Egalitarian Equivalent Capital Allocation

Abstract

We apply Moulin's notion of egalitarian equivalent cost sharing of a public good to the problem of insurance capitalization and capital allocation where the liability portfolio is fixed. We show that this approach yields overall capitalization and cost allocations that are Pareto efficient, individually rational, and, unlike other mechanisms, stable in the sense of adhering to cost monotonicity.

ACKNOWLEDGMENTS

The authors thank Daniel Bauer and Stephen Mildenhall for helpful discussions and two anonymous referees for many helpful suggestions.

FUNDING

The authors are grateful to the Insurance Risk and Finance Research Centre (IRFRC) at Nanyang Business School for its financial support.

Notes

1 A particularly vivid example of a runoff capitalization is provided by Equitas, which is in the process of discharging the liabilities of multiple Lloyd's syndicates following the 1993 market restructuring.

2 Bauer and Zanjani (2013a) provide a review of gradient methods as well as alternative approaches to capital allocation.

3 Although this article is concerned with an economic approach to capital allocation, it should also be acknowledged the economic approach, in the sense of taking profit or welfare maximization as the guiding objective, is by no means the only approach to capital allocation. Examples of optimization approaches with different objectives can be found in Dhaene et al. (2003), Laeven and Goovaerts (2004), and Dhaene et al. (2012).

4 To elaborate, feasible utility distributions are ones that can be achieved when the public good is being paid for in full. To find an egalitarian equivalent utility distribution, we go through a process of finding the highest possible level of free public good production that is associated with a feasible utility distribution. Thus, the utility distribution could be achieved in two ways: (1) by providing a free amount of the public good, where every consumer pays a cost share of zero or (2) by providing a larger amount of the public good, that is fully paid for with cost shares that are allocated to each of the consumers.

5 Note that the contracted indemnity here is taken as a given. For analysis of the optimal level of indemnity, see Zanjani (2010) and Bauer and Zanjani (2016).

6 We have expressed frictional costs as a function of assets rather than capital, but note that this form is flexible enough to capture frictional capital costs. For example, consider , where is a continuous and increasing function and capital is the difference between assets and expected liabilities . Notice that capital is a continuous and increasing function of assets, so that c(a) will inherit continuity and monotonicity as well.

7 Note that the assignments will generally depend on the utility functions, so one could write the assignments as functions of the utility functions in addition to the cost function. At the expense of precision, we choose the simpler presentation to avoid notational tedium and to underscore our interest in cost monotonicity, which will involve varying the cost function while holding the utility functions fixed.

8 This data are originally used in Bauer and Zanjani (2013b), where sample statistics for the full sample and each of the sublines can be found. The four-line aggregation used here is also taken from Bauer and Zanjani (2013b).

9 The form of exponential utility implies that one could alternatively express this as a risk aversion coefficient of 1000× 0.01 = 10 with losses expressed in billions, or a risk aversion coefficient of 1 with losses expressed in hundreds of millions, and so forth.

10 By changing the risk aversion coefficient from 0.01 to 0.02, the Pareto optimal level of asset at τ = 1% increases to 1,336 million, which falls between the 2nd and the 3rd largest aggregate loss.

11 Note that, although the EE method produces a positive premium allocation to each of the lines, a negative frictional cost allocation may be implied by the premium allocation if the allocated premium is below the actuarial cost, as seen in this example.