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INTERNACIONAL SECTION

 

A "square-root rule" for reinsurance

 

Uma "regra de raiz quadrada" para resseguro

 

 

Michael R. Powers^I; Martin Shubik^II

^IProfessor and Director of Advanta Center for Financial Services Studies Temple University - USA. E-mail: michael.powers@temple.edu
^IISeymour H. Knox. Professor of Mathematical Institutional Economics Yale University - USA. E-mail: martin.shubik@yale.edu

 

 

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ABSTRACT

In previous work, the authors derived a mathematical expression for the optimal (or "saturation") number of reinsurers for a given number of primary insurers (see Powers and Shubik, 2001). In the current article, we show analytically that, for large numbers of primary insurers, this mathematical expression provides a "square-root rule"; i.e., the optimal number of reinsurers in a market is given asymptotically by the square root of the total number of primary insurers. We note further that an analogous "fourth-root rule" applies to markets for retrocession (the reinsurance of reinsurance).

Keywords: Primary insurance, reinsurance, retrocession, market size, square-root rule.

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RESUMO

Em um estudo anterior, os autores derivaram uma expressão matemática para a quantidade ótima (ou "de saturação") de resseguradoras para um dado número de seguradoras primárias (vide Powers e Shubik, 2001). Neste trabalho, mostramos analiticamente que, para grandes quantidades de seguradoras primárias, esta expressão matemática oferece uma "regra de raiz quadrada"; i.e., o número ótimo de resseguradoras em um mercado é dado assintoticamente pela raiz quadrada do número total de seguradoras primárias. Além disso, observamos que uma "regra de raiz quarta" análoga aplica-se aos mercados para retrocessão (o resseguro do resseguro).

Palavras-chave: Seguro primário; resseguro; retrocessão; tamanho do mercado; regra de raiz quadrada.

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1 INTRODUCTION

In the past two decades, the global reinsurance market has experienced a period of rapid change. Mergers and acquisitions have led to dramatic consolidation, reflected in a 61 percent decline in the number of domestic U.S. reinsurers (from 628 to 244) over the eleven years from 1990 through 2000 (see Venezian, Viswanathan, and Jucá, 2005). Since the early 1990s, experimentation with insurance-based securities, including various property-catastrophe indexes (see Powers and Powers, 1997) and catastrophe bonds, has provided novel alternatives to traditional reinsurance products.

These changes give rise to a number of fundamental questions:

• Is a reinsurance market necessary?

• If so, what is the optimal number of reinsurers?

• Is there a role for retrocession (i.e., the insuring of reinsurers)?

• If so, is there a theoretical or practical upper bound on the number of retrocession levels?

In previous work, the authors derived a mathematical expression for the optimal (or "saturation") number of reinsurers for a given number of primary insurers (see Powers and Shubik, 2001). In the current article, we show analytically that, for large numbers of primary insurers, this mathematical expression provides a "square-root rule"; i.e., the optimal number of reinsurers in a market is given asymptotically by the square root of the total number of primary insurers.

1.1 A Primary Insurance Market

We first review the formal model of a primary insurance market presented in Powers, Shubik, and Yao (1998) and Powers and Shubik (1998). This model employs a Cournot price-formation mechanism with arbitrary numbers of buyers and sellers, so that marginal changes in insurer solvency and competitive forces can be studied directly as the numbers of players change.

Consider a primary insurance market game with players consisting of m homogeneous customers, i = 1,2, ... , m, and n₀ homogeneous insurance firms, i = 1,2, ... , n₀. At time 0, let each customer (buyer) i have initial endowment B_i(0) = V + A consisting of one unit of property with replacement value V and A(>V) dollars in cash. Furthermore, let each insurer (seller) j have initial endowment S_i(0) = R/n₀ dollars of net worth, where R is the total amount of capital supplied by investors to the insurance market.

It is assumed that, during the policy period [0, t], each customer's property is subject to a random loss with probability π, and that all losses are total. The random variable δ_i equals 1 if customer i suffers a property loss during [0, t], and equals 0 otherwise, where the δ_i ~ i.i.d. Bernoulli(π).

To insure against a potential property loss in [0, t], each customer i has the option of purchasing insurance from some insurer by making a strategic bid, χ_i [0, V], that represents the amount that he or she is willing to pay for insurance. Simultaneously, each insurer j has the option of offering to sell insurance by making a strategic offer, γ_i [0, cR/n₀], that represents the total dollar amount of risk that j is willing to assume, where c > 1 is a solvency constraint imposed by government regulators.

It is assumed that all bids and offers are submitted to a central clearinghouse that:

• calculates an average market price of insurance per exposure unit, P₀(x, y) = :¹

• collects all premium bids, x., and distributes them to the n0 insurers in proportion to the insurers' respective coverage offers, y_i (i.e., insurer j receives the premium amount y_ip₀(x, y)): and

• randomly assigns each customer i to an insurer j(i) so that each insurer ends up with the same number of customers, µ₀ (i.e., it is assumed that n₀ divides m exactly and that µ₀ = m/n₀).

Letting M_j denote the set of customers associated with insurer j, it is assumed that if customer i M_j suffers a loss in [0,t], then he or she will receive a loss payment in the amount γ_i -i.e., an amount proportional not only to i's premium bid, x_i, but also to j's coverage offer, y_i. This loss payment is bounded above by V to reduce problems of moral hazard.

To recognize the possibility of insurer insolvency during [0, t], let n_j. be a Bernoulli random variable that equals 1 if insurer j becomes insolvent, and equals 0 otherwise. If there is an insolvency, it is assumed that government guaranty funds will pay a fixed proportion g [0, 1] of all insurance claims made against the insolvent insurer.

1.2 A Market for Insurance and Reinsurance

In Powers and Shubik (2001), we extended the above model of a primary insurance market by introducing one or more levels of reinsurers. Although the second and higher levels of reinsurance are commonly referred to as "retrocession", the model denotes each level of reinsurance by its distance from the primary insurance market; thus, level "1 " denotes the reinsurance of primary insurers, level "2 " the reinsurance of level "1 " reinsurers, etc.

In essence, the insurance/reinsurance model comprises an (r + 1)-stage strategic game in which there is first an interaction between the customers and the primary insurers, then an interaction between the primary insurers and the level 1 reinsurers, etc., through r levels of reinsurance. The solution to be considered is a perfect pure strategy non-cooperative equilibrium (PSNE) - "perfect" in the sense that the equilibrium in the overall game is also an equilibrium in every sub-game.

For the insurance/reinsurance market game with one primary insurance market and R E {1 , 2, 3, ...} levels of reinsurance, the following three assumptions provide the basic analytic framework.

Assumption 1: There are

(i) m homogeneous customers (buyers) in the primary market, each with utility function u_B(w) =

(ii) n₀ homogeneous insurers (sellers) in the primary market, each with utility function u_S(w) = , and

(iii) n_λ homogeneous reinsurers at level λ [1, r], each with utility function u_s(λ)(w) = ,

where m > n₀ > n₁ > ... > n_i > 1 and β > σ₀ > σ₁ > ... > σ₁ > 0.

Assumption 2: The primary insurers make offers y_j0 and bids x , the reinsurers at level λ [1, r-1] make offers y and bids x, and the reinsurers at level r make offers y where: (1) all primary insurers and reinsurers make their offers independently of their bids, (2) price determinations, premium distributions, and random customer assignments are made at each level by a central clearinghouse, and (3) each reinsurer at level λ [1, r] ends up with the same number of customers, µ_λ (i.e., it is assumed that n_λ divides n_λ-1 exactly and that µ = n_λ-1/n_λ).

Assumption 3: Letting M denote the set of primary insurers associated with level 1 reinsurer K, it follows that if insurer j M suffers losses associated with customers h H_j M_j in [0, t], then it will receive a loss payment in the amount y (i.e., an amount proportional not only to j's premium bid x, but also to k's coverage offer, y, and that an analogous loss payment rule is applied at each higher level of reinsurance λ [2, r].

 

2 MARKET EQUILIBRIUM

For the above insurance/reinsurance market game, we were able to show the existence of a unique type-symmetric pure strategy equilibrium in which price and quantity (for the primary market and r levels of reinsurance) are given implicitly by a system of nonlinear equations. (See Theorem 1 of Powers and Shubik, 2001.) From this result, it can be shown that both price and quantity decrease over the successive insurance/reinsurance levels.

2.1 Risk-Neutral Reinsurers

For the special case in which all reinsurers at levels λ [L, r] are risk neutral (i.e., σ_λ 0 for λ [L, r]), we were able to: (1) show that equilibria do not exist for reinsurance levels λ [L + 1, r], and (2) find explicit analytical forms for price and quantity in both the primary market and for all reinsurance levels λ [1, L]. Specifically, the equilibrium price at reinsurance level λ [1,L] is given by

where 1, and

denotes the equilibrium price in the primary insurance market. (See Corollary 1 of Powers and Shubik, 2001.)

2.2 Optimal Number of Reinsurers

For the case of r = 1, we used the above results to identify conditions under which the reinsurance market is "saturated" -i.e., under which it is no longer desirable, on the margin, to introduce an additional risk-neutral reinsurer rather than an additional risk-averse primary insurer. (See Section 5.2 of Powers and Shubik, 2001.) This was done by comparing the price of insurance in the primary insurance market under two alternatives.

The first alternative, denoted by A₁, is a primary insurance market with one level of reinsurance, where the primary market has n₀ insurers, and the reinsurance market has n₁ reinsurers. The second alternative, A₂, is the same primary insurance market, except that the number of primary insurers is increased by one (to n₀ + 1), while the number of reinsurers is decreased by one (to n₁ -1). To identify the point at which the number of reinsurers has reached its optimal saturation level, we solved for the maximum value of n7 such that P < P ; i.e.,

where the existence of a unique solution n₁* {2, 3, .... n₀ -1} is guaranteed by the fact that the inequality in (1) is satisfied for n₁ = 2, but not for n₁ = n₀.

Having computed n* for values of n₀ in the interval [10,500], we presented a graph of these results, reproduced in Figure 1 .

 

 

From a tabular display of the same results (see Table 1 ) it is easy to see that the solution to (1), n*, is approximately equal to the square root of n₀ (although this observation was not made in Powers and Shubik, 2001).

 

 

3 A SQUARE-ROOT RULE

We now show analytically that, for large numbers of primary insurers, the solution to problem (1) is indeed a "square-root rule"; i.e., the optimal number of reinsurers in a market is given asymptotically by the square root of the total number of primary insurers, as stated formally in the following result.

Theorem 1: For sufficiently large n₀, there exists a unique solution n_t* = n_t*(n₀) {2, 3, nQ - 1} to problem (1), where

as n₀ ∞.

Proof: First, we extend problem (1) from the two-dimensional integer grid {(n₀, n₁): 2 < n₁ < n₀} to the corresponding two-dimensional real space {(a, x) : 2 < x < a} by considering the inequality

Apart from its points of unboundedness, (2) is equivalent to the cubic polynomial inequality

Thus, the unique solution specified by the theorem - if it exists - is given by n₁* = [x*], where x* = x* (a) (2, a) is a positive real root of f(x) such that f'(x) > 0.

For large a, one can rewrite (2) as

from which it follows that we seek the roots x(α) of

Solving (4) for a as a function of x yields

Anticipating that there exists at least one positive root x(a) = o(a^2/3) to (4), we find exactly one solution to (5); namely,

which implies

For sufficiently large a, it is clear that (a) (2, a).

To confirm that (a) is the desired root of f(x), we consider the local extrema of this polynomial, given by

f'(x) = 3(a² -a-1)x² + 2(a³ - a² + 3a + 2)x -(a³ + a² + 3a + 1) = 0

or equivalently,

As a ∞, the larger solution to (6) is given by

For sufficiently large a, it follows that (a) > x_u(a), which implies f'() > 0. Therefore, x*(a) = (a).

 

4 DISCUSSION AND CONCLUSIONS

It is interesting to note that the asymptotic square-root rule for the optimal number of reinsurers - and indeed the exact solution n₁ * given by (1) - depends only on the number of primary insurers. In a recent article, Venezian et al. (2005) tested the proposed square-root rule empirically, using the numbers of primary insurers and reinsurers from a group of eighteen to twenty different national insurance markets over a period of eleven years. Instructively, these authors found that their data are consistent with the square-root rule, and that year-to-year variations in their regression model are reasonably well explained by a measure of market risk aversion proposed by Madsen and Pedersen (2003). This suggests that, in addition to the square-root relationship, the number of reinsurers in a market during a specific time period may be associated with the risk aversion of investors during that period.

Beyond the world of ordinary reinsurance lie the misty realms of retrocession (second-order reinsurance), second-order retrocession (third-order reinsurance), and so on. Although our model in Powers and Shubik (2001) permitted an arbitrary number of reinsurance levels, we acknowledged that such clearly defined levels are not reflective of the real world. While a few specialized purveyors of retrocession do exist, higher-order reinsurance is typically provided by ordinary reinsurers through the packaging and repackaging of risk through various types of pooling arrangements.

It may be argued that the absence of distinct higher-order reinsurance markets is consistent with the analytical results of our game-theoretic model. Assuming that there exist at least two second-order reinsurers in the market, and again using the minimization of price in the primary insurance market as the optimality criterion, the expression for the optimal number of second-order reinsurers is given by

By methods similar to those employed in the proof of Theorem 1, the following result may be obtained.

Theorem 2: For sufficiently large n₀, if n_t ~ , then there exists a unique solution n₂* = n₂*(n₀) {2, 3, ... n₁ -1} to problem (7), where

as n₀ ∞

In short, the number of retrocessionaires in a market should be approximately equal to the fourth-root of the number of primary insurers, which for most national insurance markets (other than that of the U.S.) is rather small. Thus, according to the model, one should not expect to see distinct significant retrocession markets, except perhaps in the U.S. This result agrees with empirical observation.

More generally, for all financial instruments involving risk and transaction costs, the principles dictating how many levels of paper are economically optimal need to be considered. We suspect that four or five is an extreme upper bound, and our result here conforms to this.

 

References

DUBEY, P.; SHUBIK, M.. A Theory of Money and Financial Institutions. 28. The Non-Cooperative Equilibria of a Closed Trading Economy with Market Supply and Bidding Strategies. Journal of Economic Theory, vol.17, n.1, p. 1-20, February 1978.         [ Links ]

MADSEN, C. K.; PEDERSEN. H. W.. An Examination of Insurance Pricing and Underwriting Cycles. In: 13^TH INTERNATIONAL AFIR COLLOQUIUM, September 2003, Maastricht, The Netherlands.         [ Links ]

POWERS. I. Y.; POWERS, M. R.. Seeking the Perfect Catastrophe Index, Best's Review, Property/Casualty Edition, p. 101-103, December 1997.         [ Links ]

______, M. R.; SHUBIK, M.. On the Tradeoff between the Law of Large Numbers and Oligopoly in Insurance. Insurance: Mathematics and Economics, vol.23, n.2, p. 141-156, November 1998.         [ Links ]

______, Toward a Theory of Reinsurance and Retrocession. Insurance: Mathematics and Economics, vol.29, n.2, p.271 -290, October 2001.         [ Links ]

POWERS, M. R.; SHUBIK, M.; YAO, S. T.. Insurance Market Games: Scale Effects and Public Policy. Zeitschrift für Nationalökonomie (Journal of Economics), vol.67, n.2, p. 109-134, June 1998.         [ Links ]

VENEZIAN, E. C; VISWANATHAN, K. S.; JUCÁ, I. B.. A 'Square-Root Rule' for Reinsurance? Evidence from Several National Markets. Journal of Risk Finance, vol.6, n.4, p.319-334, August 2005.         [ Links ]

 

 

Received 26 August 2005
Accepted 22 September 2006

 

 

Note - address of the authors: Temple University, 481 Ritter Annex (044-00), Philadelphia - PA - USA, 19122
Yale University, PO BOX 208200, New Haven - CT - USA, 06520-8200
1 To avoid potential problems of division by zero, it is assumed (as in Dubey and Shubik, 1978) that the clearing house furnishes at least one insurer, and one customer per insurer, that must make non-zero bids/offers.