This paper examines the properties that a risk measure should satisfy in order to characterize an investor's preferences. In particular, we propose some intuitive and realistic examples that describe several desirable features of an ideal risk measure. This analysis is the first step in understanding how to classify an investor's risk. Risk is an asymmetric, relative, heteroskedastic, multidimensional concept that has to take into account asymptotic behavior of returns, inter-temporal dependence, risk-time aggregation, and the impact of several economic phenomena that could influence an investor's preferences. In order to consider the financial impact of the several aspects of risk, we propose and analyze the relationship between distributional modeling and risk measures. Similar to the notion of ideal probability metric to a given approximation problem, we are in the search for an ideal risk measure or ideal performance ratio for a portfolio selection problem. We then emphasize the parallels between risk measures and probability metrics, underlying the computational advantage and disadvantage of different approaches.

Keywords:

References

C. Acerbi, Journal of Banking and Finance 26, 1505 (2002), DOI: 10.1016/S0378-4266(02)00281-9. Crossref, Web of Science, Google Scholar
P. Artzneret al., Mathematical Finance 9, 203 (1999), DOI: 10.1111/1467-9965.00068. Crossref, Web of Science, Google Scholar
V. S. Bawa, Journal of Finance 31, 1169 (1976), DOI: 10.2307/2326281. Crossref, Web of Science, Google Scholar
V. S. Bawa, Journal of Financial and Quantitative Analysis 13, 255 (1978), DOI: 10.2307/2330386. Crossref, Web of Science, Google Scholar
L. A. Balzer, How to measure risk, Conference on Investment Performance, AIC Conferences, Sydney, Australia (1990) . Google Scholar
L. A. Balzer, Managing Downside Risk in Financial Markets: Theory Practice and Implementation, eds. F. Sortino and S. Satchell (Butterworth-Heinemann, Oxford, 2001) pp. 103–156. Crossref, Google Scholar
A. Biglovaet al., Journal of Portfolio Management 31, 103 (2004). Crossref, Web of Science, Google Scholar
R. Bordley and M. LiCalzi, Decisions in Economics and Finance 23, 53 (2000), DOI: 10.1007/s102030050005. Crossref, Web of Science, Google Scholar
E. Castagnoli, Benchmarking under risk, Atti convegno AMASES 2004, Modena . Google Scholar
E. Castagnoli and M. LiCalzi, Theory and Decision 41, 281 (1996), DOI: 10.1007/BF00136129. Crossref, Web of Science, Google Scholar
E. Castagnoli and M. LiCalzi, SZIGMA 29, 199 (1999). Web of Science, Google Scholar
E. De Giorgi, Journal of Banking and Finance 29(4), 895 (2005). Crossref, Web of Science, Google Scholar
M. Friedman and L. J. Savage, Journal of Political Economy 56, 279 (1948), DOI: 10.1086/256692. Crossref, Web of Science, Google Scholar
R. Giacometti and S. Ortobelli, A comparison between dispersion measures for the asset allocation problem, Technical Report 7, University of Bergamo (2001) . Google Scholar
G. Hanoch and H. Levy, Review of Economic Studies 36, 35 (1969). Web of Science, Google Scholar
G. A. Holton, Financial Analysts Journal 60, 19 (2004), DOI: 10.2469/faj.v60.n6.2669. Crossref, Web of Science, Google Scholar
E. Karni , Decision Making Under Uncertainty: The Case of State-Dependent Preferences ( Harvard University Press , Cambridge , 1985 ) . Crossref, Google Scholar
F. H. Knight , Risk, Uncertainty and Profit ( Hart, Schaffner and Marx , New York , 1921 ) . Google Scholar
J. Longerstaey and P. Zangari, RiskMetrics — Technical Document, 4th edn. (J.P. Morgan, New York, 1996). Google Scholar
R. A. Olsen, Financial Analysts Journal 53(2), 62 (1997), DOI: 10.2469/faj.v53.n2.2073. Crossref, Google Scholar
S. Ortobelli, Theory and Decision 51, 297 (2001), DOI: 10.1023/A:1015511211848. Crossref, Web of Science, Google Scholar
S. Ortobelliet al., International Journal of Theoretical and Applied Finance 8, 1107 (2005), DOI: 10.1142/S0219024905003402. Link, Google Scholar
S. Ortobelli, S. T. Rachev, H. Shalit and F. J. Fabozzi, Risk probability functionals and probability metrics applied to portfolio theory, Working Paper, Department of Probability and Applied Statistics, University of California, Santa Barbara, USA (2006) . Google Scholar
S. Ortobelli and S. T. Rachev, Mathematical and Computer Modelling 34, 1037 (2001). Crossref, Web of Science, Google Scholar
G. Pflug , Probabilistic Constrained Optimization: Methodology and Application , ed. S. Uryasev ( Kluwer , Dordrecht , 2000 ) . Google Scholar
S. T. Rachev and S. Mittnik , Stable Paretian Models in Finance ( Wiley & Sons , Chichester , 2007 ) . Google Scholar
S. Rachev et al. , Decisions in Banking and Finance ( 2007 ) . Google Scholar
S. Rachev , Probability Metrics and the Stability of Stochastic Models ( Wiley & Sons , Chichester , 1991 ) . Google Scholar
R. T. Rockafellar and S. Uryasev, Journal of Banking and Finance 26(7), 1443 (2002), DOI: 10.1016/S0378-4266(02)00271-6. Crossref, Web of Science, Google Scholar
R. T. Rockafellar, S. Uryasev and M. Zabarankin, Finance and Stochastics 10, 51 (2006), DOI: 10.1007/s00780-005-0165-8. Crossref, Web of Science, Google Scholar
M. Rothschild and J. Stiglitz, Journal of Economic Theory 2, 225 (1970), DOI: 10.1016/0022-0531(70)90038-4. Crossref, Web of Science, Google Scholar
A. D. Roy, Econometrica 20, 431 (1952), DOI: 10.2307/1907413. Crossref, Google Scholar
R. Ryan and F. J. Fabozzi, Journal of Portfolio Management 28(4), 7 (2002). Crossref, Web of Science, Google Scholar
G. Samorodnitsky and M. S. Taqqu , Stable Non Gaussian Random Processes: Stochastic Models with Infinite Variance ( Chapman and Hall , New York , 1994 ) . Google Scholar
M. J. Schervisch, T. Seidenfeld and J. B. Kadane, Journal of the American Statistical Association 85, 840 (1990), DOI: 10.2307/2290023. Crossref, Web of Science, Google Scholar
M. Shaked and G. Shanthikumar , Stochastic Orders and Their Applications ( Academic Press Inc. Harcourt Brace & Company , New York , 1994 ) . Google Scholar
W. F. Sharpe, Journal of Portfolio Management 45 (1994). Google Scholar
F. A. Sortino and S. Satchell , Managing Downside Risk in Financial Markets: Theory, Practice and Implementation ( Butterworth Heinemann , Oxford , 2001 ) . Google Scholar
S. Stoyanov, S. T. Rachev and F. J. Fabozzi, Applied Mathematical Finance 14, 402 (2007), DOI: 10.1080/13504860701255292. Google Scholar
S. Stoyanov, S. T. Rachev, S. Ortobelli and F. J. Fabozzi, Relative deviation metrics and the problem of strategy replication, to appear in Journal of Banking and Finance . Google Scholar
G. Szegö, Journal of Banking and Finance 26(7), 1253 (2002). Crossref, Web of Science, Google Scholar
G. Szegö (ed.) , Risk Measures for the 21st Century ( Wiley & Sons , Chichester , 2004 ) . Google Scholar
L. G. Tesler, Review of Economic Studies 23, 1 (1956). Web of Science, Google Scholar