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The Ruin Problem in Multiple Line Insurance A Simplified Model**

Published online by Cambridge University Press:  19 October 2009

Alfred E. Hofflander Jr  and
Richard M. Duvall
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Extract

Most of the insurance oriented literature contains several levels of abstraction regarding the financial solvency of insurance companies. The first level represents those advances in the actuarial profession which are destined to eventually solve the double problem of capacity and capitalization (hereinafter referred to as the Ruin Problem). Most of these works, however, are concerned with developing the tools necessary to determine the theoretical loss distribution in insurance. The emphasis, then, is upon the mean and variance of the total loss distribution as they can be derived from the distribution of frequency and severity.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 2 , Issue 2 , June 1967 , pp. 150 - 165
Copyright
Copyright © School of Business Administration, University of Washington 1967

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References

1 Capacity and capitalization problems in insurance are really analogous to the debt/equity question in corporate finance. Thus, the authors are concerned with how much the total liability of an insurer can be with a given amount of equity so that the probability of ruin is smaller than some number ɛ. Thus, capacity ≃ liabilities and capitalization ≃ equity.

2 For a description of this technique see Dickerson, O. D., Katti, S. K., and Hofflander, A. E., “Loss Distribution in Non-Life Insurance,” Journal of Risk and Insurance, Vol. 28, No. 3 (09, 1961), pp. 45–54CrossRefGoogle Scholar.

3 See Kenney, R., Fundamentals of Fire and Casualty Strength. (Dedham, Mass: 1957)Google Scholar. As Professor Pfeffer pointed out to the authors, Roger Kenney was primarily concerned with the issue of conserving policyholder's surplus and did not attempt to define the precise limits of tolerance for his various ratios. Also, it should be recognized that he was basing his ratios on pre-World War II experience. Probably a more meaningful set of ratios would be those established by the English companies and used by the surplus lines associations; specifically the cover ratio which is a ratio of total assets to premiums written. This ratio recognizes the possibility that loss reserves may not be accurate but that, whether they be excessive or inadequate, the impact of these errors will be on policyholder's surplus. Taking total assets therefore removes the problem of pinpointing the adequacy of reserves. What the cover ratio should be is a matter of individual judgment by company managements. The English feel that a cover ratio above 1.25 is increasingly desirable and a cover ratio below 1.25 is increasingly undesirable. There are no firm rules on this.

4 Houston, D. B., “Risk, Insurance, and Sampling,” Journal of Risk and Insurance, Vol. 31, No. 4 (12, 1964), p. 532CrossRefGoogle Scholar.

5 Ibid. Apparently Houston ignores the use of PHS as a cover for loss reserves.

6 Houston, op. cit., pp. 531–532.

7 The Variation and Co-variation between lines is examined in Lambert, E. W. and Hofflander, A. E., “Effects of New Multiple-Line Underwriting on Investment Portfolios of Property-Liability Insurers,” Journal of Risk and Insurance, Vol. 33, No. 2 (06 1966)CrossRefGoogle Scholar.

8 The problem here revolves around the definition of a line of business. For example, many would argue that Auto Property Damage Insurance written in New York is a different line from Auto Property Damage Insurance written in Wyoming. Obviously, this could be carried to extremes, since every sub-class of business which has either the average loss or standard deviation of losses significantly different from the average loss or standard deviation of losses of another sub-class is a separate line. Lack of usable data and time rather than theoretical considerations will limit the number of classes used by a company for its analysis.

9 The loss reserve is a liability which represents the present value of losses which have occurred but have not yet been settled.

10 Expenses and losses are included in y. For an examination of the variability of loss ratios see Lambert and Hofflander, op. cit. For an examination of the variability of Expense Ratios see Hofflander, A. E., “Expense Ratios, the Business Cycle, and the Stock Market,” The Annals of the Society of Property and Casualty Underwriters, Vol. 19, No. 2 (06, 1966) pages 51–64Google Scholar.

11 Similar to Houston, op. cit., p. 530.

12 Earned premiums represent premiums which have been earned due to the passage of time. Written premiums represent sales. If a company is not growing or declining very fast, written and earned premiums are approximately equal. For the discussion in this paper, the terms will be used interchangeably.

13 Lambert and Hofflander, op. cit.

14 Data on the relationship between loss reserves established by companies and the actual amount necessary to settle a claim are not generally available on an aggregate basis. The author is presently engaged in making an estimate of errors in Loss Reserves. The effect of Loss Reserve Margins on Calendar Year Results is investigated in an unpublished paper of the same title by R. J. Balcarek which was presented to the Casualty Actuarial Society Club in Philadelphia – Spring, 1965.

15 Best's Aggregates and Averages, 1964, p. 54–55, Grand Totals for 808 stock companies. For mutuals Loss Reserves/PHS = 1.0.

16 Source of data: Ferrari, J. Robert, “Profitability, Risk and Diversification of Property and Liability Insurance: A Markowitz Portfolio selection approach,” (unpublished manuscript)Google Scholar. These data were derived from an individual company's experience rather than from the industry experience.

17 Bicklehaupt, David L., Transition to Multiple-Line Insurance Companiesh, (Homewood, Ill. Richard D. Irwin, 1961), p. 197Google Scholar. Alternative data presented shows a drop from 1.73 to 1.22, ibid., p. 196.

18 For an interesting discussion of this problem see Otto, Ingolf H. E., “Capacity,” Journal of Insurance, Vol. XXVIII (March, 1961), pp. 53–70CrossRefGoogle Scholar.