and the expected number of deaths from the smoker group is
0.35N(1−(0.953.5)3)=0.14575N.
So the proportion of deaths expected to come from the smoker group is 0.14575N
N(0.14575+0.09271) =61.1%.
Even though the smokers comprise only 35% of the population, they are
expected to account for over 60% of the deaths.
3.11 Mortality improvement modelling
A challenge in developing and using survival models is that survival probabil- ities are not constant. Commonly, mortality experience gets lighter over time;
each generation, on average, lives longer than the previous generation. This can be explained by advances in health care and by improved standards of living.
Of course, there are exceptions, such as mortality shocks from war or from disease, or declining life expectancy in countries where access to health care worsens, often because of civil upheaval.
The changes in mortality over time are sometimes separated into three components: trend, shock and idiosyncratic. The trend describes the gradual reduction in mortality rates over time. We often refer to this as the longevity trend. The shock describes short-term jumps in mortality rates, often caused by war or pandemic disease. The idiosyncratic component describes year-to-year random variation that does not come from trend or shock, though it is often difficult to distinguish.
While the shock and idiosyncratic components are inherently unpredictable, we can identify trends by examining aggregate mortality patterns over a number of years. We can then allow for mortality improvement by using a survival model which depends on both age and calendar year. So, for example, we expect the mortality rate for lives who are aged 50 in 2015 to be different from the mortality rate for lives who are aged 50 in 2025; a life table that depends on both age and calendar year can be used to capture this.
In this section we present some models and methods for integrating mortality improvement into actuarial analysis for life contingent risks.
First, it might be valuable to demonstrate what we mean by mortality or longevity improvement. InFigure 3.3we show raw (that is, with no smoothing) mortality rates for US males aged 30–44 from 1960–2015, and for US females aged 50–69 for the same period, obtained from the Human Mortality Database (HMD). In each figure the higher lines are for the older ages, and the lower lines for the younger ages.
0.0050.0040.0030.0020.001
0.000 1960 1970 1980 1990
Year (a)
2000 2010
Mortality rates 0.0250.0200.0150.0100.005
0.000 1960 1970 1980 1990 2000
Year
2010
Mortality rates
(b)
Figure 3.3 US mortality experience 1960–2015 (from HMD): (a) males aged 30–44 and (b) females aged 50–69.
Overall, we see that, for each age, mortality rates are generally declining over time, although there are exceptional periods where the rates shift upwards.
We also note that the rates are not very smooth. There appears to be some random variation around the general trends.
When modelling mortality we generally smooth the raw data to reduce the impact of sampling variability. It is also common in longevity modelling to use heatmaps of mortality improvement to illustrate the two-dimensional data, rather than the age curves of mortality rates inFigure 3.3.
InFigure 3.4we show a plot of smoothed mortality improvement factors for US data, for 1951–2007. The mortality improvement factor is the percentage
3.11 Mortality improvement modelling 87
90
6.5%–7.0%
6.0%–6.5%
5.5%–6.0%
5.0%–5.5%
4.5%–5.0%
4.0%–4.5%
3.5%–4.0%
3.0%–3.5%
2.5%–3.0%
2.0%–2.5%
1.5%–2.0%
1.0%–1.5%
0.5%–1.0%
0.0%–0.5%
–0.5%–0.0%
–1.0%–0.5%
–1.5%–1.0%
–2.0%–1.5%
–2.5%–2.0%
–3.0%–2.5%
–3.5%–3.0%
87 84 81 78 75 72 69 66 63 60 57 54 51 48 45 42 39 36 33 30
1951 1954 1957 1960 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005
(a)
90
6.5%–7.0%
6.0%–6.5%
5.5%–6.0%
5.0%–5.5%
4.5%–5.0%
4.0%–4.5%
3.5%–4.0%
3.0%–3.5%
2.5%–3.0%
2.0%–2.5%
1.5%–2.0%
1.0%–1.5%
0.5%–1.0%
0.0%–0.5%
–0.5%–0.0%
–1.0%–0.5%
–1.5%–1.0%
–2.0%–1.5%
87 84 81 78 75 72 69 66 63 60 57 54 51 48 45 42 39 36 33 30
1951 1954 1957 1960 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005
(b)
Figure 3.4 US smoothed mortality improvement heatmaps, 1951–2007, for (a) males and (b) females.
reduction in the mortality rate for each age over each successive calendar year.
That is, if the smoothed mortality rate for agexin yeart isq(x,˜ t)then the smoothedmortality improvement factorat agexand yeartis
ϕ(x,t)=1− q˜(x,t)
˜
q(x,t−1).
The heatmaps and the curves inFigure 3.3 illustrate the following three effects.
Year effects
Calendar year effects are identified in the heatmaps with vertical patterns. For example, look at the years 1958–1970 in Figure 3.4(a).
The vertical lighter column for those years indicates that longevity improvement was paused or reversed for all ages in those years, though the impact was different for different age groups; the same phenomenon is apparent in the raw data inFigure 3.3(a),where we see gently rising mortality rates over most ages between 1960 and 1970. The next vertical section of the graph shows mortality improving again, with improvement more marked for younger lives (illustrated with the darker tones) than for older.
In Figures 3.3(a) and 3.4(a) we also see a very clear and severe deterioration in mortality between 1984 and 1991 most strongly affecting younger males. This area illustrates the impact of the HIV/AIDS epidemic on younger male mortality in the USA. In the following period, around 1993–2000, mortality in the same age range showed very strong improvement, as medical and social management of HIV/AIDS produced an extraordinary turnaround in the impact of the disease on population mortality.
Age effects
Age effects in the heatmaps are evident from horizontal patterns; in Figure 3.4there is little evidence of pure age effects that are protracted across the whole period. The most obvious impact of age in the heatmaps is in the way that different age groups are impacted differently by the calendar year effects. For example, the mortality improvement experienced by US females in the 1970s was more significant for people below 45 years old than for older lives.
In both the heatmaps, we see less intense patterns of improvement or decline at older ages. It is common to assume that we will not see any significant mortality improvement at the very oldest ages, say, beyond age 95. The idea is that, although more people are living to older ages, there is not much evidence that the oldest attainable age is increasing. This phenomenon is referred to as therectangularization of mortality, from the fact that the trend in longevity is generating more