In the papers presented here, approaches to multi-period valuation of a liability cashflow in runoff, subject to repeated capital requirements, are developed and analyzed. The valuation approaches are inspired by current risk-based regulatory frameworks for the insurance industry, and consistent with the fundamental principles underlying them. The capital requirements are partly financed by capital providers with limited liability, meaning that the capital providers cannot lose more than the provided capital. Limited liability is an essential ingredient in the considered multi-period valuation framework.

In the first paper, multi-period cost-of-capital valuation is considered. The liability value is defined in terms of the capital provider's criterion for accepting to provide capital which gives rise to a backward recursion from which the liability value can be computed. Explicit solutions to the recursion are obtained when the cashflows can be expressed in terms of multivariate Gaussian distributions.

The second paper recognizes that due to limited liability (an option to default) the cashflow to the capital provider can be seen as that of a financial derivative instrument with optionality. Arbitrage-free valuation of this cashflow, similar to the valuation of so-called American type contingent claims, forms the basis of the multi-period approach to liability cashflow valuation considered here. The issue of selection of a replicating portfolio for offsetting the hedgeable part of the liability cashflow is investigated.

The first two papers consider cashflows and valuations at a fixed set of times to be interpreted as the years from current time until the runoff of the liability is complete. In the third paper, the valuation and cashflow times are allowed to be arbitrary in the form of an arbitrary partition of the entire runoff period. The focus here is to properly define and analyze the effects of letting the mesh of the partition tend to zero, exploring the continuous-time value processes that appear.

Insurance companies are required by regulation to be in possession of liquid assets that ensure that they can meet their obligations to policyholders with high probability. The amount is usually determined by an actuarial valuation, with for instance the Solvency II regulatory framework providing standard formulae. In this thesis we investigate a valuation procedure where the value of a liability cash flow is determined via a backwards recursive relationship, meaning that the value at time t depends on the value at time t+1. The value corresponds to an amount required to be able to raise capital from an external capital provider with limited liability, in order to meet capital requirements imposed by a regulating body. 

Paper I describes the valuation philosophy that will more or less be shared by all papers in the thesis. It establishes a recursive relationship given via a mapping, that satisfy the properties of a dynamic monetary utility function. Conditions are given where finite p:th moments are preserved in the recursion and a link to the well known subject of dynamic monetary risk measures and utility functions is established. The structure of the recursion is used to find closed-form values for certain stochastic processes, most importantly in the case where we have jointly Gaussian cash flows.

Paper II explores the valuation procedure in the presence of a risk-neutral probability measure, which correctly prices the financial instruments that are priced by the financial market but is also assumed to express the risk aversion toward non-hedgeable insurance risk of the capital provider. We show that the valuation procedure is equivalent to an optimal stopping problem, giving us an alternative way to define the valuation procedure. We reproduce many of the structural results from Paper I under the assumed conditions. We also consider the choice of replicating portfolio under different criteria, especially the criterion of minimizing the need for external capital.

Paper III considers the discrete-time valuation from paper I, but where the valuation times form an arbitrary partition of the time interval on which the runoff of the liability occurs. We investigate the properties of the value as the mesh of the partition goes to zero. We define a "continuous-time value" of a liability cash flow and find closed form expressions and some structural results for classes of stochastic processes including Lévy processes and Itô diffusions.

Paper IV tackles the numerical difficulties of performing the recursive valuation procedure where a closed-form value cannot be found. Under Markovian assumptions, a so-called least-squares Monte Carlo (LSM) algorithm is investigated, a method that was developed to tackle optimal stopping problems. We show some overarching consistency results for the LSM algorithm in the general setting of dynamic monetary utility functions and also explore numeric performance for some example models.