Contingencies Online - July/August 2011

Estimated values of A1(t) and A2(t)
MALE (1950–2007) FEMALE (1950–2007)

- 9 .0
- 10

There’s uncertainty in the estimation
of parameters μ and C. It’s possible to
address parameter uncertainty using the
Markov Chain Monte Carlo (MCMC)
method. I would refer readers to Ap-
pendix B of Cairns’ 2006 article in the
Journal of Risk and Insurance (see Ref-
erences on Page 65) for further details
regarding the MCMC that I used.
Projecting future fertility rates isn’t
a straightforward process. The offi-
cial projection made by the Japanese
government is largely qualitative, with
careful consideration of cohort-specific
characteristics. Factors such as first mar-
riage age, divorce rate, and childbearing
age are taken into account. While the
official projection is carefully done, it
isn’t stochastic. I expanded the official
projection to a stochastic one, using the
following process:
1n(B(t + 1)) – 1n( 1. 26) = ∅ 1{1n(B(t)) –
1n( 1. 26)} + σ W(t + 1)
where B(t) is the total fertility rate in year
t, ∅ 1 and σ are constants, and {W(t)} is
a sequence of standard normal random
variables.
With mortality and fertility models in
hand, it’s possible to make a population
projection. First, consider the aggregate
population P(t) in year t. In year t>2005,
the total population can be expressed by
the following formula:
P(t) = P(2005) + F(t) – G(t),
where F(t) and G(t) represent births
and deaths from year 2005 to t, respec-
tively. The population in the base year
is obtained from the total population
estimated from the national census.
For ease of explanation, I won’t discuss
the migrations.

A1(t)

- 10 .0

A1(t)

- 11 - 12

- 11 .0
1950 1970 1990
MALE (1950–2007)
- 13
1950 1970 1990
FEMALE (1950–2007)
0 .120
A2(t)
0 .095
A2(t)
0.100 0.110
0 .065
1950
1970
1990
0 .090
1950
1970
1990
estimating longevity risk
My primary objective was to examine
the effects of demographic assumptions
on the model. As such, I needed to keep
all non-demographic assumptions deter-
ministic. In particular, I assumed that the
CPI growth rate was constant over time
and fixed at 1. 5 percent.