Step 2 Construct a scale function that can be applied to the base mortality rates to generate appropriate rates for future years.
3.12.1 Single-factor mortality improvement scales
The simplest scale functions depend only on age. If we denote the improve- ment factor for agexasϕx, then, fort=1, 2, 3,. . .,
q(x,t)=q(x, 0)(1−ϕx)t.
So, ifrp(x,t) denotes the probability that a life who is agedx at time t survivesryears, using the single-factor improvement model we have
rp(x,t)=p(x,t)p(x+1,t+1)p(x+2,t+2) · · · p(x+r−1,t+r−1)
=
1−q(x,t) 1−q(x+1,t+1)
· · ·
1−q(x+r−1,t+r−1)
=
1−q(x, 0)(1−ϕx)t 1−q(x+1, 0)(1−ϕx+1)t+1 . . .
1−q(x+r−1, 0)(1−ϕx+r−1)t+r−1 .
InFigure 3.5,we show a set of age-based improvement factors published by the Society of Actuaries in 1994, known as Scale AA.
0 0.005 0.01 0.015 0.02
0 10 20 30 40 50 60 70 80 90 100
Improvement factors
Age
Males Females
Figure 3.5 Scale AA mortality improvement factors.
3.12 Mortality improvement scales 91 Table 3.8 RP2000 Male healthy annuitant mortality
rates, with Scale AA improvement factors.
Agex q(x, 0) ϕx
60 0.008196 0.016
61 0.009001 0.015
62 0.009915 0.015
63 0.010951 0.014
64 0.012117 0.014
65 0.013419 0.014
66 0.014868 0.013
67 0.016460 0.013
68 0.018200 0.014
69 0.020105 0.014
70 0.022206 0.015
Example 3.15 InTable 3.8we show base mortality rates for males in the year 2000, and we show the Scale AA mortality improvement factors, denotedϕx, for the same age range.
Calculate the 10-year survival probability for a life aged 60, with and without the mortality improvement scale.
Solution 3.15 Without mortality improvement we have
10p60= 9 t=0
(1−q(60+t, 0))=0.87441.
With mortality improvement we have
10p60 = 9 t=0
(1−q(60+t,t))
= 9 t=0
1−q(60+t, 0)(1−ϕ60+t)t
=0.88277.
As we expect, the survival probability is a little higher when we allow for
mortality improvement.
The one-factor mortality improvement scales have proven too simplistic. The AA scale predicts that mortality at age 70 would improve by 1.5% per year indefinitely, but the heatmap shows improvement rates of around 2.75% in the mid 2000s. On the other hand, the heatmap shows that the higher values of the improvement factors might not persist for later cohorts.
3.12.2 Two-factor mortality improvement scales
A more robust approach to deterministic mortality improvement scales uses improvement factors that are a function of both age and calendar year. This approach is used in the MP2014 tables of the Society of Actuaries as well as in the CPM scales of the Canadian Institute of Actuaries. The two-factor mortality improvement function isϕ(x,t), where the mortality rate for a life aged xin yeart, fort = 1, 2, 3,. . ., is determined from the base mortality rates as
q(x,t)=q(x, 0) t k=1
(1−ϕ(x,k)). (3.16) This means that ther-year survival probability,rp(x,t), is now
rp(x,t)=
1−q(x,t) 1−q(x+1,t+1)
· · ·
1−q(x+r−1,t+r−1)
= 1−
q(x, 0)(1−ϕ(x, 1))(1−ϕ(x, 2))· · ·(1−ϕ(x,t))
× 1−
q(x+1, 0)(1−ϕ(x+1, 1))· · ·(1−ϕ(x+1,t+1)) ...
× 1−
q(x+r−1, 0)(1−ϕ(x+r−1, 1))· · ·(1−ϕ(x+r−1, t+r−1)) . Example 3.16 You are given mortality rates for lives aged 50 to 60 applying in 2010 inTable 3.9,and improvement factors for 2011 to 2020 inTable 3.10.
Calculate the difference between the expected number of deaths between ages 50 and 55 from 100 000 independent lives, assuming (a) mortality follows
Table 3.9 Mortality rates forExample 3.16.
x q(x, 2010)
50 0.002768
51 0.002905
52 0.003057
53 0.003225
54 0.003412
55 0.003622
56 0.003858
57 0.004128
58 0.004436
59 0.004789
60 0.005191
3.12 Mortality improvement scales 93 Table 3.10 Improvement factors forExample 3.16.
ϕ(x, 2010+t)
x t=1 2 3 4 5 6 7 8 9 10
50 0.0206 0.0227 0.0238 0.0243 0.0241 0.0233 0.0221 0.0205 0.0188 0.0170 51 0.0180 0.0205 0.0221 0.0229 0.0230 0.0226 0.0216 0.0203 0.0188 0.0171 52 0.0156 0.0181 0.0201 0.0213 0.0218 0.0217 0.0210 0.0200 0.0186 0.0171 53 0.0124 0.0148 0.0168 0.0184 0.0193 0.0195 0.0192 0.0185 0.0175 0.0162 54 0.0093 0.0115 0.0134 0.0150 0.0164 0.0170 0.0171 0.0167 0.0160 0.0151 55 0.0066 0.0085 0.0104 0.0120 0.0134 0.0145 0.0150 0.0150 0.0146 0.0140 56 0.0045 0.0061 0.0078 0.0094 0.0109 0.0121 0.0130 0.0134 0.0134 0.0131 57 0.0033 0.0045 0.0060 0.0075 0.0090 0.0103 0.0113 0.0121 0.0125 0.0124 58 0.0031 0.0037 0.0049 0.0063 0.0078 0.0091 0.0102 0.0111 0.0117 0.0120 59 0.0039 0.0039 0.0046 0.0057 0.0071 0.0084 0.0096 0.0105 0.0112 0.0117 60 0.0055 0.0049 0.0050 0.0058 0.0069 0.0082 0.0094 0.0103 0.0110 0.0115
the base table with no improvement, and (b) mortality improvement follows the age-year improvement factors in the table, and the lives are all aged 50 in 2015.
Solution 3.16 With no mortality improvement, we have 5p50 = 0.98473, which means that the expected number of deaths before age 55 from 100 000 lives aged 50 is 100 000(1−0.98473)=1527.
With mortality improvement we have
q(50, 5)=0.002768(1−0.0206)(1−0.0227)(1−0.0238)
×(1−0.0243)(1−0.0241)
=0.002768(0.889710)=0.002463,
q(51, 6)=0.002905(1−0.0180)· · ·(1−0.0226)=0.002905(0.877640)
=0.002550,
q(52, 7)=0.003057(1−0.0156)· · ·(1−0.0210)=0.003057(0.868466)
=0.002655,
q(53, 8)=0.003225(1−0.0124)· · ·(1−0.0185)=0.003225(0.869233)
=0.002803,
q(54, 9)=0.003412(1−0.0093)· · ·(1−0.0160)=0.003412(0.875102)
=0.002986.
The five-year survival probability is then
(1−q(50, 5))(1−q(51, 6))· · ·(1−q(54, 9))=0.98662,
and so the expected number of deaths before age 55 from 100 000 lives aged 50 in 2015, allowing for mortality improvement, is 1338. The difference between
the number of expected deaths with and without mortality improvement is
therefore 189.
3.12.3 Cubic spline mortality improvement scales
In this section we describe the method used to construct the age-year improve- ment factors for the US and Canadian tables, which was first proposed by the Continuous Mortality Investigation Bureau (CMIB), a standing committee of the Institute and Faculty of Actuaries in the UK.
The improvement scales are determined in three steps.
1. Determine short-term improvement factors, using regression or other smoothing techniques applied to recent experience.
2. Determine long-term improvement factors, and the time at which the long-term factors will be reached. After this time, the factors are assumed to be constant. This step is usually based on subjective judgment.
3. Determine intermediate improvement factors using smooth functions that will connect the short- and long-term factors.
For the MP2014 tables, the Society of Actuaries used the following three steps to generate past and future improvement factors, ϕ(x,t), where x is the age (integer values from 15 to 95) andtis the calendar year from 1950 forwards.
1. Improvement factors for calendar years 1950–2007 are determined by taking the raw mortality experience from the US Social Security Admin- istration (SSA) database. A two-dimensional smoothing method is applied to the logarithm of the raw mortality rates, generating smooth log-mortality rates denoteds(x,t). The two-dimensional smoothing ensures thats(x,t)is smooth across agesxand across calendar yearst. The smoothed historical mortality rates up to 2007 are then
˜
q(x,t)=es(x,t)
and the historical improvement factors for 1950–2007 are ϕ(x,t)=1− q(x,˜ t)
˜
q(x,t−1)=1−es(x,t)−s(x,t−1).
We remark that two-dimensional smoothing techniques are beyond the scope of this book; in what follows we assume that the required smoothed historical improvement factors based on smoothed mortality rates exist.
2. Long-term improvement factors were set at 1% at all ages up to age 85, decreasing linearly to 0% at age 115, for both males and females. These factors are assumed to apply from 2027.
3. Intermediate factors covering calendar years 2008–2026 are determined usingcubic splines. Two distinct approaches could be taken to determining
3.12 Mortality improvement scales 95 these intermediate factors. The first approach is age-based, under which we use historical improvement factors and the assumed long-term improvement factors for a given age to determine the intermediate factors for that age. The second approach is cohort-based, under which we use historical improvement factors and the assumed long-term improvement factors for a cohort. If we apply each approach over all ages and cohorts, we obtain two sets of intermediate improvement factors by age and by calendar year, and these sets of factors are typically different. In practice, the approach adopted was to average the factors.
A spline is a smooth function that can be used to interpolate between two other functions. In our context, we have the historical improvement factors up to 2007, and we have the assumed long-term improvement factors applying from 2027, which are assumed to be constant for each age. A cubic spline is a cubic function of time (in years) measured from 2007 which matches the improvement function values at 2007 and 2027, and also matches the gradient of the improvement function at 2007 and at 2027. The two end points joined by the spline are calledknots. Using the two knots, and the gradients at the two knots, we have four equations, which we can solve for the four parameters of the cubic function.
Theage-based cubic spline uses a fixed age for the spline function. The four equations for the function for agexare derived as follows.
1. We set 2007 as our base year, when t = 0, as this is the last year of the historic data. From the historic data, we haveϕ(x, 2007)for each agex, and this will be set to match the cubic function att=0.
2. The first year of the assumed long-term factors is 2027, soϕ(x, 2027)will be set to match the cubic function att=20.
3. The gradient att=0 will be estimated asϕ(x, 2007)−ϕ(x, 2006), and this will be matched to the first derivative of the cubic function att=0.
4. The gradient att =20 will be estimated asϕ(x, 2028)−ϕ(x, 2027), and this will be matched to the first derivative of the cubic spline att = 20.
Generallyϕ(x, 2028)−ϕ(x, 2027)will be zero, as we assume constant long- term improvement factors.
So, lettingCa(x,t)=at3+bt2+ct+drepresent the age-based cubic spline, with derivativeCa(x,t)=3at2+2bt+c, the four conditions described above give the following four equations:
Knot att=0: Ca(x, 0)=d=ϕ(x, 2007)
Knot att=20: Ca(x, 20)=203a+202b+20c+d=ϕ(x, 2027) Gradient att=0:Ca(x, 0)=c=ϕ(x, 2007)−ϕ(x, 2006)
Gradient att=20:Ca(x, 20)=3a202+40b+c=ϕ(x, 2028)−ϕ(x, 2027)
So we immediately have the values ofcanddfrom the equations fort=0, leaving two equations to be solved foraandbto give the polynomialCa.
Thecohort-based splineis similar, but it smooths the improvement factors for a cohort, starting at agex, say, in 2007, and adding one year to the age as we move across one year in time. This gives the following four conditions for the cohort-based cubic spline.
1. The left side knot for the cohort spline for a life aged x in 2007 is ϕ(x, 2007), as for the age-based spline, and again we sett=0 at 2007.
2. The right side knot for the cohort-based spline matches the cubic function att=20 with the improvement factor for a life agedx+20 in 2027, which isϕ(x+20, 2027).
3. The left side gradient for the cohort-based spline is the difference between the improvement factor for age xin 2007 and the factor for agex−1 in 2006, that is ϕ(x, 2007)−ϕ(x−1, 2006). This is matched to the first derivative of the cohort spline att=0.
4. The right side gradient for the cohort-based spline is the difference between the long-term improvement factor for age x+21 in 2028 and the long-term improvement factor for agex+20 in 2027. That is, we use ϕ(x+21, 2028)−ϕ(x+20, 2027), which is matched to the first derivative of the cohort spline att=20.
So, lettingCc(x,t) = a∗t3+b∗t2+c∗t+d∗ represent the cohort-based cubic spline, with derivativeCc(x,t)=3a∗t2+2b∗t+c∗, the four conditions described above give the following four equations:
Knot att=0: Cc(x, 0)=d∗=ϕ(x, 2007)
Knot att=20: Cc(x+20, 20)=203a∗+202b∗+20c∗+d∗
=ϕ(x+20, 2027)
Gradient att=0:Cc(x, 0)=c∗=ϕ(x, 2007)−ϕ(x−1, 2006) Gradient att=20:
Cc(x+20, 20)=3a∗202+40b∗+c∗=ϕ(x+21, 2028)−ϕ(x+20, 2027) For the US tables, the improvement factor for agexin yeartis then taken as the average of the two splines, namely
ϕ(x, 2007+t)=0.5Ca(x,t)+0.5Cc(x,t) fort=1, 2,. . ., 19.
Example 3.17 Calculate the MP2014 one-year improvement factor for a female life aged 40 in 2020, given the following values for short- and long- term improvement factors:
3.12 Mortality improvement scales 97 ϕ(40, 2006)=0.0162, ϕ(40, 2007)=0.0192, ϕ(40, 2027)=0.01,
ϕ(40, 2028)=0.01,
ϕ(26, 2006)= −0.0088, ϕ(27, 2007)= −0.0088, ϕ(47, 2027)=0.01, ϕ(48, 2028)=0.01.
Solution 3.17 The four equations for the age-based cubic spline are Knot att=0: Ca(40, 0)=d=ϕ(40, 2007)=0.0192
Knot att=20: Ca(40, 20)=8000a+400b+20c+d=0.01 Gradient att=0:Ca(40, 0)=c=ϕ(40, 2007)−ϕ(40, 2006)=0.003 Gradient att=20:Ca(40, 20)=1200a+40b+c
=ϕ(40, 2028)−ϕ(40, 2027)=0
The equations fort=0 givec=0.003 andd=0.0192, and the equations for t=20 then yielda=9.8×10−6andb= −3.69×10−4.
So for a life aged 40 in 2020 we have
Ca(40, 13)=133a+132b+13c+d=0.01737.
The four equations for the cohort-based cubic spline for a life who is aged 40 in 2020 apply to the cohort who are aged 27 in 2007, so the four equations for the spline are
Knot att=0: Cc(27, 0)=d∗=ϕ(27, 2007)= −0.0088
Knot att=20: Cc(47, 20)=8000a∗+400b∗+20c∗+d∗=0.01 Gradient att=0:Cc(27, 0)=c∗=ϕ(27, 2007)−ϕ(26, 2006)=0 Gradient att=20:Cc(47, 20)=1200a∗+40b∗+c∗
=ϕ(48, 2028)−ϕ(47, 2027)=0 Solving as for the age-based spline we obtain
a∗= −4.7×10−6, b∗=1.41×10−4, c∗=0, d∗= −0.0088.
So for a life aged 40 in 2020 we have
Cc(40, 13)=133a∗+132b∗+13c∗+d∗=0.00470, and hence the improvement factor for age 40 in 2020 is
ϕ(40, 2020)=0.5Ca(40, 13)+0.5Cc(40, 13)=0.011035.