Mathematical foundation of the replicating portfolio approach

Natolski, Jan; Werner, Ralf

doi:10.1080/03461238.2017.1388273

Cited by 4 publications

(7 citation statements)

References 16 publications

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“…Usually, cash flows are aggregated to the terminal time point, leading to the consideration of terminal values. This point of view was mathematically supported in Natolski and Werner [17].…”

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confidence: 50%

“…Recently, Cambou and Filipovic [16] proved that the matching of terminal values indeed has a mathematical foundation in the sense that a good match of the terminal values is strongly intertwined with a good approximation of the risk measure of the future MCEV (i.e., the resulting risk capital). Simultaneously, Natolski and Werner [17] demonstrated that this foundation holds for any form of matching problem in Natolski and Werner [18] (and many more), especially including all cash flow matching and all terminal value matching problems considered here. In both Cambou and Filipovic [16] and Natolski and Werner [18], it is shown that it is possible to change from the real world measure in the first period to the risk neutral measure while maintaining the strong link between objective function and error in risk capital.…”

Section: Real World Risk Neutralmentioning

confidence: 99%

“…Further, squared cash flow matching provides worse bounds than cash flow matching (to be defined below). Details on the exact relationships are provided in Natolski and Werner [17]. 1 In Natolski and Werner [18] it is argued that working with the risk neutral measure is feasible, while it is better to keep the mix of measures as indicated in Figure 1.…”

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“…As has been shown in Natolski and Werner [18], the terminal value match can be seen as a squared cash flow match plus an additional dynamic trading strategy moving cash flows optimally in time. From a theoretical perspective, both choices provide a good link between matching criterion and approximation of risk capital (see Natolski and Werner [17]), although terminal value matching dominates squared cash flow matching. Further, both problems lead to strictly convex quadratic optimization problems possessing a unique optimal solution (under rather weak assumptions).…”

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confidence: 99%

“…In Section 2, we introduce the mathematical setup for the financial market following Natolski and Werner [17], and we recall the cash flow matching problem of the Fermat-Torricelli type. The main Section 3 states and proves the aforementioned properties of the replication problem, and Section 4 concludes.…”

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Mathematical Analysis of Replication by Cash Flow Matching

Natolski

Werner

2017

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Abstract:The replicating portfolio approach is a well-established approach carried out by many life insurance companies within their Solvency II framework for the computation of risk capital. In this note, we elaborate on one specific formulation of a replicating portfolio problem. In contrast to the two most popular replication approaches, it does not yield an analytic solution (if, at all, a solution exists and is unique). Further, although convex, the objective function seems to be non-smooth, and hence a numerical solution might thus be much more demanding than for the two most popular formulations. Especially for the second reason, this formulation did not (yet) receive much attention in practical applications, in contrast to the other two formulations. In the following, we will demonstrate that the (potential) non-smoothness can be avoided due to an equivalent reformulation as a linear second order cone program (SOCP). This allows for a numerical solution by efficient second order methods like interior point methods or similar. We also show that-under weak assumptions-existence and uniqueness of the optimal solution can be guaranteed. We additionally prove that-under a further similarly weak condition-the fair value of the replicating portfolio equals the fair value of liabilities. Based on these insights, we argue that this unloved stepmother child within the replication problem family indeed represents an equally good formulation for practical purposes.

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Section: Real World Risk Neutralmentioning

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Mathematical Analysis of Replication by Cash Flow Matching

Natolski

Werner

2017

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The value of a liability cash flow in discrete time subject to capital requirements

2019

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The aim of this paper is to define the market-consistent multiperiod 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 approach presented 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.

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Fair Valuation of Insurance Liability Cash-Flow Streams in Continuous Time: Applications

Delong¹,

Dhaene²,

Barigou³

2019

ASTIN Bull.

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Delong et al. (2018) presented a theory of fair (market-consistent and actuarial) valuation of insurance liability cash-flow streams in continuous time. In this paper, we investigate in detail two practical applications of our theory of fair valuation. In the first example, we consider the fair valuation of a terminal benefit which is contingent on correlated tradeable and non-tradeable financial risks. In the second example, we consider a portfolio of unit-linked contracts contingent on a non-tradeable insurance and a tradeable financial risk. We derive partial differential equations (PDEs) which characterize the continuous-time fair valuation operators in these two examples and we find explicit solutions to these PDEs. The fair values of the liabilities are decomposed into the best estimate of the liability and a risk margin. The arbitrage-free representations of the fair values of the liabilities are derived and the dynamic hedging strategies associated with the continuous-time fair valuation operators are also established. Detailed interpretations of the results, which should be useful both for researchers and practitioners, are provided.

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Mathematical foundation of the replicating portfolio approach

Cited by 4 publications

References 16 publications

Mathematical Analysis of Replication by Cash Flow Matching

Mathematical Analysis of Replication by Cash Flow Matching

The value of a liability cash flow in discrete time subject to capital requirements

Fair Valuation of Insurance Liability Cash-Flow Streams in Continuous Time: Applications

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