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.

欧洲监管机构要求对通常无法（完全）交易的保险负债进行市场一致的估值。此类负债的一个例子是参与养老金合同，该合同通常是长期的，容易受到潜在风险驱动因素的中期动态的影响。处理这些特征需要时间一致的定价。但是，通常用作定价运算符的众所周知的非线性溢价原理并非时间一致的。基于这种动机，我们研究了具有混合收益的参与式养老金合同的时间一致性和市场一致性（TCMC）精算估值。我们使用具有保证利率的标准利润分享机制，并将其推广到具有精算和可对冲财务风险的混合利润分享机制，在合同过程中。在一个时期内，通过“两步精算估值”来保持市场一致性。时间一致性是通过在评估期的预定子时间间隔内对这些一周期两步评估进行“向后迭代”而获得的。我们使用最小二乘蒙特卡洛方法在向后迭代中实现条件运算符。我们将TCMC价格的结果与折现收益的期望值进行比较，并测量相对风险负荷和时间一致性风险溢价。此外，我们研究了随机利率与确定利率相比的影响。时间一致性是通过在评估期的预定子时间间隔内对这些一周期两步评估进行“向后迭代”而获得的。我们使用最小二乘蒙特卡洛方法在向后迭代中实现条件运算符。我们将TCMC价格的结果与折现收益的期望值进行比较，并测量相对风险负荷和时间一致性风险溢价。此外，我们研究了随机利率与确定利率相比的影响。时间一致性是通过在评估期的预定子时间间隔内对这些一周期两步评估进行“向后迭代”而获得的。我们使用最小二乘蒙特卡洛方法在向后迭代中实现条件运算符。我们将TCMC价格的结果与折现收益的期望值进行比较，并测量相对风险负荷和时间一致性风险溢价。此外，我们研究了随机利率与确定利率相比的影响。