American Institute of Mathematical Sciences

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

Optimal insurance in a changing economy

Jingzhen Liu 1, , Ka-Fai Cedric Yiu 2, , Tak Kuen Siu 3, and  Wai-Ki Ching 4,

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.
Keywords: dynamic programming, Optimal insurance, regime-switching HJB equations., regime-switching, utility maximization.
Mathematics Subject Classification: Primary: 93E20; Secondary: 49l2.
Citation: Jingzhen Liu, Ka-Fai Cedric Yiu, Tak Kuen Siu, Wai-Ki Ching. Optimal insurance in a changing economy. Mathematical Control & 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, (1981).   Google Scholar

[2]

K. J. Arrow, Uncertainty and the welfare economics of medical care,, American Economic Review, 53 (1963), 941.   Google Scholar

[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.  doi: 10.1016/j.jmateco.2007.09.004.  Google Scholar

[4]

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

[5]

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

[6]

J. Buffington and R. J. Elliott, Regime switching and European options,, Stochastic Theory and Control, (2002), 73.  doi: 10.1007/3-540-48022-6_5.  Google Scholar

[7]

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

[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.   Google Scholar

[9]

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

[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.  doi: 10.1007/s10436-005-0013-z.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1080/13504860600659222.  Google Scholar

[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.  doi: 10.1080/10920277.2000.10595899.  Google Scholar

[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.   Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.  doi: 10.1080/01621459.1958.10501484.  Google Scholar

[23]

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

[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.  doi: 10.1080/00207728008966998.  Google Scholar

[25]

H. Tong, Some comments on the Canadian lynx data (with discussion),, Journal of the Royal Statistical Society, 140 (1977), 432.   Google Scholar

[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.  doi: 10.1016/j.automatica.2010.02.027.  Google Scholar

show all references

References:
[1]

N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems,, North Holland, (1981).   Google Scholar

[2]

K. J. Arrow, Uncertainty and the welfare economics of medical care,, American Economic Review, 53 (1963), 941.   Google Scholar

[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.  doi: 10.1016/j.jmateco.2007.09.004.  Google Scholar

[4]

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

[5]

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

[6]

J. Buffington and R. J. Elliott, Regime switching and European options,, Stochastic Theory and Control, (2002), 73.  doi: 10.1007/3-540-48022-6_5.  Google Scholar

[7]

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

[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.   Google Scholar

[9]

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

[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.  doi: 10.1007/s10436-005-0013-z.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1080/13504860600659222.  Google Scholar

[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.  doi: 10.1080/10920277.2000.10595899.  Google Scholar

[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.   Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.  doi: 10.1080/01621459.1958.10501484.  Google Scholar

[23]

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

[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.  doi: 10.1080/00207728008966998.  Google Scholar

[25]

H. Tong, Some comments on the Canadian lynx data (with discussion),, Journal of the Royal Statistical Society, 140 (1977), 432.   Google Scholar

[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.  doi: 10.1016/j.automatica.2010.02.027.  Google Scholar

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