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
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
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
Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios Summary Risk Parity Notations 1 Risk Parity Application 2 Market vs Idiosyncratic Risk Parity Allocation 3 An Application on Commodities Portfolios 4 4/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions
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
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 = ( w 1 , . . . , w n ) ′ 6/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions
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 = ( w 1 , . . . , w n ) ′ The portfolio return is: n � w ′ y = w i y i i =1 6/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions
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 = ( w 1 , . . . , w n ) ′ The portfolio return is: n � w ′ y = w i y i i =1 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
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 ( λ w ) = λ R ( w ) for any positive λ 7/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions
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 ( λ w ) = λ R ( w ) for any positive λ We can derive this condition and obtain for λ = 1 n n ∂ R ( w ) � � R ( w ) = w i = R i ( w ) ∂ w i i =1 i =1 7/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions
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 ( λ w ) = λ R ( w ) for any positive λ We can derive this condition and obtain for λ = 1 n n ∂ R ( w ) � � R ( w ) = w i = R i ( w ) ∂ w i i =1 i =1 R i ( 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
Risk Parity Notations Risk Parity Application Market vs Idiosyncratic Risk Parity Allocation An Application on Commodities Portfolios Risk contributions for general risk measures • Volatility: R i ( w ) = Cov ( w i y i , w ′ y ) V ( w ′ y ) • Value-at-Risk: R i ( w ) = E [ w i y i | w ′ y = q α ( w ′ y )] • Expected Shortfall: R i ( w ) = E [ w i y i | w ′ y > q α ( w ′ y )] We obtain different structural interpretations of R i ( 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
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
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
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
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
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
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
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
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
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: R i ( w ) = π i , i = 1 , . . . , n (1) where π = ( π 1 , . . . , π n ) ′ is equivalent to a benchmark portfolio 16/32 Darolles, S., Gourieroux, C., and Jay, E. Robust Portfolio Allocation with Risk Contribution Restrictions
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
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