Optimal asset allocation: Risk and information uncertainty

In asset allocation problem, the distribution of the assets is usually assumed to be known in order to identify the optimal portfolio. In practice, we need to estimate their distribution. The estimations are not necessarily accurate and it is known as the uncertainty problem. Many researches show that most people are uncertainty aversion and this affects their investment strategy. In this article, we consider risk and information uncertainty under a common asset allocation framework. The effects of risk premium and covariance uncertainty are demonstrated by the worst scenario in a set of measures generated by a relative entropy constraint. The nature of the uncertainty and its impacts on the asset allocation are discussed.postprin

A unified 'bang-bang' principle with respect to R-invariant performance benchmarks

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Higher-Order, Polar and Sz.-Nagy’s Generalized Derivatives of Random Polynomials with Independent and Identically Distributed Zeros on the Unit Circle

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A class of nonzero-sum investment and reinsurance games subject to systematic risks

© 2016 Informa UK Limited, trading as Taylor & Francis Group. Recently, there have been numerous insightful applications of zero-sum stochastic differential games in insurance, as discussed in Liu et al. [Liu, J., Yiu, C. K.-F. & Siu, T. K. (2014). Optimal investment of an insurer with regime-switching and risk constraint. Scandinavian Actuarial Journal 2014(7), 583–601]. While there could be some practical situations under which nonzero-sum game approach is more appropriate, the development of such approach within actuarial contexts remains rare in the existing literature. In this article, we study a class of nonzero-sum reinsurance-investment stochastic differential games between two competitive insurers subject to systematic risks described by a general compound Poisson risk model. Each insurer can purchase the excess-of-loss reinsurance to mitigate both systematic and idiosyncratic jump risks of the inter-arrival claims; and can invest in one risk-free asset and one risky asset whose price dynamics follows the famous Heston stochastic volatility model [Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies6, 327–343]. The main objective of each insurer is to maximize the expected utility of his terminal surplus relative to that of his competitor. Dynamic programming principle for this class of nonzero-sum game problems leads to a non-canonical fixed-point problem of coupled non-linear integral-typed equations. Despite the complex structure, we establish the unique existence of the Nash equilibrium reinsurance-investment strategies and the corresponding value functions of the insurers in a representative example of the constant absolute risk aversion insurers under a mild, time-independent condition. Furthermore, Nash equilibrium strategies and value functions admit closed forms. Numerical studies are also provided to illustrate the impact of the systematic risks on the Nash equilibrium strategies. Finally, we connect our results to that under the diffusion-approximated model by proving explicitly that the Nash equilibrium under the diffusion-approximated model is an (Formula presented.) -Nash equilibrium under the general Poisson risk model, thereby establishing that the analogous Nash equilibrium in Bensoussan et al. [Bensoussan, A., Siu, C. C., Yam, S. C. P. & Yang, H. (2014). A class of nonzero-sum stochastic differential investment and reinsurance games. Automatica50(8), 2025–2037] serves as an interesting complementary case of the present framework

A class of non-zero-sum stochastic differential investment and reinsurance games

In this article, we provide a systematic study on the non-zero-sum stochastic differential investment and reinsurance game between two insurance companies. Each insurance company’s surplus process consists of a proportional reinsurance protection and an investment in risky and risk-free assets. Each insurance company is assumed to maximize his utility of the difference between his terminal surplus and that of his competitor. The surplus process of each insurance company is modeled by a mixed regime-switching Cramer–Lundberg diffusion approximation process, i.e. the coefficients of the diffusion risk processes are modulated by a continuous-time Markov chain and an independent market-index process. Correlation between the two surplus processes, independent of the risky asset process, is allowed. Despite the complex structure, we manage to solve the resulting non-zero sum game problem by applying the dynamic programming principle. The Nash equilibrium, the optimal reinsurance/investment, and the resulting value processes of the insurance companies are obtained in closed forms, together with sound economic interpretations, for the case of an exponential utility function.postprin

Valuing equity-linked death benefits in a regime-switching framework

© 2015 by Astin Bulletin. All rights reserved. In this article, we consider the problem of computing the expected discounted value of a death benefit, e.g. in Gerber et al. (2012, 2013), in a regime-switching economy. Contrary to their proposed discounted density approach, we adopt the Laplace transform to value the contingent options. By this alternative approach, closed-form expressions for the Laplace transforms of the values of various contingent options, such as call/put options, lookback options, barrier options, dynamic fund protection and the dynamic withdrawal benefits, have been obtained. The value of each contingent option can then be recovered by the numerical Laplace inversion algorithm, and this efficient approach is documented by several numerical illustrations. The strength of our methodology becomes apparent when we tackle the valuations of exotic contingent options in the cases when (1) the contracts have a finite expiry date; (2) when the time-until-death variable is uniformly distributed in accordance with De Moivre's law