Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T20:51:33.482Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparing the riskiness of dependent portfolios via nested L-statistics

Published online by Cambridge University Press:  08 November 2016

Ranadeera G.M. Samanthi ,
Wei Wei  and
Vytaras Brazauskas
Ranadeera G.M. Samanthi
Affiliation:
Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA
Wei Wei
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA
Vytaras Brazauskas*
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA
*
*Correspondence to: Vytaras Brazauskas, Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA. Tel: 1-414-229-5656; Fax: 1-414-229-4907; E-mail: vytaras@uwm.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A non-parametric test based on nested L-statistics and designed to compare the riskiness of portfolios was introduced by Brazauskas et al. (2007). Its asymptotic and small-sample properties were primarily explored for independent portfolios, though independence is not a required condition for the test to work. In this paper, we investigate how performance of the test changes when insurance portfolios are dependent. To achieve that goal, we perform a simulation study where we consider three different risk measures: conditional tail expectation, proportional hazards transform, and mean. Further, three portfolios are generated from exponential, Pareto, and lognormal distributions, and their interdependence is modelled with the three-dimensional t and Gaussian copulas. It is found that the presence of strong positive dependence (comonotonicity) makes the test very liberal for all the risk measures under consideration. For types of dependence that are more common in an insurance environment, the effect of dependence is less dramatic but the results are mixed, i.e., they depend on the chosen risk measure, sample size, and even on the test’s significance level. Finally, we illustrate how to incorporate such findings into sensitivity analysis of the decisions. The risks we analyse represent tornado damages in different regions of the United States from 1890 to 1999.

Keywords

CopulasDependent risksRisk measuresSimulationsStatistical tests
Type
Papers
Information
Annals of Actuarial Science , Volume 11 , Issue 2 , September 2017 , pp. 237 - 252
Copyright
© Institute and Faculty of Actuaries 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albrecht, P. (2004). Risk measures. In B. Sundt & J. Teugels (Eds.), Encyclopedia of Actuarial Science, vol. 3 (pp. 1493–1501). Wiley, London.Google Scholar
Brazauskas, V., Jones, B.L., Puri, M.L. & Zitikis, R. (2007). Nested L-statistics and their use in comparing the riskiness of portfolios. Scandinavian Actuarial Journal, 2007(3), 162–179.CrossRefGoogle Scholar
Brazauskas, V., Jones, B.L., Puri, M.L. & Zitikis, R. (2008). Estimating conditional tail expectation with actuarial applications in view. Journal of Statistical Planning and Inference, 138(11), 3590–3604.Google Scholar
Brazauskas, V. & Kaiser, T. (2004). Discussion of “Empirical estimation of risk measures and related quantities” by Jones and Zitikis. North American Actuarial Journal, 8(3), 114–117.Google Scholar
Brooks, H.E. & Doswell, C.A. III (2001). Normalized damage from major tornadoes in the United States: 1890–1999. Weather and Forecasting, 16, 168–176.2.0.CO;2>CrossRefGoogle Scholar
Chernoff, H., Gastwirth, J.L. & Jones, M.V. (1967). Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation. The Annals of Mathematical Statistics, 38(1), 52–72.Google Scholar
Darolles, S., Gourieroux, C. & Jasiak, J. (2009). L-performance with an application to hedge funds. Journal of Empirical Finance, 16(4), 671–685.Google Scholar
David, H.A. & Nagaraja, H.N. (2003). Order Statistics, 3rd edition. Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Embrechts, P., Lindskog, F. & McNeil, A. (2003). Modelling dependence with copulas and applications to risk management. In S. Rachev (Ed.), Handbook of Heavy Tailed Distributions in Finance (pp. 329–384). Elsevier, Amsterdam.CrossRefGoogle Scholar
Frees, E.W. & Valdez, E. (1998). Understanding relationships using copulas. North American Actuarial Journal, 2(1), 1–25.Google Scholar
Hosking, J.R.M. (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society: Series B, 52, 105–124.Google Scholar
Joe, H. (2014). Dependence Modeling with Copulas. Chapman and Hall, Boca Raton, FL.Google Scholar
Jones, B.L., Puri, M.L. & Zitikis, R. (2006). Testing hypotheses about the equality of several risk measure values with applications in insurance. Insurance: Mathematics and Economics, 38, 253–270.Google Scholar
Jones, B.L. & Zitikis, R. (2003). Empirical estimation of risk measures and related quantities. North American Actuarial Journal, 7(4), 44–54.CrossRefGoogle Scholar
Jones, B.L. & Zitikis, R. (2005). Testing for the order of risk measures: an application of L-statistics in actuarial science. Metron, LXIII(2), 193–211.Google Scholar
Jones, B.L. & Zitikis, R. (2007). Risk measures, distortion parameters, and their empirical estimation. Insurance: Mathematics and Economics, 41, 279–297.Google Scholar
Kaiser, T. & Brazauskas, V. (2006). Interval estimation of actuarial risk measures. North American Actuarial Journal, 10(4), 249–268.CrossRefGoogle Scholar
McNeil, A.J., Frey, R. & Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton, NJ.Google Scholar
Necir, A. & Meraghni, D. (2009). Empirical estimation of the proportional hazard premium for heavy-tailed claim amounts. Insurance: Mathematics and Economics, 45(1), 49–58.Google Scholar
Necir, A. & Meraghni, D. (2010). Estimating L-functionals for heavy-tailed distributions and application. Journal of Probability and Statistics, 2010, article id 707146, 1–34.Google Scholar
Necir, A., Meraghni, D. & Meddi, F. (2007). Statistical estimate of the proportional hazard premium of loss. Scandinavian Actuarial Journal, 2007(3), 147–161.CrossRefGoogle Scholar
Nelson, R.B. (2006). An Introduction to Copulas, 2nd edition. Springer, New York, NY.Google Scholar
Samanthi, R.G.M., Wei, W. & Brazauskas, V. (2016). Ordering Gini indexes of multivariate elliptical risks. Insurance: Mathematics and Economics, 68(3), 84–91.Google Scholar
Tapiero, C.S. (2004). Risk management: an interdisciplinary framework. In B. Sundt & J. Teugels (Eds.), Encyclopedia of Actuarial Science, vol. 3 (pp. 1483–1493). Wiley, London.Google Scholar
Young, V.R. (2004). Premium principles. In B. Sundt & J. Teugels (Eds.), Encyclopedia of Actuarial Science, vol. 3 (pp. 1322–1331). Wiley, London.Google Scholar