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...insurance contracts. In the case of having to cover frictional costs, the suggested allocation method may even lead to inappropriate insurance prices. Beside the purpose of pricing insurance contracts, capital allocation methods proposed in the literature and used in insurance practice are typically intended to help derive capital budgeting decisions in insurance companies, such as expanding or contracting lines of business. We also show that net present value analyses provide better capital budgeting decisions than capital allocation in general.
INTRODUCTION
The usefulness of capital allocation methods, i.e., ways that allocate the equity capital of an (insurance) company to different lines of business, can be assessed only in the context of the company's economic goals. Although this statement sounds so obvious, failure to consider context is precisely the current state of affairs in capital allocation discussion. Articles about capital allocation typically begin by listing certain properties that an allocation method should possess, (1) the most prominent of which are: adding-up property, no undercut, symmetry, and consistency. (2) Capital allocation is supposed to be useful in accomplishing the goals of competitive pricing of insurance contracts and making optimal capital budgeting decisions, (3) but instead of analyzing whether various allocation methods are appropriate in certain situations, the literature focuses almost exclusively on whether the proposed allocation methods encompass the above-listed "essential" properties.
Stewart Myers and James Read (4) have proposed an important capital allocation method for insurance companies. They discovered "a unique and non arbitrary" (5) allocation method that leads to an "adding-up" property; i.e., the equity capital allocated to the single lines of business "adds up" to the overall equity capital of the insurance company. Using option-pricing techniques, the allocation depends on the marginal contribution of a contract in a single line of business to the default value of the whole firm. (6) Myers and Read propose using their capital allocation method in pricing insurance contracts. In particular, they propose using it to determine correct loadings on fair premiums in cases where there are frictional costs of holding equity capital. (7)
The Myers and Read article won the 2002 ARIA best paper prize and has since been widely discussed in the academic literature. For example, Kneuer (2003), Ruhm and Mango (2003), Vrieze and Brehm (2003), and Mildenhall (2004) analyze the technical requirements, especially concerning distributional assumptions, and the practical limitations of the Myers and Read approach. Meyers (2003) argues that for the question of expanding or contracting lines of business, capital allocation, including the Myers and Read approach, is not necessary, a finding in line with Phillips, Cummins, and Allen (1998), if no frictional costs are taken into account. Because of the huge number of possible risk measures and allocation methods, Venter (2003, 2004) does not believe the approach will give clear guidance about the profitability of different lines of business or help in making capital budgeting decisions, (8) but does think the method is appropriate for the purpose of pricing insurance contracts. (9) Cummins, Lin, and Phillips (2005) find, on an empirical basis, that the Myers and Read way of allocating the frictional cost of capital is reflected in the insurance premiums observed.
The first goal of this paper is to show that capital allocation to lines of business based on the Myers and Read approach is either not necessary for insurance rate making (in the case of no frictional costs) or even leads to incorrect loadings (when frictional costs are considered). Furthermore, capital allocation techniques are proposed for making capital budgeting decisions in lines of business. We will show that these techniques lead, in principle, to wrong decisions--and not only with respect to the Myers and Read approach. Setting out the reasons for that result is our second goal.
The paper is organized as follows. In the next section, "Pricing Insurance Contracts, Risk Management Costs, and Equity Capital," we set out our arguments in a situation without frictional costs, setting the stage for the next section, "Pricing Insurance Contracts and Frictional Costs," in which we do consider frictional costs. In the section, "Performance Measurement and Optimal Capital Budgeting Decisions for Lines of Business," we discuss the main problems that arise when capital allocation methods are used for profit ranking and capital budgeting decisions. The last section summarizes the key results and concludes.
PRICING INSURANCE CONTRACTS, RISK MANAGEMENT COSTS, AND EQUITY CAPITAL
The theoretical basis of the Myers and Read capital allocation method is the contingent claims approach for insurance pricing. (10) In this framework, the fair insurance price is determined by the claims payoff distribution, the arbitrage-free valuation function, and the contract's safety level (measured by the value of the default put option). Clearly, this method of calculating competitive insurance prices does not depend on the insurance company's preexisting portfolio, which in turn means that it makes no difference to the insurance price whether the company is a single- or multi-line insurer, everything else being held equal. Thus, no allocation of equity capital to lines of business or to single insurance contracts is necessary in making the pricing decision.11 To achieve a desired safety level, the insurance company must establish certain risk management measures. Equity capital is only one of these, and can be (partially) substituted by reinsurance, alternative risk transfer, and other measures. The necessary risk management costs are covered by the insurance premiums. Let us now formalize this line of reasoning.
The one-period option-pricing framework for pricing insurance contracts used by Myers and Read was first proposed by Doherty and Garven. (12) Let [P.sup.old] indicate the competitive premium (paid at time t = 0) of the preexisting underwriting portfolio of an insurance company that consists of several lines of business. The insurance portfolio yields stochastic claims costs [L.sup.old.sub.1] at time t = 1. The present value of these claims costs is denoted by PV([L.sup.old.sub.1]). PV(*) denotes an arbitrage-free valuation function. [D.sup.old] stands for the present value of the default put option. If [E.sup.old.sub.0] indicates the initial equity capital of the company at time t = 0, and r the stochastic rate of return on its investment portfolio, then the default value [D.sup.old] is given by:
[D.sup.old] = PV(max{[L.sup.old.sub.1] - ([E.sup.old.sub.0] + [P.sup.old])(1 + r),0}). (1)
The competitive premium of the initial insurance portfolio [P.sup.old] is:
[P.sup.old] = PV([L.sup.old.sub.1]) - [D.sup.old]. (2)
Note that the premium [P.sup.old] should also be the basis for a regulated premium if the regulatory authority wants shareholders and policyholders to earn a risk adequate return on their capital.
As in the Myers and Read article, the company's safety level can be defined by the default-value-to-liability ratio: (13)
[d.sup.old] = [D.sup.old] / PV([L.sup.old.sub.1]) (3)
The objective now is to price a new contract in line i with stochastic claims costs [L.sup.new,i.sub.1]. The default put option value of the new contract is denoted as [D.sup.new,i]. If the default-value-to-liability ratio of the preexisting portfolio is to be maintained, then for the default-value-to-liability ratio of the new contract in line i,...
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