Longevity trend risk over limited time horizons

Abstract

We consider various aspects of longevity trend risk viewed through the prism of a finite time window. We show the broad equivalence of value-at-risk (VaR) capital requirements at a p-value of 99.5% to conditional tail expectations (CTEs) at 99%. We also show how deferred annuities have higher risk, which can require double the solvency capital of equivalently aged immediate anuities. However, results vary considerably with the choice of model and so longevity trend-risk capital can only be determined through consideration of multiple models to inform actuarial judgement. This model risk is even starker when trying to value longevity derivatives. We briefly discuss the importance of using smoothed models and describe two methods to considerably shorten VaR and CTE run times.

Short-Term Mortality Fluctuations and Longevity Risk-Adjusted Age: Learning the Resilience of a Country to a Health Shock

“…The factor x * is responsible for the limit of the Gompertzian regime, and the condition λ * > λ[i] is used as the constant value when chronological ages are equal to x * . These parameters are also known as the speciesspecific lifespan and the species-specific accidental mortality rate [14]. Model (3) can be linearized as:…”

A value-at-risk approach to mis-estimation risk

“…Throughout this paper, mortality improvements will be modelled up to y 1 , but no future mortality improvements after y 1 will be assumedthe mortality rates used will be those applying at y 1 with no allowance for future changes in time apart from the ageing of the lives assured. We are concerned in this paper with a value-at-risk assessment of the mis-estimation risk in V θ; y 1 only; for a value-at-risk approach to longevity trend risk after time y 1 , see Börger (2010), Plat (2011) or Richards et al (2020).…”

“…For example, for future trends there is no way of knowing if the chosen projection model is correct. This model risk in forecasting is discussed elsewhere; see for example Cairns (1998) and Richards et al (2020). However, this paper is concerned with the first basis element, i.e.…”

Some comments on “A Hermite spline approach for modelling population mortality” by Tang, Li & Tickle (2022)

“…For a long-term forecast of period mortality, one would probably use . However, with short-term value-at-risk calculations for the likes of Solvency II (Richards et al , 2020) it would be important to fully express the short-term variability in , and so one would probably use the trivariate parameterisation of Tang et al (2022) for sample paths.…”