Mathematical Analysis of Replication by Cash Flow Matching

Abstract

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.

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

“…is referred to as terminal-value matching in [15], [16] and [17]. It is a standard quadratic optimization problem with explicit solution…”

“…Then, a risk measure is applied to the approximation of the liability value yielding an approximation of the capital requirement. In [3], [15], [16] and [17] various aspects of this replicating portfolio approach to capital calculations are studied. A fact that somewhat complicates the analysis is that risk measures defining capital requirements are defined with respect to the real-world probability measure P, whereas the replication criteria are usually expressed in terms of the market's pricing measure Q.…”

“…A fact that somewhat complicates the analysis is that risk measures defining capital requirements are defined with respect to the real-world probability measure P, whereas the replication criteria are usually expressed in terms of the market's pricing measure Q. Comparisons of properties and effects of different replication criteria are presented in [15], [16] and [17]. In [3], it is shown how replicating portfolio theory can be formulated in order to allow for efficient replication of liability values exhibiting path-dependence.…”

“…In [3], it is shown how replicating portfolio theory can be formulated in order to allow for efficient replication of liability values exhibiting path-dependence. Common to the works [3], [15], [16] and [17] is that the liability value is defined as a conditional expected value of the sum of discounted liability cash flows. This is different from the approach presented here.…”

Assessing Asset-Liability Risk with Neural Networks

“…If first and second derivatives of the portfolio with respect to the underlying risk factors are accessible, the method is computationally efficient, but its accuracy depends on how well the second order Taylor polynomial approximates the true portfolio loss. Similarly, the replicating portfolio approach approximates future cashflows with a portfolio of liquid instruments that can be priced efficiently; see e.g., Wüthrich (2016), Pelsser and Schweizer (2016), Natolski and Werner (2017) or Cambou and Filipović (2018). Building on work on American option valuation (see e.g., Carriere 1996;Longstaff and Schwartz 2001;Tsitsiklis and Van Roy 2001), Broadie et al (2015) as well as Ha and Bauer (2019) have proposed to regress future cash flows on finitely many basis functions depending on state variables known at the risk horizon.…”