Inflation impact on multi-cash-flow hybrid life products

The current economic environment is frequently subject to inflationary pressures that severely erode retirees’ purchasing power. Consequently, insurers are increasingly expected to offer life annuity products that explicitly incorporate inflation protection. However, pricing these hybrid multi-cash-flow products is challenging because financial markets lack assets that perfectly replicate long-term inflation, thereby leading directly to an incomplete financial market. Since inflation-indexed annuities are also exposed to mortality risk, we need a multi-decomposition-step operator able to formally separate the hedgeable financial strategy from idiosyncratic biometric pooling and systematic risk buffering, while ensuring a market–actuarial-consistent valuation. Moreover, in practice, inflation-indexed annuities are exposed to inflation that fluctuates stochastically, which, in turn, calls for a joint stochastic treatment of nominal interest rates. Driven by the previous requirements, our first contribution is to extend the stochastic-interest-rate three-step method of Belhouari et al. (2025) to incomplete financial markets for multi-cash-flow hybrid life payments. We found that both the hedgeable component and its associated systematic financial risk depend on the hedging strategy. However, we deliberately evaluate the hedgeable component and its associated systematic financial risk such that their sum is invariant to the hedging strategy and equals the best-estimate value. Hence, our goal is to explore how the insurer can allocate the same best-estimate value between these two components depending on the hedging strategy chosen and the assets available in the hedging portfolio. First, we start by relying only on standard nominal zero-coupon bonds, which are readily available in the financial market as traded assets, and highlight that the hedge price does not benefit from the nominal interest rate–inflation correlation. We then examine cross-hedging with an inflation-correlated stock and find that it improves partial hedging and modestly reduces the hedge price, and increases the systematic financial risk premium that can be absorbed by holding additional capital.