Yehuda Kahane

Risk Considerations in Insurance Ratemaking

This paper examines insurance pricing and its regulation in the context of efficient capital markets. Starting with an aggregated model and generalizing results reported recently in the literature about “proper” underwriting profit, the paper turns to disaggregation of the model with m insurance lines. The main result is that no unique set of rates exists that regulators may impose to avoid disturbing market equilibrium. Preliminary empirical evidence presented shows that the “systematic risk” of underwriting profits approaches zero in most lines. Thus an intuitive solution for underwriting profit rates in these lines equal to minus the riskless interest rate, is reasonable.

A Portfolio Approach to the Property-Liability Insurance Industry

This paper contains an analysis of a portfolio model which simultaneously optimizes the investment and insurance portfolios of the property-liability insurance industry. The mathematical formulation is an extension of earlier approaches in that it permits the direct development of the envelope efficiency frontier for all levels of insurance leverage. Using data on nineteen insurance lines and two types of assets for the period 1956-1971, efficient portfolios for both unconstrained and also constrained solutions are obtained. In each case, some insurance lines tend to be consistently excluded from the optimal portfolios because of their risk-return characteristics. The implications of this effect on the availability of insurance and rate-making are discussed. Finally, in contrast to accepted practice and theory it is found that the investment policy of the firm need not necessarily become more conservative as the insurance portfolio becomes more risky. In analyzing the structure and performance of the property-liability insurance industry many approaches and measures have been used, each with its own merits. For example, Hensley [6] undertook a study of the competitive aspects of the insurance industry as well as other aspects of its structure and more recently these issues have been studied by Joskow

The Theory of Insurance Risk Premiums—A Re-Examination in the Light of Recent Developments in Capital Market Theory

The premium calculation principle is one of the main objectives of study for actuaries. There seems to be full agreement among the leading theoreticians in the field that the insurance premium should reflect both the expected claims and certain loadings. This is true for policy, risk or portfolio. There are three types of positive loadings: a) a loading to cover commissions, administrative costs and claim-settlement expenses; b) a loading to cover some profit (a cost-plus approach); and c) a loading for the risk taken by the insurer when underwriting the policy. The administrative costs can be considered a part of “expected gross claims†. Thus, the insurers ratemaking decision depends on his ability to estimate expected claims (including costs) and on the selection of a fair risk loading.The main concern in the literature is the appropriate measurement of the risk and the exact loading formula. BA¼hlmann [1970, ch. 5] and others identified four possible principles of risk loading, namely, the expected value principle, the standard deviation loading, the variance loading, and the loading according to the principle of constant utility. Various studies point to the advantages and disadvantages of these principles and also examine some additional loading forms—semi-variance, skewness, etc. (e.g., BA¼hlmann [1970], Benktander [1971], Berger [1972], Burness [1972], Berliner [1974], Berliner and Benktander [1976], Bohman [1976], Cooper [1974], Gerber [1975] and others). Despite different preferences in choosing the appropriate loading calculation principle, all seem to agree that the risk loading must be positive, since, otherwise, the firm would just have to wait for its ruin, that is bound to come sooner or later, according to risk theory.The purpose of this article is to re-examine the appropriate principle of premium calculation in light of the recent developments in the theory of finance and especially in the theory of capital market equilibrium. These developments may suggest a new point of view and raise a few questions regarding the loading rules.

Insurance premiums and default risk in mutual insurance

Abstract Two types of default risk are discussed in the article: The traditional “probability of ruin” (insurer being unable to meet his obligations) and a “perceived probability of ruin” (the probability of the insured being affected by ruin). The explicit relationship between these probabilities on the actuarial loading factors of a mutual insurer were developed. The explicit mathematical formulae obtained for these complex relationships were followed also by numerical results. A second concept presented in the paper is related to the idea of actuarially fair premiums. It is shown that the premium must also be a function of the payments of the other insured as well as their claim distributions, reflecting thereby the simultaneity and mutual dependence of the insured.

Concepts and Trends in the Study of Insurer’s Solvency

Insolvency, a fundamental term in the insurance literature, is not clearly defined. Basically a firm is “solvent” within a single period model if its terminal value is higher than the total obligations to its creditors. For the more realistic case of a going concern, another definition is needed: an insurer will be regarded “ruined” (bankrupt) if its liabilities exceed the value of its assets, i.e., equity is zero (negative equity cannot occur in a stock company due to the limited liability!). In this situation the value of all future income streams is lower than the value of all debts (Stiglitz [1972]).

Voluntary Insurance Coverage, Compulsory Insurance, and Risky-Riskless Portfolio Opportunities

A decision involving a portfolio of an uninsurable risk, two other risks (one of which can be insured voluntarily, while the second is covered by compulsory insurance), and a risk-free investment is examined. The focus is on the effects of compulsory insurance coverage on the demand for voluntary insurance, and extends the results of Schulenburg (1986) by elaborating upon the impact of the risk-free opportunity on the tradeoff between voluntary and compulsory coverages.

Regulation in the Insurance Industry: Determination of Premiums in Automobile Insurance

The controlled rates in the insurance industry must, on the one hand, ensure profitability and solvency of the insurance companies, and on the other hand, avoid exorbitant charges. Measuring the ex-post profit of insurance companies for rates determination is impracticable, since approximations are used in the evaluation of the outstanding claims. In addition to that, the allocation of income from investment to the business account of each field that the company deals with, is arbitrary. The authors suggest an econometric model for rates determination. They estimate the average claim as well as the probability for a claim. This method can be used in order to determine the controlled rates. Moreover, the suggested method proved to be useful in determining the rates for specific groups of policyowners as well as for selecting the desired policyowners. Companies should carry out from time to time such an empirical research in order to improve their sales policy. Automobile insurance rates must be approved by the government. Such controlled rates should, on the one hand, ensure the profitability and solvency of insurance companies, and on the other, avoid exorbitant charges. The determined rates should, of course, be revised from time to time. Theoretically, this can be done by examining the profit-and-loss accounts of the insurers: losses mean that rates should be increased, while excessive profits mean that they should be reduced. This method, however, is rendered impracticable by the accounting techniques employed in the profitand-loss accounts of insurance companies, which restrict their usefulness, Yehuda Kahane, Ph.D., served as Visiting Professor in the University of Florida 1973-74, while on leave from his position of Lecturer in The Hebrew University of Jerusalem. Haim Levy, Ph.D., is Director of the School of Business Administration and Associate Professor in The Hebrew University of Jerusalem. He is co-author of Investment and Portfolio Analysis. This paper was submitted in December, 1973. This paper summarizes the theoretical problems encountered and some of the empirical results found by the authors in a comprehensive study on rate determination for automobile insurance in Israel. This study has been financed by the Israel Union of Insurance Companies and by the Insurance and Savings Authority, The Ministry of Finance, Government of Israel. The authors wish to thank Mr. Ben Horim for his help in this study and two anonymous referees for their helpful comments.

Optimal investment-dividend policy of an insurance firm under regulation

Abstract An insurance decision model including intervention by a regulating agency is defined. The insurance firms problem is to establish an investment policy as well as a dividend strategy. Regulation is exercised by a minimal barrier policy for cash holding and penalities for violating this barrier. The joint Insurance Firm-Regulating Agency problem is discussed by using concepts drawn from Stackleberg strategies in game theory. As in the classical model of collective risk theory it is assumed that premium payments are received deterministically from policyholders at a constant rate, while the claim process is determined by a Compound Poisson process. Finally a diffusion approximation is used in order to obtain tractable results for a general claim size distribution.

Moral Hazard and Insurance Market Structure

The objective of this article is to suggest an explanation for the observed market structure of the insurance industry, which is based on three parties—insureds, insurers, and reinsurers. This structure should not be viewed simply as a special version of the common marketing system in which the reinsurer is analogous to a wholesaler and the insurers are analogous to the retailers. Careful examination reveals a much more complicated framework involving agency and information problems as well as trading of financial capacity. Despite its practical importance, little has been written about this issue and most of the articles that do appear in the literature are not really in tune with reality, since the authors restrict themselves to dealing with only two parties at a time (insured and insurer or insurer and reinsurer). Moreover, they deal with market structure and retention problems as two distinct issues.

Reinsurance contracts: A utility approach vs. insurance capacity considerations

Abstract Present explanations for the existence of reinsurance are based on utility analysis and assume some sort of risk aversion in the insurer-reinsurer interface. This analogy to the insured-insurer interface stands in contrast to capital market equilibrium theory, since, unlike the isured, the insurer cannot gain from further diversification of a risk insured at the market price of risk. The existence of reinsurers must, therefore, be explained by market imperfections. The present study explains reinsurance as an outcome of the solvency regulation and capacity considerations of insurers and reinsurers. This approach enables the handling of various types of reinsurance contracts (and therefore offers also a wider perspective than earlier studies, which focused only on proportional, quota share, agreements). Interesting relationships between reinsurance pricing, solvency and capacity are observed within our framework.