## Caveat emptor

I wrote earlier about survivor forwards as a means of transferring longevity risk. One natural question for investors to ask is: what is the likelihood of loss exceeding a given amount? The only sensible means of answering this question is to use a stochastic projection model — a deterministic model generates scenarios, but without attaching probabilities it cannot be used to approach this problem.

Consider an example where Party A (a pension scheme, say) offers a survivor forward based on males in England and Wales. Party A offers to pay a fixed rate of 0.47 per £10 million nominal for survivors between age 60 and 85. The investor, Party B, is asked to pay the floating leg, i.e. the actual survival rate, *S*. The net payment to Party A will therefore be (*S* - 0.47) × £10 million. If mortality improvements are faster than expected, *S* will be larger than expected and Party A will receive a payment from Party B. If Party A has longevity liabilities, such as a pension scheme, then this payment will offset the losses incurred from paying out more pensions than expected. Equally, if mortality improvements are slower than expected, *S* will be smaller than expected and Party A will have to pay the difference to Party B. Again, this shouldn't be too bad for Party A as the cost of paying this difference can be funded from the reduced reserves required for paying pensions. It is for these reasons that Party A would want to enter into a survivor-forward contract for hedging its longevity risk.

Party B (the investor, or buyer of survival risk) is naturally interested in knowing the size and probability of loss. For example, Party B might want to know the probability of losing £½ million or more on this £10 million survivor forward. Using a DDE (2007) model (a smoothed variant of the Lee-Carter model) and calibrating it to population data for England and Wales, Party A says the best estimate of the survival rate is 0.467. This suggests that the survivor forward is in the money for Party B to the tune of around £30,000 (= (0.47 - 0.467) × £10 million). Furthermore, Party A says the model suggests that the probability of Party B losing £½ million or more is just 0.13%.

However, Party B should be wary of accepting Party A's assertions, not least because of the phenomenon of *model risk*. Party B notes that Party A has used a drift projection, which is a strong assumption and leads to narrow confidence intervals. Furthermore, in an era of accelerating mortality improvements, drift models can under-state future improvements. Instead, Party B uses a more flexible ARIMA model, of which the drift model is a subset. Using an ARIMA(3,1,3) model for projection, Party B finds a best estimate of the survival rate to be 0.516, suggesting that the survivor forward is actually well out of the money at -£460,000 (= (0.47 - 0.516) × £10 million). Worse still, the ARIMA projection suggests that the probability of Party B losing £½ million or more is 46.3%. Unsurprisingly, Party B would decline to use Party A's model as the exclusive basis for agreeing the price of the survivor forward.

The lesson from this example is threefold. First, you can only sensibly value mortality derivatives using stochastic models. Second, model risk means that you cannot rely on a single model of any kind to project future mortality. And third, you should never rely solely on models picked by the seller.

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