Some Stochastic Optimization Problems in Reinsurance and Insurance

Insurance, which hedges against the risk of a contingent loss, is an indispensable risk management tool for both institutions and individuals. Reinsurance, namely, a form of insurance accessible to insurers, helps limit the liability of an insurer on certain set of risks and protect against catastrophic events, while various insurance products are available for individuals to cover uncertain losses from almost every aspect of their daily life. This thesis focuses on dynamically controlling the utilities of decision makers by imposing various controls, including reinsurance for insurers, and life annuity and term life insurance for individuals, either analytically or numerically. Utilizing (re)insurance to attain certain objectives has long been a central focus in the actuarial science literature. This thesis aims at making contributions in the existing literature by applying models that are more in line with reality, both in regard to the underlying dynamic models and control variables. In Chapter 3, we study the optimal reinsurance-investment strategy for dynamic contagion claims. Such a claim process no longer possesses the stationary and independent increment property, and can capture contagion due to endogenous (self-exciting) and exogenous (externally-exciting) factors. Adopting the time-consistent mean-variance criterion, we analytically solve for the equilibrium strategies and analyze the impact of some contagion factors on the resulting optimal reinsurance strategies. Chapter 4 models the basic surplus process as a spectrally negative Lévy process, and focuses on the partial information of the unobservable stock return rate to look into the optimal reinsurance-investment problem under the time-consistent mean-variance criterion. Analytical solutions are obtained by solving an extended HJB equation, and hedging demand due to partial information is carefully studied. Chapter 5 is devoted to the study of the optimal allocation of life annuity, term life insurance and consumption for an individual under a general force of mortality. In our setup, an individual's decision of life annuity, term life insurance and consumption are allowed to depend on the current wealth, existing life annuity and existing term life insurance, and realistic lump-sum purchases are considered. Assuming a CRRA preference, a penalty method is applied to numerically solve for the optimal allocations of wealth in life annuity, term life insurance and consumption. To ensure that the thesis flows smoothly, Chapter 1 introduces the background literature and main motivations of this thesis. Chapter 2 is devoted to mathematical preliminaries for the latter chapters. Finally, Chapter 6 concludes the thesis with potential directions for future research

Essays on Insurance Economics

This dissertation thesis address how aggregate shocks affect insurance firms\u27 risk management and asset investment decisions as well as the impact of these decisions on insurance prices and regulation. The first chapter develops a signaling model to examine how insurance firms choose among retention, reinsurance and securitization especially for catastrophe risks. The second chapter examines the determination of insurance prices in an integrated equilibrium framework where insurers\u27 assets may be subject to both idiosyncratic and aggregate shocks. The third chapter presents an empirical analysis of the hypothesized impacts of internal capital and asset risk on insurance prices as predicted by the results of the second chapter. The last chapter investigates the optimal design of insurance regulation to achieve the Pareto optimal asset and liquidity management by insurers as well as risk sharing between insurers and insurees. Chapter 1 provides a novel explanation for the predominance of retention and reinsurance relative to securitization in catastrophe risk transfer using a signaling model. An insurer\u27s risk transfer choice trades off the lower signaling costs of reinsurance against the additional costs of reinsurance stemming from sources such as their market power, higher cost of capital relative to capital markets, and compensation for their monitoring costs. In equilibrium, the lowest risk insurers choose reinsurance, while intermediate and high risk insurance choose partial and full securitization, respectively. An increase in the loss size increases the average risk of insurers who choose securitization. Consequently, catastrophe risks, which are characterized by low frequency-high severity losses, are only securitized by very high risk insurers. Chapter 2 develops a unified equilibrium model of competitive insurance markets where insurers\u27 assets may be exposed to idiosyncratic and aggregate shocks. We endogenize the relationship between insurance prices and insurers internal capital that potentially reconcile the conflicting predictions of previous theories that investigate the relation using partial equilibrium frameworks. Equilibrium effects lead to a non-monotonic U-shaped relation between insurance price and internal capital. Specifically, the equilibrium insurance price first decreases with a positive shock to the internal capital when it is below certain threshold level, and then increases with a positive shock the internal capital when it is above the threshold level. Further, we also derive another testable implication that an increase in the asset default risk increases the insurance price and decrease the insurance coverage. Chapter 3 studies the property and casualty insurance industry in periods from 1992 to 2012 based on the aggregate level of NAIC data. We show that the insurance price decreases with an increase in the surplus of insurance firms at the end of the previous year when the surplus is lower than 8.5 billion, and then increase when the surplus is higher than 8.5 billion. Our results provide support for the hypothesis of a U-shaped relationship between internal capital and insurance price. Our results also provide evidence for the positive relationship between asset portfolio risk and insurance price. Chapter 4 studies the effects of aggregate risk on the Pareto optimal asset and liquidity management by insurers as well as risk-sharing between insurers and insurers. When aggregate risk is low, both insurers and insurers hold no liquidity reserves, insurees are fully insured, and insurers bear all aggregate risk. When aggregate risk takes intermediate values, both insurees and insurers still hold no liquidity reserves, but insurers partially share aggregate risk with insurers. When aggregate risk is high, however, it is optimal to hold nonzero liquidity reserves, and insurees partially share aggregate risk with insurers. The efficient asset and liquidity management policies as well as the aggregate risk allocation can be implemented through a regulatory intervention policy that combines a minimum liquidity requirement when aggregate risk is high, ex post contingent on the aggregate state, comprehensive insurance policies, and reinsurance

Insuring Against International Hazards: Descriptive and Prescriptive Aspects

Today multinational firms face grave uncertainties with respect to their investment strategies in other countries. This paper stresses the importance of integrating the descriptive aspects of this problem with prescriptive recommendations. It does so by raising two broad interrelated questions: (1) How do multinational firms and insurers deal with the problems of international risk in making their decisions on what investments to undertake? (2) What role can analytic approaches, including insurance mechanisms, play in better managing risk and uncertainty in international transactions? These questions are addressed by developing a conceptual framework which emphasizes the importance of problem formulation, institutional arrangements and decision processes as a basis for prescriptive recommendations. The problem is characterized by lack of a detailed statistical data base to estimate probabilities and consequences of different types of political, economic, and social risks. Corporate planners and risk managers who have responsibility for these investment decisions are anxious to avoid uncertainty. Hence, their actions appear to be greatly influenced by past experience and personal contacts. Our prescriptive recommendations are designed to widen the statistical data base by the use of experts and Bayesian analysis as well as to broaden the responsibility for investment decisions within the organization. We also propose a jointly operated private-federal insurance program which maintains features of current government operated systems but has private firms marketing policies and settling claims. The above theoretical concepts are illustrated with a case study of Indonesia's investment evaluation problem pursuant to their decision to provide the United States with liquefied natural gas in the early 1970's. This case study illustrates the political risks of firms investing even in highly developed economies such as the United States

On Some Stochastic Optimal Control Problems in Actuarial Mathematics

The event of ruin (bankruptcy) has long been a core concept of risk management interest in the literature of actuarial science. There are two major research lines. The first one focuses on distributional studies of some crucial ruin-related variables such as the deficit at ruin or the time to ruin. The second one focuses on dynamically controlling the probability that ruin occurs by imposing controls such as investment, reinsurance, or dividend payouts. The content of the thesis will be in line with the second research direction, but under a relaxed definition of ruin, for the reason that ruin is often too harsh a criteria to be implemented in practice. Relaxation of the concept of ruin through the consideration of "exotic ruin" features, including for instance, ruin under discrete observations, Parisian ruin setup, two-sided exit framework, and drawdown setup, received considerable attention in recent years. While there has been a rich literature on the distributional studies of those new features in insurance surplus processes, comparably less contributions have been made to dynamically controlling the corresponding risk. The thesis proposes to analytically study stochastic control problems related to some "exotic ruin" features in the broad area of insurance and finance. In particular, in Chapter 3, we study an optimal investment problem by minimizing the probability that a significant drawdown occurs. In Chapter 4, we take this analysis one step further by proposing a general drawdown-based penalty structure, which include for example, the probability of drawdown considered in Chapter 3 as a special case. Subsequently, we apply it in an optimal investment problem of maximizing a fund manager's expected cumulative income. Moreover, in Chapter 5 we study an optimal investment-reinsurance problem in a two-sided exit framework. All problems mentioned above are considered in a random time horizon. Although the random time horizon is mainly determined by the nature of the problem, we point out that under suitable assumptions, a random time horizon is analytically more tractable in comparison to its finite deterministic counterpart. For each problem considered in Chapters 3--5, we will adopt the dynamic programming principle (DPP) to derive a partial differential equation (PDE), commonly referred to as a Hamilton-Jacobi-Bellman (HJB) equation in the literature, and subsequently show that the value function of each problem is equivalent to a strong solution to the associated HJB equation via a verification argument. The remaining problem is then to solve the HJB equations explicitly. We will develop a new decomposition method in Chapter 3, which decomposes a nonlinear second-order ordinary differential equation (ODE) into two solvable nonlinear first-order ODEs. In Chapters 4 and 5, we use the Legendre transform to build respectively one-to-one correspondence between the original problem and its dual problem, with the latter being a linear free boundary problem that can be solved in explicit forms. It is worth mentioning that additional difficulties arise in the drawdown related problems of Chapters 3 and 4 for the reason that the underlying problems involve the maximum process as an additional dimension. We overcome this difficulty by utilizing a dimension reduction technique. Chapter 6 will be devoted to the study of an optimal investment-reinsurance problem of maximizing the expected mean-variance utility function, which is a typical time-inconsistent problem in the sense that DPP fails. The problem is then formulated as a non-cooperative game, and a subgame perfect Nash equilibrium is subsequently solved. The thesis is finally ended with some concluding remarks and some future research directions in Chapter 7

Risk-mitigation techniques: from (re-)insurance to alternative risk transfer

Insurance risks knowledge is becoming essential for both financial stability and social security purposes, moreover in a country with a very low insurance education like Italy. In insurance industry, Solvency II requirements introduced new issues for actuarial risk management in non-life insurance, challenging the market to have a consciousness of its own risk profile, and also investigating the sensitivity of the solvency ratio depending on the insurance risks and technical results on either a short-term and medium-term perspective. For this aim, in the present thesis, firstly a partial internal model for underwriting risk is developed for multi-line non-life insurers. Specifically, the risk-mitigation and profitability impacts of traditional reinsurance in the underwriting risk model are introduced, and a global framework for a feasible application of this model consistent with a medium-term analysis is provided. Reinsurance have to be considered in the assessment of Non-Life insurers risk profile, with particular regard to the Solvency II Underwriting Risk because of its impact on business and risk strategy. Risk mitigation techniques appear as a key driver of Non-Life insurance business as they can change risk profile over either the short-term or medium-term perspective. They impact the technical result of the year in such a way that it is important to assess how reinsurance strategies decrease the volatility, reducing the capital requirements, but, on the other hand, they also change the mean of distributions in different ways according to the price for risk requested by reinsurers. At the same time, risk mitigation also influences Non-Life insurance management actions as it can improve business strategy and capital allocation (also in potential capital recovery plans). Furthermore, the analysis a medium-term capital requirement would ask insurers to have more capital than in a one-year time horizon, and in this framework risk mitigation effects linked to reinsurance strategies must be assessed on either risk/return perspective trade-off. On the other hand, (re)insurance can play an active role in mitigating physical risks, and in particular natural catastrophe risks. In this context, as well as in natural disasters, Alternative Risk Transfer (ART) is becoming a new significant actuarial and capital management tool for insurers and, potentially, for government measures in recovery actions of economic and social losses in case of natural disasters. Catastrophe Bonds are insurance-linked securities that have been increasingly used as an alternative to traditional reinsurance for two decades. In exchange for a Spread over to the risk-free rate, protection is provided against stated perils that could impact the insured portfolio. A broad literature has flourished to investigate what are the features that significantly influence the Spread, in addition to the portfolio’s expected loss. Almost all proposed models are based on multivariate linear regression, that has provided satisfactory predictive performance as well as easily interpretability. This thesis also explores the use of Machine Learning models in modeling the determinant at issuances, contrasting both their predictive performance and their interpretability with respect to traditional models. An overview of the economics of CAT bonds, on current literature and on the statistical methodologies will be provided also. Aim of this Thesis is to provide a solid framework of insurance risk transfer for both pure underwriting and catastrophe risks, investigating risk transfer practices from traditional to alternative and most innovative technique. In these fields, firstly a suitable risk model is used in order to describe main impacts on insurance business model. Then, the main innovative alternative risk transfer for catastrophe risks are illustrated and CAT Bond will be adequately described, investigating main pricing models using a machine learning approach. Finally, a possible Italian CAT Bond issuance is provided in order to investigate an integrated solution with a traditional reinsurance underlying an alternative risk transfer in order to achieve a public-private partnership to natural catastrophe

How does the life insurance business perform and behave: the case of the UK industry

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonThis thesis reviews the UK life insurance industry comprehensively in terms of performance and business behaviour. One major contribution of the thesis is to challenge the conventional view on evaluation of investment funds from a shareholders’ perspective. The accounting valuation techniques to evaluate investment from the policyholder’s perspective have not been advanced to the same extent as methods designed to evaluate investment from the shareholder’s perspective, due partly to the accounting complexity of the investment management. Against this context, the thesis develops a valuation method on the basis that policyholders’ basic expectation that their saved funds shall be invested with value growth higher than inflation in the real goods market, and the thesis takes this as the benchmark to assess the reported value of policyholders’ assets. The thesis employs this valuation to assess the performance of different life assurance products (conventional vs. modern) and examine whether the transformation (from conventional to modern) has any impact on insurer performance and behaviour. The thesis also examines whether product diversification impacts realised and unrealised investment income homogenously; the result suggests that the effect of product diversification on performance varies across different measurements of realised and unrealised gain. The second major contribution of the thesis is to test the validity of different output proxies and compare efficiency scores based ranking for competitive firms to the value creation based ranking. Overall, the thesis suggests that different output proxies give consistently similar ranking for competitive firms, and cost efficiency based on different proxies are closely related to conventional measurers of firm performance and value creation in terms of value and ranking

The Impact of Medical Spending Growth on Guaranteed Renewable Health insurance

I examine the problem of writing guaranteed renewable health insurance in the presence of medical spending growth. Prior research suggests that the growth and difficulty in forecasting future medical costs is an impediment to multiperiod health insurance, where contract reserves are used to pay a portion of the benefits in later years of the contract. Medical spending growth is an input to calculating the magnitude of premiums and reserves, so setting up reserves to pay future claims involves forecasting spending growth. Hedging assets can ameliorate the investment problem by providing assets that automatically adjust to unexpected shocks in spending growth. I expand an existing model of guaranteed renewability in an economy with risk to show the specific ways that medical spending growth enters the premium and reserve functions. I treat stochastic trend as a factor the insurance company can predict with error. I utilize aggregate and individual level insurance spending data and financial returns data to analyze whether medical trend can be hedged with existing assets. I separate trend into predictable and error components and analyze the correlation between the error component and return on assets. I find that medical spending growth is predictable with error over short and medium time horizons. I find that there is no significant correlation between asset returns and forecast errors across several broad asset classes. The combination of partially predictable spending growth and the absence of a hedging asset imply that insurers should be using reserves to manage the macroeconomic risk of spending growth. The load for reserving for trend is an up-front cost in addition to the up-front expense of guaranteed renewability. Insurers should use a diversified investment strategy for reserves rather than one targeted at trying to match spending growth. I conclude by noting the positive and negative effects of the newly passed health reform law (PPACA) on guaranteed renewable health insurance and other health insurance arrangements that require contract reserves and policies that shift health care spending onto public plans