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Multi-period valuation of insurance liabilities subject to capital requirements
Stockholm University, Faculty of Science, Department of Mathematics.
2018 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2018. , p. 115
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:su:diva-161949OAI: oai:DiVA.org:su-161949DiVA, id: diva2:1262564
Presentation
2018-12-03, Hus 6, rum 306, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Available from: 2018-11-14 Created: 2018-11-12 Last updated: 2018-11-14Bibliographically approved
List of papers
1. Insurance valuation: A computable multi-period cost-of-capital approach
Open this publication in new window or tab >>Insurance valuation: A computable multi-period cost-of-capital approach
2017 (English)In: Insurance, Mathematics & Economics, ISSN 0167-6687, E-ISSN 1873-5959, Vol. 72, p. 250-264Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Valuation of insurance liabilities, Multi-period valuation, Market-consistent valuation, Cost of capital, Risk margin, Dynamic risk measurement
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-140138 (URN)10.1016/j.insmatheco.2016.12.002 (DOI)000393534100020 ()
Available from: 2017-02-28 Created: 2017-02-28 Last updated: 2019-12-04Bibliographically approved
2. The value of a liability cash flow in discrete time subject to capital requirements
Open this publication in new window or tab >>The value of a liability cash flow in discrete time subject to capital requirements
2020 (English)In: Finance and Stochastics, ISSN 0949-2984, E-ISSN 1432-1122, Vol. 24, no 1, p. 125-167Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Market-consistent valuation, Replicating portfolios, Capital requirements
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:su:diva-161902 (URN)10.1007/s00780-019-00408-0 (DOI)000511748200004 ()
Available from: 2018-11-12 Created: 2018-11-12 Last updated: 2020-03-02Bibliographically approved
3. Continuous-time limits of multi-period cost-of-capital valuation
Open this publication in new window or tab >>Continuous-time limits of multi-period cost-of-capital valuation
2018 (English)Report (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2018. p. 37
Series
Research report / Mathematical Statistics, ISSN 1650-0377 ; 2018:11
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-161906 (URN)
Available from: 2018-11-12 Created: 2018-11-12 Last updated: 2019-12-04Bibliographically approved

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