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.     

Applications of Fuzzy Regression in Actuarial Analysis

In this article, we propose several applications of fuzzy regression techniques for actuarial problems. Our main analysis is motivated, on the one hand, by the fact that several articles in the financial and actuarial literature suggest using fuzzy numbers to model interest rate uncertainty but do not explain how to quantify these rates with fuzzy numbers. Likewise, actuarial literature has recently focused some of its attention in analyzing the Term Structure of Interest Rates (TSIR) because this is a key instrument for pricing insurance contracts. With these two ideas in mind, we show that fuzzy regression is suitable for adjusting the TSIR and discuss how to apply a fuzzy TSIR when pricing life insurance contracts and property‐liability policies. Finally, we reflect on other actuarial applications of fuzzy regression and develop with this technique the London Chain Ladder Method for obtaining Incurred But Not Reported Reserves.     

“Analysis of Incremental Returns of Canadian Mutual Funds”, J. F. Carrière; C. F. Hill,April 2001

Abstract

The determination and allocation of economic capital is important for pricing, risk management, and related insurer financial decision making. This paper considers the allocation of economic capital to lines of business in insurance. We show how to derive closed-form results for the complete markets, arbitrage-free allocation of the insurer default option value, or insolvency exchange option, to lines of business for an insurer balance sheet. We assume that individual lines of business and the surplus ratio are joint log-normal although the method we adopt allows other assumptions. The allocation of the default option value is required for fair pricing in the multiline insurer. We discuss and illustrate other methods of capital allocation, including Myers-Read, and give numerical examples for the capital allocation of the default option value based on explicit payoffs by line.     

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.     

Real Longevity Insurance with a Deductible: Introduction to Advanced-Life Delayed Annuities (ALDA)

Abstract

This paper explores the financial properties of a concept product called an advanced-life delayed annuity (ALDA). The ALDA is a variant of a pure deferred annuity contract that is acquired by installments, adjusted for consumer price inflation, and pays off toward the end of the human life cycle. The ALDA concept is aimed at the growing population of North Americans without access to a traditional defined benefit (DB) pension plan and the implicit longevity insurance that a DB plan contains. I show that under quite reasonable pricing assumptions, a consumer can invest or allocate $1 per month, while saving for retirement, and receive between $20 and $40 per month in benefits, assuming the deductible in this insurance policy is set high enough. The ALDA concept might go a long way in mitigating the psychological barrier to voluntary lump-sum annuitization.     

Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility

We introduce an expected utility approach to price insurance risks in a dynamic financial market setting. The valuation method is based on comparing the maximal expected utility functions with and without incorporating the insurance product, as in the classical principle of equivalent utility. The pricing mechanism relies heavily on risk preferences and yields two reservation prices - one each for the underwriter and buyer of the contract. The framework is rather general and applies to a number of applications that we extensively analyze.     

The Application of Corporate Governance Indicators With XBRL Technology to Financial Crisis Prediction

ABSTRACT

The widespread adoption of eXtensible Business Reporting Language (XBRL) suggests that intelligent software agents can now use financial information disseminated on the Web with high accuracy. Financial data have been widely used by researchers to predict financial crises; however, few studies have considered corporate governance indicators in building prediction models. This article presents a financial crisis prediction model that involves using a genetic algorithm for determining the optimal feature set and support vector machines (SVMs) to be used with XBRL. The experimental results show that the proposed model outperforms models based on only one type of information, either financial or corporate governance. Compared with conventional statistical methods, the proposed SVM model forecasts financial crises more accurately.     

A general class of distortion operators for pricing contingent claims with applications to CAT bonds

ABSTRACT

The current paper provides a general approach to construct distortion operators that can price financial and insurance risks. Our approach generalizes the (Wang 2000) transform and recovers multiple distortions proposed in the literature as particular cases. This approach enables designing distortions that are consistent with various pricing principles used in finance and insurance such as no-arbitrage models, equilibrium models and actuarial premium calculation principles. Such distortions allow for the incorporation of risk-aversion, distribution features (e.g. skewness and kurtosis) and other considerations that are relevant to price contingent claims. The pricing performance of multiple distortions obtained through our approach is assessed on CAT bonds data. The current paper is the first to provide evidence that jump-diffusion models are appropriate for CAT bonds pricing, and that natural disaster aversion impacts empirical prices. A simpler distortion based on a distribution mixture is finally proposed for CAT bonds pricing to facilitate the implementation.     

Pricing credit derivatives under incomplete information: a nonlinear-filtering approach

This paper considers a general reduced-form pricing model for credit derivatives where default intensities are driven by some factor process X. The process X is not directly observable for investors in secondary markets; rather, their information set consists of the default history and of noisy price observations for traded credit products. In this context the pricing of credit derivatives leads to a challenging nonlinear-filtering problem. We provide recursive updating rules for the filter, derive a finite-dimensional filter for the case where X follows a finite-state Markov chain, and propose a novel particle-filtering algorithm. A numerical case study illustrates the properties of the proposed algorithms.     

Underwriting profit margin of P/L insurance in the fuzzy-ICAPM

The ICAPM is used to study the underwriting profit margin of the P/L insurance company, including the insurances of automobile damage, automobile liability and fire, in which the parameters are the symmetric or non-symmetric triangular fuzzy numbers. From the ten-year data of a company in Taiwan we determine the lower and upper limits associated with the various α-level of the fuzzy numbers. Our results show that the best-fitting parameters of the model from our data are the asymmetric triangular fuzzy numbers. The skew factors in each insurance are determined, which could be used to perform the forecasting of the underwriting profit margin. Our results show that the systematic risk in the fuzzy environment (with best-fitting value of skew factor) becomes larger than that in the crisp environment. However, the insurance underwriting leverage and insurance financial leverage in the fuzzy environment are smaller than those in the crisp environment. JEL Classification G22 · G32 The author is grateful to the two anonymous reviewers for The Geneva Risk and Insurance Review for their comments on an earlier version of this paper.     

A decision theoretic formulation of insurance risk theory

Abstract

In a number of papers Borch has shown how certain insurance problems can be formulated using the concept of utility. (See Borch [3], [4], [5], [6], [7] and [8].) Borch's work is used as a building block in Part I of this report, which presents a Bayesian decision theoretic formulation of some of the main aspects of insurance risk theory. Part I makes use of the concepts of utility and subjective probability. It is admitted that these concepts are more commonly associated with individuals rather than groups of individuals such as insurance companies. However, in this report, we will refer to an insurance company as an individual (albeit a neuter one) and assume that it can quantify its preferences for consequences and its opinions about the occurrence of events. Further, we assume that a company “behaves” according to certain rules of consistent behavior which imply that when presented with several risky courses of action, the company will take the action which has the greatest expected utility. Formal treatments of assumptions that lead to this mode of behavior can be found in Savage [17] and Pratt, Raiffa, and Schlaifer [15].     

Author’s Reply: Skewness and Stock Option Prices - Discussion by Terence Chan,Michel Jacques,Heinz H. Mu¨ller,Geérard Pafumi,Elias S.W. Shiu,and Serena Tiong,July 1997

Abstract

In recent years, the combined effects of deregulation in financial services, along with advances in telecommunications and information technology, are forcing far-reaching changes upon the insurance industry. The result is the industry is becoming more competitive. The emerging role of electronic commerce (e-commerce) is particularly important and interesting to study.

I offer a brief survey of the role of e-commerce in the insurance industry. The paper is organized in the following manner: Section 1 summarizes Internet trends and discusses various related public policy issues; Section 2 addresses online insurance supply and demand; Section 3 discusses the economics of disintermediation and reintermediation and explains how this applies to e-commerce in the insurance industry. Finally, Section 4 offers a set of concluding remarks.     

A quarterly time-series classifier based on a reduced-dimension generated rules method for identifying financial distress

Predicting financial distress has been and will remain an important and challenging issue. Many methods have been proposed to predict bankruptcies and detect financial crises, including conventional approaches and techniques involving artificial intelligence (AI). Financial distress information influences investor decisions, and investors depend on analysts’ opinions and subjective judgements in assessing such information, which sometimes results in investors making mistakes. In the light of the foregoing, this paper proposes a novel quarterly time series classifier, which reduces the sheer volume of high-dimensional data to be analysed and provides decision-makers with rules that can be used as a reference in assessing the financial situation of a company. This study employs the following six attribute selection methods to reduce the high-dimensional data: (1) the chi-square test, (2) information gain, (3) discriminant analysis, (4) logistic regression (LR) analysis, (5) support vector machine (SVM) and (6) the proposed Join method. After selecting attributes, this study utilises the rough set classifier to generate the rules of financial distress. To verify the proposed method, an empirically collected financial distress data-set is employed as the experimental sample and is compared with the decision tree, multilayer perceptron and SVM under Type I error, Type II error and accuracy criteria. Because financial distress data are quarterly time series data, this study conducts non-time series and time series (moving windows) experiments. The experimental results indicate that the LR and chi-square attribute selection combined with the rough set classifier outperform the listing methods under Type I, Type II error and accuracy criteria.     

Valuation of Equity-Linked Insurance and Annuity Products with Binomial Models

Abstract

In this paper we develop a valuation method for equity-linked insurance products. We assume that the premium information of term life insurances, pure endowment insurances, and endowment insurances at all maturities is obtainable within a company or from the insurance market. Using a method similar to that of Jarrow and Turnbull (1995), we derive three martingale probability measures associated with these basic insurance products. These measures are agedependent, include an adjustment for the mortality risk, and reproduce the premiums of the respective insurance products. We then extend the martingale measures to include the financial market information using copulas and use them to evaluate equity-linked insurance contracts and equity-indexed annuities in particular. This is different from the traditional approach under which diversification of mortality risk is assumed. A detailed numerical analysis is performed for various existing equity-indexed annuities in the North American market.     

Two Paradigms for The Market Value of Liabilities

Abstract

Asset/liability management (ALM) theory and practices of insurers have matured and developed from early applications to guaranteed investment contracts (GICs) to all annuity and insurance products today. An important and logical next step of inquiry is the definition of, and calculation procedures for, the market value of an insurance liability. Because all ALM strategies have as their goal the management of some value of assets in relation to some value of liabilities, this inquiry will provide at last a canonical basis for ALM: the management of relative market values.

To set the stage for this exploration, the theory and application of pricing in a complete market are reviewed, as are the practical limitations of this theory in the real, and far from complete, financial markets. The notion of an ad hoc pricing model is developed, and examples are reviewed and critiqued. These models, though imperfect compared with pricing in a complete market, bridge the gap between pricing theory and practice.

The current state of the liabilities market is also discussed, and this market is seen to naturally split into a “long” and a “short” submarket. Of particular interest is the theoretical possibility of these markets becoming broad-based, deep and active, and the conclusions are relevant to the issue of long/short price equalization.

Two paradigms are then explored for defining and subsequently calculating an insurance liability market value. A “paradigm” is a generalized model or framework for accomplishing the task at hand. Each paradigm reflects observable market trading activity, however infrequent, and each is based on methods of valuation consistent with finance-theoretic approaches that are routinely used for the market valuation of assets.

In addition, each paradigm allows for a sequence of ad hoc valuation methodologies, which differ in the extent to which various risks are explicitly modeled versus judgmentally reflected in a risk spread. These paradigms are discussed and contrasted, and arguments made for the potential evolution of the respective values if a “liability” market began trading actively. Practical constraints on the realization of this evolution are also noted.

The last section of this paper discusses a host of considerations related to the application of option-pricing theory to insurance company liabilities.     

How Does Information Disclosure Affect Bank Systemic Risk in the Presence of a Deposit Insurance System?

ABSTRACT

This article establishes a dynamic game with incomplete information to theoretically analyze the influence mechanism of information disclosure on systemic risk in the presence of a deposit insurance system. To verify the mechanism, we use panel data on 247 global banks in 41 countries during the period 2006 to 2015 in an empirical analysis. Our article finds that a high degree of information disclosure can reduce deposit insurance premiums and weaken the negative incentive from a bailout by regulatory authorities. Moreover, the effect of deposit insurance on financial stability is not apparent, but the synergistic effect of deposit insurance and information disclosure reduces bank systemic risk. Furthermore, different deposit insurance designs affect bank behavior, so it is crucial for bank supervisors to create proper deposit insurance systems, which are helpful in strengthening market discipline and preventing moral hazard thus contributing to a stable financial environment. Therefore, under the deposit insurance system, regulatory authorities should strive to improve the standard of information disclosure to ensure systemic stability.     

A Fuzzy Set Approach for Generalized CRR Model: An Empirical Analysis of S&;P 500 Index Options

This paper applies fuzzy set theory to the Cox, Ross and Rubinstein (CRR) model to set up the fuzzy binomial option pricing model (OPM). The model can provide reasonable ranges of option prices, which many investors can use it for arbitrage or hedge. Because of the CRR model can provide only theoretical reference values for a generalized CRR model in this article we use fuzzy volatility and fuzzy riskless interest rate to replace the corresponding crisp values. In the fuzzy binomial OPM, investors can correct their portfolio strategy according to the right and left value of triangular fuzzy number and they can interpret the optimal difference, according to their individual risk preferences. Finally, in this study an empirical analysis of S&P 500 index options is used to find that the fuzzy binomial OPM is much closer to the reality than the generalized CRR model.This project has been supported by NSC 93-2416-H-009-024.JEL Classification:     

Zur Entscheidungsnützlichkeit eines IFRS-Abschlusses von Versicherern – Analyse der angedachten Regelungen zum endgültigen IFRS 4

In its Discussion Paper from May 2007 for the final IFRS 4 (“Insurance Contracts”), the IASB planned the “Current Exit Value” (CEV) to evaluate insurance liabilities. The aim of this article is to examine the impacts of the planned Current-Exit-Value-Approach on the decision usefulness of insurers' financial statements. In order to evaluate accounting rules, the criterion decision usefulness is divided into two sub-criteria – relevance and reliability. We found out that the Current-Exit-Value-Approach for insurance contracts in conjunction with fair value accounting of relating assets actually generates relevant information for users of insurers' financial statements. But we have serious concerns about a cost-adequate implementation of CEV approach. Furthermore, the reliability of accounting information based on a synthetic fair value is strongly questionable. Only extensive disclosure requirements can bring transparency to users and therefore assure reliable accounting information. Besides, there are other topics that should be reviewed before releasing a final standard (e.?g. customer relationship, credit characteristics of insurance liabilities, unit of account).     

Perspectives of Risk Sharing

In this paper we present an overview of the standard risk sharing model of insurance. We discuss and characterize a competitive equilibrium, Pareto optimality, and representative agent pricing, including its implications for insurance premiums. We only touch upon the existence problem of a competitive equilibrium, primarily by presenting several examples. Risk tolerance and aggregation is the subject of one section. Risk adjustment of the probability measure is one topic, as well as the insurance version of the capital asset pricing model. The competitive paradigm may be a little demanding in practice, so we alternatively present a game theoretic view of risk sharing, where solutions end up in the core. Properly interpreted, this may give rise to a range of prices of each risk, often visualized in practice by an ask price and a bid price. The nice aspect of this is that these price ranges can be explained by "first principles", not relying on transaction costs or other frictions. We also include a short discussion of moral hazard in risk sharing between an insurer and a prospective insurance buyer. We end the paper by indicating the implications of our results for a pure stock market. In particular we find it advantageous to discuss the concepts of incomplete markets in this general setting, where it is possible to use results for closed, convex subspaces of an L 2 -space to discuss optimal risk allocation problems in incomplete financial markets.