions Interpretation of risk contributions Application to non-normal risk measures Non-normal value-at-risk and expected shortfall Historical value-at-risk Analysis of risk budgeting portfolios Definition of a risk budgeting portfolio The right specification of the RB portfolio Solving the non-linear system of risk budgeting contraints Some properties of the RB portfolio Particular solutions with the volatility risk measure Existence and uniqueness of the RB portfolio Optimality of the risk budgeting portfolio Stability of the risk budgeting approach Special case: the ERC portfolio The two-asset case (n 2) The general case (n > 2) Optimality of the ERC portfolio Back to the notion of diversification Diversification index Concentration indices Difficulty of reconciling the different diversification concepts Risk budgeting versus weight budgeting Comparing weight budgeting and risk budgeting portfolios New construction of the minimum variance portfolio Using risk factors instead of assets Pitfalls of the risk budgeting approach based on assets Duplication invariance property 135

4 Polico invariance property Impact of the reparametrization on the asset universe Risk decomposition with respect to the risk factors Some illustrations Matching the risk budgets Minimizing the risk concentration between the risk factors Solving the duplication and polico invariance properties 146 II Applications of the Risk Parity Approach Risk-Based Indexation Capitalization-weighted indexation Theory support Constructing and replicating an equity index Pros and cons of CW indices Alternative-weighted indexation Desirable properties of AW indices Fundamental indexation Risk-based indexation The equally weighted portfolio The minimum variance portfolio The most diversified portfolio The ERC portfolio Comparison of the risk-based allocation approaches Some illustrations Simulation of risk-based indices Practical issues of risk-based indexation Findings of other empirical works What is the best alternative-weighted indexation? Style analysis of alternative-weighted indexation Application to Bond Portfolios Some issues in bond management Debt-weighted indexation Yield versus risk Bond portfolio management Term structure of interest rates Pricing of bonds Without default risk 197 ix

5 X With default risk Risk management of bond portfolios Using the yield curve as risk factors Taking into account the default risk Some illustrations Managing risk factors of the yield curve Managing sovereign credit risk Measuring the credit risk of sovereign bond portfolios Comparing debt-weighted, gdp-weighted and risk-based indexations Risk Parity Applied to Alternative Investments Case of commodities Why investing in commodities is different Commodity futures markets How to define the commodity risk premium Designing an exposure to the commodity asset class Diversification return Comparing EW and ERC portfolios Hedge fund strategies Position sizing Portfolio allocation of hedge funds Choosing the risk measure Comparing ERC allocations Budgeting the risk factors Limiting the turnover Portfolio Allocation with Multi-Asset Classes Construction of diversified funds Stock/bond asset mix policy Growth assets versus hedging assets Are bonds growth assets or hedging assets? Analytics of these results Risk-balanced allocation Pros and cons of risk parity funds Long-term investment policy Capturing the risk premia Strategic asset allocation Allocation between asset classes Asset classes or risk factor classes Allocation within an asset class Risk budgeting with liability constraints Absolute return and active risk parity 294

6 Conclusion 299 A Technical Appendix 301 A.l Optimization problems 301 A. 1.1 Quadratic programming problem 301 A. 1.2 Non-linear unconstrained optimization 303 A. 1.3 Sequential quadratic programming algorithm 306 A. 1.4 Numerical solutions of the RB problem 307 A.2 Copula functions 308 A.2.1 Definition and main properties 308 A.2.2 Parametric functions 312 A.2.3 Simulation of copula models 314 A Distribution approach 314 A Simulation based on conditional copula functions 315 A.2.4 Copulas and risk management 316 A.2.5 Multivariate survival modeling 319 A.3 Dynamic portfolio optimization 322 A.3.1 Stochastic optimal control 322 A Bellman approach 322 A Martingale approach 323 A.3.2 Portfolio optimization in continuous-time 324 A.3.3 Some extensions of the Merton model 326 A Lifestyle funds 326 A Lifecycle funds 329 A Liability driven investment 332 B Tutorial Exercises 337 B.l Exercises related to modern portfolio theory 337 B.l.l Markowitz optimized portfolios 337 B.l.2 Variations on the efficient frontier 338 B.l.3 Sharpe ratio 339 B.1.4 Beta coefficient 341 B.l.5 Tangency portfolio 342 B.l.6 Information ratio 343 B.l.7 Building a tilted portfolio 344 B.l.8 Implied risk premium 345 B.l.9 Black-Litterman model 346 B.l. 10 Portfolio optimization with transaction costs 347 B.l. 11 Impact of constraints on the CAPM theory 348 B.l.12 Generalization of the Jagannathan-Ma shrinkage approach 349 B.2 Exercises related to the risk budgeting approach 351 B.2.1 Risk measures 351 B.2.2 Weight concentration of a portfolio 352 xi

7 xii B.2.3 ERC portfolio 353 B.2.4 Computing the Cornish-Fisher value-at-risk 354 B.2.5 Risk budgeting when risk budgets are not strictly positive 355 B.2.6 Risk parity and factor models 356 B.2.7 Risk allocation with the expected shortfall risk measure 358 B.2.8 ERC optimization problem 359 B.2.9 Risk parity portfolios with skewness and kurtosis B.3 Exercises related to risk parity applications 362 B.3.1 Computation of heuristic portfolios 362 B.3.2 Equally weighted portfolio 362 B.3.3 Minimum variance portfolio 363 B.3.4 Most diversified portfolio 365 B.3.5 Risk allocation with yield curve factors 366 B.3.6 Credit risk analysis of sovereign bond portfolios B.3.7 Risk contributions of long-short portfolios 370 B.3.8 Risk parity funds 371 B.3.9 The Frazzini-Pedersen model 372 B.3.10 Dynamic risk budgeting portfolios 374 Bibliography 377 Subject Index 399 Author Index 405