## Haircut or hedge-trim?

Richard Willet's observation last year on the restatement of population estimates was picked up again recently by the BBC.  Amongst the implications of the missing nonagenarians are some potentially interesting consequences for index-based longevity hedges. These are derivative contracts based on population mortality data. The idea is that an organisation holding longevity risk, such as an insurer or pension fund, would buy or sell an appropriate instrument to transfer risk to an investor willing to take it. The portfolio being hedged will not have the same mortality dynamics as the population — so-called basis risk — but the idea is that the hedge will provide at least partial protection.

Imagine you are a holder of longevity risk and a few years ago you entered into a hedge based on the survival probability from age 85 to age 100 (not a very likely choice of age range, but bear with me for a moment). After you signed the contract and handed over some money, the national statistics agency announces that population estimates were over-stated.  As a result, mortality rates for people in their nineties are in fact 15% higher than previously thought, and that future mortality calculations will now reflect this going forward. Depending on the precise clauses of your contract, your hedge might have immediately turned into a liability if it were a q- or S-forward. If it were a q- or S-option then it would have moved further out of the money and become substantially less valuable. Of course, for every loser in the forward game there is a winner: your counter-party would be smiling at their unexpected good fortune.

Now, a more realistic hedge would involve ages 65–90 (say), so any real-life hedges based on ONS data over this age range aren't nearly as badly affected as my contrived example. But who's to say they might not be in the future?  After all, nobody expected the population estimates at ages 90–99 to be revised this dramatically.

One consequence of this will be to make people think very carefully about the clauses in any contracts based on population mortality indices.  Another consequence might be to make people that bit keener on indemnity reinsurance: this is a reinsurance contract which specifically covers your portfolio's longevity experience.  The possibility of the population mortality rates being unexpectedly restated is in fact an unusual form of basis risk, i.e. where you try to measure or manage your risk using a proxy instead of the actual portfolio risk itself.

Of course, it may well be that index-based derivatives contracts have always had clauses covering just this kind of eventuality.  If they didn't, I imagine they will now!

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