An optimal stopping problem in risk theory

Abstract

In classical risk theory often stationary premium and claim processes are considered. In some cases it is more convenient to model non-stationary processes which describe a movement from environmental conditions, for which the premiums were calculated, to less favorable circumstances. This is done by a Markov-modulated Poisson claim process. Moreover the insurance company is allowed to stop the process at some random time, if the situation seems unfavorable, in order to calculate new premiums. This leads to an optimal stopping problem which is solved explicitly to some extent.     

Risk-adjusted credibility premiums using distorted probabilities

Abstract

Denneberg (1990) and Wang (1996a) propose that one calculate risk-adjusted insurance premiums as the expectation with respect to a distorted probability measure, a non-additive set function. This premium principle is supported by the theories of decision making of Yaari (1987) and of Schmeidler (1989). Denneberg (1994a) presents three conditioning rules for updating non-additive set functions in light of available information. In this work, we show how to apply these three update rules to calculate a risk-adjusted credibility premium and, thereby, combine credibility theory with this relatively new premium principle. Our main result is that, for some pairs of distortion function and update rule, one gets the same risk-adjusted credibility premium by distorting the predictive probability distribution, as required by the theory of Yaari, or by updating the distorted probability, as required by the theory of Schmeidler.     

Upper bounds on stop-loss premiums under constraints on claim size distribution

Abstract

The problem of maximal stop-loss premium under prescribed constraints on claim size distribution is taken up again. The methods of linear programming are used to show that the recent results of others are intuitively obvious. These results are then extended by the linear programming technique to cases of more general constraints, e.g. prescribed claim size variance, or prescribed minimum frequency of excess claims. In particular it is shown that, typically, the upper bound on stop-loss premiums is generated by a claim size distribution which has all its mass concentrated at very few points. In contrast with the results obtained by others recently, it is seen that the claim size distribution which produces the maximal stop-loss premium is not generally independent of the excess. Some numerical examples are given showing that the methods used here can sometimes improve considerably the recent results of others. The case of a compound Poisson distribution is treated briefly.     

Credibility for Severity Revisited

Abstract

It is basic actuarial knowledge that the pure premium of an insurance contract can be written as the product of the expected claim number and the expected claim amount. Actuaries use credibility theory to incorporate the contract’s individual experience into this calculation in a statistically optimal way. For many years, however, the use of credibility was limited to the frequency component. Starting with the paper by Hewitt (1971), there have been various suggestions as to how credibility theory also can be applied to the severity component of the pure premium. The latest such suggestion, Frees (2003), revived the interest in the problem.

In this paper, we review four different formulas incorporating frequency and severity into credibility calculations. We then compare by simulation which one is most accurate at predicting a contract’s next-year outcome. It is found that the classical formula of Bühlmann (1967) is as good as the other ones in many cases. Alternatives, however, may offer easier analysis of the separate effects of frequency and severity on the premium.

We also show that all the formulas reviewed in this paper stem from the same minimization problem, and we present a general, integrated, solution. At the same time, we complete Gerber (1972) by providing a proof to the main result of this paper and by stating required additional assumptions.     

Bayesian Premium Rating with Latent Structure

We propose a fully Bayesian approach to non-life risk premium rating, based on hierarchical models with latent variables for both claim frequency and claim size. Inference is based on the joint posterior distribution and is performed by Markov Chain Monte Carlo. Rather than plug-in point estimates of all unknown parameters, we take into account all sources of uncertainty simultaneously when the model is used to predict claims and estimate risk premiums. Several models are fitted to both a simulated dataset and a small portfolio regarding theft from cars. We show that interaction among latent variables can improve predictions significantly. We also investigate when interaction is not necessary. We compare our results with those obtained under a standard generalized linear model and show through numerical simulation that geographically located and spatially interacting latent variables can successfully compensate for missing covariates. However, when applied to the real portfolio data, the proposed models are not better than standard models due to the lack of spatial structure in the data.     

Sequential Quasi-Credibility for Scale Dispersion Models

The sequential approach to credibility, developed by Landsman and Makov [(1999a) On stochastic approximation and credibility. Scand. Actuarial J. 1, 15-31; (1999b) Sequential credibility evaluation for symmetric location claim distributions. Insurance: Math. Econ. 24, 291-300] is extended to the scale dispersion family, which contains distributions often used in actuarial science: log-normal, Weibull, Half normal, Stable, Pareto, to mention only a few. For members of this family a sequential quasi-credibility formula is devised, which can also be used for heavy tailed claims. The results are illustrated by a study of log-normal claims.     

Regression Tree Credibility Model

This article applies machine learning techniques to credibility theory and proposes a regression-tree-based algorithm to integrate covariate information into credibility premium prediction. The recursive binary algorithm partitions a collective of individual risks into mutually exclusive subcollectives and applies the classical Bühlmann-Straub credibility formula for the prediction of individual net premiums. The algorithm provides a flexible way to integrate covariate information into individual net premiums prediction. It is appealing for capturing nonlinear and/or interaction covariate effects. It automatically selects influential covariate variables for premium prediction and requires no additional ex ante variable selection procedure. The superiority in prediction accuracy of the proposed algorithm is demonstrated by extensive simulation studies. The proposed method is applied to the U.S. Medicare data for illustration purposes.     

E. J. Moorhead (1910 – 2004)

Abstract

As is well known in actuarial practice, excess claims (outliers) have a disturbing effect on the ratemaking process. To obtain better estimators of premiums, which are based on credibility theory, Künsch and Gisler and Reinhard suggested using robust methods. The estimators proposed by these authors are indeed resistant to outliers and serve as an excellent example of how useful robust models can be for insurance pricing. In this article we further refine these procedures by reducing the degree of heuristic arguments they involve. Specifically we develop a class of robust estimators for the credibility premium when claims are approximately gamma-distributed and thoroughly study their robustness-efficiency trade-offs in large and small samples. Under specific datagenerating scenarios, this approach yields quantitative indices of estimators’ strength and weakness, and it allows the actuary (who is typically equipped with information beyond the statistical model) to choose a procedure from a full menu of possibilities. Practical performance of our methods is illustrated under several simulated scenarios and by employing expert judgment.     

AN ECONOMETRIC ANALYSIS OF EQUITY COSTS AND RISK PREMIUMS IN THE ELECTRIC UTILITY INDUSTRY: 1971–1985

This paper develops a varying parameter econometric model that estimates the cost of equity of individual utility firms from 1971 to 1985. The equity costs estimated in this framework can be analyzed in terms of their statistical precision. The paper also examines, theoretically and empirically, the relationship between the econometric estimates of the equity risk premiums and the risk-free interest rates. The data do not support the hypothesis that risk premiums are independent of interest rates. Also, the relationship appears to vary over time. These results invalidate the risk premium approach in which equity costs are estimated by adding a constant, historical average risk premium to the prevailing interest rates.     

An empirical study of the value-relevance of using proportionate consolidation accounting for investments in joint ventures

This research examines bond risk premiums to determine whether creditors of companies with investments in joint ventures reflect legal or implicit measures of the debts of joint ventures. The legal view suggests that the amount of potential loss from an investment in a joint venture is limited to the investment. The implicit view suggests that the operations of the joint venture and the venturer are interdependent. Equity method accounting reflects the legal view and proportionate consolidation reflects the implicit view.The study examines whether bond risk premiums are more highly associated with accounting numbers from proportionate consolidation than equity method accounting. The study uses data from 10Ks, the Wall Street Journal, and Moody's Bond Record from May 1, 1995 through April 30, 1998. These 4 years are used because US interest rates were fairly stable during this period, which is an important factor when examining bond risk premiums. Additionally, the companies in the study needed to remain stable across the window of study – no mergers, acquisitions, buy-outs, or liquidations – in order to maintain a comparative sample over the entire time period. The risk premium model uses measures of default that change between equity method accounting and proportionate consolidation. Differences in the explanatory power of the model determine how creditors view the joint venture debts.The study shows that approximately half of equity investments represent investments in joint ventures. Furthermore, the average joint venture uses debt to finance about two-thirds of the assets. The results show that proportionate consolidation fails to improve the explanatory power of the model when examining the entire set of companies that invest in joint ventures. However, the data reject the null hypothesis of no improvement with proportionate consolidation when examining companies who guarantee the debt of their joint venture. The policy implication of this study indicates that a change to proportionate consolidation would provide more value-relevant information to creditors when companies guarantee the debt of the joint venture.     

From T-Bills to Stocks: Seasonal Anomalies Revisited

This paper employed eleven data series which consist of stocks, bonds, bills, equity premiums, term premiums, and various default premiums to investigate whether January seasonality reported in existing literature is robust across different states of the economy as this has important trading implications. For the periods 1926–1990, small stocks, small stock premiums, low grade bonds, and default premiums (spread between high grade, low grade and government bonds) reveal January seasonality and that the seasonality is robust across different states of the economy except for low grade bond returns and default premiums. January seasonality for low grade bond returns and low grade bond default premiums are primarily driven by results found during periods of economic expansion. Overall, January seasonality is more evident during the economic expansion periods although the magnitude of default premiums is larger during periods of economic contraction. Furthermore, prior findings of strong summer equity returns are primarily driven by the results found during the periods of economic contraction. It is also found that equity returns are generally higher during periods of economic expansion.     

R&D expenditures and implied equity risk premiums

This study investigates the relationship between research and development (R&D) expenditures and risk premiums implied in the costs of equity capital. We posit that R&D expenditures represent an information risk factor resulting from both information asymmetry about R&D between investors and managers and low-quality R&D reporting that impairs the coordination between investors and managers with respect to managers’ investment decisions. Our results support our position by showing a positive association between R&D expenditures and implied equity risk premiums. From this research along with prior studies, investors can have better knowledge about the risky nature of R&D expenditures that drive up implied risk premiums and at the same time provide opportunities to earn excess returns in a short to long horizon. Accounting standard setters can benefit from this study’s findings that R&D expenditures represent an off-balance-sheet risk factor and thus warrant reconsidering SFAS No. 2 for potential capitalization of R&D expenditures.     

Case Studies Using Panel Data Models

Abstract

In this paper, we examine case studies from three different areas of insurance practice: health care, workers’ compensation, and group term life. These different case studies illustrate how the broad class of panel data models can be applied to different functional areas and to data that have different features. Panel data, also known as longitudinal data, models are regression-type models that have been developed extensively in the biological and economic sciences. The data features that we discuss include heteroscedasticity, random and fixed effect covariates, outliers, serial correlation, and limited dependent variable bias. We demonstrate the process of identifying these features using graphical and numerical diagnostic tools from standard statistical software.

Our motivation for examining these cases comes from credibility rate making, a technique for pricing certain types of health care, property and casualty, workers’ compensation, and group life coverages. It has been a part of actuarial practice since Mowbray’s (1914) fundamental contribution. In earlier work, we showed how many types of credibility models could be expressed as special cases of panel data models. This paper exploits this link by using tools developed in connection with panel data models for credibility rate-making purposes. In particular, special routines written for credibility rate-making purposes are not required.     

V. E. Gamborg

Abstract

A glance at the numerous papers dealing with the influence of the rate of interest on the value of premiums will show that most authors aim at computing annuity values for a new rate of interest without first re-calculating the commutation columns, It is only in exceptional cases that they derive premiums or policy-values directly, i.e. without first finding annuity values.² Apart from the fact, that both premiums and policyvalues are implicitly given by a set of annuity values, the reason for the usual procedure lies in the type of calculations which is contemplated, since they cannot conveniently be applied to such ratios as premiums and policy-values. In the following lines we show how the method developed by A. J. Lotka ³ for the calculation of the rate of increase of a stable population is capable of generalisation and of successful application to our problem; thus the detour via the calculation of annuity values can be avoided.     

International asset returns and exchange rates

We present a consumption-based international asset-pricing model to study global equity premiums, the US riskfree rate and the cross section of international asset returns. The model entails idiosyncratic, country-specific consumption risk, which helps explain the magnitude of global equity premiums. It also features country-specific habit formation, which helps explain the level of the interest rate on the US short-term Treasury bills traded by domestic and international investors. We find that the model explains approximately 40–50% of the cross section of currency and equity premiums as well as expected returns from value and growth portfolios of at least a dozen countries. Changes in real exchange rates are responsible for explaining approximately half of the cross section of international asset returns.     

Club deals in leveraged buyouts

We analyze the pricing and characteristics of club deal leveraged buyouts (LBOs)—those in which two or more private equity partnerships jointly conduct an LBO. Using a comprehensive sample of completed LBOs of U.S. publicly traded targets conducted by prominent private equity firms, we find that target shareholders receive approximately 10% less of pre-bid firm equity value, or roughly 40% lower premiums, in club deals compared to sole-sponsored LBOs. This result is concentrated before 2006 and in target firms with low institutional ownership. These results are robust to controls for target and deal characteristics, including size, Q, measures of risk, and time and industry fixed effects. We find little support for benign motivations for club deals based on capital constraints, diversification motives, or the ability of clubs to obtain favorable debt amounts or prices, but it is possible that the lower pricing of club deals is an inadvertent byproduct of an unobserved benign motivation for club formation.     

Social connections,reference point and acquisition premium

This paper examines the impact of acquirer-target social connections along with the target 52-week high (Baker et al., 2012) on acquisition premiums. We show that acquisition premium is more sensitive to first-degree connection than the reference point, suggesting that information is the main driving force for determining acquisition premiums. The findings also indicate that connected directors are more likely to favour firms where they hold higher positions and negotiate favourable premiums. Acquirers pay lower premiums when target directors are retained in the new entity. Connected acquirers are also more likely to finance their deals with equity. Overall, this paper provides support to the information flow hypothesis that acquirers with social connections have better access to target information and enhanced bargaining power in negotiations.     

Comoment risk and stock returns

We estimate investable comoment equity risk premiums for the US markets. The stock's contribution to the asymmetry and the fat tails of the market portfolio's payoff are priced into a coskewness premium and a cokurtosis premium. We construct zero-investment strategies that are long and short in coskewness and cokurtosis equity risks; we infer from the spread the returns attached to a unit exposure to US equity coskewness and cokurtosis. The coskewness and cokurtosis premiums present positive monthly average returns of 0.27% and 0.14% from January 1959 to December 2011. Comoment risks appear to be significantly priced within the US stock market and display significant explanatory power regarding the US size and book-to-market effects. The premiums do not subsume, but rather complement the empirical capital asset pricing model. Our analysis relies on data collected from CRSP (Chicago Research Center for Security Prices) over December 1955 to December 2011. To our knowledge, the paper is the first to propose investable higher-moment risk factors over such an extensive time period.     

Bayesian Modelling of Outstanding Liabilities Incorporating Claim Count Uncertainty

Abstract

This paper deals with the prediction of the amount of outstanding automobile claims that an insurance company will pay in the near future. We consider various competing models using Bayesian theory and Markov chain Monte Carlo methods. Claim counts are used to add a further hierarchical stage in the model with log-normally distributed claim amounts and its corresponding state space version. This way, we incorporate information from both the outstanding claim amounts and counts data resulting in new model formulations. Implementation details and illustrations with real insurance data are provided.     

On time-heterogeneous credibility estimation

Abstract

Let χ_i be the total claim amount of an insurance policy in calendar year i. We assume that the χ_i's are conditionally independent given an unknown random parameter ø, and that for all i. In the present paper it is under these assumptions shown how to calculate the credibility estimator of m(ø) by recursive updating. We also give estimators for the unknown parameters α_i, β_i, and ?_i based on portfolio data. Finally we mention some related models.