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.

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The regulator in Europe calls for the market-consistent valuation of the insurance liabilities that usually are not (fully) tradable. An example of such liabilities is the participating pension contract that is generally longdated and vulnerable to the medium-time dynamics of the underlying risk drivers. Dealing with these characteristics requires time-consistent pricing. However, the well-known non-linear premium principles, often used as pricing operators, are not time-consistent. Based on this motivation, we study the time-consistent and market-consistent (TCMC) actuarial valuation of the participating pension contracts with hybrid payoff. We use a standard profit-sharing mechanism with guaranteed interest rate, and generalize it to a hybrid profit-sharing mechanism with the actuarial and hedgeable financial risks, over the course of the contract. Market-consistency is maintained by "two-step actuarial valuation" in a one-period setting. Time-consistency is obtained by a "backward iteration" of these one-period two-step valuations over the predetermined sub-intervals of the valuation period. We use the Least-Square Monte-Carlo method to implement the conditional operators in the backward iteration. We compare the results of TCMC price to the expected value of the discounted payoff and measure the relative risk loading and time-consistency risk premium. Besides, we investigate the effect of the stochastic interest rate as compared to the deterministic one.

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. This method is applied to an insurance risk process in the form of a diffusion process and a jump process in order to capture the heavy tailed nature of insurance liabilities. We show that in the case of the diffusion process, the one-period time-consistent Variance premium principle converges to the non-linear exponential indifference price. Furthermore, we show that the Standard-Deviation and the Cost-of-Capital principle converge to the same price limit. Adding the jump risk gives a more realistic picture of the price. Furthermore, we no longer observe that the different premium principles converge to the same limit since each principle reflects the effect of the jump differently. In the Cost-of-Capital principle, in particular the VaR operator fails to capture the jump risk for small jump probabilities, and the time-consistent price depends on the distribution of the premium jump. * The research leading to these results has received funding from the European Union Seventh Framework Programme ([FP7