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

Introduction to Risk Parity and Budgeting

Appendix A – Technical Appendix

c Thierry Roncalli† & CRC Press

†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)

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Table A.1, Page 312

Table: Examples of Archimedean copula functions

Copula ϕ (u) C(u1,u2) C ⊥ −lnu u1u2 GumbelGumbel copula (−lnu)θ exp

• −
• ˜

uθ

1 + ˜

uθ

2

1/θ FrankFrank copula −ln e−θu−1

e−θ −1

− 1

θ ln

• 1+ (e−θu1 −1)(e−θu2 −1)

e−θ −1

• Joe

−ln

• 1−(1−u)θ

1−

• ¯

uθ

1 + ¯

uθ

2 − ¯

uθ

1 ¯

uθ

2

1/θ ClaytonClayton copula u−θ −1

• u−θ

1

+u−θ

2

−1 −1/θ

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

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Figure A.6, Page 318

Figure: Quantile-quantile dependence measure for the t1 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, ρS,B = 20%)

Profile α⋆

C

α⋆

B

α⋆

S

ˆ γ Safety −0.03 99.88 0.15 867 Defensive −3.02 85.52 17.50 7.20 Balanced −8.55 59.07 49.48 2.55 Dynamic −14.07 32.61 81.46 1.55

Table: Calibration of the lifestyle fund profiles (T = 10 years, ρS,B = −20%)

Profile α⋆

C

α⋆

B

α⋆

S

ˆ γ Safety −0.08 99.93 0.15 867 Defensive −10.59 90.62 19.98 6.72 Balanced −26.16 76.82 49.34 2.72 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

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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

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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