Too good to be true?

People in poor health don't live as long as their healthier colleagues. This obvious fact underpins the existence of the enhanced annuity market in the United Kingdom. Retirees who can demonstrate a health problem can often get a higher pension at retirement by going to a specialist annuity provider.

The same phenomenon works in reverse — for a given size of pension, the cost of providing it would be lower for those in poor health.  A recent article described how this could supposedly be used to the advantage of an employer with a defined-benefit pension scheme.  Under certain circumstances, the cost of buying out an ill-health retiree's benefits can be less than the notional reserve held for those benefits. The member benefits from insurance-company guarantees on the pension, while the scheme improves its solvency position. What's not to like?

The answer lies in the phenomenon of selection, which here is an actuarial variation of there being no such thing as a free lunch.  Basically, it means that after buying out the benefits for those in poorer health, the pension scheme should toughen its mortality assumptions as those remaining are in better-than-average health.  The scheme will therefore be left no better off in reality if the insurer has priced carefully.  Indeed, if the insurer has correctly priced all risks, the scheme might be left slightly worse off in reality as insurance-company guarantees have a cost not usually reflected in pension-scheme funding.

To illustrate this, consider a simple example of ten male lives aged 65 following 100% of S1PA "in aggregate".  Using a 3% discount rate, the reserve for a pension of £5,000 p.a. is around £671,000, i.e. £67,100 for each member on average.  However, assume that nine of the lives are healthy and follow 90% of S1PA, while one of them is unhealthy and follows 262% of S1PA.  An enhanced annuity is purchased for this unhealthy life for £47,200, which superficially looks like the scheme has "saved" nearly twenty thousand pounds on this one member (£19,900 = £67,100 - £47,200).  This appears to amount to a "saving" of around 3% of scheme liabilities (3.0% = £19,900 / £671,000).

However, the above calculation crucially ignores the effect of adverse selection.  The remaining members are now by definition healthier — the "average" mortality is now 90% of S1PA, not 100%.  The average reserve for each of the remaining nine members therefore climbs from £67,100 to £69,300, giving a total reserve of around £623,700.  The difference between this and the starting reserve of £671,000 is essentially equal to the premium charged by the life insurer.  In short, if both the insurer and the pension scheme are properly reserving, there is negligible benefit from selectively buying out ill-health lives.

In practice, the scheme could even leave itself worse off.  For example, an insurer should normally price conservatively, to cover expenses and to make a profit, so it might have charged more than the reserve-neutral figure of £47,200.  Furthermore, with a smaller membership the newly shrunk scheme will also be exposed to increased volatility of mortality-related cashflows.

Towards the end of the article, an anonymous but skeptical actuary is quoted as saying "Like anything, if it sounds too good to be true, it probably is". Nothing further need be said!

Assume we have a random variable, $$X$$, with expected value ... Read more