Introduction to Risk Parity and Budgeting

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1 Introduction to Risk Parity and Budgeting Chapter 1 Modern Portfolio Theory c Thierry Roncalli & CRC Press Evry University & Lyxor Asset Management, France Instructors may find the description of the book at the following addresses: May 22, 2013 Introduction to Risk Parity and Budgeting Modern Portfolio Theory 1 / 40

2 Figure 1.1, Page 6 Figure: Optimized Markowitz portfolios Introduction to Risk Parity and Budgeting Modern Portfolio Theory 2 / 40

3 Figure 1.2, Page 8 Figure: The efficient frontier of Markowitz Introduction to Risk Parity and Budgeting Modern Portfolio Theory 3 / 40

4 Table 1.1, Page 7 Table: Solving the φ-problem φ x x x x µ (x ) σ (x ) Introduction to Risk Parity and Budgeting Modern Portfolio Theory 4 / 40

5 Tables 1.2 & 1.3, Page 8 Table: Solving the unconstrained µ-problem µ x x x x σ (x ) φ Table: Solving the unconstrained σ-problem σ x x x x µ (x ) φ Introduction to Risk Parity and Budgeting Modern Portfolio Theory 5 / 40

6 Figure 1.3, Page 10 Figure: The efficient frontier with some weight constraints Introduction to Risk Parity and Budgeting Modern Portfolio Theory 6 / 40

7 Table 1.4, Page 10 Table: Solving the σ-problem with weight constraints x i R x i 0 0 x i 40% σ x x x x µ (x ) φ Introduction to Risk Parity and Budgeting Modern Portfolio Theory 7 / 40

8 Figure 1.4, Page 13 Figure: The capital market line Introduction to Risk Parity and Budgeting Modern Portfolio Theory 8 / 40

9 Figure 1.5, Page 15 Figure: The efficient frontier with a risk-free asset Introduction to Risk Parity and Budgeting Modern Portfolio Theory 9 / 40

10 Tables 1.5 & 1.6, Pages 17 & 18 Table: Computation of the beta Portfolio µ (y) β (y x ) π (y x ) e e e e x ew Table: Computation of the beta with a constrained tangency portfolio Portfolio µ (y) β (y x ) π (y x ) e e e e x ew Introduction to Risk Parity and Budgeting Modern Portfolio Theory 10 / 40

11 Figure 1.6, Page 20 Figure: The efficient frontier with a benchmark Introduction to Risk Parity and Budgeting Modern Portfolio Theory 11 / 40

12 Figure 1.7, Page 22 Figure: The tangency portfolio with respect to a benchmark Introduction to Risk Parity and Budgeting Modern Portfolio Theory 12 / 40

13 Table 1.7, Page 26 Table: Black-Litterman portfolios #0 #1 #2 #3 #4 #5 x x x x σ (x x 0 ) Introduction to Risk Parity and Budgeting Modern Portfolio Theory 13 / 40

14 Result (1.16), Page 28 Ĉ = Introduction to Risk Parity and Budgeting Modern Portfolio Theory 14 / 40

15 Figure 1.8, Page 30 Figure: Trading hours of asynchronous markets (UTC time) t m 1 t m 1:00 7:00 1:00 7:00 Topix 14:30 21:00 14:30 21:00 S&P 500 8:00 16:30 8:00 16:30 Eurostoxx Introduction to Risk Parity and Budgeting Modern Portfolio Theory 15 / 40

16 Figure 1.9, Page 31 Figure: Density of the estimator ˆρ with asynchronous returns Introduction to Risk Parity and Budgeting Modern Portfolio Theory 16 / 40

17 Figure 1.10, Page 33 Figure: Hayashi-Yoshida estimator Introduction to Risk Parity and Budgeting Modern Portfolio Theory 17 / 40

18 Figure 1.11, Page 35 Figure: Cumulative weight W m of the IGARCH model Introduction to Risk Parity and Budgeting Modern Portfolio Theory 18 / 40

19 Figure 1.12, Page 36 Figure: Estimation of the S&P 500 volatility Introduction to Risk Parity and Budgeting Modern Portfolio Theory 19 / 40

20 Figure 1.13, Page 38 Figure: Density of the uniform correlation estimator Introduction to Risk Parity and Budgeting Modern Portfolio Theory 20 / 40

21 Result (1.19), Page 39 Ĉ = Introduction to Risk Parity and Budgeting Modern Portfolio Theory 21 / 40

22 Result (1.21), Page 39 Ĉ = Introduction to Risk Parity and Budgeting Modern Portfolio Theory 22 / 40

23 Table 1.8, Page 45 Table: Sensitivity of the MVO portfolio to input parameters ρ 70% 90% 90% σ 2 18% 18% µ 1 9% x x x Introduction to Risk Parity and Budgeting Modern Portfolio Theory 23 / 40

24 Figure 1.16, Page 46 Figure: Uncertainty of the efficient frontier Introduction to Risk Parity and Budgeting Modern Portfolio Theory 24 / 40

25 Figure 1.17, Page 48 Figure: Resampled efficient frontier Introduction to Risk Parity and Budgeting Modern Portfolio Theory 25 / 40

26 Result (1.23), Page 49 Ĉ = Introduction to Risk Parity and Budgeting Modern Portfolio Theory 26 / 40

27 Result (1.24), Page 51 Ĉ = Introduction to Risk Parity and Budgeting Modern Portfolio Theory 27 / 40

28 Table 1.9, Page 53 Table: Solutions of penalized mean-variance optimization MVO Ridge Lasso (NC) (C) (S) (D) (S) (D) x x x x Introduction to Risk Parity and Budgeting Modern Portfolio Theory 28 / 40

29 Figure 1.18, Page 54 Figure: Weights of penalized MVO portfolios (in %) Introduction to Risk Parity and Budgeting Modern Portfolio Theory 29 / 40

30 Tables 1.10 & 1.11, Page 55 Table: Principal component analysis of the covariance matrix Σ Asset / Factor % 72.29% 22.43% % 69.06% 20.43% % 2.21% 95.29% Eigenvalue 8.31% 0.84% 0.26% % cumulated 88.29% 97.20% % Table: Principal component analysis of the information matrix I Asset / Factor % 72.29% 65.35% % 69.06% 69.38% % 2.21% 30.26% Eigenvalue % cumulated 74.33% 97.65% % Introduction to Risk Parity and Budgeting Modern Portfolio Theory 30 / 40

31 Figure 1.19, Page 56 Figure: PCA applied to the stocks of the FTSE index (June 2012) Introduction to Risk Parity and Budgeting Modern Portfolio Theory 31 / 40

32 Table 1.12, Page 57 Table: Effect of deleting a PCA factor x MV λ 1 = 0 λ 2 = 0 λ 3 = 0 λ 4 = 0 λ 5 = 0 λ 6 = 0 x x x x x x Introduction to Risk Parity and Budgeting Modern Portfolio Theory 32 / 40

33 Table 1.13, Page 60 Table: Limiting the turnover of MVO portfolios τ x 0 x x x x µ (x ) σ (x ) Introduction to Risk Parity and Budgeting Modern Portfolio Theory 33 / 40

34 Page 62 Number of solved QP problems n b n x Heuristic Backward Forward Introduction to Risk Parity and Budgeting Modern Portfolio Theory 34 / 40

35 Table 1.14, Page 63 Table: Sampling the SX5E index with the heuristic algorithm k Stock b i σ ( x (k) b ) 1 Nokia Carrefour Repsol Unibail-Rodamco Muenchener Rueckver RWE Koninklijke Philips Generali CRH Volkswagen LVMH Telefonica Bayer Vinci BBVA Sanofi Allianz Total Siemens Introduction to Risk Parity and Budgeting Modern Portfolio Theory 35 / 40

36 Table 1.15, Page 63 Table: Sampling the SX5E index with the backward elimination algorithm k Stock b i σ ( x (k) b ) 1 Iberdrola France Telecom Carrefour Muenchener Rueckver Repsol BMW Generali RWE Koninklijke Philips Air Liquide GDF Suez Bayer BNP Paribas Total LVMH Allianz Sanofi BBVA Siemens Introduction to Risk Parity and Budgeting Modern Portfolio Theory 36 / 40

37 Table 1.16, Page 64 Table: Sampling the SX5E index with the forward selection algorithm k Stock b i σ ( x (k) b ) 1 Siemens Banco Santander Bayer Eni Allianz LVMH France Telecom Carrefour BMW Société Générale CRH Air Liquide RWE Nokia Unibail-Rodamco Repsol Essilor Muenchener Rueckver Iberdrola Introduction to Risk Parity and Budgeting Modern Portfolio Theory 37 / 40

38 Figure 1.20, Page 65 Figure: Sampling the SX5E and SPX indices Introduction to Risk Parity and Budgeting Modern Portfolio Theory 38 / 40

39 Tables 1.17 & 1.18, Pages 68 & 69 Table: Minimum variance portfolio when x i 10% x i λ i λ + i σ i ρ i,j Table: Minimum variance portfolio when 10% x i 40% x i λ i λ + i σ i ρ i,j Introduction to Risk Parity and Budgeting Modern Portfolio Theory 39 / 40

40 Tables 1.19 & 1.20, Pages 69 & 70 Table: Mean-variance portfolio when 10% x i 40% and µ = 6% x i λ i λ + i σ i ρ i,j Table: MSR portfolio when 10% x i 40% x i λ i λ + i σ i ρ i,j Introduction to Risk Parity and Budgeting Modern Portfolio Theory 40 / 40

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