The DB Underpin Hybrid Pension Plan

Abstract

In this paper the defined benefit underpin guarantee is valued as a financial option, within the traditional funding paradigms of actuarial science. Assuming fixed interest rates, and assuming that salaries can be treated as a tradable asset, we value the guarantee using fair value principles. Contribution rates are developed for the Entry Age Normal, Projected Unit Credit, and Traditional Unit Credit funding methods. In addition, for the accruals methods, we demonstrate the implied hedging strategy. The traditional unit credit offers the best method of these three, as it is consistent with the principles of financial economics, and the resulting contributions more naturally follow the cost of the emerging benefit, without creating expensive barriers to new hires. The method generates significant contribution volatility, and we demonstrate how this can be reduced with suitable benefit design and ongoing risk management.     

Financial Economics and Actuarial Practice

Abstract

Starting in the United Kingdom and continuing through the U.S. and Canadian actuarial professions, proponents of financial economics have been forcefully promoting a review of traditional actuarial practices and training. In particular, the financial theories first proposed by Modigliani and Miller and subsequently developed by others have been used to highlight serious weaknesses in typical actuarial thinking. In summary, it is claimed that much actuarial advice wrongly specifies value, that guidelines and standards need radical revision, and that traditional actuarial intuition suffers in comparison to newer modes of thought adopted by other professions.

This paper examines concepts from both financial economics and actuarial science as applied to defined benefit schemes using a simple discounted cash-flow framework as a reference point. The general finding is that many standard modes of actuarial thought are, in fact, indefensible when examined with the tools and techniques of financial economics. The call for revision of actuarial training and practices is credible and necessary.

However, the paper also touches upon areas where a heavy-handed application of finance theory could be misguided due to limitations in the simple financial economic models presented. It concludes that financial economics should be carefully integrated into actuarial thought rather than appended to existing actuarial theory or inserted as a wholesale replacement.     

VaR and expected shortfall: a non-normal regime switching framework

We have developed a regime switching framework to compute the Value at Risk and Expected Shortfall measures. Although Value at Risk as a risk measure has been criticized by some researchers for lack of subadditivity, it is still a central tool in banking regulations and internal risk management in the finance industry. In contrast, Expected Shortfall is coherent and convex, so it is a better measure of risk than Value at Risk. Expected Shortfall is widely used in the insurance industry and has the potential to replace Value at Risk as a standard risk measure in the near future. We have proposed regime switching models to measure value at risk and expected shortfall for a single financial asset as well as financial portfolios. Our models capture the volatility clustering phenomenon and variance-independent variation in the higher moments by assuming the returns follow Student-t distributions.     

Pricing Weather Derivatives Using the Indifference Pricing Approach

Abstract

This paper adopts an incomplete market pricing model–the indifference pricing approach–to analyze valuation of weather derivatives and the viability of the weather derivatives market in a hedging context. It incorporates price risk, weather/quantity risk, and other risks in the financial market. In a mean-variance framework, the relationship between the actuarial price and the indifference price of weather derivatives is analyzed, and conditions are obtained concerning when the actuarial price does not provide an appropriate valuation for weather derivatives. Conditions for the viability of the weather derivatives market are examined. This paper also analyzes the effects of partial hedging, natural hedges, basis risk, quantity risk, and price risk on investors’ indifference prices by examining the distributional impacts of the stochastic variables involved.     

Översikt av utländska aktuarietidskrifter

Abstract

In this paper a continuous-time model of a reinsurance market is presented, which contains the principal components of uncertainty transparent in such a market: Uncertainty about the time instants at which accidents take place, and uncertainty about claim sizes given that accidents have occurred.

Due to random jumps at random time points of the underlying claims processes, the absence of arbitrage opportunities is not sufficient to give unique premium functionals in general. Market preferences are derived under a necessary condition for a general exchange equilibrium. Information constraints are found under which premiums of risks are determined. It is demonstrated how general reinsurance treaties can be uniquely split into proportional contracts and nonproportional ones.

Several applications to reinsurance markets are given, and the results are compared to the corresponding theory of the classical one-period model of a reinsurance syndicate.

This paper attempts to reach a synthesis between the classical actuarial risk theory of insurance, in which virtually no economic reasoning takes place but where the net reserve is represented by a stochastic process, and the theory of partial equilibrium price formation at the heart of the economics of uncertainty.     

Relative Importance of Risk Sources in Insurance Systems

Abstract

Actuaries, and other managers of uncertainty, identify factors in modeling insurance risks because they believe (1) that these factors affect the outcome of a risk or (2) that the factors can be managed, thus allowing analysts a degree of control over the insurance system. This paper shows how to use a statistical measure, the coefficient of determination, for quantifying the relative importance of a source of uncertainty. With a quantitative measure of relative importance, risk managers can sharpen their intuition about the relative importance of risk factors and become better custodians of financial security systems.

This paper shows that the coefficient of determination is intuitively appealing in assessing the effectiveness of basic risk management techniques including risk exchange, pooling, and financial risk management. A single source common to all risks reduces the effectiveness of a pool; the risk measure quantifies the relative importance of this common source. The coefficient of determination is shown to have roots in the economics as well as the statistics literature. This connection provides further motivation for using the coefficient of determination and also suggests alternative measures for quantifying relative importance. The risk measure is useful in multivariate situations in which several factors affect a risk simultaneously. The paper illustrates this usefulness by considering a pool of policies that is subject to mortality, a common disaster, and a common investment environment.     

Model Selection and Averaging in Financial Risk Management

Simulated asset returns are used in many areas of actuarial science. For example, life insurers use them to price annuities, life insurance, and investment guarantees. The quality of those simulations has come under increased scrutiny during the current financial crisis. When simulating the asset price process, properly choosing which model or models to use, and accounting for the uncertainty in that choice, is essential. We investigate how best to choose a model from a flexible set of models. In our regime-switching models the individual regimes are not constrained to be from the same distributional family. Even with larger sample sizes, the standard model-selection methods (AIC, BIC, and DIC) incorrectly identify the models far too often. Rather than trying to identify the best model and limiting the simulation to a single distribution, we show that the simulations can be made more realistic by explicitly modeling the uncertainty in the model-selection process. Specifically, we consider a parallel model-selection method that provides the posterior probabilities of each model being the best, enabling model averaging and providing deeper insights into the relationships between the models. The value of the method is demonstrated through a simulation study, and the method is then applied to total return data from the S&P 500.     

Measuring Terminable Postretirement Obligations

Abstract

New approaches are needed to value benefit plans subject to unilateral changes or termination. The paper focuses on postretirement health benefits, but the thesis may be relevant to any flow not guaranteed by law or accumulating funds.

Retiree health benefits have usually been extended to participating active employees only in concert with a reserved right by the plan sponsor to control the design and, by implication, the cash flow. Over the course of the last fifteen years, this reserved extension of benefits has almost invariably led to reductions in benefits, when compared to the plan of benefits at an earlier period. In most cases, such reductions were anticipated under the circumstances that came to prevail (high health care cost increases), but were not taken into account by most of the projection and discounting methods of the time.

Current actuarial and accounting methods generate present values for terminable retiree health plans that have little credibility as measures of the beneficiary’s asset or the sponsor’s liability. Improvements are needed that will expand the actuarial toolbox and provide solutions to economic and accounting problems.

This paper provides a basis for discussion of assumptions that are appropriate when the plan sponsor can unilaterally and dramatically change future cash flows. The paper discusses how actuaries might best approach measurement situations where further plan reductions, or outright terminations, are to be anticipated. It introduces refinements and briefly discusses how each would fit with the usual actuarial model and how differences might affect behavior. These ideas are related to financial economics and the Bader-Gold paper “Reinventing Pension Actuarial Science.” Before concluding, the wider topic of discount rate selection is briefly addressed.     

The frequency of regime switching in financial market volatility

The mechanism of risk responses to market shocks is considered as stagnant in recent financial literature, whether during normal or stress periods. Since the returns are heteroskedastic, a little consideration was given to volatility structural breaks and diverse states. In this study, we conduct extensive simulations to prove that the switching regime GARCH model, under the highly flexible skewed generalized t (SGT) distribution, is remarkably efficient in detecting different volatility states. Next, we examine the switching regime in the S&P 500 volatility for weekly, daily, 10-minute and 1-minute returns. Results show that the volatility switches regimes frequently, and differences between the distributions of the high and low volatility states become more accentuated as the frequency increases. Moreover, the SGT is highly preferable to the usually employed skewed t distribution.     

Lognormal Mixed Models for Reported Claims Reserves

Abstract

Traditional claims-reserving techniques are based on so-called run-off triangles containing aggregate claim figures. Such a triangle provides a summary of an underlying data set with individual claim figures. This contribution explores the interpretation of the available individual data in the framework of longitudinal data analysis. Making use of the theory of linear mixed models, a flexible model for loss reserving is built. Whereas traditional claims-reserving techniques don’t lead directly to predictions for individual claims, the mixed model enables such predictions on a sound statistical basis with, for example, confidence regions. Both a likelihood-based as well as a Bayesian approach are considered. In the frequentist approach, expressions for the mean squared error of prediction of an individual claim reserve, origin year reserves, and the total reserve are derived. Using MCMC techniques, the Bayesian approach allows simulation from the complete predictive distribution of the reserves and the calculation of various risk measures. The paper ends with an illustration of the suggested techniques on a data set from practice, consisting of Belgian automotive third-party liability claims. The results for the mixed-model analysis are compared with those obtained from traditional claims-reserving techniques for run-off triangles. For the data under consideration, the lognormal mixed model fits the observed individual data well. It leads to individual predictions comparable to those obtained by applying chain-ladder development factors to individual data. Concerning the predictive power on the aggregate level, the mixed model leads to reasonable predictions and performs comparable to and often better than the stochastic chain ladder for aggregate data.     

Credibility Using Copulas

Abstract

Credibility is a form of insurance pricing that is widely used, particularly in North America. The theory of credibility has been called a “cornerstone” in the field of actuarial science. Students of the North American actuarial bodies also study loss distributions, the process of statistical inference of relating a set of data to a theoretical (loss) distribution. In this work, we develop a direct link between credibility and loss distributions through the notion of a copula, a tool for understanding relationships among multivariate outcomes.

This paper develops credibility using a longitudinal data framework. In a longitudinal data framework, one might encounter data from a cross section of risk classes (towns) with a history of insurance claims available for each risk class. For the marginal claims distributions, we use generalized linear models, an extension of linear regression that also encompasses Weibull and Gamma regressions. Copulas are used to model the dependencies over time; specifically, this paper is the first to propose using a t-copula in the context of generalized linear models. The t-copula is the copula associated with the multivariate t-distribution; like the univariate tdistributions, it seems especially suitable for empirical work. Moreover, we show that the t-copula gives rise to easily computable predictive distributions that we use to generate credibility predictors. Like Bayesian methods, our copula credibility prediction methods allow us to provide an entire distribution of predicted claims, not just a point prediction.

We present an illustrative example of Massachusetts automobile claims, and compare our new credibility estimates with those currently existing in the literature.     

Effects of Risk Management on Cost Efficiency and Cost Function of the U.S. Property and Liability Insurers

Abstract

This paper adopts the one-step stochastic frontier approach to investigate the impact of risk management tools of derivatives and reinsurance on cost efficiency of U.S. property-liability insurance companies. The stochastic frontier approach considers both the mean and variance of cost efficiency. The sample includes both stock and mutual insurers. Among the findings, the cost function of the entire sample carries the concavity feature, and insurers tend to use financial derivatives for firm value creation. The results also show that for the entire sample the use of derivatives enhances the mean of cost efficiency but accompanied with larger efficiency volatility. Nevertheless, the utilization of financial derivatives mitigates efficiency volatility for mutual insurers. This research provides important insights for the practice of risk management in the property-liability insurance industry.     

Spatial modelling of claim frequency and claim size in non-life insurance

In this paper, models for claim frequency and average claim size in non-life insurance are considered. Both covariates and spatial random effects are included allowing the modelling of a spatial dependency pattern. We assume a Poisson model for the number of claims, while claim size is modelled using a Gamma distribution. However, in contrast to the usual compound Poisson model, we allow for dependencies between claim size and claim frequency. A fully Bayesian approach is followed, parameters are estimated using Markov Chain Monte Carlo (MCMC). The issue of model comparison is thoroughly addressed. Besides the deviance information criterion and the predictive model choice criterion, we suggest the use of proper scoring rules based on the posterior predictive distribution for comparing models. We give an application to a comprehensive data set from a German car insurance company. The inclusion of spatial effects significantly improves the models for both claim frequency and claim size, and also leads to more accurate predictions of the total claim sizes. Further, we detect significant dependencies between the number of claims and claim size. Both spatial and number of claims effects are interpreted and quantified from an actuarial point of view.     

Bounds for Ratios of Posterior Expectations: Applications in the Collective Risk Model

This paper considers the collective risk model for the insurance claims process. We will adopt a Bayesian point of view, where uncertainty concerning the specification of the prior distribution is a common question. The robust Bayesian approach uses a class of prior distributions which model uncertainty about the prior, instead of a single distribution. Relatively little research has dealt with robustness with respect to ratios of posterior expectations as occurs with the Esscher and Variance premium principles. Appropriate techniques are developed in this paper to solve this problem using the k -contamination class in the collective risk model.     

The decision to outsource risk management services

We apply transaction cost economics to identify factors influencing companies’ decision to internally generate or outsource risk management services. A unique sample is used which combines publicly available data with private information supplied by 281 Australian listed companies. We find that expenditure on research and development, greater technological uncertainty, more competitive environments, more overseas sales and transaction frequency are associated with less outsourcing of risk management services. Uncertainty due to environmental diversity is associated with more outsourcing of risk management services. Companies that outsource risk management services also have lower staff turnover and provide more specialised training and longer contracts for risk management suppliers.     

Strategies for Dividend Distribution: A Review

Abstract

In today’s world of financial uncertainty, one major public concern is to assess (and possibly improve) the stability of companies that take on risks. Actuaries have been aware of that issue for a very long time and have a great experience in modeling the activity of a risk business. During the first part of the twentieth century, they focused on the probability of ruin to assess the stability of their company. In his seminal paper of 1957 Bruno de Finetti criticized this approach and laid the foundations of what would become an increasingly popular topic: the study of dividend strategies. The contributions made by actuaries in that field constitute a substantial body of knowledge, whose interest is relevant not only to insurance but also to a much broader range of areas of practice. In this paper we aim at a taxonomical synthesis of the 50 years of actuarial research that followed de Finetti’s original paper.     

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 Effect of Market Regimes on Style Allocation

We analyse time-varying risk premia and the implications for portfolio choice. Using Markov Chain Monte Carlo (MCMC) methods, we estimate a multivariate regime-switching model for the Carhart (1997) four-factor model. We find two clearly separable regimes with different mean returns, volatilities, and correlations. In the High-Variance Regime, only value stocks deliver a good performance, whereas in the Low-Variance Regime, the market portfolio and momentum stocks promise high returns. Regime-switching induces investors to change their portfolio style over time depending on the investment horizon, the risk aversion, and the prevailing regime. Value investing seems to be a rational strategy in the High-Variance Regime, momentum investing in the Low-Variance Regime. An empirical out-of-sample backtest indicates that this switching strategy can be profitable, but the overall forecasting ability for the regime-switching model seems to be weak compared to the iid model.     

Markov Chain Monte Carlo Methods in Financial Econometrics

Markov Chain Monte Carlo (MCMC) methods have become very popular in financial econometrics during the last years. MCMC methods are applicable where classical methods fail. In this paper, we give an introduction to MCMC and present recent empirical evidence. Finally, we apply MCMC methods to portfolio choice to account for parameter uncertainty and to incorporate different degrees of belief in an asset pricing model.     

Fuzzy Financial Pricing of Property-Liability Insurance

Abstract

This paper uses fuzzy set theory (FST) to solve a problem in actuarial science, the financial pricing of property-liability insurance contracts. The fundamental concept of FST is the alternative formalization of membership in a set to include the degree or strength of membership. FST provides consistent mathematical rules for incorporating vague, subjective, or judgmental information into complex decision processes. It is potentially important in insurance pricing because much of the information about cash flows, future economic conditions, risk premiums, and other factors affecting the pricing decision is subjective and thus difficult to quantify by using conventional methods. To illustrate the use of FST, we “fuzzify” a well-known insurance financial pricing model, provide numerical examples of fuzzy pricing, and propose rules for project decision-making using FST. The results indicate that FST can lead to significantly different decisions than the conventional approach.