## Moment problem and its applications to risk assessment.(English)Zbl 1414.91236

Summary: This article discusses how to assess risk by computing the best upper and lower bounds on the expected value $$\mathrm{E}[\phi(X)]$$, subject to the constraints $$\mathrm{E}[X^i]=\mu_i$$ for $$i=0,1,2,\dots, n$$. $$\phi(x)$$ can take the form of the indicator function $$\phi (x)=\mathbb{I}_{(-\infty -K]}(x)$$ in which the bounds on $$\mathrm{Pr}(X\leq K)$$ are calculated and the form $$\phi(x)=(\varphi (x)-K)_+$$ in which the bounds on financial payments are found. We solve the moment bounds on $$\mathrm{E} [\mathbb{I}_{(-\infty,K]}(X)]$$ through three methods: the semidefinite programming method, the moment-matching method, and the linear approximation method. We show that for practical purposes, these methods provide numerically equivalent results. We explore the accuracy of bounds in terms of the number of moments considered. We investigate the usefulness of the moment method by comparing the moment bounds with the “point” estimate provided by the Johnson system of distributions. In addition, we propose a simpler formulation for the unimodal bounds on $$\mathrm{E} [\mathbb{I}_{(-\infty,K]}(X)]$$ compared to the existing formulations in the literature. For those problems that could be solved both analytically and numerically given the first few moments, our comparisons between the numerical and analytical results call attention to the potential differences between these two methodologies. Our analysis indicates the numerical bounds could deviate from their corresponding analytical counterparts. The accuracy of numerical bounds is sensitive to the volatility of $$X$$. The more volatile the random variable $$X$$ is, the looser the numerical bounds are, compared to their closed-form solutions.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 90C22 Semidefinite programming

### Software:

Sostools; YALMIP; SeDuMi; GloptiPoly
Full Text:

### References:

 [1] Alexander, C., Market Risk Analysis [2] Bertsimas, D.; Popescu, I., On the Relation between Option and Stock Prices: An Optimization Approach, Operations Research, 50, 358-374, (2002) · Zbl 1163.91382 [3] Bertsimas, D.; Popescu, I., Optimal Inequalities in Probability Theory: A Convex Optimization Approach, SIAM Journal on Optimization, 15, 3, 780-804, (2005) · Zbl 1077.60020 [4] Brockett, P.; Cox, S., Insurance Calculation Using Incomplete Information, Scandinavian Actuarial Journal, 94-108, (1985) · Zbl 0601.62131 [5] Ĉerbáková, J.; Haasis, (H.-D.; Kopfer, H.; Schönberger, J., Operations Research Proceedings 2005, XXIII, Worst-Case VaR and CVaR, 817-822, (2005), New York: Springer, New York [6] Courtois, C.; Denuit, M., Bounds on Convex Reliability Functions with Known First Moments, European Journal of Operational Research, 177, 365-377, (2006) · Zbl 1111.90027 [7] Courtois, C.; Denuit, M., Convex Bounds on Multiplicative Processes, with Applications to Pricing in Incomplete Markets, Insurance: Mathematics and Economics, 42, 95-100, (2008) · Zbl 1141.91498 [8] Cox, S. H., Bounds on Expected Values of Insurance Payments and Option Prices, Transactions of the Society of Actuaries, XLIII, 231-260, (1991) [9] Cox, S. H.; Lin, Y.; Tian, R.; Zuluaga, L. F., Bounds for Probabilities of Extreme Events Defined By Two Random Variables, Variance: Advancing the Science of Risk, 4, 47-65, (2009) [10] De Schepper, A.; Heijnen, B., Distribution-Free Option Pricing, Insurance: Mathematics and Economics, 40, 179-199, (2007) · Zbl 1141.91434 [11] De Schepper, A.; Heijnen, B., How to Estimate the Value at Risk Under Incomplete Information, Journal of Computational and Applied Mathematics, 233, 2213-2226, (2010) · Zbl 1182.91099 [12] Eddy, A.; Rivero, C., Multiplicative Structure of the Resolvent Matrix for the Truncated Hausdorff Matrix Moment Problem, Operator Theory: Advances and Applications, 266, 193-210, (2012) · Zbl 1262.44004 [13] Ghaoui, L. E.; Oks, M.; Oustry, F., Worst-Case Value-at-Risk and Robust Portfolio Optimization: A Conic Programming Approach, Operations Research, 51, 4, 543-556, (2003) · Zbl 1165.91397 [14] Goovaerts, M.; Haezendonck, J.; DeVylder, F., Numerical Best Bounds on Stop-loss Premiums, Insurance: Mathematics and Economics, 1, 287-302, (1982) · Zbl 0498.62089 [15] Goovaerts, M.; Kaas, R., Application of the Problem of Moment to Derive Bounds on Integrals with Integral Constraints, Insurance: Mathematics and Economics, 4, 99-111, (1985) · Zbl 0559.62086 [16] Heijnen, B., Best Upper and Lower Bounds on Modified Stop-Loss Premiums in Case of Known Range, Mode, Mean and Variance of the Original Risk, Insurance: Mathematics and Economics, 9, 207-220, (1990) · Zbl 0717.62102 [17] Henrion, D.; Lasserre, J. B., GloptiPoly: Global Optimization over Polynomials with Matlab and SeDuMi, ACM Transactions on Mathematical Software, 29, 2, 165-194, (2003) · Zbl 1070.65549 [18] Hilbert, D., Über die darstellung definiter formen als summe von formen-quadraten, Mathematische Annalen, 32, 342-350, (1888) · JFM 20.0198.02 [19] Hurlimann, W., Analytical Bounds for Two Value-at-Risk Functions, ASTIN Bulletin, 32, 2, 235-265, (2002) · Zbl 1094.91032 [20] Johnson, N. L., Systems of Frequency Curves Generated By Methods of Translation, Biometrika, 36, 1-2, 149-176, (1949) · Zbl 0033.07204 [21] Johnson, N. L., Systems of Frequency Curves Derived from the First Law of Laplace, Trabajos de Estadístics, 5, 283-291, (1954) · Zbl 0058.35102 [22] Kaas, R.; Goovaerts, M., Best Bounds for Positive Distributions with Fixed Moments, Insurance: Mathematics and Economics, 5, 87-92, (1986) · Zbl 0593.62112 [23] Kaas, R.; Goovaerts, M., Extremal Values of Stop-Loss Premiums Under Moment Constraints, Insurance: Mathematics and Economics, 5, 279-283, (1986) · Zbl 0609.62134 [24] Karlin, S.; Studden, W., Pure and Applied Mathematics, XV, Tchebycheff Systems: With Applications in Analysis and Statistics, (1966), New York: John Wiley and Sons, New York · Zbl 0153.38902 [25] Khintchine, A. Y., On Unimodal Distributions, In Izv. Nauchno. Issled. Inst. Mat. Mech. Tomsk. Gos. Univ. 2, 1-7, (1938) [26] Klugman, S. A.; Panjer, H. H.; Wilmott, G. E., Loss Models: From Data to Decisions, (2008), New York: John Wiley & Sons, New York · Zbl 1159.62070 [27] Kolmogorov, A.; Yushkevich, A., Mathematics of the 19th Century: Function Theory According to Chebyshev Ordinary Differential Equations Calculus of Variations Theory of Finite Differences, (1998), Basel: Springer, Basel · Zbl 0892.01004 [28] Lasserre, J. B.; Toh, K.-C.; Yang, S., A Bounded Degree SOS Hierarchy for Polynomial Optimization, EURO Journal on Computational Optimization, 5, 1, 87-117, (2017) · Zbl 1368.90132 [29] Lo, A. W., Semi-Parametric Upper Bounds for Option Prices and Expected Payoffs, Journal of Financial Economics, 19, 373-387, (1987) [30] Löfberg, J., Proceedings of the CACSD Conference, Yalmip: A Toolbox for Modeling and Optimization in MATLAB, (2004), Taipei, Taiwan [31] Petrov, V. V., On Lower Bounds for Tail Probabilities, Journal of Statistical Planning and Inference, 137, 2703-2705, (2007) · Zbl 1120.60016 [32] Popescu, I., A Semidefinite Programming Approach to Optimal Moment Bounds for Convex Classes of Distributions, Mathematics of Operations Research, 30, 3, 632-657, (2005) · Zbl 1082.60011 [33] Prajna, S.; Papachristodoulou, A.; Parrilo, P. A., Proceedings of the 41st IEEE Conference on Decision and Control, Introducing SOSTOOLS: A General Purpose Sum of Squares Programming Solver, 741-746, (2002), Las Vegas, NV [34] Rivero, A. E. C.; Dyukarev, Y. M.; Fritzsche, B.; Kirstein, B., A Truncated Matricial Moment Problem on a Finite Interval—The Case of an Odd Number of Prescribed Moments, Operator Theory: Advances and Applications, 176, 99-164, (2007) · Zbl 1122.44004 [35] Royden, H., Bounds on a Distribution Function When Its First n Moments are Given, Annals of Mathematical Statistics, 24, 3, 361-376, (1953) · Zbl 0051.33701 [36] Simonato, J. G., The Performance of Johnson Distributions for Computing Value at Risk and Expected Shortfall, Journal of Derivatives, 19, 1, 7-24, (2011) [37] Smith, J. E., Moment Methods for Decision Analysis, (1990) [38] Valdez, E. A.; Dhaene, J.; Maj, M.; Vanduffel, S., Bounds and Approximations for Sums of Dependent Log-Elliptical Random Variables, Insurance: Mathematics and Economics, 44, 385-397, (2009) · Zbl 1162.91440 [39] Vandenberghe, L.; Boyd, S.; Comanor, K., Generalized Chebyshev Bounds Via Semidefinite Programming, SIAM Review, 49, 1, 52-64, (2007) · Zbl 1151.90512 [40] Van Parys, B. P. G.; P. J., Goulart; D., Kuhn, Generalized Gauss Inequalities Via Semidefinite Programming, Mathematical Programming, 156, 1, 271-302, (2016) · Zbl 1351.90125 [41] Wong, M. H.; Zhang, S., Computing Best Bounds for Nonlinear Risk Measures with Partial Information, Insurance: Mathematics and Economics, 52, 2, 204-212, (2013) · Zbl 1284.91278 [42] Ynduráin, F. J.; Graves-Morris, P., Padé Approximants, The Moment Problem and Applications, 45-63, (1973), London: Institute of Physics, London [43] Zagorodnyuk, S. M., A Description of All Solutions of the Matrix Hamburger Moment Problem in a General Case, (2009) · Zbl 1224.44006
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