Skip to main content Accessibility help
×
Home

CAPITAL ALLOCATION WITH MULTIVARIATE RISK MEASURES: AN AXIOMATIC APPROACH

  • Linxiao Wei (a1) and Yijun Hu (a2)

Abstract

Capital allocation is of central importance in portfolio management and risk-based performance measurement. Capital allocations for univariate risk measures have been extensively studied in the finance literature. In contrast to this situation, few papers dealt with capital allocations for multivariate risk measures. In this paper, we propose an axiom system for capital allocation with multivariate risk measures. We first recall the class of the positively homogeneous and subadditive multivariate risk measures, and provide the corresponding representation results. Then it is shown that for a given positively homogeneous and subadditive multivariate risk measure, there exists a capital allocation principle. Furthermore, the uniqueness of the capital allocation principe is characterized. Finally, examples are also given to derive the explicit capital allocation principles for the multivariate risk measures based on mean and standard deviation, including the multivariate mean-standard-deviation risk measures.

Copyright

References

Hide All
1Ahmadi-Javid, A. (2012). Entropic value-at-risk: a new coherent risk measure. Journal of Optimization Theorey and Application 155: 11051123.
2Artzner, P., Delbaen, F., Eber, J.M. & Heath, D. (1997). Thinking coherently. Risk 10: 6871.
3Artzner, P., Delbaen, F., Eber, J.M. & Heath, D. (1999). Coherent measures of risk. Mathematical Finance 9(3): 203228.
4Buch, A. & Dorfleitner, G. (2008). Coherent risk measures, coherent capital allocations and the gradient allocation principle. Insurance: Mathematics and Economics 42: 235242.
5Burgert, C. & Rüschendorf, L. (2006). consistent risk measures for portfolio vectors. Insurance: Mathematics and Economics 38: 289297.
6Delbaen, F. (2002). Coherent risk measures on general probability spaces. In Advances in Finance and Stochastics, Essays in Honour of Dieter Sondermann. Berlin: Springer-Verlag, 137.
7Denault, M. (2001). Coherent allocation of risk capital. Journal of Risk 4(1): 721.
8Deprez, O. & Gerber, H.U. (1985). On convex principles of premium calculation. Insurance: Mathematics and Economics 4(3): 179189.
9Fischer, T. (2003). Risk capital allocation by coherent risk measures based on one-sided moments. Insurance: Mathematics and Economics 32: 135146.
10Föllmer, H. & Schied, A. (2002). Convex measures of risk and trading constraints. Finance and Stochastics 6: 429447.
11Föllmer, H. & Schied, A. (2004). Stochastic finance: An introduction in discrete time, 2nd ed., De Gruyter Studies in Mathematics, Vol. 27. Berlin: Walter de Gruyter.
12Frittelli, M. & Rosazza Gianin, E. (2002). Putting order in risk measures. Journal of Banking Finance 26: 14731486.
13Kalkbrener, M. (2005). An axiomatic approach to capital allocation. Mathematical Finance 15(3): 425–47.
14Kalkbrener, M. (2009). An axiomatic characterization of capital allocations of coherent risk measures. Quantitative Finance 9(8): 961965.
15Karoui, N.E. & Ravanelli, C. (2009). Cash subadditive risk measures and interest rate ambiguity. Mathematical Finance 19(4): 561590.
16Markowitz, H. (1952). Portfolio selection. Journal of Finance 7: 7791.
17Rüschendorf, L. (2013). Mathematical Risk Analysis. Berlin: Springer.
18Tsanakas, A. (2004). Dynamic capital allocation with distortion risk measures. Insurance: Mathematics and Economics 35: 223243.
19Tsanakas, A. (2008). Risk measurement in the presence of background risk. Insurance: Mathematics and Economics 42: 520528.
20Tsanakas, A. (2009). To split or not to split: Capital allocation with convex risk measures. Insurance: Mathematics and Economics 44: 268277.
21Wei, L. & Hu, Y. (2014). Coherent and convex risk measures for portfolios with applications. Statistics and Probability Letters 90: 114120.

Keywords

CAPITAL ALLOCATION WITH MULTIVARIATE RISK MEASURES: AN AXIOMATIC APPROACH

  • Linxiao Wei (a1) and Yijun Hu (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed