If the standard deviations are large, we should doubt the accuracy of the values,
or we should add a margin for conservatism. If the standard deviations
are much smaller than we expect, then we should suspect that the calculation
is giving too much weight to an arbitrary value and too little to experience.
fIgURE 2: Average Mortality rates over All years
Age Actual Scenario 1 Scenario 2 Scenario 3
100 0.327 0.325 0.324 0.328
101 0.341 0.339 0.338 0.344
102 0.356 0.353 0.352 0.361
103 0.371 0.367 0.366 0.378
104 0.387 0.381 0.380 0.396
105 0.402 0.395 0.393 0.414
Source: Author
been the underlying mortality rates of the population, but
I haven’t attempted to reflect real experience closely.
The rates for each scenario are remarkably close to the “
actual.” Note that the rates at the higher ages are further from the
actual. Even though the absolute values vary, the shape of the
tables is quite consistent.
But did I just get lucky with the three scenarios that I chose to
show? A better test is to look at the mean and standard deviation
of the graduated mortality rates over all 100 scenarios. Because
I had the rates at each age and each scenario, it was a simple matter to calculate mean and standard deviation by age across the
scenarios. The results are given in Figure 3. The table shows ages
95 to 110, all of the ages used in the graduation. (I had data for
higher ages, but it seemed there were not enough data to warrant use in the graduation.) The table also includes the “actual”
mortality rates for comparison.
Because the standard deviation in the mortality rate is under
1 percent of the mortality rate for ages under 102, I think I had
pretty good accuracy in the method. The variability is more at the
higher ages—just over 5 percent at age 110—but I think we can accept that at those high ages.
It’s also possible to test using actuarial values. The present
value of a 16-year temporary life annuity at age 95 of 1 per annum
is 2.165 at 5 percent interest. The mean of the present values of
the annuities using the simulated mortality rates is 2.171 with a
standard deviation of 0.014 (0.6 percent of the mean).
fIgURE 3: Mean and Standard deviation of
Mortality rates From Simulated deaths
Standard
Deviation
0.0021
0.0016
0.0016
0.0019
0.0022
0.0025
0.0030
0.0037
0.0049
0.0065
0.0085
0.0111
0.0141
0.0177
0.0217
0.0263
Age
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
Actual
0.253
0.268
0.282
0.297
0.312
0.327
0.341
0.356
0.371
0.387
0.402
0.417
0.433
0.449
0.465
0.485
Mean
0.253
0.267
0.282
0.296
0.311
0.325
0.340
0.355
0.370
0.385
0.400
0.416
0.432
0.448
0.464
0.481
Source: Author
for suspecting that there’s a bias in the method being tested.
If the standard deviations are large, we should doubt the accuracy
of the values, or we should add a margin for conservatism. If the
standard deviations are much smaller than we expect, then we
should suspect that the calculation is giving too much weight to
an arbitrary value and too little to experience.
BOB HOWARD is a fellow of the Society of Actuaries and
of the Canadian Institute of Actuaries. this article is excerpted
from a monograph he presented at the SoA 2011 living to
100 and beyond International Symposium, which was held
Jan. 5–7, 2011, in orlando, Fla. to read the monograph,
go to http://www.soa.org/library/monographs/life/living-
to-100/2011/mono-li11-5b-howard.pdf.
Confidence in the Work
It’s possible, by random simulations, to get a sense of how much
variability should be expected in an actuarial table you have
developed. That knowledge is important for knowing how accurate the actuarial values are likely to be.
Well-behaved means and low standard deviations should give
us confidence in our work. If mean mortality rates are materially
different from those used to simulate deaths, there’s good reason
free Tools
you can download free tools for Whittaker-Henderson graduation and for generating
random variants on a number of distributions at
http://www.howardfamily.ca/~bob. these tools
can be easily integrated into Microsoft excel
and a variety of programming languages.
