×

zbMATH — the first resource for mathematics

Risk- and value-based management for non-life insurers under solvency constraints. (English) Zbl 1403.91194
Summary: The aim of this paper is to study optimal risk- and value-based management decisions regarding a non-life insurer’s investment strategy by maximizing shareholder value based on preference functions, while simultaneously controlling for the ruin probability. We thereby extend previous work by deriving analytical solutions and by explicitly accounting for the policyholders’ willingness to pay depending on their risk sensitivity based on the insurer’s reported solvency status, which will be of great relevance under Solvency II. We further investigate the impact of the risk-free interest rate, (non-linear) dependencies between assets and liabilities, distributional assumptions as well as reinsurance. One main finding is that the consideration of default-risk-driven premiums is vital for optimal management decisions, since, e.g., the target ruin probability implying a higher shareholder value differs for various risk sensitivities of the policyholders. Furthermore, in the present setting, proportional reinsurance increases shareholder value only for non-risk sensitive policyholders.

MSC:
91B30 Risk theory, insurance (MSC2010)
Software:
BRENT; copula; copula; DEoptim; QRM
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albrecher, H.; Teugels, J. L., Exponential behavior in the presence of dependence in risk theory, Journal of Applied Probability, 43, 1, 257-273, (2006) · Zbl 1097.62110
[2] Andersen, E. S., On the collective theory of risk in case of contagion between claims, Bulletin of the Institute of Mathematics and its Applications, 12, 275-279, (1957)
[3] Ardia, D.; Mullen, K. M.; Peterson, B. G.; Ulrich, J., (2016), “DEoptim“: Differential Evolution in “R”. R package version 2.2-4
[4] Asmussen, S.; Albrecher, H., Ruin probabilities, (2010), World Scientific Singapore · Zbl 1247.91080
[5] Avanzi, B.; Gerber, H. U.; Shiu, E. S., Optimal dividends in the dual model, Insurance: Mathematics and Economics, 41, 1, 111-123, (2007) · Zbl 1131.91026
[6] Bai, L.; Zhang, H., Dynamic mean-variance problem with constrained risk control for the insurers, Mathematical Methods of Operations Research, 68, 1, 181-205, (2008) · Zbl 1156.93037
[7] Bankovsky, D.; Klüppelberg, C.; Maller, R., On the ruin probability of the generalised Ornstein-Uhlenbeck process in the cramér case, Journal of Applied Probability, 48, A, 15-28, (2011) · Zbl 1238.60079
[8] Bäuerle, N., Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62, 1, 159-165, (2005) · Zbl 1101.93081
[9] Beard, R.; Pentikäinen, T.; Pesonen, E., Risk theory: the stochastic basis of insurance, (2013), Springer New York · Zbl 0405.62079
[10] Bi, J.; Guo, J., Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, Journal of Optimization Theory and Applications, 157, 1, 252-275, (2012) · Zbl 1266.91093
[11] Braun, A.; Schmeiser, H.; Schreiber, F., On consumer preferences and the willingness to pay for term life insurance, European Journal of Operational Research, 253, 3, 761-776, (2017) · Zbl 1346.91131
[12] Braun, A.; Schmeiser, H.; Schreiber, F., Portfolio optimization under solvency II: implicit constraints imposed by the market risk standard formula, Journal of Risk and Insurance, 84, 1, 177-207, (2017)
[13] Brent, R., Algorithms for minimization without derivatives, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0245.65032
[14] Brito, N. O., Marketability restrictions and the valuation of capital assets under uncertainty, Journal of Finance, 32, 4, 1109-1123, (1977)
[15] Chen, Y., The finite-time ruin probability with dependent insurance and financial risks, Journal of Applied Probability, 48, 4, 1035-1048, (2011) · Zbl 1230.91069
[16] Chen, Y.; Yuen, K. C., Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stochastic Models, 25, 1, 76-89, (2009) · Zbl 1181.62011
[17] Collamore, J. F., Random recurrence equations and ruin in a Markov-dependent stochastic economic environment, The Annals of Applied Probability, 19, 4, 1404-1458, (2009) · Zbl 1176.60018
[18] Cramér, H., On the mathematical theory of risk, (1930), Skandia Jubilee Stockholm, Vol · JFM 56.1098.05
[19] Cheung, E. C.; Wong, J. T., On the dual risk model with Parisian implementation delays in dividend payments, European Journal of Operational Research, 257, 1, 159-173, (2017) · Zbl 1394.91204
[20] Diasparra, M. A.; Romera, R., Bounds for the ruin probability of a discrete-time risk process, Journal of Applied Probability, 46, 1, 99-112, (2009) · Zbl 1159.91404
[21] Diasparra, M.; Romera, R., Inequalities for the ruin probability in a controlled discrete-time risk process, European Journal of Operational Research, 204, 3, 496-504, (2010) · Zbl 1189.91071
[22] Dimitrova, D. S.; Kaishev, V. K.; Zhao, S., On finite-time ruin probabilities in a generalized dual risk model with dependence, European Journal of Operational Research, 242, 1, 134-148, (2015) · Zbl 1341.91090
[23] Eling, M.; Gatzert, N.; Schmeiser, H., Minimum standards for investment performance: A new perspective on non-life insurer solvency, Insurance: Mathematics and Economics, 45, 1, 113-122, (2009) · Zbl 1231.91181
[24] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling extremal events: for insurance and finance, (2013), Springer New York
[25] EY (2016).: Sub-Saharan Africa — the evolution of insurance regulation. http://www.ey.com/Publication/vwLUAssets/EY-sub-saharan-africa-the-evolution-of-insurance-regulation/\(FILE/EY-sub-saharan-africa-the-evolution-of-insurance-regulation.pdf\)
[26] Fischer, K.; Schlütter, S., Optimal investment strategies for insurance companies when capital requirements are imposed by a standard formula, The Geneva Risk and Insurance Review, 40, 1, 15-40, (2015)
[27] Gatzert, N.; Holzmüller, I.; Schmeiser, H., Creating customer value in participating life insurance, Journal of Risk and Insurance, 79, 3, 645-670, (2012)
[28] Gatzert, N.; Kellner, R., The effectiveness of gap insurance with respect to basis risk in a shareholder value maximization setting, Journal of Risk and Insurance, 81, 4, 831-860, (2014)
[29] Hao, X.; Tang, Q., Asymptotic ruin probabilities for a bivariate Lévy-driven risk model with heavy-tailed claims and risky investments, Journal of Applied Probability, 49, 4, 939-953, (2012) · Zbl 1255.91180
[30] Heyde, C. C.; Wang, D., Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims, Advances in Applied Probability, 41, 1, 206-224, (2009) · Zbl 1162.60014
[31] Hofert, M.; Kojadinovic, I.; Maechler, M.; Yan, J., (2017), “copula”: Multivariate Dependence with Copulas. R package version 0.999-17
[32] Huang, Y.; Yang, X.; Zhou, J., Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, Journal of Computational and Applied Mathematics, 296, 443-461, (2016) · Zbl 1331.91097
[33] Hult, H.; Lindskog, F., Ruin probabilities under general investments and heavy-tailed claims, Finance and Stochastics, 15, 2, 243-265, (2011) · Zbl 1303.91091
[34] Jin, C.; Li, S.; Wu, X., On the occupation times in a delayed sparre Andersen risk model with exponential claims, Insurance: Mathematics and Economics, 71, 304-316, (2016) · Zbl 1371.91094
[35] Kabanov, Y.; Pergamenshchikov, S., In the insurance business risky investments are dangerous: the case of negative risk sums, Finance and Stochastics, 20, 2, 355-379, (2016) · Zbl 1342.60105
[36] Klüppelberg, C.; Kostadinova, R., Integrated insurance risk models with exponential Lévy investment, Insurance: Mathematics and Economics, 42, 2, 560-577, (2008) · Zbl 1152.60325
[37] Kroll, Y.; Levy, H.; Markowitz, H. M., Mean‐variance versus direct utility maximization, Journal of Finance, 39, 1, 47-61, (1984)
[38] Kurowicka, D.; Joe, H., Dependence modeling: vine copula handbook, (2011), World Scientific Publishing Singapore
[39] Li, J., Asymptotics in a time-dependent renewal risk model with stochastic return, Journal of Mathematical Analysis and Applications, 387, 2, 1009-1023, (2012) · Zbl 1230.91076
[40] Li, S.; Garrido, J., On a class of renewal risk models with a constant dividend barrier, Insurance: Mathematics and Economics, 35, 3, 691-701, (2004) · Zbl 1122.91345
[41] Li, J.; Tang, Q., Interplay of insurance and financial risks in a discrete-time model with strongly regular variation, Bernoulli, 21, 3, 1800-1823, (2015) · Zbl 1336.91048
[42] Liang, Z.; Bayraktar, E., Optimal reinsurance and investment with unobservable claim size and intensity, Insurance: Mathematics and Economics, 55, 156-166, (2014) · Zbl 1296.91161
[43] Liang, Z.; Yuen, K. C.; Guo, J., Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance: Mathematics and Economics, 49, 2, 207-215, (2011) · Zbl 1218.91084
[44] Liang, Z.; Yuen, K. C.; Cheung, K. C., Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump‐diffusion risk model, Applied Stochastic Models in Business and Industry, 28, 6, 585-597, (2012) · Zbl 1286.91068
[45] Lin, X.; Dongjin, Z.; Yanru, Z., Minimizing upper bound of ruin probability under discrete risk model with Markov chain interest rate, Communications in Statistics-Theory and Methods, 44, 4, 810-822, (2015) · Zbl 1338.60184
[46] Loistl, O., Does mean-variance portfoliomanagement deserve expected Utility’s approximative affirmation, European Journal of Operational Research, 247, 2, 676-680, (2015) · Zbl 1346.91214
[47] Lorson, J.; Schmeiser, H.; Wagner, J., Evaluation of benefits and costs of insurance regulation - A conceptual model for solvency II, Journal of Insurance Regulation, 31, 1, 125-156, (2012)
[48] Lundberg, F., Approximerad framställning afsannollikhetsfunktionen: II. återförsäkring af kollektivrisker, (1903), Almqvist & Wiksells Boktr Uppsala
[49] Markowitz, H., Mean-variance approximations to expected utility, European Journal of Operational Research, 234, 2, 346-355, (2014) · Zbl 1304.91203
[50] Markowitz, H., Reply to Professor loistl, European Journal of Operational Research, 247, 2, 680-681, (2015) · Zbl 1346.91216
[51] Mayers, D.; Smith, C. W., Contractual provisions, organizational structure, and conflict control in insurance markets, Journal of Business, 54, 3, 407-434, (1981)
[52] McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative risk management: concepts, techniques and tools, (2005), Princeton University Press Princeton · Zbl 1089.91037
[53] Mikosch, T., Non-life insurance mathematics: an introduction with the Poisson process, (2009), Springer Berlin · Zbl 1166.91002
[54] Nyrhinen, H., On the ruin probabilities in a general economic environment, Stochastic Processes and their Applications, 83, 2, 319-330, (1999) · Zbl 0997.60041
[55] Paulsen, J., Risk theory in a stochastic economic environment, Stochastic processes and their Applications, 46, 2, 327-361, (1993) · Zbl 0777.62098
[56] Paulsen, J., Ruin models with investment income, Probability Surveys, 5, 416-434, (2008) · Zbl 1189.91077
[57] Promislow, D. S.; Young, V. R., Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9, 3, 110-128, (2005) · Zbl 1141.91543
[58] Ramsden, L.; Papaioannou, A. D., Asymptotic results for a Markov-modulated risk process with stochastic investment, Journal of Computational and Applied Mathematics, 313, 38-53, (2017) · Zbl 1410.91285
[59] Romera, R.; Runggaldier, W., Ruin probabilities in a finite-horizon risk model with investment and reinsurance, Journal of Applied Probability, 49, 4, 954-966, (2012) · Zbl 1255.91185
[60] Schäl, M., On discrete-time dynamic programming in insurance: exponential utility and minimizing the ruin probability, Scandinavian Actuarial Journal, 2004, 3, 189-210, (2004) · Zbl 1141.91031
[61] Schlütter, S., Capital requirements or pricing constraints? an economic analysis of measures for insurance regulation, The Journal of Risk Finance, 15, 5, 533-554, (2014)
[62] Schmidli, H., Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001, 1, 55-68, (2001) · Zbl 0971.91039
[63] Schmidli, H., On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 12, 3, 890-907, (2002) · Zbl 1021.60061
[64] Segerdahl, C. O., Über einige risikotheoretische fragestellungen, Scandinavian Actuarial Journal, 1942, 1-2, 43-83, (1942) · JFM 68.0311.02
[65] Shen, X. M.; Lin, Z. Y.; Zhang, Y., Uniform estimate for maximum of randomly weighted sums with applications to ruin theory, Methodology and Computing in Applied Probability, 11, 4, 669-685, (2009) · Zbl 1177.60026
[66] Swiss Re (2016). Sigma Nr. 1/2016. Natural catastrophes and man-made disasters in 2015: Asia suffers substantial losses. http://media.swissre.com/documents/sigma1_2016_en.pdf.
[67] Tang, Q.; Tsitsiashvili, G., Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Processes and their Applications, 108, 2, 299-325, (2003) · Zbl 1075.91563
[68] Tang, Q.; Tsitsiashvili, G., Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments, Advances in Applied Probability, 36, 4, 1278-1299, (2004) · Zbl 1095.91040
[69] Tobin, J., Liquidity preference as behavior toward risk, Review of Economic Studies, 25, 2, 65-86, (1958)
[70] Wakker, P.; Thaler, R.; Tversky, A., Probabilistic insurance, Journal of Risk and Uncertainty, 15, 1, 7-28, (1997) · Zbl 0883.90052
[71] Weng, C.; Zhang, Y.; Tan, K. S., Ruin probabilities in a discrete time risk model with dependent risks of heavy tail, Scandinavian Actuarial Journal, 3, 205-218, (2009) · Zbl 1224.91093
[72] Wu, R.; Lu, Y.; Fang, Y., On the gerber-shiu discounted penalty function for the ordinary renewal risk model with constant interest, North American Actuarial Journal, 11, 2, 119-134, (2007)
[73] Yang, Y.; Hashorva, E., Extremes and products of multivariate AC-product risks, Insurance: Mathematics and Economics, 52, 2, 312-319, (2013) · Zbl 1284.60108
[74] Yang, Y.; Konstantinides, D. G., Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks, Scandinavian Actuarial Journal, 2015, 8, 641-659, (2015) · Zbl 1401.91205
[75] Yang, Y.; Leipus, R.; Šiaulys, J., Tail probability of randomly weighted sums of subexponential random variables under a dependence structure, Statistics & Probability Letters, 82, 9, 1727-1736, (2012) · Zbl 1334.62029
[76] Yang, H.; Zhang, L., Ruin problems for a discrete time risk model with random interest rate, Mathematical Methods of Operations Research, 63, 2, 287-299, (2006) · Zbl 1115.60084
[77] Yao, D.; Yang, H.; Wang, R., Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European Journal of Operational Research, 211, 3, 568-576, (2011) · Zbl 1237.91143
[78] Yi, L.; Chen, Y.; Su, C., Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation, Journal of Mathematical Analysis and Applications, 376, 1, 365-372, (2011) · Zbl 1206.60039
[79] Yu, T. Y.; Tsai, C.; Huang, H. T., Applying simulation optimization to the asset allocation of a property-casualty insurer, European Journal of Operational Research, 207, 1, 499-507, (2010) · Zbl 1205.91167
[80] Zhang, Y.; Shen, X.; Weng, C., Approximation of the tail probability of randomly weighted sums and applications, Stochastic processes and their applications, 119, 2, 655-675, (2009) · Zbl 1271.62030
[81] Zimmer, A.; Schade, C.; Gründl, H., Is default risk acceptable when purchasing insurance? experimental evidence for different probability representations, reasons for default, and framings, Journal of Economic Psychology, 30, 1, 11-23, (2009)
[82] Zimmer, A.; Gründl, H.; Schade, C.; Glenzer, F., An incentive-compatible experiment on probabilistic insurance and implications for an Insurer’s solvency level, (2014), Goethe University of Frankfurt, Working Paper
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.