Time-consistent and market-consistent actuarial valuation of the participating pension contract

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 long-dated 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.