Introduction Risk Diversification Measures Empirical Applications Conclusion References

Risk Diversification

Matteo Malavasi1,2 Sergio Ortobelli2,3 Stefan Truck1

1Department of Actuarial Studies and Business Analytics, Macquarie University,

2Department of Management, Economics and Quantitative Methods, Universityof Bergamo,

3Department of Finance, University of Ostrava

iPARM Australia 2019Sydney, November 26, 2019

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Introduction Risk Diversification Measures Empirical Applications Conclusion References

Outline

Introduction

Risk Diversification Measures

Empirical Applications

Conclusion

References

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Introduction Risk Diversification Measures Empirical Applications Conclusion References

Introduction

Portfolio diversification is often measured based on the number ofassets with non zero weights in a portfolio:

• Diversification orderings (Marshall 1979; Wong 2007).

• Statistics of two portfolios’ weight vectors (Egozcue 2010;Ortobelli 2018)

• Herfindal index based on portfolio weights.

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Introduction Risk Diversification Measures Empirical Applications Conclusion References

Introduction

There are many alternative approaches to portfolio diversification:

• Correlation based diversification (Levy, 1970; Silvapulle, 2001;Dopfel, 2003)

• Return Gap (Statman, 2004)

• Diversification Ratio (Choueifaty and Coignard, 2008)

• Diversification Delta (Vermorken et al., 2014; Salazar et al.,2018)

• Portfolio Diversification Index (Rudin, 2006; Meucci, 2009).

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Introduction Risk Diversification Measures Empirical Applications Conclusion References

Motivation

• Define a new class of functionals, which we called riskdiversification measures (RDMs) and coherent riskdiversification measures (CRDMs).

• We show that under elliptical distributed returns, any CRDMdepends on the first two moments of the portfolio and assetreturns distribution.

• We constrict so-called mean-risk diversification efficientfrontiers (similar to the mean-variance efficient frontier).

• We want to examine the performance of risk diversificationoptimal portfolios during periods of financial distress.

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Risk Diversification Measures: Definition

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.

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Conditional Value at Risk (Expected Shortfall)

V@R CV@R0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

(1- )% of Probability

Figure: Example of CV@R

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CV@Rs vs Weighted Average of CV@Rs

V@R90% CV@R90%

0

0.1

0.2

0.3

0.4

0.5

10% of Probability

V@R95% CV@R95%

0

0.1

0.2

0.3

0.4

0.5

5% of Probability

V@R99% CV@R99%

0

0.1

0.2

0.3

0.4

0.5

1% of Probability

Figure: CV@Rs at different αs

CV@R90% CV@R95% CV@R99%

0

0.005

0.01

0.015

0.02

0.025

0.03

Weighted Average of CV@RsPortfolio CV@R

Figure: CV@Rs vs weightedaverage of CV@Rs

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Risk Diversification Measures

Diversification Ratio:

Dσ(P) = 1− σ(P)∑Ni=1 wiσ(Xi )

Diversification Conditional Value at Risk, i.e. DCV@Rα

DCV@Rα(P) = 1− CV@Rα(P)∑Ni=1 wiCV@Rα(Xi )

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Dataset

• 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.

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Mean Risk Diversification Efficient Frontier (1)

Similarly to the mean variance efficient frontier, we solve thefollowing optimization problem:

maxw

E[w ′X

]s.t. Dν(w ′X ) ≥ d

n∑i=1

wi = 1 , 0 ≤ wi ≤ 1, i = 1, . . . , n

where d is a desired level of risk diversification. The solution thendepends on the values d and on the choice of the risk measure ν.

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Mean Risk Diversification Efficient Frontier (2)

To compute the mean risk diversification efficient frontiers weproceed similarly to the mean variance framework. For eachmeasure:

• We compute first the portfolio with maximum riskdiversification.

• Then, we select 100 equally spaced points in the intervalbetween 0 and the maximum level of risk diversification.

• We solve the optimization problem for each of the 100 pointsin the interval.

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Mean-Risk Diversification Efficient Frontiers (3)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

D

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

E[P

]

10-3

Portfolio 1Portfolio 2Portfolio 3Portfolio 4

Figure: Mean-Dσ EfficientFrontiers

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

DCV@R99%

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

E[P

]

10-3

Portfolio 1Portfolio 2Portfolio 3Portfolio 4

Figure: Mean-DCV@R99%Efficient

Frontiers

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Table: Average statistics, sum of squared weights and number of investedassets.

Area RDM Mean St. Dev. Skewness Kurtosis∑

i w2i ] Assets

1

Dσ 0.00268 0.0108 2.651 22.543 0.9186 2.166DCV@R90%

0.00267 0.0109 2.667 22.782 0.9264 2DCV@R95%

0.00268 0.0109 2.720 23.302 0.9345 2.636DCV@R99%

0.00269 0.0109 2.726 23.307 0.9454 2.666

5

Dσ 0.00214 0.0069 1.182 8.7405 0.3031 6.33DCV@R90%

0.00222 0.0078 2.078 16.404 0.4401 7DCV@R95%

0.00227 0.0082 2.217 17.109 0.4952 4.818DCV@R99%

0.00237 0.0089 2.207 16.699 0.5972 2.3

7

Dσ 0.00165 0.0049 0.666 4.8942 0.1381 13.19DCV@R90%

0.00195 0.0062 1.449 9.0999 0.2709 6.545DCV@R95%

0.00201 0.0068 1.517 8.8859 0.3373 5.363DCV@R99%

0.00205 0.0072 1.505 8.4170 0.3986 3.6

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Facing Period of Financial Distress (1)

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.

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Facing Period of Financial Distress (2)

3 Jan 2006 15 Sep 2007 15 Sep 2008 15 Sep 2009 7 Oct 20140.5

1

1.5

2

2.5

3Ex-post Wealth

DCV@R

90%

DCV@R

95%

DCV@R

99%

GMVD

EWMSR

Figure: Ex post wealth evolution

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Table: Performances of DCV@R90%, DCV@R95%, DCV@R99% Dσ, GMVand EW.

Strategies µann σann SR(X ) ] Assets Transaction Cost

Crisis PeriodDCV@R90%

0.10068 0.3916 0.0154 9.416 0,0009DCV@R95%

0.02101 0.4173 0.0031 10 0.0010DCV@R99%

0.15317 0.4074 0.0220 7.416 0.0013Dσ 0.07286 0.4377 0.0101 15.08 0.0005GMV -0.1338 0.3025 -0.0299 28 0.0003EW 0.07024 0.4305 0.0099 28 0

After CrisisDCV@R90%

0.15598 0.1312 0.0695 10.930 0.0009DCV@R95%

0.12196 0.1366 0.0530 8.662 0.0010DCV@R99%

0.12235 0.1481 0.0491 5.825 0.0011Dσ 0.16513 0.1323 0.0727 13.80 0.0004GMV 0.08984 0.1114 0.0486 28 0.0004EW 0.13829 0.1462 0.0558 28 0

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Weights Evolution

1 2 3 4 5 6 7 8 9 10 11 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure: Portfolio weightsevolution of DCV@R99%

.

1 2 3 4 5 6 7 8 9 10 11 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure: Portfolio weightsevolution of Dσ

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Conclusion

• 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.

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References

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.

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