Time-consistent actuarial valuations

Abstract

Recent theoretical results establish that time-consistent valuations (i.e. pricing operators) can be created by backward iteration of one-period valuations. In this paper we investigate the continuous-time limits of well-known actuarial premium principles when such backward iteration procedures are applied. We show that the one-period variance premium principle converges to the non-linear exponential indifference valuation. Furthermore, we study the convergence of the one-period standard-deviation principle and establish that the Cost-of-Capital principle, which is widely used by the insurance industry, converges to the same limit as the standard-deviation principle. Finally, we study the connections between our time-consistent pricing operators, Good Deal Bound pricing and pricing under model ambiguity.

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

“…The risk margin can be derived from a subjective actuarial valuation or from a cost-of-capital approach. Similar to our paper, the continuous-time limit of the iterative multiperiod valuation operator with the time step h → 0 is studied in Pelsser (2010), Pelsser and Stadje (2014), Pelsser and Ghalehjooghi (2016) and Engsner et al (2017). In particular, Pelsser (2010), Pelsser and Stadje (2014) and Pelsser and Ghalehjooghi (2016) also use PDEs to describe the continuous-time valuation operator in their model.…”

“…Pelsser 2010and Pelsser and Ghalehjooghi (2016) propose a completely different one-period valuation operator , called a two-step valuation operator. Dhaene et al (2017) and Barigou and Dhaene (2019) use the one-period valuation operator (3.11) but they consider static hedging strategies.…”

“…Fair valuation of insurance liabilities, which combines market-consistent and actuarial valuations, is also investigated in Pelsser (2010), Pelsser and Stadje (2014), Pelsser and Ghalehjooghi (2016), Happ et al (2015), Engsner et al (2017), , , Möhr (2011), Dhaene et al (2017), Barigou and Dhaene (2019) and Barigou et al (2018). In all these papers, a multi-period model and an iterative valuation operator are considered.…”

Affordable and adequate annuities with stable payouts: Fantasy or reality?

“…For an introduction to the valuation of contingent claims consisting of actuarial (diversifiable) and financial (nondiversifiable) risks, we refer toTsanakas et al (2013),Stadje and Pelsser (2014),Pelsser and Salahnejhad (2016), andDhaene et al (2017).26 If g j and r tj depend on past stock return shocks, then we are typically still able to derive closed-form expressions for the annuity price and the hedging strategy. Incorporating a stochastic interest rate and a stochastic volatility will require us to determine adequate stochastic processes for these variables as well as a realistic dependence model describing the interactions between the different processes.…”

Affordable and Adequate Annuities with Stable Payouts: Fantasy or Reality?

“…For an introduction to the valuation of contingent claims consisting of actuarial (diversifiable) and financial (nondiversifiable) risks, we refer toTsanakas et al (2013),Stadje and Pelsser (2014),Pelsser and Salahnejhad (2016), and Dhaene et al(2017).26 If g j and r tj depend on past stock return shocks, then we are typically still able to derive closed-form expressions for the annuity price and the hedging strategy. Incorporating a stochastic interest rate and a stochastic volatility will require us to determine adequate stochastic processes for these variables as well as a realistic dependence model describing the interactions between the different processes.…”