An Economic Approach to Capital Allocation

ABSTRACT This article starts with primitive assumptions on preferences and risk. It then derives prices consistent with a social optimum within an insurance company and the consumer-level capital allocation implied therein. The allocation "adds up" to the total capital of the firm (a result echoing findings in the congestion pricing literature--where optimal tolls exactly cover the rental cost of the highway). The allocation follows each consumer's share of recoveries in states of insurer default, weighted by the severity of the default in terms of welfare impact. However, the article argues that an economic approach technically restricts only the capital allocated to marginal units of coverage: inframarginal units could in principle receive different allocations. INTRODUCTION Financial institutions hold capital to absorb risks inherent in their contracts. If capital is scarce (i.e., costly to hold), then the institution must allocate capital to contract for purposes of pricing and performance measurement. For example, in an insurance company, policies cannot be priced properly without a charge for the company's costs arising from the holding of capital. However, it is not obvious how capital should be allocated to contract when the contracts are heterogeneous in risk. Indeed, as shown in Merton and Perold (1993), any attempt to do so is fraught with peril. In general, a fixed allocation is an unreliable guide for decisions involving large changes to the company portfolio (such as, e.g., exiting a business line). Nevertheless, the problem has received considerable attention over the past decade. Given the limitation outlined in Merton and Perold (1993), allocation methods have focused on the margin of the company; that is, although allocation methods may fail when analyzing large supramarginal and inframarginal changes to a contract portfolio, they can be used effectively to analyze small, marginal changes. Contributions to this "marginal approach" include early work by Tasche (2000) and Myers and Read (2001), as well as subsequent generalizations (e.g., Kalkbrener, 2005; Mildenhall, 2006). Applied theory on financial institutions has also embraced the marginal approach (e.g., Perold, 2005; Stoughton and Zechner, 2007). (1) The method in this literature is to start with a target value for a differentiable risk measure and to compute the marginal capital increase required to maintain the risk measure at the target value as a particular risk exposure within the portfolio is expanded. Under some conditions, marginal capital requirements calculated in this manner have the attractive property of "adding up" to the total capital of the firm when summed over all exposures (2) and, more broadly, can satisfy the coherence principles outlined by Denault (2001). (3) The marginal approach thus provides a logical and practical method for the allocation of capital. The allocation approach, however, is ad hoc in the sense that the risk measure is chosen arbitrarily. Since the risk measure is not connected to the underlying preferences of claimants, derived prices will be economically optimal only in situations where the firm faces a binding regulatory constraint based on a risk measure, or in situations where claimant preferences reference the risk measure target rather than actual outcomes. In this sense, the marginal approach--as practiced to date--lacks an economic foundation. This article provides such a foundation by solving the allocation problem in the context of an insurance company with techniques originally used for the analysis of public goods and common pool resources. In this context, the shared resource is capital--with the company's policyholders acting as rival claimants on the shared, nonexcludable resource. We use a social planning problem as a starting point, with the consequent assumption that consumers care about the distribution of their own recoveries rather than an average financial target for the company. …