Insurance Valuation: A Two-Step Generalised Regression Approach

Abstract

Current approaches to fair valuation in insurance often follow a two-step approach, combining quadratic hedging with application of a risk measure on the residual liability, to obtain a cost-of-capital margin. In such approaches, the preferences represented by the regulatory risk measure are not reflected in the hedging process. We address this issue by an alternative two-step hedging procedure, based on generalised regression arguments, which leads to portfolios that are neutral with respect to a risk measure, such as Value-at-Risk or the expectile. First, a portfolio of traded assets aimed at replicating the liability is determined by local quadratic hedging. Second, the residual liability is hedged using an alternative objective function. The risk margin is then defined as the cost of the capital required to hedge the residual liability. In the case quantile regression is used in the second step, yearly solvency constraints are naturally satisfied; furthermore, the portfolio is a risk minimiser among all hedging portfolios that satisfy such constraints. We present a neural network algorithm for the valuation and hedging of insurance liabilities based on a backward iterations scheme. The algorithm is fairly general and easily applicable, as it only requires simulated paths of risk drivers.

The 3-step hedge-based valuation: fair valuation in the presence of systematic risks

“…It is straightforward to see that the valuation introduced in Example 2 is a 3-step hedge-based valuation. Below, we show that the quantile hedging valuation of Barigou et al (2021) is also an example of a 3-step hedge-based valuation.…”

“…Example 3 (The quantile hedging valuation of Barigou et al (2021)). In Barigou et al (2021), the authors start from the hedge-based valuation, but propose to use a quantile hedging approach for the residual part of the claim. Assume that i ∈ (0, 1) is the cost-of-capital rate.…”

“…Below, we show that the quantile hedging valuation of Barigou et al . (2021) is also an example of a 3-step hedge-based valuation.…”

“…(2017) and Barigou et al . (2021), who also define a general hybrid valuation principle but mainly focus on product claims of the form (2.7). In this case, conditioning on the vectors and results in a diversifiable claim, and we have an actuarial valuation together with a corresponding risk management strategy that we want to employ for such a diversifiable claim.…”

“…Market-consistent valuation requires a combination of actuarial and financial valuation methods, as was first pointed out by Brennan and Schwartz (1976a), who considered the valuation of guarantees in unit-linked insurance contracts; see also Embrechts (2000). Recent approaches to define fair valuations are Pelsser and Stadje (2014), Pelsser and Schweizer (2016), Wuthrich (2016), Dhaene et al (2017), Delong et al (2019a), , , Barigou et al (2021), Deelstra et al (2020), Bacinello et al (2021), Chen et al (2021).…”