Introduction to Risk Parity and Budgeting Appendix A Technical - PowerPoint PPT Presentation

Introduction to Risk Parity and Budgeting Appendix A – Technical Appendix � Thierry Roncalli † & CRC Press c † Evry University & Lyxor Asset Management, France Instructors may find the description of the book at the following addresses: http://www.crcpress.com/product/isbn/9781482207156 http://www.thierry-roncalli.com/RiskParityBook.html May 22, 2013 Introduction to Risk Parity and Budgeting Technical Appendix 1 / 13

Figure A.1, Page 309 Figure: Example of building a bivariate probability distribution with a copula function Introduction to Risk Parity and Budgeting Technical Appendix 2 / 13

Figure A.2, Page 311 Figure: Level curves of bivariate distributions (Frank copula) Introduction to Risk Parity and Budgeting Technical Appendix 3 / 13

Figure A.3, Page 311 Figure: Level curves of bivariate distributions (Gumbel copula) Introduction to Risk Parity and Budgeting Technical Appendix 4 / 13

Table A.1, Page 312 Table: Examples of Archimedean copula functions ϕ ( u ) C ( u 1 , u 2 ) Copula C ⊥ − ln u u 1 u 2 � � 1 / θ � ( − ln u ) θ � u θ u θ ˜ 1 + ˜ GumbelGumbel copula exp − 2 � � 1 + ( e − θ u 1 − 1 )( e − θ u 2 − 1 ) − ln e − θ u − 1 − 1 FrankFrank copula θ ln e − θ − 1 e − θ − 1 � 1 − ( 1 − u ) θ � � 1 / θ u θ u θ u θ u θ � ¯ 1 + ¯ 2 − ¯ 1 ¯ Joe − ln 1 − 2 � − 1 / θ � u − θ − 1 u − θ + u − θ ClaytonClayton copula − 1 1 2 Introduction to Risk Parity and Budgeting Technical Appendix 5 / 13

Figure A.4, Page 313 Figure: Comparison of normal and t copulas Introduction to Risk Parity and Budgeting Technical Appendix 6 / 13

Figure A.5, Page 318 Figure: Quantile-quantile dependence measure for the normal copula Introduction to Risk Parity and Budgeting Technical Appendix 7 / 13

Figure A.6, Page 318 Figure: Quantile-quantile dependence measure for the t 1 copula Introduction to Risk Parity and Budgeting Technical Appendix 8 / 13

Tables A.2 & A.3, Page 328 Table: Calibration of the lifestyle fund profiles ( T = 10 years, ρ S , B = 20 % ) α ⋆ α ⋆ α ⋆ ˆ Profile γ C B S Safety − 0 . 03 99 . 88 0 . 15 867 Defensive − 3 . 02 85 . 52 17 . 50 7 . 20 − 8 . 55 59 . 07 49 . 48 2 . 55 Balanced − 14 . 07 32 . 61 81 . 46 1 . 55 Dynamic Table: Calibration of the lifestyle fund profiles ( T = 10 years, ρ S , B = − 20 % ) α ⋆ α ⋆ α ⋆ ˆ Profile γ C B S Safety − 0 . 08 99 . 93 0 . 15 867 − 10 . 59 90 . 62 19 . 98 6 . 72 Defensive − 26 . 16 76 . 82 49 . 34 2 . 72 Balanced Dynamic − 41 . 72 63 . 03 78 . 70 1 . 71 Introduction to Risk Parity and Budgeting Technical Appendix 9 / 13

Figure A.7, Page 329 Figure: Sensitivity of the equity allocation α ⋆ S (in %) in lifestyle funds Introduction to Risk Parity and Budgeting Technical Appendix 10 / 13

Figure A.8, Page 332 Figure: Influence of the parameters on the glide path of target-date funds Introduction to Risk Parity and Budgeting Technical Appendix 11 / 13

Figure A.9, Page 335 Figure: Example of the LDI utility function Introduction to Risk Parity and Budgeting Technical Appendix 12 / 13

Figure A.10, Page 335 Figure: Optimal exposure α ⋆ ( t ) (in %) in the LDI portfolio Introduction to Risk Parity and Budgeting Technical Appendix 13 / 13