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Financial Pricing of Insurance in the Multiple Line Insurance Company

  • Richard D. Phillips
  • J. David Cummins
  • Franklin Allen

A limitation of the existing financial pricing models is the implicit or explicit assumption that insurers produce only one type of insurance, even though most insurers produce multiple types of coverage with differing risk characteristics and liability growth rates. The purpose of this paper is to remedy this deficiency in the existing literature by providing a theoretical and empirical analysis of insurance pricing in a multiple line firm. The authors adopt an option pricing approach to model the insurer’s default risk. The standard Black-Scholes model is generalized to incorporate more than one class of liabilities, and pricing formulae are generated for each liability class. The theoretical predictions of the model are tested using data on an extensive sample of publicly traded U. S. property-liability insurers. The authors note there have been only a few prior papers on insurance pricing in a multiple line firm, mostly in the actuarial literature. Nearly all have approached the problem by assuming that the insurer’s equity capital is allocated among the lines of business, usually in proportion to each line’s share of the insurer’s liability. Prices for a given line of insurance then incorporate an aggregate profit charge equal to the assumed cost of capital for the line multiplied by its assigned equity. This approach lacks theoretical foundation and the allocation rules tend to be ad hoc. In addition, little progress has been made in estimating relative costs of capital by line. A more appropriate model of multiple line insurance pricing has been developed by Allen (1993), who shows that it is incorrect to allocate capital by line when computing insurance prices because the capital is present to back all of the company’s policies and thus, is inherently indivisible. Allen’s model offers important insights into the multi-line pricing problem. However, it does not incorporate default risk. The theoretical development in the present paper combines the option pricing approach with insights drawn from the Allen model to derive a pricing model for a multiple line firm subject to default risk. The authors’ model implies that it is not appropriate to allocate capital by line. Rather, the price of insurance by line is determined by the overall risk of the firm and the line-specific liability growth rates. The authors predict that insurance groups in which liabilities are widely dispersed among subsidiaries will command lower prices than unaffiliated single insurers or insurance groups where business is heavily concentrated among the principal affiliates. The empirical results support the hypotheses: prices vary across firms depending upon overall-firm default risk and the concentration of business among subsidiaries; but within a given firm, prices do not vary by line after adjusting for line-specific liability growth rates. Using an option pricing framework, the authors show that the informationally-efficient, competitive market price of insurance for a given line of business depends on the overall risk of the firm rather than the risk of the individual line being priced. This result is due to the fact that it is not the equity of the insurer but rather the expected cost of insolvency that should be allocated to the various divisions of the firm. The model yields two empirical predictions: 1) the price of insurance should be inversely related to firm default risk; and, 2) the price of insurance across lines of business for a given insurer should be equal after controlling the default risk and line-specific liability growth rates. The empirical results support the predictions of the model. The empirical evidence is broadly consistent with the view that insurance markets are informationally efficient and competitive and that price regulation has an adverse effect on the relative prices among lines of insurance. This provides further support for the argument that regulation is likely to have adverse effects on resource allocation and the quality and availability of insurance. The results also suggest that there is likely to be a market reward for the development and adoption of improved risk management techniques that enable insurers to efficiently reduce their default risk.

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Paper provided by Wharton School Center for Financial Institutions, University of Pennsylvania in its series Center for Financial Institutions Working Papers with number 96-09.

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Date of creation: Mar 1996
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Handle: RePEc:wop:pennin:96-09
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  1. Lo, Andrew W. & Craig MacKinlay, A., 1990. "An econometric analysis of nonsynchronous trading," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 181-211.
  2. Merton, Robert C., 1977. "An analytic derivation of the cost of deposit insurance and loan guarantees An application of modern option pricing theory," Journal of Banking & Finance, Elsevier, vol. 1(1), pages 3-11, June.
  3. Harrington, Scott E, 1987. "A Note on the Impact of Auto Insurance Rate Regulation," The Review of Economics and Statistics, MIT Press, vol. 69(1), pages 166-70, February.
  4. Anne Gron, 1994. "Capacity Constraints and Cycles in Property-Casualty Insurance Markets," RAND Journal of Economics, The RAND Corporation, vol. 25(1), pages 110-127, Spring.
  5. Merton, Robert C, 1974. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 29(2), pages 449-70, May.
  6. Black, Fischer & Cox, John C, 1976. "Valuing Corporate Securities: Some Effects of Bond Indenture Provisions," Journal of Finance, American Finance Association, vol. 31(2), pages 351-67, May.
  7. Fuller, Russell J & Kerr, Halbert S, 1981. "Estimating the Divisional Cost of Capital: An Analysis of the Pure-Play Technique," Journal of Finance, American Finance Association, vol. 36(5), pages 997-1009, December.
  8. Kraus, Alan & Ross, Stephen A, 1982. " The Determination of Fair Profits for the Property-Liability Insurance Firm," Journal of Finance, American Finance Association, vol. 37(4), pages 1015-28, September.
  9. Winter Ralph A., 1994. "The Dynamics of Competitive Insurance Markets," Journal of Financial Intermediation, Elsevier, vol. 3(4), pages 379-415, September.
  10. Cummins, J David, 1988. " Risk-Based Premiums for Insurance Guaranty Funds," Journal of Finance, American Finance Association, vol. 43(4), pages 823-39, September.
  11. Franklin Allen, . "Estimating Divisional Cost of Capital for Insurance Companies (Reprint 037)," Rodney L. White Center for Financial Research Working Papers 30-91, Wharton School Rodney L. White Center for Financial Research.
  12. Pennacchi, George G, 1987. "A Reexamination of the Over- (or Under-) Pricing of Deposit Insurance," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 19(3), pages 340-60, August.
  13. Ronn, Ehud I. & Verma, Avinash K., 1989. "Risk-based capital adequacy standards for a sample of 43 major banks," Journal of Banking & Finance, Elsevier, vol. 13(1), pages 21-29, March.
  14. Fischer, Stanley, 1978. "Call Option Pricing when the Exercise Price Is Uncertain, and the Valuation of Index Bonds," Journal of Finance, American Finance Association, vol. 33(1), pages 169-76, March.
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