We study market-consistent valuation of liability cash flows motivated by current regulatory frameworks for the insurance industry. The value assigned to an insurance liability is the consequence of (1) considering a hypothetical transfer of an insurance company's liabilities, and financial assets intended to hedge these liabilities, to an empty corporate entity, and (2) considering the circumstances under which a capital provider would want to achieve and maintain ownership of this corporate entity given limited liability for the owner and that capital requirements have to be met at any time for continued ownership.

We focus on the consequences of the capital provider assessing the value of continued ownership in terms of a least favorable expectation of future dividends, meaning that we consider expectations with respect to probability measures in a set of equivalent martingale measures. We show that natural conditions on the set of probability measures imply that the value of a liability cash flow is given in terms of a solution to a backward recursion. Through a life and a non-life insurance example we demonstrate how to make the valuation approach operational.

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.

We consider multi-period cost-of-capital valuation of a liability cash flow subject to repeated capital requirements that are partly financed by capital injections from capital providers with limited liability. Limited liability means that, in any given period, the capital provider is not liable for further payment in the event that the capital provided at the beginning of the period turns out to be insufficient to cover both the current-period payments and the updated value of the remaining cash flow. The liability cash flow is modeled as a continuous-time stochastic process on [0,T]. The multi-period structure is given by a partition of [0,T] into subintervals, and on the corresponding finite set of times, a discrete-time cost-of-capital-margin process is defined. Our main objective is the analysis of existence and properties of continuous-time limits of discrete-time cost-of-capital-margin processes corresponding to a sequence of partitions whose meshes tend to zero. Moreover, we provide explicit expressions for the limit processes when cash flows are given by Itô diffusions and processes with independent increments.

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.

We consider multi-period cost-of-capital valuation of a liability cashflow subject to repeated capital requirements that are partly financed by capital injections from capital providers with limited liability. Limited liability means that, in any given period, the capital provider is not liable for further payment in the event that the capital provided at the beginning of the period turns out to be insufficient to cover both the current-period payments and the updated value of the remaining cash flow. The liability cash flow is modeled as a continuous-time stochastic process on [0, T]. The multi-period structure is given by apartition of [0, T] into subintervals, and on the corresponding finite set of times a discrete-time value process is defined. Our main objectiveis the analysis of existence and properties of continuous-time limits of discrete-time value processes corresponding to a sequence of partitions whose meshes tend to zero. Moreover, we provide explicit and interpretable valuation formulas for a wide class of cash flow models.

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.

We present an approach to market-consistent multi-period valuation of insurance liability cash flows based on a two-stage valuation procedure. First, a portfolio of traded financial instrument aimed at replicating the liability cash flow is fixed. Then the residual cash flow is managed by repeated one-period replication using only cash funds. The latter part takes capital requirements and costs into account, as well as limited liability and risk averseness of capital providers. The cost-of-capital margin is the value of the residual cash flow. We set up a general framework for the cost-of-capital margin and relate it to dynamic risk measurement. Moreover, we present explicit formulas and properties of the cost-of-capital margin under further assumptions on the model for the liability cash flow and on the conditional risk measures and utility functions. Finally, we highlight computational aspects of the cost-of-capital margin, and related quantities, in terms of an example from life insurance.

In this paper we explore ways of numerically computing recursive dynamic monetary risk measures and utility functions. Computationally, this problem suffers from the curse of dimensionality and nested simulations are unfeasible if there are more than two time steps. The approach considered in this paper is to use a Least Squares Monte Carlo (LSM) algorithm to tackle this problem, a method which has been primarily considered for valuing American derivatives, or more general stopping time problems, as these also give rise to backward recursions with corresponding challenges in terms of numerical computation. We give some overarching consistency results for the LSM algorithm in a general setting as well as explore numerically its performance for recursive Cost-of-Capital valuation, a special case of a dynamic monetary utility function.