When British mathematician and actuary Benjamin Gompertz presented his law of human mortality in 1825, his basic assumption was simply that a person becomes more likely to die while growing older.
This may have rung true to Gompertz, then a 46-year-old man already five years beyond the average life expectancy of his time. However, his simple linear equation neglected to factor in one of the important age-period effects of his time. Throughout the 19th century, infant and maternal mortality were strikingly high and, of those children who survived childbirth, almost half died before the age of five.
Though many contributions in the field of mortality law were made throughout the latter half of the 19th century, actuarial mortality modeling is a relatively new science. Largely forgotten over much of the 20th century, it became an existential question towards the end of the 1990s, when – at about the same time the dot-com bubble burst – actuaries discovered that the pensionable population was significantly outliving their models’ predictions.
Given that liabilities increase by 3% for every year a 65-year-old pensioner outlives the modeled life expectancy, the impact of lower mortality rates at higher ages was significant, with companies opting out of pension provisioning to rid themselves of legacy liabilities. “When put to the test of real-life events, existing mortality models often proved faulty, and demographers amended them with ad hoc fixes as they went along. To avoid this happening in the future, we offer a new approach for designing a mortality model from scratch,” David Blake, professor of economics at the Cass Business School in London, tells PROJECT M. Together with PhD candidate Andrew Hunt, he developed a ‘general procedure’ (GP) for constructing mortality models.
Moving away from current one-size-fits-all mortality modeling, Blake and Hunt’s new approach uses a “combination of statistical methods and expert judgment to identify sequentially every significant demographic feature in the data and give it a specific functional form.
“The GP follows individuals through time as they age. This is crucial because, as we know, life expectancy depends on year of birth,” Blake points out. “Our point was not to introduce yet another new model, but to suggest a standardized approach that could become the basis for building all country-specific models in the future.”
ACCOUNTING FOR NATIONAL DIFFERENCES
This is crucial, since we are currently talking about the wrong type of guarantees David Blake
Current models attempt to forecast mortality by imposing a fixed structure across three key features known to influence mortality: age, period and cohort (or year of birth). The GP begins by identifying all the significant age-period terms according to each country’s unique attributes. In Western countries, for instance, the mortality rate in the first six months is high and then falls off only to rise again during the teenage years when reckless behavior – particularly among males – causes the mortality rate to increase.
Once this ‘accident hump’ has been crossed, the mortality rate falls again until – at around 45 – it begins its slow, fairly linear climb. The significance of the GP is its acknowledgement of national differences. Where seven age-period terms may be identified for one country, in another it might be six; the weighting of each of the terms might also be different.
For instance, when Blake and Hunt applied the GP to UK data, they identified seven age-period terms: the general level of mortality (modeled as a constant); increasing mortality with age (modeled as an upward sloping straight line); young adult mortality, which is a humped-shaped function centered at age 25; childhood mortality (a put option); postponement of middle-age mortality to old-age mortality (a Rayleigh function); peak of the accident hump around ages 18-19 (a lognormal function); and middle-age deaths between ages 55-65 (a normal function).
IDENTIFYING COHORT EFFECTS
In comparing these results with those of the United States, they found that although the first three age-period effects were the same as those of the United Kingdom, they did not have the same relative significance. Once all the age-period terms have been identified, any significant structure remaining in the data can be associated with the cohort effect. One of the key discoveries from implementing the GP was the risk of wrongly allocating an age-period effect to a cohort effect. Since a cohort effect “follows” each cohort as it ages, whereas an age-period effect does not, then such a misallocation leads to increasingly poor mortality-rate projections over time.
The most prominent cohort effect in datasets across the world was triggered by what became known as the Spanish flu. Ravaging populations in countries around the world from 1918 to 1920, reports on this unusually deadly influenza epidemic were not censored in Spain, creating the false illusion that the country was particularly hard hit. Worldwide, 50 million to 100 million people died. Those who survived had significantly higher life expectancy than those born a few years earlier or a few years later.
When Blake and Hunt expanded their modeling approach to countries outside the United Kingdom, their results indicated that some countries didn’t have a strong cohort effect. The GP picked this up, whereas a standard age-period-cohort model applied to these countries might well estimate a statistically significant cohort effect, although the relationship would be spurious since what was being picked up was in reality another age-period effect.
For instance, whereas obesity – which has almost doubled in the United States since the early 1960s – has a cohort effect there, as described by US demographer S. Jay Olshansky, it doesn’t in Japan. “The strongest cohort effect historically recorded was triggered by the so-called Spanish flu epidemic at the end of World War I.
This is still visible in datasets today. Babies that survived the pandemic were much stronger and have had noticeably lighter mortality every year since, compared with neighboring birth cohorts.” Blake also points to increasing numbers of women picking up the bad habits of men – smoking and drinking – as another cohort effect.
According to Blake, a mortality model generated by the GP will accurately describe features observed in the past and be easy to calibrate and explain to other stakeholders. It will also capture specific mortality features for different birth years and project them as individuals age, and provide reliable forecasts of mortality rates at specific ages for longevity risk management strategies using, say, longevity swaps.
“We believe the general procedure will help demographers to look out for certain features in a dataset, whilst recognizing that these features might occur at different times in different countries, or not at all.” Apart from a more effective looking glass for demographers, Blake believes the GP will also be helpful in pricing guarantees on retirement income streams more accurately. “This is crucial, since we are currently talking about the wrong type of guarantees.
“Instead of accumulation-phase guarantees (such as guaranteed minimum returns), we should focus on trying to guarantee income during the decumulation phase. Since these guarantees are very far in the future, a good mortality forecasting model, specifically designed for the population of interest, is critical to the commercial success of those providing these guarantees.”