Quantitative assessment of common practice procedures in the fair evaluation of embedded options in insurance contracts

Abstract

This work analyses the common industry practice used to evaluate financial options written on with-profit policies issued by European insurance companies. In the last years regulators introduced, with the Solvency II directive, a market consistent valuation framework for determining the fair value of asset and liabilities of insurance funds. A relevant aspect is how to deal with the estimation of sovereign credit and liquidity risk, that are important components in the valuation of the majority of insurance funds, which are usually heavily invested in treasury bonds. The common practice is the adoption of the certainty equivalent approach (CEQ) for the risk neutral evaluation of insurance liabilities, which results in a deterministic risk adjustment of the securities cash flows. In this paper, we propose an arbitrage free stochastic model for interest rate, credit and liquidity risks, that takes into account the dependences between different government bond issuers. We test the impact of the common practice against our proposed model, via Monte Carlo simulations. We conclude that in the estimation of options whose payoff is determined by statutory accounting rules, which is often the case for European traditional with-profit insurance products, the deterministic adjustment for risk of the securities cash flows is not appropriate, and that a more complete model such as the one described in this article is a viable and sensible alternative in the context of market consistent evaluations. JEL classification codes: C63.

A market-consistent framework for the fair evaluation of insurance contracts under Solvency II

“…where Y I is a standard Brownian motion, assumed to be independent from the Brownian motions driving the dynamics of the risk-free rate and the credit spread and ϕ I (t) is a deterministic function of time that allows a perfect fitting of the initial ZCB prices of the specific issuer. As highlighted in [Gambaro et al, 2017], the possibility of having a negative liquidity spread is relevant for capturing the so called fly-to-liquidity effects. If we assume that the liquidity spread is independent from both the risk free rate and the credit spread, then, simply, the ZCB formula contains a second multiplicative adjustment factor in closed form.…”

“…These market-consistent evaluations are mostly carried out in practice by applying the Certainty Equivalent approach (CEQ) in discounting expected cash flows, which implies adjusting contractual cash flows for the implied risk premium above the risk-free rate provided by EIOPA (see [CFO-Forum, 2016a] and [EIOPA, 2017]). The paper [Gambaro et al, 2017] has shown how this approach can lead to biased results in the valuation of contractual financial options in traditional insurance products where the pay-off is determined by statutory accounting rules. Another important aspect affecting market consistent valuations, is the rigid interpretation of market-consistency, which very often restricts the instruments to which stochastic models are calibrated to few categories of options such as interest rate swaps and equity indexes.…”

“…The proposed model is applied to the valuation of embedded options in life insurance with-profit contracts (Value Of Guarantees, or VOG). As explained in details in [Gambaro et al, 2017], a with-profit (or participating) policy is a life insurance contracts offering certain guarantees to the policyholder, as a minimum rate of return. The policy is written on a segregated fund that is owned by the life insurance company, and that must be kept separate from the company's other assets.…”

“…For instance, the risk of a (fixed notional) callable bond can be hedged using a Bermuda swaption, hence a calibration to quoted swaption prices of the interest rate model is obviously desirable. Moreover, as already discussed in [Gambaro et al, 2017] the traditional with-profit insurance funds have a complex financial structure due to several factors such as accounting rules, management actions, insurance risks, as for instance the mortality risk, and a bonus rate mechanism that depends on the history of fund returns. Hence, hedging is an open problem in this setting (see [Luciano and Regis, 2014] for a discussion of hedging strategies in presence of mortality risk for a simplified insurance product).…”

An Options Pricing Method Based on the Atmospheric Environmental Health Index

“…In theory, the Esscher transform accounts for the basic hypothesis of options, which is the possibility of risk-free arbitrage. This method can be used to simulate a variety of statistical distributions, which makes options pricing more widely applicable [23].…”