Skip to main content
SpringerLink
Log in

Optimisation in the presence of tail-dependence and tail risk: A heuristic approach for strategic asset allocation

  • Paper
  • Published:
Journal of Asset Management Aims and scope Submit manuscript

Abstract

This paper presents a method to overcome the classical drawbacks of the Monte Carlo methods for the asset allocation, that is, resampling, deeply dependent on the multinormal assumption. This approach allows to set a derivative-free barrier against joint extreme negative returns (tail-dependence or contagion) and extreme (negative) returns (univariate tail risk) not considered in the multinormal framework. This barrier is set through an extensive use of copulas and Extreme Value Theory. The model has been applied on a sample of 11 euro-denominated asset classes with historical inputs. The weights have been tested on simulated (multivariate Student's t) returns and with real out-of-the sample returns. A comparison has been performed with the asset allocation given by the resampling method. The results provide evidence of a barrier against extreme negative returns occurring simultaneously. Furthermore, the model is totally distribution-free and therefore it does not involve any a priori decision on the marginal distributions for asset returns. The cost of this approach (loss of Sharpe ratio), in our example, is negligible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6

Similar content being viewed by others

Notes

  1. A wide reference on these studies can be found in Jondeau and Rockinger (2006).

  2. A wide reference to copula can be found in Joe (1997) and Nelsen (1998). More recently, an excellent introduction to copulas and several financial applications can be found in McNeil et al. (2005), Rob van den Goorbergh (2004) and Embrechts et al. (2002).

  3. A list of known ϕ(.) functions is available in Frees and Valdez (1998) and Nelsen (1998).

  4. As noted by Bouyè et al. (2001), Archimedean copula significantly reduces the number of parameters to be estimated. Implicit (Gaussian and t) copula provides information about dependence between each pair of random variables and, therefore, for N variables, N(N−1)/2 parameters have to be estimated. For Archimedean copula, the dependence is characterised only by N−1 parameters.

  5. Table 1 highlights the relationships between ρ τ and α in the case of Archimedean copulas.

  6. It is really easy to verify this fallacy by drawing two correlated (ρ) vectors of normally distributed random variables (X 1 and X 2) and then taking Y 1=Exp(X 1) and Y 2=Exp(X 2). This transformation is nonlinear strictly increasing. ρ(Y 1; Y 2)≠ρ(X 1; X 2) whereas ρ τ (Y 1; Y 2)=ρ τ (X 1; X 2).

  7. A sample of bivariate Student's t(ν) is simulated in two steps (see algorithm 3.10 in McNeil et al., 2005): (1) simulation of multivariate normal variates using Cholesky decomposition; (2) introduction of a mixing variable W drawn from an Inverse Gaussian — IG — distribution (W∼IG(1/2ν, 1/2ν)).

  8. Again, simulate two random vectors of correlated normally distributed random variables (X 1 and X 2), take the Exp(X 1) and Exp(X 2) and then calculate the Pearson's rho and Kendall's tau in both cases (for an analytical proof, see Proposition 5.29 of McNeil et al., 2005).

  9. Notice that the closed formula is available for Pearson's rho (ρ). For the sake of comparison, however, we derive direct relationships with Kendall's tau(ρ τ ). The relationship between Spearman's rho (ρ S ) and ρ for multivariate Student's t is not available. This is another reason for considering only Kendall's tau(ρ τ ). It is also worth highlighting that Archimedean copulas can also be rotated.

  10. For example the relationship between t-copula and Spearman's rho is still unknown (Hult and Lindskog, 2002). Moreover, we do not have a direct formula to link the parameter α of the Gumbel copula to the Spearman's rho (Frees and Valdez, 1998).

  11. A detailed description of this Monte Carlo simulation is reported in the Appendix of Longin (2005).

  12. Notice that the Gaussian distribution can be viewed as a special case of the Stable Paretian. In particular, Gaussian distribution corresponds to the S(2, 0, σ, μ) distribution. α is the index of stability or tail exponent and controls the decay in the tails of the distribution. The other parameters (σ, β, μ) control, respectively, skewness, scale and location. Notice also that the Stable Paretian has known variance if and only if α=2 (Gaussian case). With 0<α<2 (the classical case in portfolio theory reveals 1<α<2) and both skewness and location parameter equal to zero, closed formulas for optimal portfolios are available minimising the scale parameter σ equivalent to the classical risk parameter (standard deviation) in the Normal case.

  13. For a large sample of equities, indexes and bonds, the authors find the supremacy of logistic, gamma, lognormal and extreme value distributions.

  14. Notice that σ (scale parameter) in (10) is not required as in this conditional variance approach we do not directly simulate the returns but we simulate specific standardised innovations, given τ i opt. In an unconditional approach, we could need random variates drawn from a GPD; therefore, the scale parameter σ should also be estimated.

  15. Note that this approach, known as the ‘Method of Moments’, is appropriate in the case of few observations. Otherwise, the classical MLE estimators for the parameter α can be applied but it requires a huge amount of data, rarely available with a monthly framework.

  16. The asymptotic distribution may be used for calculating critical values for hypothesis test if n>2,000. In this case, the authors provide the quantiles of the test statistics, dependent only on the level of confidence p.

  17. Recall that the P-test is ‘the sample is peaked as a normal distribution’; the T-test is ‘the sample has tails comparable with the normal distribution; and the L-test is ‘the sample is leptokurtic as a normal distribution’.

  18. See Von Eye A. and Bogat A. (2004) for further details on Mardia's test.

  19. Notice that a similar two-step GARCH-EVT approach has been applied in the risk management field by McNeil and Frey (2000) for a more reliable estimation of VaR and Expected Shortfall (ES). The authors first filter the returns in order to have i.i.d. innovations. Second, they fit a GPD distribution by maximum likelihood estimators for the innovations and then estimate VaR and ES.

  20. A similar approach involves the bootstrapping of the standardised innovations parameterised using the α-Clayton copula and the replacement of these innovations in the filtering process estimated in (13b). This method has been extensively applied in the field of the Filtered Historical Simulation (FHS) proposed by Giannopoulos and Tunaru (2005), Barone-Adesi and Giannopoulos (2001) to estimate different risk measures for a portfolio consisting of linear and nonlinear securities. This methodology, however, requires a huge amount of historical data to incorporate all possible movements of asset returns. This is not the case when we deal with monthly observations but could be appropriate with daily returns. An alternative approach could be to fit a multivariate GARCH model, for example the DCC-MVGARCH model proposed by Engle in 2002. The DCC model is a multivariate GARCH model with time-varying correlations. It assumes a (conditionally) joint normal distribution for return innovations. Note that this assumption implies normal conditional margins, and a normal conditional copula, which is fully explained by the correlation coefficients. The DCC-MVGARCH model estimates simultaneously both the margins and the dynamic matrix of correlations. Van den Goorbergh (2004) demonstrated that a copula-based approach systematically overperforms the easier DCC-MVGARCH approach in the risk management framework.

  21. We decide to simulate a matrix with 60 rows (five years of monthly returns).

  22. Recall that Student's t(ν) has at least ν moments; thus, imposing ν>2, the variance always exists and the minimisation approach à laMarkowitz is valid.

References

  • Athayde, G. M. and Flôres, R. G. (2004) ‘Finding a Maximum Skewness Portfolio — A General Solution to the Three-Moments Portfolio Choice’, Journal of Economic Dynamics and Control (28).

  • Baierl, G. T. and Chen, P. (2000) ‘Choosing Managers and Funds’, Journal of Portfolio Management (Winter).

  • Balkema, A. A. and De Haan, L. (1974) ‘Residual Life Time at Great Age’, Annals of Probability (2).

  • Barone-Adesi, G. and Giannopoulos, K. (2001) ‘Non Parametric VaR Techniques: Myths and Realities’, Economic Notes 30 (July).

  • Black, F. and Litterman, R. (1992) ‘Global Portfolio Optimization’, Financial Analysts Journal (September).

  • Bouyè, E., Gaussel, N. and Salmon, N. (2001) ‘Investigating Dynamic Dependence Using Copulae’, European Financial Management Association.

  • Bradley, B. O. and Taqqu, M. S. (2002) ‘Financial Risk and Heavy Tails’, in S.T. Rachev (ed.), Heavy-Tailed Distributions in Finance, North Holland, Amsterdam.

    Google Scholar 

  • Campbell, J. Y., Huisman, R. and Koedijk, K. (2001) ‘Optimal Portfolio Selection in a Value-at-Risk framework’, Journal of Banking and Finance, 25.

  • Chamberlain, G. (1983) ‘A Characterization of the Distributions that Imply Mean–Variance Utility Functions’, Journal of Economic Theory (29).

  • Danielsson, J., De, Haan L., Peng, L. and De Vries, C. G. (2000) ‘Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation’, Econometric Institute Report, no. 197, Erasmus University Rotterdam.

  • Degen, M., Embrechts, P. and Lambrigger, D. D. (2006) ‘The Quantitative Modeling of Operational Risk: Between g-and-h and EVT’, Working Paper ETH, Preprint, Zurich, December.

  • Dutta, A. and Perry, G. (2006) ‘A Tale of Tails: An Empirical Analysis of Loss Distribution Models for Estimating Operational Risk Capital’, Working Paper no. 6-13 Federal Reserve Bank of Boston, July.

  • Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance, Springer.

  • Embrechts, P., McNeil, A. J. and Straumann, D. (2002) ‘Correlation and Dependency in Risk Management: Properties and Pitfalls’, in M. Dempster (ed.), Risk Management: Value-at-Risk and Beyond, Cambridge University Press, Cambridge.

    Google Scholar 

  • Engle, R. F. (2002) ‘Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models’, Journal of Business and Economic Statistics, 20 (3).

  • Frees, E. and Valdez, E. (1998) ‘Understanding Relationships Using Copulas’, North American Actuarial Journal (2).

  • Genest, C. and Rivest, L. (1993) ‘Statistical Inference Procedures for Bivariate Archimedean Copulas’, Journal of the American Statistical Association (88).

  • Giannopoulos, K. and Tunaru, R. (2005) ‘Coherent Risk Measures Under Filtered Historical Simulation’, Journal of Banking and Finance, 29.

  • Harvey, C. R and Siddique, A. (2000) ‘Conditional Skewness in Asset Pricing Tests’, Journal of Finance (55).

  • Hogg, R. V and Lenth, R. V. (1984) ‘A Review of Some Adaptive Statistical Techniques’, Communications in Statistics — Theory and Methods (17).

  • Hult, H. and Lindskog, F. (2002) ‘Multivariate Extremes, Aggregation and Dependence in Elliptical Distributions’, Advanced in Applied Probability (34).

  • Jansen, D. W. and De Vries, C. G. (1991) ‘On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspectives’, The Review of Economic and Statistics (73).

  • Jobson, J. D. and Korkie, B. (1981) ‘Putting Markowitz theory to work’, Journal of Portfolio Management (Summer).

  • Jobst, A. (2007) ‘Operational Risk-the Sting is Still in the Tail But the Poison Depends on the Dose’, Journal of Operational Risk, 2 (2).

  • Joe, H. (1997) ‘Multivariate Models and Dependence Concepts’, in Monographs on Statistics and Applied Probability, Vol. 73, Chapman & Hall, London.

    Google Scholar 

  • Jondeau, E. and Rockinger, M. (2005) ‘Conditional Asset Allocation under Non-Normality: How Costly Is the Mean-Variance Criterion’, Working Paper, Institute of Banking and Finance, HEC Lausanne.

  • Jondeau, E. and Rockinger, M. (2006) ‘Optimal Portfolio Allocation under Higher Moments’, European Financial Management.

  • Jorion, P. (1992) ‘Portfolio Optimization in Practice’, Financial Analysts Journal (January–February).

  • Leibowitz, M. L., Bader, L. N. and Kogelman, S. (1996a) Return Targets and Shortfall Risks — Studies in Strategic Asset Allocation, Irwin Professional Publishing, New York.

    Google Scholar 

  • Leibowitz, M. L, Bader, L. N. and Kogelman, S. (1996b) ‘Asset Allocation under Shortfall Constraints’, The Journal of Portfolio Management (Winter).

  • Levy, H. and Duchin, R. (2004) ‘Asset Return Distributions and the Investment Horizon’, The Journal of Portfolio Management (Spring).

  • Longin, F. (1996) ‘The Asymptotic Distribution of Extreme Stock Market Returns’, The Journal of Business, 69 (3).

  • Longin, F. (2005) ‘The Choice of the Distribution of Asset Returns: How EVT Can Help’, The Journal of Banking and Finance, 29.

  • Longin, F. and Solnik, B. (2001) ‘Extreme Correlation and International Equity Markets’, Journal of Finance, 56.

  • Lucas, L. and Riepe, M. W. (1996) ‘The Role of Returns-Based Style Analysis: Understanding, Implementing, and Interpreting the Technique’, Ibbotson Research, Working Paper.

  • McNeil, J. A. and Frey, R. (2000) ‘Estimation of Tail-Related Risk Measures for Heteroskedastic Financial Time Series: An Extreme Value Approach’, Journal of Empirical Finance (7).

  • McNeil, J. A., Frey, R. and Embrechts, P. (2005) ‘Quantitative Risk Management: Concepts, Techniques and Tools’, Princeton Series in Finance. Princeton University Press, Princeton and Oxford.

  • Michaud, R. O. (1998) Efficient Asset Management, Harvard Business School Press, Boston, MA.

    Google Scholar 

  • Nelsen, I. G. (1998) An Introduction to Copulas, Lecture Notes in Statistics, Springer-Verlag, New York.

    Google Scholar 

  • Peiro, A. (1999) ‘Skewness in Financial Returns’, The Journal of Banking and Finance, 23 (6).

  • Pickands, J. (1975) ‘Statistical Inference Using Extreme Order Statistics’, Annals of Statistics (3).

  • Resnick, S. I. and Starica, C. (1997) ‘Smoothing the Hill Estimator’, Advanced Applied Probability (29).

  • Rockafellar, R. T. and Uryasev, S. (1999) ‘Optimization of Conditional Value-at-Risk’, Working Paper University of Washington, September.

  • Rosenberg, J. V. and Schuermann, T. (2006) ‘A General Approach to Integrated Risk Management with Skewed, Fat-Tailed Risks’, Journal of Financial Economics, 79 (3).

  • Schmid, F. and Trede, M. (2004) ‘Simple Test for Peakedness, Fat Tails and Leptokurtosis Based on Quantiles’, Computational Statistics and Data Analysis (43).

  • Sentana, E. (2001) ‘Mean-Variance Portfolio Allocation with a Value at Risk Constraint’, Discussion Paper, London School of Economics, May.

  • Sharpe, W. (1992) ‘Asset Allocation: Management Style and Performance Measurement’, The Journal of Portfolio Management (December).

  • Sklar, A. (1996) ‘Random Variables, Distribution Functions and Copulas — A Personal Look Backward and Forward’, in L. Rüschendor, B. Schweizer and M. Taylor (eds.), Distributions with Fixed Marginals and Related Topics. Institute of Mathematical Statistics: Hayward, CA, pp. 1–14.

    Chapter  Google Scholar 

  • Valdez, E. A. and Chernih, A. (2003) ‘Wang's Capital Allocation Formula for Elliptically Contoured Distributions’, Insurance: Mathematics and Economics (33).

  • Van den Goorbergh, R. (2004) ‘A Copula-Based Autoregressive Conditional Dependence Model of International Stock Markets’, Working Paper no. 22, De Nederlandsche Bank.

  • Vaz De Melo Mendes, B. and Martins De Souza, M. (2004) ‘Measuring Financial Risks With Copulas’, International Review of Financial Analysis (13).

  • Von Eye, A. and Bogat, A. (2004) ‘Testing the Assumption of Multivariate Normality’, Psychology Science, 46 (2).

  • Ward, L. S. and Lee, D. H. (2002) ‘Practical Applications of the Risk-Adjusted Return on Capital Framework’, Dynamic Financial Analysis Discussion Paper, Casualty Actuarial Society Forum, Summer.

  • Waring, B., Whitney, D., Pirone, J. and Castille, C. (2000) ‘Optimizing Manager Structure and Budgeting Manager Risk’, The Journal of Portfolio Management (Spring).

  • Williams, J. O. (1997) ‘Maximizing the Probability of Achieving the Investment Goals’, The Journal of Portfolio Management (Fall).

  • Yamai, Y and Yoshiba, T. (2005) ‘Value-at-Risk Versus Expected Shortfall: A Practical Perspective’, The Journal of Banking and Finance, 29.

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Economiche e Aziendali, Università Milano-Bicocca, Piazza dell'Ateneo Nuovo 1, Milano, 20126, Italy

    Francesco Paolo Natale

Authors
  1. Francesco Paolo Natale
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Paolo Natale.

Appendix A

Appendix A

See Tables A1 and A2.

Table a1 Pre-filtering Ljung–Box–Pierce Q-test and Engle's ARCH test
Full size table
Table a2 Post-filtering Ljung–Box–Pierce Q-test and Engle's ARCH test
Full size table

Rights and permissions

Reprints and permissions

About this article

Cite this article

Paolo Natale, F. Optimisation in the presence of tail-dependence and tail risk: A heuristic approach for strategic asset allocation. J Asset Manag 8, 374–400 (2008). https://doi.org/10.1057/palgrave.jam.2250083

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1057/palgrave.jam.2250083

Keywords

Search

Navigation