Longevity Basis Risk

Definition

Longevity basis risk is the risk that the mortality improvements of a specific insured portfolio diverge from those of the general population index underlying a standardized longevity hedge, so that the hedge fails to offset the actual liability movements. It arises from a mismatch between the population whose mortality the hedging instrument tracks and the population whose mortality the hedger is actually exposed to.

Key Ideas

Standardized longevity hedges (e.g., longevity swaps linked to a national mortality index) are cheaper and more liquid than bespoke hedges, but they expose the hedge purchaser to the differential mortality trend of their specific portfolio.
Longevity basis risk is not just a theoretical concern: the UK insured lives population has historically shown lower mortality and faster improvement than the England & Wales general population.
Multi-population mortality models are required to quantify and manage longevity basis risk; single-population models are blind to cross-population divergence.
Under Solvency II, longevity basis risk manifests as a higher risk margin: even small differences in mortality trend assumptions can generate 10–50% increases in the risk-adjusted price of a longevity hedge.

How It Works

Definition of Basis Risk in This Context (Zhou et al.)

Longevity basis risk is measured as the difference between:

The price of a longevity swap under a multi-population model — which jointly forecasts mortality for both the general and sub-population, capturing the differential trend.
The price under a single-population model — which applies the general population trend to the sub-population's current mortality level, ignoring the differential.

The "price" is the Solvency II risk margin:

$\text{Risk Margin} = 6\% \times \sum_t \mathrm{SCR}_t \cdot d_t$

where $\mathrm{SCR}_t$ is the Solvency Capital Requirement for longevity risk at time $t$ (= 99.5th-percentile stressed net asset value (NAV) change) and $d_t$ is the risk-free discount factor.

Solvency Capital Requirements for Longevity

Under the Solvency II internal model approach, the $\mathrm{SCR}_{t=0}$ for longevity risk is approximated as:

$\mathrm{SCR}_{t=0} \approx {}_{t=0}V^{\text{stressed}} - {}_{t=0}V^{\text{best estimate}}$

where ${}_{t=0}V^{\text{stressed}}$ is the present value of future annuity liabilities under the 99.5th-percentile mortality scenario, and ${}_{t=0}V^{\text{best estimate}}$ is the present value under the median scenario. The stressed scenario is derived from the tail of the stochastic mortality distribution.

Model Sensitivity (Zhou et al. Results)

For a longevity swap on a single 65-year-old male annuitant (GBP 1M annual benefit), using UK data:

Model	Best-estimate annuity	Risk margin	Risk margin vs. single-pop
Single population	£16.79M	£1.21M	—
RWAR (Random Walk AR)	£16.83M	£1.78M	+47%
VAR	£16.93M	£1.31M	+9%
VECM	£17.32M	£1.53M	+27%

The best-estimate annuity values are nearly identical across models (multi-population models marginally project lower mortality), but the risk margins differ substantially because the models assign different long-run volatilities to $\kappa_t^{(2)}$ : $\sigma(\kappa_{2025}^{(2)}) = 2.95$ (RWAR), $2.70$ (vector error correction model (VECM)), $2.16$ (vector autoregression (VAR)). Higher volatility $\to$ higher 99.5th-percentile stress $\to$ higher SCR $\to$ higher risk margin.

Note: these scenarios exclude parameter uncertainty and model risk; actual risk margins would be higher.

VECM as a Tool for Quantifying Basis Risk

The VECM specification for two Lee-Carter $\kappa_t$ factors captures basis risk through:

Error-correction terms $\rho^{(i)}(\kappa_{t-1}^{(1)} - \kappa_{t-1}^{(2)})$ : explicitly model the mean-reversion of the mortality differential, quantifying the tendency for the two populations to converge.
Symmetric structure: neither population is assumed dominant, so the model correctly represents the possibility that the sub-population leads the general population (as in the UK insured lives example).
Cointegration: the long-run constraint $\kappa_t^{(1)} - \kappa_t^{(2)} \to$ constant is automatically enforced, ensuring non-divergence without ad hoc parameter restrictions.

See Cointegration and Lee-Carter Model for details.

Why It Matters

Longevity risk is the dominant liability risk for pension funds and annuity writers; basis risk determines how much residual risk remains after hedging.
Solvency II mandates explicit quantification of longevity risk capital, making multi-population models commercially important for reinsurers pricing longevity swaps.
The difference in risk margins across models (9–47% above single-population) illustrates that model choice is not just a statistical question — it has direct financial consequences for pricing and solvency.

Open Questions

Parameter uncertainty and model risk are excluded from standard stochastic scenarios but are acknowledged to be important — how to incorporate them into risk margin calculations is an open problem.
When the mortality differential is non-stationary (diverging populations, not just mean-reverting), VECM breaks down; models with no-arbitrage restrictions on the divergence are needed.
Optimal choice of the hedging instrument in the presence of basis risk (partial hedges, customized instruments) is an active area of actuarial research.