Introduction Risk Diversification Measures Empirical Applications Conclusion References Risk Diversification Matteo Malavasi 1,2 Sergio Ortobelli 2,3 Stefan Tr¨ uck 1 1 Department of Actuarial Studies and Business Analytics, Macquarie University, 2 Department of Management, Economics and Quantitative Methods, University of Bergamo, 3 Department of Finance, University of Ostrava iPARM Australia 2019 Sydney, November 26, 2019 1 / 20
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Risk Diversi cation - IBR Conferences Australia... · 0 be a vector of returns, w = [w 1;:::;w n]0 be a vector of portfolio weights and : 2L(;F;P) !R be a risk measure. A risk diversi
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DefinitionLet X = [X1, . . . ,Xn]′ be a vector of returns, w = [w1, . . . ,wn]′ bea vector of portfolio weights and ν : Λ ∈ L (Ω,F ,P)→ R be a riskmeasure. A risk diversification measure for a portfolio P = w ′X isa functional of the from:
Dν(P) = 1− ν(P)∑ni=1 wiν(Xi )
A portfolio P1 presents higher Risk Diversification, with respect tothe risk measure ν than a portfolio P2, if Dν(P1) ≥ Dν(P2). Whenν is a coherent risk measure, then Dν is called Coherent RiskDiversification Measure.
• We consider a market composed by assets belonging to theDow Jones Industrial Average index (DJIA) from January 3,2005 to October 13, 2017. We consider only the assetspresent in the index for the entire period.
• In particular, assets belonging to the DJIA index are tradedvery regularly and the index itself represents a reasonablydiversified market portfolio.
• Moreover, the DJIA exhibits increasing correlation underperiods of financial distress, implying that diversificationbenefit decreases when needed the most.
In order to test the performance of risk diversification measuresduring period of financial distress we proceed with a rolling windowtype of analysis.
• Window: 1 year of daily observations.
• Re-balancing: monthly, i.e. approximately every 21 tradingdays.
• Starting date: 3 January 2005.
At every step of the rolling window, for each risk measure wecompute the portfolio with the maximum risk diversification.Then we compare the out-of sample performance of theconstructed portfolios.
• Risk Diversification: depends on a given risk measure,isdefined as the ratio between portfolio risk and the weightedrisk of a portfolio’s individual components.
• RDMs can be interpreted as the percentage of idiosyncraticrisk diversified in the portfolio.
• Mean-Risk Diversification Efficient Frontier: each of theefficient frontiers exhibits concavity w.r.t. risk diversification,as risk diversification increases, expected return decreases.
• Risk diversification increases with risk aversion, whileconcentration increases.
• Our results suggest that optimal risk diversification portfoliosare well able to cope with period of financial distress.
Artzner, P. and Delbaen, F. and Eber, J. and Heath, D. (1999). Coherentmeasures of risk. Mathematical finance, 3, p.203.
Choueifaty, Y., Coignard, Y. (2008). Toward maximum diversification.Journal of Portfolio Management, 35(1), 40.
Egozcue, M., Wong, W. K. (2010). Gains from diversification on convexcombinations: A majorization and stochastic dominance approach.European Journal of Operational Research, 200(3), 893-900.
Vermorken, M. A., Medda, F. R., Schroder, T. (2012). The diversificationdelta: A higher-moment measure for portfolio diversification. Journal ofPortfolio Management, 39(1), 67.
Salazar, Y., Bianchi, R. J., Drew, M. E., Truck, S. (2017). TheDiversification D elta: A Different Perspective. The Journal of PortfolioManagement, 43(4), 112-124.