...points an interesting future scenario. We find an equilibrium in a market of mutual marine insurers, in which some smaller clubs, having operating costs above average, may grow larger relative to the other clubs in order to become more cost effective, and where medium to larger cost-efficient clubs may stay unchanged or some even downsize relative to the others. Some of the very large clubs suffering from diseconomies of scale may have a motive to further increase relative to the other clubs. According to observations, most clubs have, during the last decade, expanded significantly in size measured by gross tonnage of entered ships, some clubs have merged, but very few seem to have decreased their underwriting activity, in particular none of the really large ones. The analysis points to the following future scenario: The small and the medium to large clubs converge in size, while there is a possibility for some very large clubs to be present as well.

INTRODUCTION

The article takes as given two scale effects observed in the marine mutual insurance industry, see e.g., Li and Shan (2004), and investigates if this is consistent with a partial equilibrium model. While the insurance products offered by the Protection and Indemnity (P&I) Clubs are rather similar, the ability to lower costs seems like an important factor for the competitiveness of these clubs. This can be achieved by utilizing scale economies.

Economies of scale due to uncertainty apparently exist in the marine mutual industry, as has been documented by several authors, e.g., by Katrishen and Scordis (1998). Skogh (1982) and Borch (1990) also suggest that the costs of an insurance firm increase at a lower rate than its output. The explanation for this, shared by many different insurance markets, is that of diversification. The more ships in a club, the less is the risk per ship, as follows essentially from the Law of Large Numbers. Even if the liabilities are not independently and identically distributed, diversification has the effect of lowering the overall risk in the portfolio in relation to its size, also observed in other industries, like banking (e.g., Baltensperger, 1972). The effect this has on the operating cost per ship is typically that of lowering this type of cost as a function of the number of ships in a club. These costs involve opportunity costs from holding reserves to deal with future claims payments. Since the uncertainty of cash flows decreases relatively with the number of policies, the costs caused by holding cash reserves and capital accounts also decrease relatively with size.

Next are the effects on the operating costs that are not directly related to uncertainty of the insurance portfolio. These can either decrease or increase relative to an increase in size. For mutual marine clubs the operating costs have been documented to increase at a lower rate than their output. However, multinational insurers can only achieve this effect up to a certain size, and those firms with size above a particular level typically display diseconomies of scale in their operating costs. This may be due to the fact that an increase in size is accompanied by an increase in the complexity of firms' operations and the cost of coordinating those operations. It may come to the point where this increase is also offsetting the scale economies due to diversification, in which case the overall operating costs may display diseconomies of scale, as found by Li and Shan (2004). These authors looked at data from thirteen major P&I Clubs for the years 2002 and 2003 and found a U-shaped operating cost function, with a turning point at about 80 million gross tons.

We should also mention that there seems to be mixed evidence on the existence of economies of scale for insurers in general, as reported in Katrishen and Scordis (1998, p. 307), and this issue does not seem to have been settled yet, in particular, for multinational insurers.

The objective of this article is to investigate if such findings are consistent with an economic equilibrium. And if so, can we learn from the nature of the competitive equilibrium in what direction this market is likely to move in the future?

The article is organized as follows: In section A P&I-Club Considered as a Syndicate we formulate the theory of syndicates for a single P&I Club consisting of a certain number of ship owners. In section Equilibrium in a Market of P&I Clubs we consider a market of such syndicates and find an optimal risk-sharing arrangement in this market when each P&I Club has a operating cost depending on its size. Section Feasibility of Equilibria discusses existence of various types of equilibriums in such markets. In section Computation of Risk Premia for Various Probability Distributions, we develop market insurance premiums for different loss distributions. In section Implications of the Economic Analysis, we provide three different interpretations of an equilibrium in the market of P&I Clubs, in increasing order of realism. Here we present estimates of the risk tolerances of the various clubs in the International Group. Section Discussion and Implications for Future Research concludes.

A P&I-CLUB CONSIDERED AS A SYNDICATE

If a group of businessmen is not satisfied with the offers received from insurance companies, the members of the group could set up a mutual insurance scheme of their own. All that is needed is really an informal agreement on how losses caused by specific random events and hitting some member shall be shared by all. A P&I Club, which offers "Protection and Indemnity" to ship owners, is such an arrangement. The old standard marine policy of Lloyds covered only three-fourths of the liability which a ship could incur in a collision, and left a number of other liabilities completely uncovered. To cover these risks ship owners formed the P&I Clubs, which in reality are mutual insurance companies.

It should be possible to include the risks covered by the P&I policy in an ordinary marine insurance contract, but it seems that the clubs have certain advantages. The members are ship owners, and their number is, even on a world wide basis, fairly limited. This means that some simplifications and some informality is possible. The premium paid is proportional to the gross tonnage of the vessel, and the number of votes a member can cast is proportional to the number of gross tons he has registered in the club.

Based on these observations, let us first formulate a model for such a club. Since it is a mutual arrangement, the theory initiated by Karl Borch (1960a), (1960b), and (1962) seems appropriate. It goes as follows:

Given is a group of I ship owners, each one facing a certain risk represented by a random variable [U.sub.i], i [member of] I = {1,2,..., I}. We model ship owner i's random endowments [X.sub.i] by

[X.sub.i] = [W.sub.i] - [U.sub.i], (1)

where [W.sub.i] = ship owner i's wealth, assumed to be a constant here, and [U.sub.i] is the potential loss facing ship owner i. The representation (1) is supposed to hold after the ship owner has insured his fleet in the commercial marine insurance market. The residual risk [U.sub.i] can be viewed as the risk not covered above some cap, or reinsurance layer, often found in XL-reinsurance contracts.

The ship owners are assumed risk averse with marginal utility functions [u'.sub.i](x)= [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i [member of] I, and therefore they seek further insurance. Not being able to obtain this in the commercial marine insurance market, these individuals are then forced to share these residual risks between themselves, rather than facing them in splendid isolation. The random endowment of ship owner i is denoted by Yi after the exchange has taken place. Let us denote the sum of the initial endowments [X.sub.i] by [X.sub.M], [X.sub.M] = [[summation].sup.I.sub.j=1] [X.sub.j]. Then the Pareto optimal sharing rules, also known to be the partial equilibrium allocations of the ship owners, are known to have the following form:

[Y.sub.i] = [a.sub.1]/[alpha] [X.sub.M] + [b.sub.i], where [b.sub.i] = [a.sub.i] ln [[lambda].sub.i] - [a.sub.i] k/[alpha], i [member of] I. (2)

This follows from the first order conditions of optimal risk exchange, given here by equating the marginal utilities multiplied by positive constants [[lambda].sub.i]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [xi] is the state price deflator, or marginal utility of the representative agent. After taking logarithms in this relation, and summing over i, market clearing implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](4)

Thus, the optimal sharing rules are affine in [X.sub.M]. The constants of proportionality [a.sub.i]/[alpha] are simply equal to to each ship owner's risk tolerance, measured relative to the other members. In order to compensate for the fact that the least risk-averse ship owner will hold the larger proportion of the total fleet, zero-sum side payments occur between the ship owners, here represented by the terms [b.sub.i]. Without these side payments, a ship owner, with a small initial wealth but with a large risk tolerance would end up with a large final wealth, but this could not possibly be consistent with his budget constraint.

In order to determine the ray [lambda] = ([[lambda].sub.1],...,[[lambda].sub.I]), we employ precisely the said budget constraints:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](5)

from which side payments bi are found as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](6)

Now the optimal sharing rules [Y.sub.i] are completely determined in terms of the given primitives of the model.

The ray [lambda] can also be determined modulo a normalization. Letting k = [[summation].sup.I.sub.j=1] [a.sub.j] ln [[lambda].sub.j] denote this normalization, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i [member of] I.

If we impose that E{[xi]} = 1/(1 + r) where r is the risk-free interest rate, we obtain [e.sup.-k/[alpha]] = (1 + r)E{[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]}, in which case the constants [lambda] are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this model market prices are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](7)

where Z is any risk having a finite variance, i.e., being in the set [L.sup.2] for short. Alternatively this can be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The last term in the expression (8) is the risk premium, which would disappear under risk neutrality.

The results related to the explicit form of the side payments given in (6) can be found in Aase (1993), where the pricing rule (7) is also derived. Further results regarding this type of models can be found in Aase (2002a,b).

EQUILIBRIUM IN A MARKET OF P&I CLUBS

The model of the previous section can be interpreted along the following lines. The group of ship owners have formed a syndicate, and the objective function of this syndicate is given by [u.sub.[lambda]](x), where [u.sub.[lambda]' (x) = [e.sup.-x/[alpha]]. It should be noted that this function has the same form as the ship owners' individual marginal utility functions, being of the negative exponential type, and hence displaying constant absolute risk aversion. Hence, the function [u.sub.[lambda]] (x) can be interpreted as the objective function of the P&I Club, which is really a syndicate.

Notice that we do not impose a "utility" function on the clubs. Rather the objective function [u.sub.[lambda]](x) is endogenously determined through the formation of the syndicate. When we later talk about the "risk aversion of the P&I Club," what we mean is the parameter (1/[alpha]), and likewise for the reciprocal "risk tolerance of the P&I Club" [alpha]. As indicated above, [alpha] - [[summation.sup.1.sub.i=1] [a.sub.i], so the syndicate's risk tolerance is the sum of the risk tolerances of the individual ship owners that constitute this P&I Club. When the ship owners in a club are all risk averse, so is the P&I Club, and when the ship owners are all risk tolerant, so is the P&I Club.

Consider a market of N different P&I Clubs formed this way, indexed by n [member of] N = {1, 2, ..., N}, having "objective functions" Vn(X) of the forms [v'.sub.n](x) = [math?????], n [member of] N. That is, the objective for each Club n is to solve

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

By an equilibrium we mean the simultaneous determination of a linear price functional [pi](.), and optimal portfolios ([Y.sub.1], [Y.sub.2],...

NOTE: All illustrations and photos have been removed from this article.




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