that life expectancy would level off or
reach a limit. In the 1960s, for example,
the consensus was that human beings
were unlikely to live beyond age 85. The
life expectancies for females in certain
countries, such as Japan and Hong Kong,
already have surpassed 85 years.
To account for increasing life expectancies, stochastic mortality models have
become a popular tool in calculating the
value of annuity products. The life expectancies in these models will continue
to increase, similar to what I found in my
study. But doubts still exist about using
them to model annuity products, particularly in relation to pricing. Because
there are insufficient data samples for
people aged 90 and over, modeling mortality rates of the elderly usually relies on
extrapolation methods, such as the Gom-pertz-Makeham law of mortality. This
means that we are not sure about the right
tail of the survival distribution and depend
on other assumptions to price annuity
products. This makes it difficult to evaluate
the risk of using stochastic mortality
models to price annuity products.
If mortality compression does take
place, we can use it to modify mortality
models and evaluate the risk. But,
as I found, the concept of mortality
compression isn’t always valid. If the
variance of the death age distribution
is decreasing, we would have more
confidence in pricing annuity products.
But for the historical data from Japan and
the United States, the standard deviations
and extremely high age survival
probabilities behave like constants. This
indicates that there is a non-negligible
probability that the right tail of the survival distribution is still unknown—or
without enough observations.
The probability of Survival Beyond a High Age
probability beyond M+2s is around 2
percent, for example, it’s possible that
the death age distribution is credible on
the left-hand side of M+s. One possibility for dealing with longevity risk is to
concentrate on the components that are
more credible. This would allow us to
focus more on products like annuity-cer-tain, designing annuities that are payable
up to the age of M+2s and using other
tools to cope with ages beyond M+2s.
(I would note that the age of M+2s is
approximately 100 years old, and it
would be sufficient for most people.)
There are several ways to tackle the
coverage of survival beyond the age
M+2s, but first we need to determine
the role of annuity products in planning
for retirement. If annuity products
aren’t the primary source of income,
then annuity coverage up to age M+2s
is probably enough. If the insured relies
solely on annuity products, then we need
to deal with estimating the probability of
surviving beyond age M+2s. It’s unfortu-
nate that at this time the data to estimate
the probability remain scarce.
The Theory of Life
Without a Limit
Because the concept of mortality
compression doesn’t eliminate doubts
about using stochastic mortality models,
the results of my study might lead to
other approaches. Since the survival
JACK C. YUE is a professor in the
Department of Statistics at national
Chengchi university, Taipei, Taiwan.
Cheung, Siu Lan Karen, Jean-Marie Robine,
Edward Jow-Ching Tu, and Graziella Caselli,
“Three Dimensions of the Survival Curve:
Horizontalization, Verticalization, and
Longevity Extension Source,” Demography, Vol.
42( 2), 243-258, 2005.
Fries, James F., “Aging, Natural Death, and
the Compression of Morbidity,” New England
Journal of Medicine, Vol. 303( 3), 130-135, 1980.
Kannisto, Väinö , “Measuring the Compression
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SOuRCE: JACk C. yuE
This article is adapted from a
presentation given at the SOA 2011
Living to 100 and Beyond international
Symposium, which was held Jan. 5-7,
2011 in Orlando, Fla. (http://livingto100.