Robust Portfolio Allocation with Risk Contribution Restrictions - - PowerPoint PPT Presentation

▶
robust portfolio allocation with risk contribution
SMART_READER_LITE
LIVE PREVIEW

Robust Portfolio Allocation with Risk Contribution Restrictions - - PowerPoint PPT Presentation

Dec 27, 2022 201 likes •700 views

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios Robust Portfolio Allocation with Risk Contribution Restrictions Darolles, S., Gourieroux, C., and Jay, E.

slide-1
SLIDE 1

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Robust Portfolio Allocation with Risk Contribution Restrictions

Darolles, S., Gourieroux, C., and Jay, E. Risk Based and Factor Investing Conference London, 5 November, 2015

1/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-2
SLIDE 2

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Published in Risk Based and Factor Investing Conference (2015), edited by Emmanuel Jurczenko:

  • We extend Risk Parity portfolios to any type of risk

measure: Value-at-Risk, Expected Shortfall, not only Variance

  • We propose a dynamic implementation of Risk Parity

portfolios with a control of transaction costs

  • We consider market risk restrictions in the

definition of Risk Parity portfolios - This presentation

2/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-3
SLIDE 3

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Our results:

  • We stabilize portfolio allocations over time and

reduce the turnover

  • We built Risk Parity portfolios with different market

neutrality levels

  • In a regulatory perspective, we control separately

market and idiosyncratic risks - This presentation

3/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-4
SLIDE 4

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Summary

1

Risk Parity Notations

2

Risk Parity Application

3

Market vs Idiosyncratic Risk Parity Allocation

4

An Application on Commodities Portfolios

4/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-5
SLIDE 5

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

  • 1. Risk Parity Notations

5/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-6
SLIDE 6

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Risk measure We consider a portfolio allocation w in n risky assets: w = (w1, . . . , wn)′

6/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-7
SLIDE 7

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Risk measure We consider a portfolio allocation w in n risky assets: w = (w1, . . . , wn)′ The portfolio return is: w′y =

n

  • i=1

wiyi

6/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-8
SLIDE 8

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Risk measure We consider a portfolio allocation w in n risky assets: w = (w1, . . . , wn)′ The portfolio return is: w′y =

n

  • i=1

wiyi The risk of this portfolio is measured by a scalar R(w)

6/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-9
SLIDE 9

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Risk contributions All the usual risk measures satisfy the homogeneity condition: R(λw) = λR(w) for any positive λ

7/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-10
SLIDE 10

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Risk contributions All the usual risk measures satisfy the homogeneity condition: R(λw) = λR(w) for any positive λ We can derive this condition and obtain for λ = 1 R(w) =

n

  • i=1

wi ∂R(w) ∂wi =

n

  • i=1

Ri(w)

7/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-11
SLIDE 11

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Risk contributions All the usual risk measures satisfy the homogeneity condition: R(λw) = λR(w) for any positive λ We can derive this condition and obtain for λ = 1 R(w) =

n

  • i=1

wi ∂R(w) ∂wi =

n

  • i=1

Ri(w) Ri(w) is the risk contribution of asset i to R(w)

7/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-12
SLIDE 12

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Risk contributions for general risk measures

  • Volatility: Ri(w) = Cov(wiyi, w′y)

V (w′y)

  • Value-at-Risk: Ri(w) = E[wiyi|w′y = qα(w′y)]
  • Expected Shortfall: Ri(w) = E[wiyi|w′y > qα(w′y)]

We obtain different structural interpretations of Ri(w) and can choose which risk measure is the most appropriate one

8/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-13
SLIDE 13

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

  • 2. Risk Parity Application

9/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-14
SLIDE 14

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

The investment universe Futures contracts on physical commodities, split into five sectors : Energy (4) Grains & Seeds (5) Softs (5) Live Stock (2) Metals (5) Total : 21 series Daily close from 14 May 1990 to 24 September 2012

10/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-15
SLIDE 15

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

”Grains & Seeds” Sector

11/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-16
SLIDE 16

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

VaR Parity portfolio (re)allocation We compare the VaR Parity optimal portfolio with 3 benchmark portfolios:

  • Equal-weighted portfolio
  • Minimum Variance portfolio
  • Volatility Parity Portfolio

12/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-17
SLIDE 17

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

VaR Parity portfolio (re)allocation We compare the VaR Parity optimal portfolio with 3 benchmark portfolios:

  • Equal-weighted portfolio
  • Minimum Variance portfolio
  • Volatility Parity Portfolio

For the 4 portfolios, we provide:

  • Figure 1: the weight of each asset in the portfolio
  • Figure 2: the contribution of each asset to the VaR

12/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-18
SLIDE 18

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios 13/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-19
SLIDE 19

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios 14/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-20
SLIDE 20

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

  • 3. Market vs Idiosyncratic Risk Parity Allocation

15/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-21
SLIDE 21

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Total risk parity portfolio The idea is to control the risk contributions of the portfolio allocations The corresponding allocation solves the system: Ri(w) = πi, i = 1, . . . , n (1) where π = (π1, . . . , πn)′ is equivalent to a benchmark portfolio

16/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-22
SLIDE 22

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Market and idiosyncratic risks BUT financial regulation allocates differently the Capital Requirement for the systematic (or market) component of the risk and for its idiosyncratic component

17/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-23
SLIDE 23

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Market and idiosyncratic risks BUT financial regulation allocates differently the Capital Requirement for the systematic (or market) component of the risk and for its idiosyncratic component Why ? Whereas the second component can be easily diversified, the first component cannot be eliminated by diversification

17/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-24
SLIDE 24

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Market and idiosyncratic risks BUT financial regulation allocates differently the Capital Requirement for the systematic (or market) component of the risk and for its idiosyncratic component Why ? Whereas the second component can be easily diversified, the first component cannot be eliminated by diversification Then ? Similarly, a portfolio manager might consider differently these two components of the risk

17/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-25
SLIDE 25

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Let us consider a one-factor model: yi = βif + σiui, i = 1, . . . , n f : common factor βi : sensitivity of the asset return to the common factor ui : idiosyncratic component

18/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-26
SLIDE 26

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Let us consider a one-factor model: yi = βif + σiui, i = 1, . . . , n f : common factor βi : sensitivity of the asset return to the common factor ui : idiosyncratic component At the portfolio level: w′y = n

  • i=1

wiβi

  • f +

n

  • i=1

wiσiui

18/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-27
SLIDE 27

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

The Risk Parity decomposition principle can be extended to disentangle the systematic and the idiosyncratic component: Ri(w) = Ris(w) + Riu(w) This decomposition can be aggregated to get: R(w) = Rs(w) + Ru(w)

19/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-28
SLIDE 28

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Market and idiosyncratic risk contributions Assets Market Idiosyncratic Total contrib contrib contrib 1 R1(w) . . . . . . i Ris(w) Riu(n) Ri(w) . . . . . . n Rn(w) Total Rs(w) Ru(w) R(w)

20/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-29
SLIDE 29

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

How to derive this new risk decomposition in practice? A virtual portfolio investing: wis in βif , wiu in σiui

21/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-30
SLIDE 30

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

How to derive this new risk decomposition in practice? A virtual portfolio investing: wis in βif , wiu in σiui The Euler decomposition of the risk measure can be applied to this virtual portfolio to get:

˜ R(ws, wu) =

n

  • i=1

wi,s ∂ ˜ R ∂wi,s (ws, wu) +

n

  • i=1

wi,u ∂ ˜ R ∂wi,u (ws, wu)

21/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-31
SLIDE 31

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

How to derive this new risk decomposition in practice? A virtual portfolio investing: wis in βif , wiu in σiui The Euler decomposition of the risk measure can be applied to this virtual portfolio to get:

˜ R(ws, wu) =

n

  • i=1

wi,s ∂ ˜ R ∂wi,s (ws, wu) +

n

  • i=1

wi,u ∂ ˜ R ∂wi,u (ws, wu)

For wu = wu = w, we get :

R(w) = ˜ R(w, w) =

n

  • i=1

wi ∂ ˜ R ∂wi,s (w, w) +

n

  • i=1

wi ∂ ˜ R ∂wi,u (w, w) =

n

  • i=1

Ris(w) +

n

  • i=1

Riu(w)

21/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-32
SLIDE 32

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Portfolios with constraint on the market risk contribution In standard RP portfolios, constraints are written on he basic assets Now we can write constraints on the idiosyncratic and systematic components of the portfolio

22/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-33
SLIDE 33

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Portfolios with constraint on the market risk contribution In standard RP portfolios, constraints are written on he basic assets Now we can write constraints on the idiosyncratic and systematic components of the portfolio A constrained optimization problem: min

w R(w) subject to Rs(w) = πR(w)

22/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-34
SLIDE 34

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Portfolios with constraint on the market risk contribution In standard RP portfolios, constraints are written on he basic assets Now we can write constraints on the idiosyncratic and systematic components of the portfolio A constrained optimization problem: min

w R(w) subject to Rs(w) = πR(w)

This problem is equivalent to: min

w R2(w) + δ[(1 − π)Rs(w) − πRu(w)]2

22/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-35
SLIDE 35

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

How to choose these 2 control parameters ?

23/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-36
SLIDE 36

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

How to choose these 2 control parameters ?

  • π is a market neutrality parameter:

Market neutral or α−portfolio when π → 0 Index or β−portfolio when π → 1

23/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-37
SLIDE 37

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

How to choose these 2 control parameters ?

  • π is a market neutrality parameter:

Market neutral or α−portfolio when π → 0 Index or β−portfolio when π → 1

  • δ is a smoothing parameter:

Minimum VaR portfolio when δ → 0 Market controled portfolio when δ → ∞

23/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-38
SLIDE 38

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

  • 4. An Application on Commodities Portfolios

24/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-39
SLIDE 39

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

We use a single factor model: the DJUBS index return We consider a mix between: the risk minimization the systematic risk contribution restriction for the VaR5% risk measure

25/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-40
SLIDE 40

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

We use a single factor model: the DJUBS index return We consider a mix between: the risk minimization the systematic risk contribution restriction for the VaR5% risk measure The corresponding program is: minw VaR(w)2 + δ[(1 − π)VaRS(w) − πVaRu(w)]2

25/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-41
SLIDE 41

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Empirical Market and idiosyncratic risk contributions For the last available date (September 24, 2012) based on the 252 days preceding the computation date Assets Beta Weight Market Idio. Total contrib contrib contrib Corn 1.09 11.9% 9.3% 6.9% 16.2% Rice 0.35 34.4% 8.7% 18.2% 26.8%

  • Soy. Oil

0.76 24.4% 13.2% 8.8% 22.1% Soybeans 0.87 16.3% 10.2% 8.0% 18.2% Wheat 1.12 13.0% 10.4% 6.3% 16.7% Total 100% 51.8% 48.3% 100%

26/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-42
SLIDE 42

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Market controlled Parity portfolios We compare 9 (3 x 3) optimal portfolios computed as functions of δ and π for the sector ”Grains & Seeds”

27/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-43
SLIDE 43

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Market controlled Parity portfolios We compare 9 (3 x 3) optimal portfolios computed as functions of δ and π for the sector ”Grains & Seeds” For the 9 portfolios, we provide:

  • Figure 1: the weight of each asset in the portfolio
  • Figure 2: the market risk contribution of each asset

Each row is for a fixed value of π ∈ (0%, 20%, 50%) Each column is for a fixed value of δ ∈ (10, 50, 100)

27/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-44
SLIDE 44

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=0% ; δ=10 ; λ=0.01

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=0% ; δ=50 ; λ=0.01

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=0% ; δ=100 ; λ=0.01

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=20% ; δ=10 ; λ=0.01

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=20% ; δ=50 ; λ=0.01

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=20% ; δ=100 ; λ=0.01

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=50% ; δ=10 ; λ=0.01

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=50% ; δ=50 ; λ=0.01

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=50% ; δ=100 ; λ=0.01

corn rice soybeanoil soybeans wheat

28/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-45
SLIDE 45

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=0% ; δ=10 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=0% ; δ=50 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=0% ; δ=100 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=20% ; δ=10 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=20% ; δ=50 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=20% ; δ=100 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=50% ; δ=10 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=50% ; δ=50 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst/Idio PTF for grains − π=50% ; δ=100 ; λ=1

corn rice soybeanoil soybeans wheat

29/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-46
SLIDE 46

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst. Contrib to Total for grains − π=0% ; δ=10 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst. Contrib to Total for grains − π=0% ; δ=50 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst. Contrib to Total for grains − π=0% ; δ=100 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst. Contrib to Total for grains − π=20% ; δ=10 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst. Contrib to Total for grains − π=20% ; δ=50 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst. Contrib to Total for grains − π=20% ; δ=100 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst. Contrib to Total for grains − π=50% ; δ=10 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst. Contrib to Total for grains − π=50% ; δ=50 ; λ=1

corn rice soybeanoil soybeans wheat Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Min−VaR(5%) Syst. Contrib to Total for grains − π=50% ; δ=100 ; λ=1

corn rice soybeanoil soybeans wheat

30/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-47
SLIDE 47

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

Concluding Remarks

31/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions

slide-48
SLIDE 48

Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios

It is possible to manage in an appropriate way the market component of Risk Parity portfolios:

  • To be in line with the financial regulation
  • To define portfolios in terms of market neutrality
  • To stabilize budget allocations over time

32/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions