Unraveling the time dynamics of life annuities

Mortality and interest rates evolve jointly, dynamically influencing the values of life annuities and actuarial reserves. Building on established results from actuarial science and demography, this paper introduces a set of differential equations that simultaneously quantify (i) changes in mortality and interest rates, and (ii) the sensitivity of annuity and reserve values to these changes, measured through entropies and durations.We illustrate the practical application of our framework by examining the long-term development of life annuity values using data for the United Kingdom from 1841 to 2021. The analysis reveals how financial and longevity risks have evolved over time, uncovering detailed patterns across interest rate terms, age groups, and causes of death. It also highlights a clear interplay between the financial and longevity risk – where the latter is at times masked by periods of elevated financial volatility.Our equations explicitly capture the joint dynamics of mortality and interest rates, to better understand the changing economic-demographic environment and its impact on life annuity values. The main contribution of this paper is to provide actuaries with practical, data-driven tools to evaluate and manage evolving risks in life annuity portfolios and actuarial reserves.