An Actuarial Premium Pricing Model for Nonnormal Insurance and Financial Risks in Incomplete Markets

Abstract

A model for pricing insurance and financial risks, based on recent developments in actuarial premium principles with elliptical distributions, is developed for application to incomplete markets and heavy-tailed distributions. The pricing model involves an application of a generalized variance premium principle from insurance pricing to the pricing of a portfolio of nontraded risks relative to a portfolio of traded risks. This pricing model for a portfolio of insurance or financial risks reflects preferences for features of the distributions other than mean and variance, including kurtosis. The model reduces to the Capital Asset Pricing Model for multinormal portfolios and to a form of the CAPM in the case where the traded and nontraded risks have the same elliptical distribution.     

The Impact of Disability Insurance on a Portfolio of Life Insurances

The aim of this article is to study the impact of disability insurance on an insurer's risk situation for a portfolio that also consists of annuity and term life contracts. We provide a model framework using discrete time nonhomogeneous bivariate Markov renewal processes and in a simulation study focus on diversification benefits as well as potential natural hedging effects (risk-minimizing or risk-immunizing portfolio compositions) that may arise within the portfolio because of the different types of biometric risks. Our analyses emphasize that disability insurances are a less efficient tool to hedge shocks to mortality and that their high sensitivity toward shocks to disability risks cannot be easily counterbalanced by other life insurance products. However, the addition of disability insurance can still considerably lower the overall company risk.     

“The Future of Risk Classification in the Age of Predictive DNA-Based Testing”, Donald C. Chambers,January 1999

Abstract

The correlation among multiple lines of business plays an important role in quantifying the uncertainty of loss reserves for insurance portfolios. To accommodate correlation, most multivariate loss-reserving methods focus on the pairwise association between corresponding cells in multiple run-off triangles. However, such practice usually relies on the independence assumption across accident years and ignores the calendar year effects that could affect all open claims simultaneously and induce dependencies among loss triangles. To address this issue, we study a Bayesian log-normal model in the prediction of outstanding claims for dependent lines of business. In addition to the pairwise correlation, our method allows for an explicit examination of the correlation due to common calendar year effects. Further, different specifications of the calendar year trend are considered to reflect valuation actuaries’ prior knowledge of claim development. In a case study, we analyze an insurance portfolio of personal and commercial auto lines from a major U.S. property-casualty insurer. It is shown that the incorporation of calendar year effects improves model fit significantly, though it contributes substantively to the predictive variability. The availability of the realizations of predicted claims permits us to perform a retrospective test, which suggests that extra prediction uncertainty is indispensable in modern risk management practices.     

The effectiveness of the VaR-based portfolio insurance strategy: An empirical analysis

This paper proposes an approach to constructing the insured portfolios under the VaR-based portfolio insurance strategy (VBPI) and provides a comprehensive analysis of its hedging effectiveness in comparison with the buy-and-hold (B&H) as well as the constant proportion portfolio insurance (CPPI) strategies in the context of the Chinese market. The results show that both of the insurance strategies are able to limit the downward returns while retaining certain upside returns, and their capabilities of reshaping the return distributions increase as the guarantee or the confidence level rises. In general, the VBPI strategy tends to outperform the CPPI strategy in terms of both the degree of downside protection and the return performance.     

On the optimality of proportional reinsurance

Proportional reinsurance is often thought to be a very simple method of covering the portfolio of an insurer. Theoreticians are not really interested in analysing the optimality properties of these types of reinsurance covers. In this paper, we will use a real-life insurance portfolio in order to compare four proportional structures: quota share reinsurance, variable quota share reinsurance, surplus reinsurance and surplus reinsurance with a table of lines.     

Skewed distributions in finance and actuarial science: a review

That the returns on financial assets and insurance claims are not well described by the multivariate normal distribution is generally acknowledged in the literature. This paper presents a review of the use of the skew-normal distribution and its extensions in finance and actuarial science, highlighting known results as well as potential directions for future research. When skewness and kurtosis are present in asset returns, the skew-normal and skew-Student distributions are natural candidates in both theoretical and empirical work. Their parameterization is parsimonious and they are mathematically tractable. In finance, the distributions are interpretable in terms of the efficient markets hypothesis. Furthermore, they lead to theoretical results that are useful for portfolio selection and asset pricing. In actuarial science, the presence of skewness and kurtosis in insurance claims data is the main motivation for using the skew-normal distribution and its extensions. The skew-normal has been used in studies on risk measurement and capital allocation, which are two important research fields in actuarial science. Empirical studies consider the skew-normal distribution because of its flexibility, interpretability, and tractability. This paper comprises four main sections: an overview of skew-normal distributions; a review of skewness in finance, including asset pricing, portfolio selection, time series modeling, and a review of its applications in insurance, in which the use of alternative distribution functions is widespread. The final section summarizes some of the challenges associated with the use of skew-elliptical distributions and points out some directions for future research.     

On the Demand for Portfolio Insurance

While insurers manage underwriting risk with various methods including reinsurance, insurers increasingly manage asset risk with options, futures, and other derivatives. Previous research shows that buyers of portfolio insurance pay considerably for downside protection. We add to this literature by providing the first evidence on the cost of portfolio insurance, the payoff to portfolio insurance, and the relative demand for portfolio insurance across VIX levels. We find that the demand for portfolio insurance is relatively high at low levels of VIX, suggesting purchasers demand more downside protection when this protection is cheap on an absolute basis (but expensive on a relative basis). We also provide the first evidence on the hedging behavior of specific investor classes and show that the demand for portfolio insurance is driven by retail investors (individuals) who buy costly insurance from institutional investors. Results are consistent with other types of paradoxical insurance‐buying behavior.     

Evaluating Liquidation Strategies for Insurance Companies

In this article, we examine liquidation strategies and asset allocation decisions for property and casualty insurance companies for different insurance product lines. We propose a cash‐flow‐based liquidation model of an insurance company and analyze selling strategies for a portfolio with liquid and illiquid assets. Within this framework, we study the influence of different bid‐ask spread models on the minimum capital requirement and determine a solution set consisting of an optimal initial asset allocation and an optimal liquidation strategy. We show that the initial asset allocation, in conjunction with the appropriate liquidation strategy, is an important tool in minimizing the capital committed to cover claims for a predetermined ruin probability. This interdependence is of importance to insurance companies, stakeholders, and regulators.     

Portfolio insurance and prospect theory investors: Popularity and optimal design of capital protected financial products

Portfolio insurance strategies are used on both the institutional and the retail side of the asset management industry. While standard utility theory struggles to provide an explanation, this study justifies the popularity of portfolio insurance strategies in a behavioral finance context. We run Monte Carlo simulations as well as historical simulations for popular portfolio insurance strategies and benchmark strategies in order to evaluate the outcomes using cumulative prospect theory. Our simulation results indicate that most portfolio insurance strategies are the preferred investment strategy for a prospect theory investor. Moreover, the analysis provides insights into how portfolio insurance products should be designed and structured to meet the preferences of prospect theory investors as accurately as possible.     

Estimating the Cost of Equity for Property-Liability Insurance Companies

Due to the highly skewed and heavy‐tailed distributions associated with the insurance claims process, we evaluate the Rubinstein‐Leland (RL) model for its ability to improve the cost of equity estimates of insurance companies because of its distribution‐free feature. Our analyses show that there is as large as a 94‐basis‐point difference in the estimated cost of insurance equity between the RL model and the capital asset pricing model (CAPM) for the sample of property‐liability insurers with more severe departures from normality. In addition, consistent with our hypotheses, significant differences in the market risk estimates are found for insurers with return distributions that are asymmetrically distributed, and for small insurers. Third, we find significant performance improvements from using the RL model by showing smaller values of excess return of the expected return of the portfolio to the model return for a portfolio of insurers with returns that are more skewed and for a portfolio of small insurers. Finally, our panel data analysis shows the differences in the market risk estimates are significantly influenced by firm size, degree of leverage, and degree of asymmetry. The implication is that insurers should use the RL model rather than the CAPM to estimate its cost of capital if the insurer is small (assets size is less than $2,291 million), and/or its returns are not symmetrical (the value of skewness is greater than 0.509 or less than ?0.509).     

A bootstrap-based comparison of portfolio insurance strategies

This study presents a systematic comparison of portfolio insurance strategies. We implement a bootstrap-based hypothesis test to assess statistical significance of the differences in a variety of downside-oriented risk and performance measures for pairs of portfolio insurance strategies. Our comparison of different strategies considers the following distinguishing characteristics: static versus dynamic protection; initial wealth versus cumulated wealth protection; model-based versus model-free protection; and strong floor compliance versus probabilistic floor compliance. Our results indicate that the classical portfolio insurance strategies synthetic put and constant proportion portfolio insurance (CPPI) provide superior downside protection compared to a simple stop-loss trading rule and also exhibit a higher risk-adjusted performance in many cases (dependent on the applied performance measure). Analyzing recently developed strategies, neither the TIPP strategy (as an ‘improved’ CPPI strategy) nor the dynamic VaR-strategy provides significant improvements over the more traditional portfolio insurance strategies.     

A dynamic bivariate common shock model with cumulative effect and its actuarial application

Abstract

Standard actuarial theory of multiple life insurance traditionally postulates independence for the remaining lifetimes mainly due to computational convenience rather than realism. In this paper, we propose a general common shock model for modelling dependent coupled lives and apply it to a life insurance model. In the proposed shock model, we consider not only simultaneous deaths of the coupled members due to a single shock (e.g. a critical accident), but also cumulative effect in the mortality rate when they survive shocks. Under the model, we derive a bivariate lifetime distribution and its marginal distributions in closed forms. We study the bivariate ageing property, dependence structure and the dependence orderings of the lifetime distribution. Based on it, we investigate the influence of dependence on the pricings of insurance policies involving multiple lives which are subject to common shocks. Furthermore, we discuss relevant useful stochastic bounds.     

Fair valuation of cliquet-style return guarantees in (homogeneous and) heterogeneous life insurance portfolios

Participating life insurance contracts allow the policyholder to participate in the annual return of a reference portfolio. Additionally, they are often equipped with an annual (cliquet-style) return guarantee. The current low interest rate environment has again refreshed the discussion on risk management and fair valuation of such embedded options. While this problem is typically discussed from the viewpoint of a single contract or a homogeneous* insurance portfolio, contracts are, in practice, managed within a heterogeneous insurance portfolio. Their valuation must then – unlike the case of asset portfolios – take account of portfolio effects: Their premiums are invested in the same reference portfolio; the contracts interact by a joint reserve, individual surrender options and joint default risk of the policy sponsor. Here, we discuss the impact of portfolio effects on the fair valuation of insurance contracts jointly managed in (homogeneous and) heterogeneous life insurance portfolios. First, in a rather general setting, including stochastic interest rates, we consider the case that otherwise homogeneous contracts interact due to the default risk of the policy sponsor. Second, and more importantly, we then also consider the case when policies are allowed to differ in further aspects like the guaranteed rate or time to maturity. We also provide an extensive numerical example for further analysis.     

危机前后美国寿险公司不动产投资研究

基于ISIS全球保险公司数据库收录的美国寿险公司数据资料,本文研究了美国寿险公司的不动产投资情况.研究发现,从美国寿险公司的资产组合构成来看,不动产投资所占的比重还很低,且在次贷危机前后不动产投资在整个资产组合中所占的比重没有很大的变动.在美国房地产市场顶峰期的2005年,寿险公司所持有的不动产投资反而是最低的.但是,...     

Demand for insurance in a portfolio setting

This paper takes an additional step toward analyzing the demand for insurance in the context of a portfolio model. An investor is endowed with a portfolio containing a risky and riskless asset that can be augmented by purchasing insurance. Here, insurance is paid for by reducing the quantity of the risky insurable asset, holding the quantity of the riskless asset fixed. In the standard insurance demand model, insurance is paid for by reducing the amount of the riskless asset. This distinction leads to a different insurance demand function because the opportunity cost of purchasing insurance is now random.     

Portfolio Insurance Strategies when Hedging Affects Share Prices

Option-based portfolio insurance can result in coordinated buying and selling, which destabilizes markets such that hedgers fail to achieve their objective. Gennotte and Leland (1990) show portfolio insurance strategies can have an impact on price movements. Ramanlal and Mann (1996) show how price movements, in turn, can alter hedging strategies. In this paper, we combine these separate effects and develop an equilibrium, executable hedging strategy. This hedging strategy requires less rebalancing than traditional portfolio insurance; more important, it achieves downside protection with a less destabilizing impact on security prices.     

Long-term optimal portfolios with floor

Long-term risk-sensitive portfolio optimization is studied with floor constraint. A simple rule to characterize its solution is mentioned under a general setting. Following this rule, optimal portfolios are constructed in several ways, using the optimal portfolio without floor constraint, combined with ideas of dynamic portfolio insurance, such as CPPI (constant proportion portfolio insurance), OBPI (option-based portfolio insurance), and DFP (dynamic fund protection). In addition, examples are presented with explicit computations of solutions.     

The dynamics of premium reserves of life annuities under the regime of Solvency II

The article considers the solvency requirements for a whole portfolio of annuities under the regime of Solvency II. More precisely, the following question is investigated: Which demand of interest on the initial capital – the Solvency II premium reserves – is needed in order to fit the balance for Solvency II capital requirements in the next year? It turns out, that even for a model portfolio of simple annuities with say guaranteed interest rate of 1,25% the demand of interest in one year is greater than 3%. So even if a life insurance company fulfill the capital requirements of Solvency II in 2016 the mentioned effect causes eventually problems in future times.     

Uniqueness of the fair premium for equity-linked life insurance contracts

An equity-linked life insurance contract combines an endowment life insurance and an investment strategy with a minimum guarantee. The benefit of this contract is determined by the guaranteed amount plus a bonus equal to a call on the portfolio. This bonus is similar to an Asian option. This article analyzes the relationship between the periodic insurance premium and its proportional share invested into the portfolio. For a general model of the financial risks we show the existence and uniqueness of an insurance premium. Furthermore the premium is strictly increasing and convex as a function of the share invested.     

Gini-type measures of risk and variability: Gini shortfall,capital allocations,and heavy-tailed risks

We introduce and explore Gini-type measures of risk and variability, and develop the corresponding economic capital allocation rules. The new measures are coherent, additive for co-monotonic risks, convenient computationally, and require only finiteness of the mean. To elucidate our theoretical considerations, we derive closed-form expressions for several parametric families of distributions that are of interest in insurance and finance, and further apply our findings to a risk portfolio of a bancassurance company.