Optimal proportional reinsurance with common shock dependence to minimise the probability of drawdown

Abstract In this paper, we study the optimal proportional reinsurance problem in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component, and the criterion is to minimise the probability of drawdown, namely, the probability that the value of the surplus process reaches some fixed proportion of its maximum value to date. By the method of maximising the ratio of drift of a diffusion divided to its volatility squared, and the technique of stochastic control theory and the corresponding Hamilton–Jacobi–Bellman equation, we investigate the optimisation problem in two different cases. Furthermore, we constrain the reinsurance proportion in the interval [0,1] for each case, and derive the explicit expressions of the optimal proportional reinsurance strategy and the minimum probability of drawdown. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results.

[1]  Junna Bi,et al.  Optimal mean–variance reinsurance and investment in a jump-diffusion financial market with common shock dependence , 2016, Math. Methods Oper. Res..

[2]  Erhan Bayraktar,et al.  Optimal investment to minimize the probability of drawdown , 2015, 1506.00166.

[3]  Bin Li,et al.  On minimizing drawdown risks of lifetime investments , 2015 .

[4]  Haluk Yener.  Maximizing survival, growth and goal reaching under borrowing constraints , 2012, 1209.6385.

[5]  Zhibin Liang,et al.  Optimal mean–variance investment and reinsurance problems for the risk model with common shock dependence , 2016 .

[6]  Pablo Azcue,et al.  Minimizing the ruin probability allowing investments in two assets: a two-dimensional problem , 2013, Math. Methods Oper. Res..

[7]  K. S. Tan,et al.  AGGREGATION OF CORRELATED RISK PORTFOLIOS: MODELS AND ALGORITHMS , 1999 .

[8]  N. Bäuerle,et al.  A note on applications of stochastic ordering to control problems in insurance and finance , 2012, 1210.3800.

[9]  Reaching goals by a deadline: digital options and continuous-time active portfolio management , 1999, Advances in Applied Probability.

[10]  M. Milevsky.  Real Longevity Insurance with a Deductible: Introduction to Advanced-Life Delayed Annuities (ALDA) , 2005 .

[11]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[12]  Sid Browne,et al.  Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark , 1999, Finance Stochastics.

[13]  Sanford J. Grossman,et al.  OPTIMAL INVESTMENT STRATEGIES FOR CONTROLLING DRAWDOWNS , 1993 .

[14]  Leonard J. Savage,et al.  How to Gamble If You Must: Inequalities for Stochastic Processes , 1965 .

[15]  Maximizing the utility of consumption with commutable life annuities , 2012 .

[16]  K. Yuen,et al.  Optimal proportional reinsurance with common shock dependence , 2015 .

[17]  Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin , 2004 .

[18]  Junyi Guo,et al.  On the first time of ruin in the bivariate compound Poisson model , 2006 .

[19]  V. Young,et al.  Optimal commutable annuities to minimize the probability of lifetime ruin , 2012 .

[20]  Junyi Guo,et al.  On a correlated aggregate claims model with Poisson and Erlang risk processes , 2002 .

[21]  V. Young,et al.  Optimal and Simple, Nearly Optimal Rules for Minimizing the Probability Of Financial Ruin in Retirement , 2006 .

[22]  H. Schmidli,et al.  On minimizing the ruin probability by investment and reinsurance , 2002 .

[23]  Minimizing the Probability of Ruin When Consumption is Ratcheted , 2008, 0806.2358.

[24]  Erhan Bayraktar,et al.  Minimizing the Probability of Lifetime Drawdown Under Constant Consumption , 2015, 1507.08713.

[25]  Erhan Bayraktar,et al.  Optimal reinsurance and investment with unobservable claim size and intensity , 2014 .

[26]  Sid Browne,et al.  Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time , 1996, Math. Oper. Res..

[27]  Hanspeter Schmidli,et al.  Optimal Proportional Reinsurance Policies in a Dynamic Setting , 2001 .

[28]  Virginia R. Young,et al.  Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift , 2005 .

[29]  Zhibin Liang,et al.  Optimal dynamic reinsurance with dependent risks: variance premium principle , 2016 .

[30]  Sid Browne,et al.  Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin , 1995, Math. Oper. Res..

[31]  Romuald Elie,et al.  Optimal lifetime consumption and investment under a drawdown constraint , 2008, Finance Stochastics.

[32]  Jun Cai,et al.  Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting , 2013 .

[33]  J. Grandell.  Aspects of Risk Theory , 1991 .

[34]  William D. Sudderth,et al.  Continuous-Time Red and Black: How to Control a Diffusion to a Goal , 1985, Math. Oper. Res..

[35]  Jakša Cvitanić,et al.  On portfolio optimization under "drawdown" constraints , 1994 .

[36]  Ananda Weerasinghe,et al.  Controlling a Process to a Goal in Finite Time , 1989, Math. Oper. Res..