Abstract
The aim of this paper is to define the market-consistent multi-period value of an insurance liability cash flow in discrete time subject to repeated capital requirements, and explore its properties. In line with current regulatory frameworks, the presented approach is based on a hypothetical transfer of the original liability and a replicating portfolio to an empty corporate entity, whose owner must comply with repeated one-period capital requirements but has the option to terminate the ownership at any time. The value of the liability is defined as the no-arbitrage price of the cash flow to the policyholders, optimally stopped from the owner’s perspective, taking capital requirements into account. The value is computed as the solution to a sequence of coupled optimal stopping problems or, equivalently, as the solution to a backward recursion.
Highlights
The aim of this paper is to define the market-consistent multi-period value of a liability cash flow in discrete time subject to repeated capital requirements in accordance with current regulatory frameworks, and explore its properties
Given an optimally selected replicating portfolio, the externally imposed capital requirements define the market-consistent value of a liability as the value it would have if it were transferred to an empty corporate entity, called a reference undertaking, whose owner has the option to terminate the ownership
In [10], the replicating portfolio was assumed to be given and the analysis only focused on the multi-period valuation of the liability cash flow
Summary
The aim of this paper is to define the market-consistent multi-period value of a liability cash flow in discrete time subject to repeated capital requirements in accordance with current regulatory frameworks, and explore its properties. Limited liability for the owner of the reference undertaking in our setting implies that the dynamic valuation mappings appearing here will in general be nonlinear, nonconvex and nonconcave regardless of any additional structure imposed on the conditional risk measures defining the capital requirements. Structural results for the selection of replicating portfolios are enabled by assuming that the conditional risk measures satisfy so-called positive homogeneity; see Theorems 2.28 and 2.30 Another approach to market-consistent liability valuation is presented by Pelsser and Stadje [23], combining no-arbitrage valuation and actuarial valuation into a general framework. The liability is valued by applying a linear pricing operator to the cumulative cash flow that the policyholders will receive, taking capital requirements and the option to default of the owner of the reference undertaking into account.
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