June  2014, 4(2): 187-202. doi: 10.3934/mcrf.2014.4.187

Optimal insurance in a changing economy

1. 

School of Insurance, Central University Of Finance and Economics, Beijing 100081

2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

3. 

Cass Business School, City University London, London, EC1Y 8TZ, United Kingdom

4. 

Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong, China

Received  March 2012 Revised  April 2013 Published  February 2014

We discuss a general problem of optimal strategies for insurance, consumption and investment in a changing economic environment described by a continuous-time regime switching model. We consider the situation of a random investment horizon which depends on the force of mortality of an economic agent. The objective of the agent is to maximize the expected discounted utility of consumption and terminal wealth over a random future lifetime. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution related to the optimal consumption, investment and insurance is provided. In the cases of a power utility and an exponential utility, we derive analytical solutions to the optimal strategies. Numerical results are given to illustrate the proposed model and to document the impact of switching regimes on the optimal strategies.
Citation: Jingzhen Liu, Ka-Fai Cedric Yiu, Tak Kuen Siu, Wai-Ki Ching. Optimal insurance in a changing economy. Mathematical Control and Related Fields, 2014, 4 (2) : 187-202. doi: 10.3934/mcrf.2014.4.187
References:
[1]

N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, North Holland, New York-Amsterdam, 1981.

[2]

K. J. Arrow, Uncertainty and the welfare economics of medical care, American Economic Review, 53 (1963), 941-973, available at http://www.aeaweb.org/aer/top20/53.5.941-973.pdf.

[3]

C. Blanchet-Scalliet, N. E. Karoui, M. Jeanblanc and L. Martellini, Optimal investment and consumption decisions when time-horizon is uncertain, Journal of Mathematical Economics, 44 (2008), 1100-1113. doi: 10.1016/j.jmateco.2007.09.004.

[4]

B. Bouchard and H. Pham, Wealth-path dependent utility maximization in incomplete markets, Finance Stochast, 8 (2004), 579-603. doi: 10.1007/s00780-004-0125-8.

[5]

E. Briys, Insurance and consumption: The continuous-time case, Journal of Risk and Insurance, 53 (1986), 718-723. doi: 10.2307/252972.

[6]

J. Buffington and R. J. Elliott, Regime switching and European options, Stochastic Theory and Control, LNCIS 280, (ed. B. Pasik-Duncan), LNCIS 280, 73-82, Springer-Verlag, Berlin, Heidelberg, 2002. doi: 10.1007/3-540-48022-6_5.

[7]

J. Buffington and R. J. Elliott, American options with regime switching, International Journal of Theoretical and Applied Finance, 5 (2002), 497-514. doi: 10.1142/S0219024902001523.

[8]

L. Delong, Optimal investment and consumption in the presence of default on a financial market driven by a Levy noise, Ann. Univ. Mariae Curie-Sk?odowska Sect. A, 60 (2006), 1-15.

[9]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Imation and Control, Applications of Mathematics (New York), 29. Springer-Verlag, New York, 1995.

[10]

R. J. Elliott, L. L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432. doi: 10.1007/s10436-005-0013-z.

[11]

R. J. Elliott and J. Hinz, Portfolio analysis, hidden Markov models and chart analysis by PF-Diagrams, International Journal of Theoretical and Applied Finance, 5 (2002), 385-399.

[12]

R. J. Elliott, T. K. Siu and L. L. Chan, Pricing volatility swaps under Heston's stochastic volatility model with regime switching, Applied Mathematical Finance, 14 (2007), 41-62. doi: 10.1080/13504860600659222.

[13]

H. U. Gerber and E. W. Shiu, Investing for retirement: Optimal capital growth and dynamic asset allocation (with discussions), North American Actuarial Journal, 4 (2000), 42-62. doi: 10.1080/10920277.2000.10595899.

[14]

S. M. Goldfeld and R. E. Quandt, The estimation of structural shifts by switching regressions, Annals of Economic and Social Measurement, 2 (1973), 475-485.

[15]

C. Gollier, Insurance and precautionary capital accumulation in a continuous-time model, Journal of Risk and Insurance, 61 (1994) 78-95. doi: 10.2307/253425.

[16]

X. Guo, Information and option pricings, Quantitative Finance, 1 (2001), 38-44. doi: 10.1080/713665550.

[17]

J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.

[18]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257. doi: 10.2307/1926560.

[19]

R. C. Merton, Optimal consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X.

[20]

J. Mossin, Aspects of rational insurance purchasing, Journal of Political Economy, 76 (1968), 553-568. doi: 10.1086/259427.

[21]

K. S. Moore and V. R. Young, Optimal insurance in a continuous-time model, Insurance Mathematics and Economics, 39 (2006), 47-68. doi: 10.1016/j.insmatheco.2006.01.009.

[22]

R. E. Quandt, The estimation of the parameters of a linear regression system obeying two separate regimes, Journal of the American Statistical Association, 53 (1958), 873-880. doi: 10.1080/01621459.1958.10501484.

[23]

H. Schlesinger and C. Gollier, Second-best insurance contract design in an incomplete market, Scandinavian Journal of Economics, 97 (1995), 123-135.

[24]

K. L. Teo, D. W. Reid and I. E. Boyd, Stochastic optimal control theory and its computational method, International Journal on Systems Science, 11 (1980), 77-95. doi: 10.1080/00207728008966998.

[25]

H. Tong, Some comments on the Canadian lynx data (with discussion), Journal of the Royal Statistical Society, Series A, General, 140 (1977), 432-436.

[26]

K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint, Automatica, 46 (2010), 1979-989. doi: 10.1016/j.automatica.2010.02.027.

show all references

References:
[1]

N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, North Holland, New York-Amsterdam, 1981.

[2]

K. J. Arrow, Uncertainty and the welfare economics of medical care, American Economic Review, 53 (1963), 941-973, available at http://www.aeaweb.org/aer/top20/53.5.941-973.pdf.

[3]

C. Blanchet-Scalliet, N. E. Karoui, M. Jeanblanc and L. Martellini, Optimal investment and consumption decisions when time-horizon is uncertain, Journal of Mathematical Economics, 44 (2008), 1100-1113. doi: 10.1016/j.jmateco.2007.09.004.

[4]

B. Bouchard and H. Pham, Wealth-path dependent utility maximization in incomplete markets, Finance Stochast, 8 (2004), 579-603. doi: 10.1007/s00780-004-0125-8.

[5]

E. Briys, Insurance and consumption: The continuous-time case, Journal of Risk and Insurance, 53 (1986), 718-723. doi: 10.2307/252972.

[6]

J. Buffington and R. J. Elliott, Regime switching and European options, Stochastic Theory and Control, LNCIS 280, (ed. B. Pasik-Duncan), LNCIS 280, 73-82, Springer-Verlag, Berlin, Heidelberg, 2002. doi: 10.1007/3-540-48022-6_5.

[7]

J. Buffington and R. J. Elliott, American options with regime switching, International Journal of Theoretical and Applied Finance, 5 (2002), 497-514. doi: 10.1142/S0219024902001523.

[8]

L. Delong, Optimal investment and consumption in the presence of default on a financial market driven by a Levy noise, Ann. Univ. Mariae Curie-Sk?odowska Sect. A, 60 (2006), 1-15.

[9]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Imation and Control, Applications of Mathematics (New York), 29. Springer-Verlag, New York, 1995.

[10]

R. J. Elliott, L. L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432. doi: 10.1007/s10436-005-0013-z.

[11]

R. J. Elliott and J. Hinz, Portfolio analysis, hidden Markov models and chart analysis by PF-Diagrams, International Journal of Theoretical and Applied Finance, 5 (2002), 385-399.

[12]

R. J. Elliott, T. K. Siu and L. L. Chan, Pricing volatility swaps under Heston's stochastic volatility model with regime switching, Applied Mathematical Finance, 14 (2007), 41-62. doi: 10.1080/13504860600659222.

[13]

H. U. Gerber and E. W. Shiu, Investing for retirement: Optimal capital growth and dynamic asset allocation (with discussions), North American Actuarial Journal, 4 (2000), 42-62. doi: 10.1080/10920277.2000.10595899.

[14]

S. M. Goldfeld and R. E. Quandt, The estimation of structural shifts by switching regressions, Annals of Economic and Social Measurement, 2 (1973), 475-485.

[15]

C. Gollier, Insurance and precautionary capital accumulation in a continuous-time model, Journal of Risk and Insurance, 61 (1994) 78-95. doi: 10.2307/253425.

[16]

X. Guo, Information and option pricings, Quantitative Finance, 1 (2001), 38-44. doi: 10.1080/713665550.

[17]

J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.

[18]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257. doi: 10.2307/1926560.

[19]

R. C. Merton, Optimal consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X.

[20]

J. Mossin, Aspects of rational insurance purchasing, Journal of Political Economy, 76 (1968), 553-568. doi: 10.1086/259427.

[21]

K. S. Moore and V. R. Young, Optimal insurance in a continuous-time model, Insurance Mathematics and Economics, 39 (2006), 47-68. doi: 10.1016/j.insmatheco.2006.01.009.

[22]

R. E. Quandt, The estimation of the parameters of a linear regression system obeying two separate regimes, Journal of the American Statistical Association, 53 (1958), 873-880. doi: 10.1080/01621459.1958.10501484.

[23]

H. Schlesinger and C. Gollier, Second-best insurance contract design in an incomplete market, Scandinavian Journal of Economics, 97 (1995), 123-135.

[24]

K. L. Teo, D. W. Reid and I. E. Boyd, Stochastic optimal control theory and its computational method, International Journal on Systems Science, 11 (1980), 77-95. doi: 10.1080/00207728008966998.

[25]

H. Tong, Some comments on the Canadian lynx data (with discussion), Journal of the Royal Statistical Society, Series A, General, 140 (1977), 432-436.

[26]

K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint, Automatica, 46 (2010), 1979-989. doi: 10.1016/j.automatica.2010.02.027.

[1]

Jiaqin Wei. Time-inconsistent optimal control problems with regime-switching. Mathematical Control and Related Fields, 2017, 7 (4) : 585-622. doi: 10.3934/mcrf.2017022

[2]

Wensheng Yin, Jinde Cao, Yong Ren. Inverse optimal control of regime-switching jump diffusions. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021034

[3]

Jiapeng Liu, Ruihua Liu, Dan Ren. Investment and consumption in regime-switching models with proportional transaction costs and log utility. Mathematical Control and Related Fields, 2017, 7 (3) : 465-491. doi: 10.3934/mcrf.2017017

[4]

Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072

[5]

Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048

[6]

Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042

[7]

Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control and Related Fields, 2020, 10 (1) : 189-215. doi: 10.3934/mcrf.2019036

[8]

Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control and Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237

[9]

Engel John C Dela Vega, Robert J Elliott. Conditional coherent risk measures and regime-switching conic pricing. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 267-300. doi: 10.3934/puqr.2021014

[10]

Jun Li, Fubao Xi. Exponential ergodicity for regime-switching diffusion processes in total variation norm. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021309

[11]

Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132

[12]

Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2415-2433. doi: 10.3934/jimo.2021074

[13]

Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control and Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21

[14]

Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial and Management Optimization, 2020, 16 (2) : 531-551. doi: 10.3934/jimo.2018166

[15]

Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079

[16]

Meiqiao Ai, Zhimin Zhang, Wenguang Yu. Valuing equity-linked death benefits with a threshold expense structure under a regime-switching Lévy model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022007

[17]

Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100

[18]

Ishak Alia, Mohamed Sofiane Alia. Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022048

[19]

Nguyen Huu Du, Nguyen Thanh Dieu, Tran Dinh Tuong. Dynamic behavior of a stochastic predator-prey system under regime switching. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3483-3498. doi: 10.3934/dcdsb.2017176

[20]

Dragos-Patru Covei, Elena Cristina Canepa, Traian A. Pirvu. Stochastic production planning with regime switching. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022013

2021 Impact Factor: 1.141

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (0)

[Back to Top]