VARIATIONS OF THE LINEAR LOGARITHM
HAZARD TRANSFORM FOR MODELLING
COHORT MORTALITY RATES
by
Qian Wang
B.Econ.,Shanghai University of Finance and Economics, 2011
a Project submitted in partial fulfillment
of the requirements for the degree of
Master of Science
in the
Department of Statistics and Actuarial Science
Faculty of Sciences
c Qian Wang 2014
SIMON FRASER UNIVERSITY
Spring 2014
All rights reserved.
However, in accordance with the Copyright Act of Canada, this work may be
reproduced without authorization under the conditions for “Fair Dealing.”
Therefore, limited reproduction of this work for the purposes of private study,
research, criticism, review and news reporting is likely to be in accordance
with the law, particularly if cited appropriately.
APPROVAL
Name:
Qian Wang
Degree:
Master of Science
Title of Project:
Variations of the linear logarithm hazard transform
for modelling cohort mortality rates
Examining Committee:
Dr. Tim Swartz
Professor
Chair
Dr. Cary Chi-Liang Tsai
Associate Professor
Senior Supervisor
Simon Fraser University
Dr. Yi Lu
Associate Professor
Supervisor
Simon Fraser University
Dr. Michelle Zhou
Assistant Professor
Examiner
Simon Fraser University
Date Approved:
23 January 2014
ii
Partial Copyright Licence
iii
Abstract
Observing that there is a linear relationship between two sequences of the logarithm
of the forces of mortality (hazard rates of the future lifetime) for two years, two
variations of the linear logarithm hazard transform (LLHT) model are proposed in
this project. We first regress the sequence of the logarithm of the forces of mortality
for a cohort in year y on that for a base year. Next, we repeat the same procedure
a number of times with y increased by one and the base year unchanged each time,
and produce two sequences of slope and intercept parameters which both look linear.
Then the simple linear regression and random walk with drift model are applied to
each of these two parameter sequences. The fitted parameters can be used to forecast cohort mortality rates. Deterministically and stochastically forecasted cohort
mortality rates with the two LLHT-based approaches, and the Lee-Carter and CBD models are presented, and their corresponding forecasted errors and associated
confidence intervals are calculated for comparing the forecasting performances. Applications in pricing term life insurance and annuities are also given for illustration.
keywords: linear logarithm hazard transform; Lee-Carter model; CBD model; simple
linear regression; random walk with drift
iv
To my parents!
v
Acknowledgments
I would like to take this opportunity to express my deepest gratitude to my supervisor Dr. Cary Tsai for his helpful guidance, deep understanding and his great
mentor throughout my graduate studies. Without his enlightening instruction, impressive kindness and patience, I could not have completed my project. I also want
to give my sincere thanks to the Department of Statistics and Actuarial Science for
providing me a friendly and encouraging working environment through these years.
My sincere thanks should also be extended to Dr. Yi Lu and Dr. Michelle Zhou
for their patient review and constructive comments on my project. In addition, all
courses I have taken have stimulated my strong interest in actuarial science and
statistics and thus provided me with a solid knowledge and a huge incentive to start
my research on the current topic, thus, I would like to thank all professors who have
taught me.
I am also grateful to all my fellow postgraduate students, without their encouragement, understanding and support, this journey would have been more tough and
less fun for me. My thanks also extend to my supervisor at Munich Re, Ling Guo
and all my colleagues for the help and the memorable moments they brought me.
They are all the most fantastic teammates to work with.
At last but not least, I would like to thank my family for their unconditional
love and support.
vi
Contents
Approval
ii
Partial Copyright License
iii
Abstract
iv
Dedication
v
Acknowledgments
vi
Contents
vii
List of Tables
ix
List of Figures
x
1 Introduction
1
1.1
The motivation of this project . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Outline of this project . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2 Literature review
3
3 The models and assumptions
7
3.1
Notation preliminary
. . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2
The Lee-Carter model . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.3
The CBD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.4
Linear logarithm hazard transform . . . . . . . . . . . . . . . . . . .
10
vii
4 Mortality projection
4.1
4.2
4.3
15
The Lee-Carter model . . . . . . . . . . . . . . . . . . . . . . . . . .
15
4.1.1
Deterministic mortality projection . . . . . . . . . . . . . . .
15
4.1.2
Confidence intervals under the Lee-Carter model . . . . . . .
16
The CBD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4.2.1
Deterministic mortality projection . . . . . . . . . . . . . . .
18
4.2.2
Confidence intervals under the CBD model . . . . . . . . . .
20
The LLHT model and its variations . . . . . . . . . . . . . . . . . .
21
4.3.1
Deterministic mortality projection . . . . . . . . . . . . . . .
21
4.3.2
Confidence intervals under the LLHT-based models . . . . . .
24
5 Numerical Illustrations
5.1
28
Deterministic mortality projection . . . . . . . . . . . . . . . . . . .
29
5.1.1
Parameter fitting . . . . . . . . . . . . . . . . . . . . . . . . .
29
5.1.2
Accuracy of mortality projection . . . . . . . . . . . . . . . .
29
5.2
Stochastic mortality projection . . . . . . . . . . . . . . . . . . . . .
36
5.3
Errors in pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
6 Conclusion
47
Bibliography
49
viii
List of Tables
5.1
Fitted parameters with the LLHT model for the UK males . . . . .
30
5.2
Projected parameters with the LLHT models for the UK males . . .
31
5.3
Overall average projection errors over 11 age groups . . . . . . . . .
36
ix
List of Figures
3.1
Illustration of qx0 ,t0 ,n , qxc 0 ,t0 +1,n and qxc 0 ,t0 +m,n . . . . . . . . . . . . .
3.2
µ25,1990,60 , µ25,1994,60 and µ25,1999,60 against µ25,1989,60 for the USA
males
3.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
ln(µ25,1990,60 ), ln(µ25,1994,60 ) and ln(µ25,1999,60 ) against ln(µ25,1989,60 )
for the USA males . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
8
13
ln(µc55,1956,30 ), ln(µc55,1960,30 ) and ln(µc55,1965,30 ) against ln(µ55,1955,30 )
for the USA males . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
4.1
The fitted values k̂t for the Japan females from q25,1969,30 , . . ., q25,1989,30 17
4.2
The fitted values κ̂1t for the UK males from q25,1979,30 , . . ., q25,1999,30
4.3
4.4
The fitted values
κ̂2t
for the UK males from q25,1979,30 , . . ., q25,1999,30
The fitted values α̂t for the UK males from regressing
The fitted values β̂t for the UK males from regressing
Q-Q plots of all fitted errors by regressing
22
ln(µc45,1956,35 ), . . . , ln(µc45,1975,35 )
on ln(µ45,1955,35 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
20
ln(µc45,1956,35 ), . . . , ln(µc45,1975,35 )
on ln(µ45,1955,35 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
19
23
ln(µc1956,45,35 ), . . . , ln(µc1960,45,35 )
on ln(µ1955,45,35 ) for the USA population . . . . . . . . . . . . . . . .
25
5.1
Projected errors against x0 for the USA population . . . . . . . . . .
33
5.2
Projected errors against x0 for the UK population . . . . . . . . . .
34
5.3
Projected errors against x0 for the JAP population . . . . . . . . . .
35
5.4
95% confidence intervals on
5.5
95% confidence intervals on
5.6
95% confidence intervals on
5.7
c
q45,1975,35
c
q45,1975,35
c
q45,1975,35
for the USA population . . . .
38
for the UK population . . . .
39
for the JAP population . . . .
40
Average relative errors on premiums for the USA population . . . .
44
x
5.8
Average relative errors on premiums for the UK population . . . . .
45
5.9
Average relative errors on premiums for the Japan population . . . .
46
xi
Chapter 1
Introduction
1.1
The motivation of this project
For life insurers, accurate forecast of mortality rates are essential in pricing and
reserving of life insurance products. Underestimating mortality rates results in insureds living shorter than expected, while overestimating mortality rates leads to
longevity risk - insureds live longer than expected. Life insurers are trying to reduce
mortality risk and annuity providers hope that longevity risk can be under control.
Mortality risk makes life insurers pay more death benefits to policyholders while
longevity risk increases the survival payments to annuitants for annuity providers.
In reality, mortality risk usually comes from catastrophic events, like earthquake,
flood and war. No mortality model can foresee sudden catastrophic events or how
much death rates will increase year by year. While longevity risk results from mortality improvements due to, for example, better nutrition supply, changes of lifestyle
and advanced medical techniquess.
It is crucial for those who bear mortality and longevity risks in business to
manage and control them. To hedge or reduce these risks, a few possible ways
can be adopted, for example, transferring risks to or sharing them with reinsures,
and purchasing mortality-linked securities. However, most of these ways of hedging
rely on future mortality assumptions. Without sound projection of mortality rates,
pricing of either life insurance and annuity products or mortality-linked securities
are not accurate and fair.
1
CHAPTER 1. INTRODUCTION
2
Since the law of mortality was proposed by Gompertz (1825), numerous mortality
models have been proposed. Researchers are still trying to find good ones. A sound
mortality projection model is able to provide accurate forecast of future mortality
rates, and capture the trend of mortality changes, especially improvement over time.
The main topic of this project is to project cohort mortality rates; a few mortality
projection methods will be introduced and compared.
1.2
Outline of this project
This project is organized as follows. In Chapter 2 previous works on mortality models
are reviewed, including the Lee-Carter model, the Cairns-Blake-Dowd (CBD) model,
the linear hazard transform (LHT) model and the linear logarithm hazard transform
(LLHT) model.
Chapter 3 gives more detailed introductions to the models mentioned in Chapter 2. Mathematical expressions of the models are derived, and estimation of the
parameters for each model is presented.
In Chapter 4, applying the models to forecast mortality rates is discussed, and
two variations of the original LLHT models are proposed. These two variations are
compared with the Lee-Carter model and the CBD model based on the performances
of mortality projection.
In Chapter 5, three well-developed countries, the United States of America, the
United Kingdom and Japan, are selected to conduct the analysis for comparing all
the underlying models. For each country and gender, real cohort mortality rates
from the Human Mortality Database (HMD) for certain age span in a period are
used to fit the models and estimate the associated parameters. Then the mortality
rates for some cohorts are forecasted for a future period. Comparisons between the
forecasted and real mortality rates among all models are presented in this chapter.
Furthermore, the confidence intervals on the projected mortality rates are computed
and displayed. Applications in pricing term life insurance and annuity policies are
carried out for further comparisons among all models.
Chapter 2
Literature review
Ever since the law of mortality was given by Gompertz (1825), numerous mortality
projection models have been proposed. All mortality models proposed are deterministic at the beginning. Deterministic here means projecting mortality rates without
any uncertainty. However, with the development of statistical methodologies, especially time series, stochastic models involving uncertainty become the most popular
ones nowadays. Stochastic models have the advantage that it can produce a probability distribution rather than a deterministic point for the forecasted mortality
rates; see Booth and Tickle (2008). In this chapter, a few influential articles regarding stochastic mortality models will be reviewed.
As summarized by Tabeau (2001), mortality risks are broken down by demographic variables only: usually sex and age, often cause of death (COD), and sometimes other characteristics as well. Since time is a critical factor in any projection
model, period and cohort are two ways of expressing time. Usually, male and female
have different mortality tables. Thus, for each gender, age, period and cohort are
the factors we should consider. Mortality models are classified as zero, one, two
or three-factor ones. Zero-factor models treat mortality rates as age-specific, and
there is no underlying model now. For one-factor models, mortality rates rely on age
but treat period or cohort as a function of age. As for two-factor models, besides
age, cohort or period is chosen as the other factor. Three-factor mortality models
consider all three factors, age, period and cohort.
The Lee-Carter model was proposed by Lee and Carter (1992) and has been
3
CHAPTER 2. LITERATURE REVIEW
4
proved to be an elegant and effective method of forecasting mortality rates; it is
also a representative of two-factor models. This model is widely cited and has
served as a benchmark for mortality projections. Among mortality models, discrete
time models are preferred as mortality rates are measured at most once a year and
are easy to implement. The Lee-Carter model is among the earliest discrete time
models. It models the logarithm of central death rates to be linearly correlated
with a time-varying mortality factor and adjust age-specific effects using two sets
of coefficients depending on age. Thus, the age and period effects are captured and
the development of mortality curves are well described. The Lee-Carter model has
significant advantages: the associated parameters are easy to interpret and estimate,
and selecting random walk with drift for modelling the time-dependent factors is
quite appropriate to capture the time effect. It can produce stochastic forecasts
with probabilistic prediction intervals but involving minimal subjective judgement,
as summarized by Booth and Tickle (2008).
A number of variations of the Lee-Carter model have been proposed. Booth et al.
(2002) extended the Lee-Carter model to involve higher order terms to increase the
flexibility of change in forecasting, while Renshaw and Harberman (2003a) modeled
and forecasted using univariate autoregressive integrated moving average (ARIMA)
processes. Renshaw and Harberman (2003b) also developed a model which is paralleled to the Lee-Carter model and is based on a generalized linear model (GLM).
Later, Renshaw and Harberman (2006) extended the Lee-Carter model to include
the cohort effect, which is a three-factor model and called age-period-cohort (APC)
model. Despite the disadvantage that cohort models have demand on heavy data,
the APC model successfully captures all three main effects in the UK mortality,
representing a significant improvement over the age-period and age-cohort models
(see Booth and Tickle (2008)). Furthermore, except for extending the classic LeeCarter model, improvements on the estimation of its parameters have developed
since early 1990s. Brouhns, Denuit and Vermunt (2002) resorted to a Poisson logbilinear regression model to build projected lifetables. The approach was inspired by
a comment made by Alho (2000), purposed to avoid some drawbacks of the classic
Lee-Carter model that the errors are assumed to be homoskedastic, which is unrealistic. Moreover, Li and Chan (2005) proposed an outlier adjusted model which
CHAPTER 2. LITERATURE REVIEW
5
strengthens the estimation when outliers exist in the mortality index time series.
Also, Chen and Cox (2009) incorporated a jump process into the Lee-Carter model
and compared a transitory jump process with a permanent jump process. Chen
and Cox (2009) suggested that a regime-switching model could be used to catch the
extreme movements of mortality, while Hainaut (2011) proposed a multidimensional
Lee-Carter model. Most recently, Deng et al. (2012) proposed a stochastic model
with a double-exponential jump diffusion process that captures both asymmetric
rate jumps up and down and also cohort effect in mortality trends.
Besides the Lee-Carter model, there are some models similar to two-factor models. Among them, the most famous model is the one proposed by Cairns, Blake
and Dowd (2006), abbreviated as the CBD model. This model emphasizes older
age mortality projection, especially post-age-60 population. In terms of post-age-60
population in the UK, this model performs well. The two factors link mortality rates
to age and time, or period, more precisely. As a two-factor model, one factor models
the mortality trend over time and equally affects mortality rates at all ages, whereas
the other effect on mortality rates is proportional to age; thus, it has much more
effect at higher ages than lower ages. Later, an additional age-period effect which is
quadratic in age was added to the classical CBD model in Cairns et al. (2007). Also,
the CBD model was extended to include a cohort effect by Cairns et al. (2007), and
a number of more complex versions of the CBD model were evaluated. Cairns et al.
(2008) redefined the cohort effect to be more complex.
Assuming that there exists a linear relationship between the two sequences of
the forces of mortality, Tsai and Jiang (2010) proposed the linear hazard transform
(LHT) model. This model is easy to understand and implement since it is based
on simple linear regression. When comparing performances of mortality fitting and
forecasting with the Lee-Carter and CBD models, empirical results show that the
LHT model outperforms the other two model for certain data sets. To forecast
future mortality rates, arithmetic and geometric growth methods were proposed
by assuming arithmetic and geometric growth of the slope parameter in the simple
linear regression. In addition to the accuracy of mortality projections, the significant
advantage over the Lee-Carter and CBD models is that the LHT model only requires
mortality rates for two years to estimate the parameters. However, the LHT model
CHAPTER 2. LITERATURE REVIEW
6
has a major flaw that some fitted or forecasted death probabilities could fall below
zero, which is against the law of probability. To solve this problem, Tsai (2012)
substituted the natural logarithm of mortality hazard rates by the mortality hazard
rates, and renamed it as the linear logarithm hazard transform (LLHT) model. The
arithmetic and geometric methods to project mortality rates can still be applied in
the LLHT model, which were named as the LLHT A and LLHT G models. This
model remains as good as the LHT model in forecasting performance without its
shortcoming. Yu (2013) proposed two new approaches to project mortality rates;
one was named the LLHT C model which assumed that the change in the logarithm
of the force of mortality from time −t to 0 is the same as that from time 0 to
time t, and the other was called the LLHT T model assuming that the change in
the logarithm of the forces of mortality for a sequence of the past periods of equal
length can be used to predict the change in the logarithm of the force of mortality
for the next period. Both methods show good performance, and the LLHT T model
is the overall best among the LLHT A, LLHT G, LLHT C, LLHT T, Lee-Carter
and CBD models in selected data sets.
Chapter 3
The models and assumptions
As mentioned before, the Lee-Carter model has been a popular mortality projection
model for a long time, while the CBD model proposed a few years ago is good at
mortality modelling for old age spans. In this chapter, these two models will be
briefly introduced and a new model will be proposed.
3.1
Notation preliminary
Among all measurements of mortality rate, the central death rate m, the mortality
rate q and the force of mortality µ are commonly used.
Denote mx,t the central death rate for age x in year t, and it is defined as follows,
mx,t =
number of deaths aged x last birthday in year t
.
average population aged x last year in year t
While mortality rate qx,t is the probability that an individual aged x in year t will die
between t and t + 1. The force of mortality µx,t represents the instantaneous rate of
mortality for age x in year t. The probability that death will happen between t and
t + ∆t is approximately µx,t × ∆t for small ∆t. It can be shown that mx,t =
qx,t
R1
0 s px,t ds
where s px,t is the probability that an individual aged x in year t will survive s years.
Under the assumption of constant force of mortality, that is, the force of mortality
is a constant over [x, x + 1) × [t, t + 1), or µx+r,t+s , µx,t for r, s ∈ [0, 1), the three
measurements has the following relationships:
qx,t = 1 − e−
R1
0
µx,t (s)ds
7
= 1 − exp(−µx,t )
CHAPTER 3. THE MODELS AND ASSUMPTIONS
8
and
mx,t = R 1
qx,t
0 s px,t ds
= R1
0
qx,t
psx,t ds
= −ln(px,t ) = µx,t .
The one-year survivor probability px,t can be obtained through qx,t by px,t = 1−qx,t .
More details of these actuarial concepts can be found in Bowers et al. (1997). Also
denote ux0 ,t0 ,n as a sequence of mortality rates (u can be m, q or µ) in year t0 ,
where x0 represents the starting age, and n stands for the length of the sequence.
Mathematically, this sequence can be written as
ux0 ,t0 ,n = {ux0 ,t0 , ux0 +1,t0 , . . . , ux0 +n−1,t0 }, u ∈ {m, q, µ}.
Here, the sequence is the mortality rates of individuals for the age span [x0 , x0 +n−1]
in year t0 . Another kind of sequence ucx0 ,t0 ,n is also defined as follows,
ucx0 ,t0 ,n = {ux0 ,t0 , ux0 +1,t0 +1 , . . . , ux0 +n−1,t0 +n−1 }, u ∈ {m, q, µ}.
This sequence indicates a series of mortality rates in an age span [x0 , x0 + n − 1] of
a cohort starting from x0 years old in year t0 . As displayed in Figure 3.1, sequences
of qx0 ,t0 ,n , qxc 0 ,t0 +1,n and qxc 0 ,t0 +m,n are labeled for better understanding.
Figure 3.1: Illustration of qx0 ,t0 ,n , qxc 0 ,t0 +1,n and qxc 0 ,t0 +m,n
𝒕𝟎
𝒕𝟎 + 𝟏 ⋯ 𝒕𝟎 + 𝒎 𝒕𝟎 + 𝒎 + 𝟏
𝑞𝑥0 ,𝑡0 +1 ⋯ 𝑞𝑥0 ,𝑡0 +𝑚
Age\Year
𝒙𝟎
𝑞𝑥0 ,𝑡0
𝒙𝟎 + 𝟏
𝑞𝑥0 +1,𝑡0
⋱
𝑞𝑥0 +2,𝑡0
q𝑐𝑥0 ,𝑡0 +1,𝑛
𝒙𝟎 + 𝟐
⋮
𝑞𝑥0 ,𝑡0 ,𝑛
𝒙𝟎 + 𝒏 − 𝟏 𝑞𝑥0 +𝑛−1,𝑡0
⋯
𝒕𝟎 + 𝒏
⋯ 𝒕𝟎 + 𝒎 + 𝒏 − 𝟏
⋱
⋱
⋱
q𝑐𝑥0 ,𝑡0 +𝑚,𝑛
⋱
⋱
𝑞𝑥0 +𝑛−1,𝑡0 +𝑛
𝑞𝑥0 +𝑛−1,𝑡0 +𝑚+𝑛−1
CHAPTER 3. THE MODELS AND ASSUMPTIONS
3.2
9
The Lee-Carter model
The Lee-Carter model was proposed in year 1992; ever since then, it has been widely
used as a benchmark of mortality projection. This model decomposes mx,t into ax ,
bx and kt , which are two sequences of age-dependent constants and one time series.
Mathematically, the Lee-Carter model is given by
ln(mx,t ) = ax + bx kt + x,t ,
x = x0 , . . . , x0 + n − 1; t = t0 + 1, . . . , t0 + m,
where the natural logarithm of the central death rate is related to the age-varying
constants ax and bx and the time-specific factor kt . In this model, ax represents the
age shift effect, exp(ax ) explains the general shape across the age of the mortality
schedule as mentioned in Lee and Carter (1992), and bx implies how sensitive that
ln(mx,t ) would be to the changes in kt for each age. The time series {kt } here
reveals the level of mortality in specific time t. For each fixed x, the error terms x,t ,
t = t0 + 1, . . . , t0 + m, are assumed independent and identically normally distributed
with mean zero and variance σ2x , which reflect the age’s residual effect that is not
reflected by kt .
The parameters of the Lee-Carter model are subject to the following two conP
P
straints, t kt = 0 and x bx = 1. The singular value decomposition (SVD) method
can be used to find the estimate of parameters. However, the estimation of parameters can also be done by other methods. For example, given m consecutive sequences
mx0 ,t0 +1,n , . . . , mx0 ,t0 +m,n , from the first constraint, we have
t0 +m
1 X
ln(mx,t ),
âx =
m
x = x0 , . . . , x0 + n − 1.
t=t0 +1
While the second constraint implies that kt must equal to the sum of (ln(mx,t ) − ax )
over all ages, which is
k̂t =
x0X
+n−1
[ln(mx,t ) − âx ],
t = t0 + 1, . . . , t0 + m.
x=x0
With the estimated k̂t , b̂x can be easily computed by regressing [ln(mx,t ) − âx ] on
k̂t without the constant term for each age x.
In this project, a variation of the original Lee-Carter model is conducted for comparing with the proposed new model. Instead of using sequences mx0 ,t0 +1,n , . . . , mx0 ,t0 +m,n ,
CHAPTER 3. THE MODELS AND ASSUMPTIONS
10
consecutive sequences of central death rates mcx0 ,t0 +1,n , . . . , mcx0 ,t0 +m,n representing
m cohorts are used. The method of estimating parameters is the same as the original Lee-Carter model. Intuitively, the central death rates in a parallelogram-shaped
area of data which consists m diagonal sequences shown in Figure 3.1 are used in
the fitting of parameters.
3.3
The CBD model
The CBD model proposed by Cairns et.al (2006) is commonly set as a benchmark
of mortality projection for older ages. It relates the logit of mortality rate qx,t with
a slope term and an intercept term. The mathematical expression commonly used
nowadays comes from M5 model in Cairns et.al (2009); it is given as follows:
logit(qx,t ) = κ1t + κ2t (x − x̄) + x,t ,
x = x0 , . . . , x0 + n − 1; t = t0 + 1, . . . , t0 + m,
q
where logit(qx,t ) = log( 1−qx,tx,t ) is the logit of mortality rate qx,t for an individual
P 0 +n−1
aged x in year t; x̄ = n1 xx=x
x is the average of all ages x used in fitting; κ1t
0
and κ2t are two time series of intercept and slope terms; for each fixed x, the error
terms x,t , t = t0 + 1, . . . , t0 + m, are assumed independent and identically normally
distributed with mean zero and variance σ2x , reflecting the unexplained effect of the
model.
The estimation of the parameters κ1t and κ2t in the CBD model is quite straightforward - regressing logit(qx,t ) on (x− x̄) for each year t. Originally, the m sequences
qx0 ,t0 +1,n , . . . , qx0 ,t0 +m,n are used in the CBD model to fit the model and estimate
its parameters. However, they are replaced with qxc 0 ,t0 +1,n , . . . , qxc 0 ,t0 +m,n to do the
estimation of parameters in this project in consistence with other models.
3.4
Linear logarithm hazard transform
The force of mortality (a hazard rate) can also well represent the level of a life risk.
Tsai and Jiang (2010) proposed that there is a linear relationship between hazard
rates of two life risks. Given two sequences of hazard rates of two life risks, µx,t1 ,n
CHAPTER 3. THE MODELS AND ASSUMPTIONS
11
and µx,t2 ,n , there is a linear relationship revealed by the linear hazard transform
(LHT) model, that is,
µx,t2 ,n = α × µx,t1 ,n + β + x .
Here, the error terms x of all ages are assumed independent and identically normally
distributed with mean zero and variance σ2 . The estimation of parameters α and β
comes from the least square error method of the simple linear regression.
As stated in Tsai and Jiang (2010), the LHT model outperforms the Lee-Carter
and CBD models for some mortality data sets. However, there is a major drawback
of this model, which is that some fitted mortality q̂x,t2 ,n might be negative for some
rare cases. In case of this situation, the hazard rates are replaced by the logarithm
of the hazard rates to solve this major weakness, which is called the linear logarithm
hazard transform (LLHT) model. Mathematically, this model can be expressed as
follows:
ln(µx,t2 ,n ) = α × ln(µx,t1 ,n ) + β + x
for any pair of (t1 , t2 ). Since qx,t = 1 − exp(−exp(ln(µx,t ))), exp(−exp(ln(µx,t )))
ensures that qx,t must fall within (0, 1).
Figure 3.2 displays the linear relationships between three pairs of the force of
mortality sequences, which are sequences µ25,1990,60 , µ25,1994,60 and µ25,1999,60 against
sequence µ25,1989,60 . Obvious linear trends can be observed no matter how many
years the two sequences are apart. Similar linear relationships can be observed in
Figure 3.3 for the case of the logarithm of the force of mortality.
In this project, instead of using sequences ln(µx0 ,t2 ,n ) and ln(µx0 ,t1 ,n ), the LLHT
model will be applied to two sequences ln(µcx0 ,t2 ,n ) (the logarithm of the force of
mortality for a cohort) and ln(µx0 ,t1 ,n ) (the logarithm of the force of mortality in
year t1 ) as follows:
ln(µcx0 ,t2 ,n ) = α × ln(µx0 ,t1 ,n ) + β + x .
Empirical evidence can be found in Figure 3.4 for plots ln(µc55,1956,60 ), ln(µc55,1960,60 )
and ln(µc55,1965,60 ) against ln(µ55,1955,60 ), where the linear relationships are quite
obvious.
The main advantage of adopting this variation of the LLHT model is that it
can successfully capture the mortality improvement for a certain cohort. For life
CHAPTER 3. THE MODELS AND ASSUMPTIONS
12
Figure 3.2: µ25,1990,60 , µ25,1994,60 and µ25,1999,60 against µ25,1989,60 for the USA males
0.14
μ in years 1990,1994 and 1999
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
μ in year 1989
1990
1994
1999
Linear (1990)
Linear (1994)
Linear (1999)
0.14
CHAPTER 3. THE MODELS AND ASSUMPTIONS
13
Figure 3.3: ln(µ25,1990,60 ), ln(µ25,1994,60 ) and ln(µ25,1999,60 ) against ln(µ25,1989,60 ) for
the USA males
-1.5
-7.5
-6.5
-5.5
-4.5
-3.5
-2.5
-1.5
ln(μ) in years 1990,1994 and 1999
-2.5
-3.5
-4.5
-5.5
-6.5
-7.5
ln(μ) in year 1989
1990
1994
1999
Linear (1990)
Linear (1994)
Linear (1999)
CHAPTER 3. THE MODELS AND ASSUMPTIONS
14
insurance companies, if n-year term life insurance is priced based on mortality rates
for a certain age in some year t without considering mortality improvement, then
c
the mortality rates qx,t,n will be applied. But once qx,t,n
is adopted, pricing term
life insurance takes into account the future mortality rates for the cohort that the
policyholder belongs to. Mortality models with this kind of variation can forecast
future mortality rates for a cohort, which is quite useful for more accurate pricing
and reserving for life insurance products.
Figure 3.4: ln(µc55,1956,30 ), ln(µc55,1960,30 ) and ln(µc55,1965,30 ) against ln(µ55,1955,30 ) for
the USA males
-1.5
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
ln(μ) for cohort in years 1956,1960 and 1965
-2.0
-2.5
-3.0
-3.5
-4.0
-4.5
ln(μ) in year 1955
1956
1960
1965
Linear (1956)
Linear (1960)
Linear (1965)
Chapter 4
Mortality projection
In Chapter 3, mathematical forms and parameter estimations for two classic mortality models and a new one with two variations are introduced. In this chapter,
mortality projection methods used in each of the two classical models will be studied in detail along with two variations of the LLHT model in deterministic and
stochastic views.
4.1
4.1.1
The Lee-Carter model
Deterministic mortality projection
As introduced in the preceding chapter, the Lee-Carter model relates the logarithm
of central death rates with two age-dependent sequences of constants ax and bx along
with a time series kt . Hence this time series is the only factor links mortality rates
with time, which makes it the key factor for future mortality projections. Empirical
evidence shows that the factor kt is always decreasing over time, which explains the
trend of mortality improvements. Lee and Carter (1992) tried to use an ARIMA
time series model for k̂t , and eventually the random walk with drift model was
preferred. Mathematically, it can be expressed by
k̂t = k̂t−1 + c + et ,
where c represents a constant and the random errors et are assumed independent and
identically normally distributed with mean zero and variance σe2 . The fitted values
15
CHAPTER 4. MORTALITY PROJECTION
16
k̂t with 21 consecutive mortality sequences m25,1969,30 , . . ., m25,1989,30 for Japan
females from Human Mortality Database (http://www.mortality.org/) are plotted
in Figure 4.1. It is observed that random walk with drift is a proper model for the
time series k̂t . With m consecutive mortality sequences mx0 ,t0 ,n , . . ., mx0 ,t0 +m,n , c
is estimated by
ĉ =
1
m−1
t0 +m−1
X
t=t0 +1
(k̂t+1 − k̂t ) =
1
(k̂t +m − k̂t0 +1 ).
m−1 0
Thus, estimation of the constant c implies that the future values of k̂t can be predicted based on historical values. To project future mortality rates for an individual
aged x in year t0 + m + τ from year t0 + m, the logarithm of the projected central
death rate will be
˜
ln(m̂x,t0 +m+τ ) = âx + b̂x × k̂t0 +m+τ .
˜
where k̂t0 +m+τ = k̂t0 +m + τ × ĉ. Transforming it to the predicted one-year death
rate q̂x,t gives
˜
q̂x,t0 +m+τ = 1 − exp{−exp{ln(m̂x,t0 +m+τ )}} = 1 − exp{−exp{âx + b̂x × k̂t0 +m+τ }}.
However, as mentioned before, a variation of the Lee-Carter model is applied
in this project, and sequences mcx0 ,t0 +1,n , . . ., mcx0 ,t0 +m,n for m different cohorts are
used to estimate the parameters. We test with empirical data and find that {k̂t }
estimated from cohort mortality sequences still follows a random walk with drift.
4.1.2
Confidence intervals under the Lee-Carter model
In the Lee-Carter model, random terms exist to reflect the possible volatility of
projected mortality rates, which can be used to construct a confidence interval on
each predicted value. A good confidence interval is the one which is narrow and
can cover the real mortality rate. The characteristics of confidence intervals are also
evaluated when comparing among mortality projection models.
Standard deviation must be calculated at first to construct a confidence interval.
In the Lee-Carter model, the random errors come from two possible sources, one
from et for the time series kt , and the other from x,t which reflects the age’s residual
CHAPTER 4. MORTALITY PROJECTION
17
-10
-5
0
kt
5
10
Figure 4.1: The fitted values k̂t for the Japan females from q25,1969,30 , . . ., q25,1989,30
1970
1975
1980
1985
year
effect. The following unbiased sample variances are used to estimate the variances
of the two error terms,
σ̂e2 =
and
σ̂2x =
1
m−2
t0 +m−1
X
[k̂t+1 − k̂t − ĉ]2 ,
t=t0 +1
tX
0 +m
1
[ln(mx,t ) − âx − b̂x × k̂t ]2
m−2
t=t0 +1
for each age x, where âx , b̂x and k̂t represent the estimated parameters of ax , bx and
kt , respectively.
Combining the two error terms, the estimate of the variance of the logarithm of
the predicted central death rate is
σ̂ 2 (ln(m̂x,t0 +m+τ )) = τ × b̂2x × σ̂e2 + σ̂2x .
Since we assume that both error terms are independent and normally distributed,
the confidence interval with significance level α on the one-year death probability
CHAPTER 4. MORTALITY PROJECTION
18
qx,t0 +m+τ can be derived as follows:
qx,t0 +m+τ ∈ 1 − exp{−exp{ln(m̂x,t0 +m+τ ) ± z(1− α2 )% × σ̂(ln(m̂x,t0 +m+τ ))}}
where z(1− α2 )% is the (1 −
α
2 )%th
percentile of the standard normal distribution.
Obtaining the confidence interval helps evaluate a model effectively, which will be
discussed later.
4.2
The CBD model
4.2.1
Deterministic mortality projection
Similar to the Lee-Carter model, the CBD model also assumes that the fitted time
series factors κ̂1t and κ̂2t follow the random walk with drift. Specifically,
κ̂t = κ̂t−1 + c + V Zt
where
• κ̂t = (κ̂1t , κ̂2t )0 ,
• c is a 2 × 1 vector (c1 , c2 )0 ,
• V is a 2 × 2 matrix, consisting of variance and covariance of κ̂1t and κ̂2t , given
by
σ 2 (κ̂1 )
cov(κ̂1t , κ̂2t )
t
V =
cov(κ̂1t , κ̂2t )
σ 2 (κ̂2t )
,
• Zt is a 2 × 1 standard normal random vector.
We test with empirical data and find that using random walk with drift for κ̂1t and
κ̂2t is appropriate as shown in Figures 4.2 and 4.3, where q25,1979,30 , . . ., q25,1999,30
for UK males from Human Mortality Database are used to estimate parameters.
Parameters of the two-dimensional random walk with drift can be estimated in a
similar way to the method used in the Lee-Carter model. That is,
ĉ =
1
m−1
t0 +m−1
X
t=t0 +1
(κ̂t+1 − κ̂t ) =
1
(κ̂t +m − κ̂t0 +1 ).
m−1 0
CHAPTER 4. MORTALITY PROJECTION
19
Future time-varying factors κ̂t can be extrapolated based on historical factors for
mortality projections. The logit of the projected one-year death rate can be obtained
for lives aged x in year t0 + m + τ , that is,
˜1
˜2
logit(q̂x,t0 +m+τ ) = κ̂
t0 +m+τ + κ̂t0 +m+τ (x − x̄);
equivalently, q̂x,t0 +m+τ can be obtained by
q̂x,t0 +m+τ = 1 −
1
.
1 + exp(logit(q̂x,t0 +m+τ ))
−9.4
−9.8
−9.6
kt1
−9.2
−9.0
Figure 4.2: The fitted values κ̂1t for the UK males from q25,1979,30 , . . ., q25,1999,30
1980
1985
1990
1995
year
Note that in this project, the same mortality sequences for m cohorts are used
in fitting the model and estimating the parameters as the Lee-Carter model. The
assumption that {κ̂1t } and {κ̂2t } follow random walk with drift still remain valid
based on empirical data.
CHAPTER 4. MORTALITY PROJECTION
20
0.080
0.070
kt2
0.090
Figure 4.3: The fitted values κ̂2t for the UK males from q25,1979,30 , . . ., q25,1999,30
1980
1985
1990
1995
year
4.2.2
Confidence intervals under the CBD model
Acquiring the confidence interval on mortality rates is one of ways to evaluate the
performance of predictions. Thus, for the CBD model, the confidence interval can
be derived from the standard deviation of the logit of one-year death rate qx,t .
Similarly, the total error contains two parts, one from the two time series, and the
other from the residual error which cannot be reflected by the model. With m
consecutive mortality sequences qx0 ,t0 +1,n , . . ., qx0 ,t0 +m,n used in the fitting stage,
when predicting mortality rates for τ years later from year t0 + m, the variance of
logit of qx,t can be expressed as follows,
σ 2 (logit(q̂x,t )) = τ × σκ21 + τ × σκ22 × (x − x̄)2 + σ2x ,
t
t
where σ2x can be estimated by
σ̂2x
tX
0 +m
1
=
[logit(qx,t ) − κ̂1t − κ̂2t × (x − x̄)]2
m−2
t=t0 +1
CHAPTER 4. MORTALITY PROJECTION
21
for each age x, and
σ̂κ21
t
1
=
m−2
and
σ̂κ22 =
t
1
m−2
t0 +m−1
X
[κ̂1t+1 − κ̂1t − ĉ1 ]2 ,
t=t0 +1
t0 +m−1
X
[κ̂2t+1 − κ̂2t − ĉ2 ]2 .
t=t0 +1
After determining the standard deviation of logit(qx,t ), the confidence interval with
significance level α on one-year death probability for age x in year t0 + m + τ can
be computed as
qx,t0 +m+τ ∈ 1 −
4.3
1
1 + exp[logit(q̂x,t0 +m+τ ) ± z(1− α2 )% × σ̂(logit(q̂x,t0 +m+τ ))]
.
The LLHT model and its variations
4.3.1
Deterministic mortality projection
Recall there is a linear relationship between two sequences of the logarithm of forces
of mortality, ln(µx0 ,t1 ,n ) and ln(µcx0 ,t2 ,n ); when regressing ln(µcx0 ,t2 ,n ) on ln(µx0 ,t1 ,n ),
the slope and intercept parameters α and β can be obtained. When m consecutive
sequences ln(µcx0 ,t0 +t,n ), t = 1, 2, . . . , m, for m different cohorts are regressed on
ln(µx0 ,t0 ,n ), respectively, then m pairs of the fitted parameters [α̂t , β̂t ] are acquired.
That is,
ln(µcx0 ,t0 +t,n ) = αt × ln(µx0 ,t0 ,n ) + βt + t ,
t = 1, 2, . . . , m,
where t = {x0 ,t , . . . , x0 +n−1,t } and x0 ,t , . . . , x0 +n−1,t are independent and identically normally distributed with mean zero and variance σ2t for each fixed t. Modelling the trend of parameters can project ln(µcx0 ,t0 +t,n ) for year t ≥ m + 1. In
this section, two assumptions of modelling α and β are proposed, along with the
confidence intervals accordingly.
Figures 4.4 and 4.5 display the time series α̂t and β̂t for UK males from Human
Mortality Database by regressing ln(µc45,1956,35 ), . . ., ln(µc45,1975,35 ) on ln(µ45,1955,35 ).
Empirical evidence shows that each of α̂t and β̂t follows an approximately straight
line against time index t. We will make two assumptions about the two factors
CHAPTER 4. MORTALITY PROJECTION
22
standing for the time trend; one is the linear model, the other is the random walk
with drift model.
0.84
0.80
alpha
0.88
0.92
Figure 4.4:
The fitted values α̂t for the UK males from regressing
c
ln(µ45,1956,35 ), . . . , ln(µc45,1975,35 ) on ln(µ45,1955,35 )
5
10
15
20
t
LLHT-LR
Assuming that each of the fitted parameter sequences {α̂t } and {β̂t } has a linear
relationship against the time sequence {1, . . . , m}. This variation of the LLHT model
is called LLHT-LR in short. More specifically,
α̂t = a × t + b + eα,t ,
t = 1, 2, . . . , m
β̂t = c × t + d + eβ,t ,
t = 1, 2, . . . , m,
and
where a, b, c and d are all constants, and eγ,t , t = 1, 2, . . . , m, γ = {α, β}, are the
error terms assumed independent and identically normally distributed with mean
zero and variance σe2γ , γ = {α, β}. The least square error method is applied to
ˆ the estimates of a, b, c and d.
obtain â, b̂, ĉ and d,
CHAPTER 4. MORTALITY PROJECTION
23
-0.8
-1.2
-1.0
beta
-0.6
-0.4
Figure 4.5:
The fitted values β̂t for the UK males from regressing
c
ln(µ45,1956,35 ), . . . , ln(µc45,1975,35 ) on ln(µ45,1955,35 )
5
10
15
20
t
To project mortality rates for the cohorts born later, the corresponding α̂ and β̂
can be predicted by the regression line. Thus, the logarithm of the forecasted force
of mortality for the cohorts m + τ years later from year t0 would be
˜ m+τ × ln(µx,t ) + β̂˜m+τ ,
ln(µ̂cx,t0 +m+τ ) = α̂
0
x = x0 , . . . , x0 + n − 1; τ = 1, 2, . . . ,
where
˜ m+τ = â × (m + τ ) + b̂
α̂
and
˜
ˆ
β̂m+τ = ĉ × (m + τ ) + d.
LLHT-RW
Another way of projecting mortality rates is to apply a time series model to
the fitted parameters in the LLHT model, that is, each of {αt } and {βt } is timedependent and follows a time series. We assume a random walk with drift for each
of {αt } and {βt } as follows:
α̂t = α̂t−1 + f + eα,t
CHAPTER 4. MORTALITY PROJECTION
24
and
β̂t = β̂t−1 + g + eβ,t ,
where f and g are drift constants and the error terms {eγ,t } are assumed independent
and identically normally distributed with mean zero and variance σe2γ , γ = {α, β}.
Briefly, this variation of the LLHT model is called the LLHT-RW model. The
constants f and g can be estimated by the same method mentioned in the LeeCarter model. That is,
fˆ =
m−1
1
1 X
(α̂t+1 − α̂t ) =
(α̂m − α̂1 )
m−1
m−1
t=1
and
ĝ =
m−1
1 X
1
(β̂t+1 − β̂t ) =
(β̂m − β̂1 ).
m−1
m−1
t=1
Thus, the logarithm of the forecasted force of mortality can be derived similarly as
˜ m+τ × ln(µx ,t ,n ) + β̂˜m+τ ,
ln(µ̂cx0 ,t0 +m+τ,n ) = α̂
0 0
where
˜ m+τ = τ × fˆ + α̂m ,
α̂
and
˜
β̂m+τ = τ × ĝ + β̂m .
Transforming ln(µ̂x0 ,t0 +m+τ ) obtained by either LLHT-LR or LLHT-RW into
c
the one-year death probability q̂x,t
yields
0 +m+τ
q̂xc 0 ,t0 +m+τ = 1 − exp{−exp{ln(µ̂cx0 ,t0 +m+τ )}}.
4.3.2
Confidence intervals under the LLHT-based models
Confidence intervals are essential for evaluating models. The methods and expressions related to confidence intervals based on the LLHT model will be discussed in
this section.
LLHT-LR
CHAPTER 4. MORTALITY PROJECTION
25
0.00
Sample Quantiles
-0.05
0.00
-0.10
-0.10
-0.05
Sample Quantiles
0.05
0.05
0.10
0.10
Figure
4.6:
Q-Q
plots
of
all
fitted
errors
by
regressing
ln(µc1956,45,35 ), . . . , ln(µc1960,45,35 ) on ln(µ1955,45,35 ) for the USA population
-2
-1
0
1
2
Quantiles in standard normal distribution
-2
-1
0
1
2
Quantiles in standard normal distribution
(a) USA Male
(b) USA Female
The variance for the LLHT-LR model comes from two sources: one is the error
term reflecting the part that the whole model cannot explain, and the other comes
from the variance of the predicted future parameters.
Recall that for each year t the error terms x,t , x = x0 , . . . , x0 +n−1, in the LLHT
model are assumed independent and identically normally distributed with mean zero
and variance σ2t . Since empirical data show that the variance of x,t does not increase
in t, we assume that all x,t , x = x0 , . . . , x0 + n − 1, t = t0 + 1, . . . , t0 + m, are i.i.d.
random variables with mean zero and variance σ2 . To support this assumption, the
Q-Q plots of all fitted errors (the logarithm of the real forces of mortality minus
the logarithm of the fitted ones) by regressing ln(µc1956,45,35 ), . . . , ln(µc1960,45,35 ) on
ln(µ1955,45,35 ) are shown in Figure 4.6. Observing from these two Q-Q plots, we
see that empirical values of x,t are in approximate line with a normal distribution.
Thus, the assumption that all x,t , x = x0 , . . . , x0 + n − 1, t = t0 + 1, . . . , t0 + m, are
i.i.d. random variables is reasonable.
The variance of the error terms, σ2 , can be estimated easily by averaging σ̂2t
CHAPTER 4. MORTALITY PROJECTION
26
over t = 1, . . . , m as
Pm 2
Pm Pn
(ln(µx0 +k−1,t0 +t ) − α̂t × ln(µx0 +k−1,t0 ) − β̂t )2
t=1 σ̂t
2
σ̂ =
= t=1 k=1
.
m
m × (n − 2)
Recall that if for a simple linear relationship y = β0 + β1 x + , the estimate of
the standard deviation of the predicted ŷ is
s
1
(ξ − x̄)2
σ̂(ŷ|x = ξ) = σ̂ 1 + + Pn
2
n
i=1 (xi − x̄)
where ξ is not in the original fitting data set, and σ̂ is the standard variance of the
error terms. Thus, the standard deviations of the predicted α̂m+τ and β̂m+τ can be
calculated as
s
1+
(m + τ − t̄)2
1
+ Pm
2
m
t=1 (t − t̄)
1+
1
(m + τ − t̄)2
,
+ Pm
2
m
t=1 (t − t̄)
σ̂(α̂m+τ ) = σ̂eα
and
s
σ̂(β̂m+τ ) = σ̂eβ
where
m
σ̂e2α
1 X
=
[α̂t − â × t − b̂]2
m−2
t=1
and
m
σ̂e2β
1 X
ˆ 2.
=
[β̂t − ĉ × t − d]
m−2
t=1
LLHT-RW
With the assumption that each of {αt } and {βt } follows a random walk with drift,
the standard deviation is slightly different from above. In this case, the variances of
the predicted α̂m+τ and β̂m+τ are now calculated by
σ̂ 2 (α̂m+τ ) = σ̂e2α × τ
and
σ̂ 2 (β̂m+τ ) = σ̂e2β × τ,
where
σ̂e2α =
m−1
1 X
[α̂t+1 − α̂t − fˆ]2
m−2
t=1
CHAPTER 4. MORTALITY PROJECTION
and
σ̂e2β
27
m−1
1 X
[β̂t+1 − β̂t − ĝ]2 .
=
m−2
t=1
Note that the variance of remains the same as the LLHT-LR model.
Similarly in the two variations of the LLHT model, combining the two sources of
variances gives the total variance of the logarithm of the forecasted force of mortality
as
σ̂ 2 (ln(µ̂cx,t0 +m+τ,n )) = σ̂ 2 (α̂m+τ ) × [ln(µx,t0 ,n )]2 + σ̂ 2 (β̂m+τ ) + σ̂2 .
Then we may construct the confidence interval with significance level α on the oneyear death probability as
c
q̂x,t
∈ 1 − exp{−exp{ln(µ̂cx,t0 +mτ ) ± z(1− α2 )% × σ̂(ln(µ̂cx0 ,t0 +m+τ , n))}.
0 +m+τ
Note that the variances of the predicted parameters σ̂ 2 (α̂m+τ ) and σ̂ 2 (β̂m+τ ) are
different in the LLHT-LR and LLHT-RW models.
Chapter 5
Numerical Illustrations
In this chapter, empirical mortality data are used to fit and forecast mortality rates
with the Lee-Carter, CBD, LLHT-LR and LLHT-RW models mentioned in the preceding two chapters. To compare the accuracy of the projected mortalities among
models, some statistical measures are proposed and calculated; also the confidence
intervals on the forecasted mortality rates are computed and compared in this chapter.
In this project, the mortality data used to estimate parameters in the models
come from Human Mortality Database (HMD). Three developed countries, the United States of America, the United Kingdom and Japan, are selected in this project.
All the three countries are good representatives for their advanced medical techniques and well-organized social systems with own characteristic. The USA has the
most advanced medical techniques but has no nationwide care system, while the UK
has its own National Health Service which has worked for over half a century, and
Japan is famous for its longevity which is the trend for worldwide demographics.
For the Lee-Carter and CBD models, 5 sequences qxc 0 ,1956,35 , . . ., qxc 0 ,1960,35 are used
for fitting (m = 5, the number of years for fitting), and 15 sequences qxc 0 ,1961,35 , . . .,
qxc 0 ,1975,35 are predicted (M = 15, the number of years for predicting). For the LLHT
models, qx0 ,1955,35 for the base year is also used for estimating the parameters. Thus,
the mortality rates for 15 cohorts are predicted for the age span [x0 , x0 +35−1]. The
latest HMD contains mortality data up to 2009 for Japan; the predicted mortality
rates will be compared with actual ones, and some statistics will be calculated to
28
CHAPTER 5. NUMERICAL ILLUSTRATIONS
29
quantify the errors.
5.1
5.1.1
Deterministic mortality projection
Parameter fitting
Table 5.1 shows parameters α̂t and β̂t estimated with the mortality data for the
UK males.
Here, five sequences ln(µc40,1956,35 ), . . ., ln(µc40,1960,35 ) are regressed
on ln(µ40,1955,35 ), respectively, which produces five pairs of parameters labeled as
[α̂1 , β̂1 ], . . . , [α̂5 , β̂5 ]. The trend of sequences of the fitted slope and intercept parameters reveals information on mortality improvement or deterioration. Since
ln(µcx0 ,t0 +t,n ) = αt × ln(µx0 ,t0 ,n ) + βt + t ,
t = 1, 2, . . . , m,
the fitted one-year survival probability can be derived as
p̂cx0 ,t0 +t,n = exp{−exp{α̂t × ln(µx0 ,t0 ,n ) + β̂t }}.
Thus, the decreasing trend of α̂t and β̂t in Table 5.1 gives increasing survival probabilities which is interpreted as mortality improvement.
Applying the two assumptions made in Chapter 4, linear regression and random
walk with drift are conducted, respectively, on those pairs of parameters; then fifteen
pairs of parameters are predicted for the consecutive 15 sequences ln(µc40,1961,35 ), . . .,
ln(µc40,1975,35 ). Table 5.2 shows the forecasted slope and intercept parameters. Thus,
the logarithm of the predicted forces of mortality can be calculated by ln(µ40,1955,35 )
and the predicted parameters, which give the one-year death probabilities for the 15
cohorts. As observed from the predicted parameters, the decreasing trend of αt and
βt implies decreasing logarithm of forces of mortality or increasing one-year survival
probabilities. This trend is consistent with the reality as younger cohorts tend to
have lower mortality rates.
5.1.2
Accuracy of mortality projection
Recall that mortality rates of different forms, ln(mx,t ), logit(qx,t ) and ln(µx,t ), are
modelled by the Lee-Carter, CBD and LLHT models, respectively, in this project.
CHAPTER 5. NUMERICAL ILLUSTRATIONS
30
Table 5.1: Fitted parameters with the LLHT model for the UK males
base year 1955
fitting year
α
β
1956
0.91683 -0.46815
1957
0.91199 -0.48971
1958
0.90369 -0.56344
1959
0.89757 -0.65329
1960
0.89381 -0.54010
To evaluate and compare these three models, the three forms of mortality rates are
unified into the one-year death probability, qx,t . Also, three statistics are introduced
to calculate the errors for evaluating accuracy of projection, which are the mean
absolute error (M AE), the root mean square error (RM SE) and the mean absolute
percentage error (M AP E). All those statistics measure the level of deviation of
the projected mortality rates form the real ones. M AE and RM SE measure the
absolute deviation while M AP E uses a percentage to represent relative deviance.
Denote q̂x,t,n and qx,t,n the projected and real mortality rate sequences, t =
t0 + m + 1, ..., t0 + m + M . The three types of errors can be calculated over the age
span of one cohort, or over all forecast years with the same age. The M AEx , RM SEx
and M AP Ex for age x in all cohorts over M forecast years t0 +m+1, . . . , t0 +m+M
are defined by
t +m+M
1 0X
|q̂x,t − qx,t |,
M AEx =
M
t=t0 +m+1
v
u
X
u 1 t0 +m+M
RM SEx = t
[q̂x,t − qx,t ]2
M
t=t0 +m+1
and
1
M AP Ex =
M
t0 +m+M
X
t=t0 +m+1
|
q̂x,t − qx,t
| × 100%.
qx,t
c
The M AEt , RM SEt and M AP Et for qx,t,n
over the age span [x0 , x0 + n − 1] are
given by
n
M AEt =
1X
|q̂x+k−1,t+k−1 − qx+k−1,t+k−1 |,
n
k=1
CHAPTER 5. NUMERICAL ILLUSTRATIONS
31
Table 5.2: Projected parameters with the LLHT models for the UK males
base year 1955
forecast year
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
LLHT-LR
α
β
0.88665 -0.63518
0.88060 -0.66593
0.87456 -0.69667
0.86851 -0.72742
0.86247 -0.75817
0.85642 -0.78892
0.85038 -0.81966
0.84433 -0.85041
0.83829 -0.88116
0.83224 -0.91190
0.82620 -0.94265
0.82015 -0.97340
0.81411 -1.00415
0.80807 -1.03489
0.80202 -1.06564
LLHT-RW
α
β
0.88806 -0.55809
0.88231 -0.57608
0.87655 -0.59406
0.87080 -0.61205
0.86504 -0.63004
0.85929 -0.64802
0.85354 -0.66601
0.84778 -0.68400
0.84203 -0.70199
0.83628 -0.71997
0.83052 -0.73796
0.82477 -0.75595
0.81901 -0.77393
0.81326 -0.79192
0.80751 -0.80991
v
u n
u1 X
RM SEt = t
[q̂x+k−1,t+k−1 − qx+k−1,t+k−1 ]2
n
k=1
and
n
M AP Et =
1 X q̂x+k−1,t+k−1 − qx+k−1,t+k−1
|
| × 100%.
n
qx+k−1,t+k−1
k=1
However, when comparing the errors, the overall M AE and M AP E over the age
span [x0 , x0 + n − 1] and all projection years t0 + m + 1, . . . , t0 + m + M are computed
by taking average as follows:
M AE =
1
M
and
1
M AP E =
M
t0 +m+M
X
t=t0 +m+1
t0 +m+M
X
t=t0 +m+1
M AEt =
x +n−1
1 0X
M AEx
n x=x
0
x +n−1
1 0X
M AP Et =
M AP Ex .
n x=x
0
For RM SE, the two ways of getting the overall RM SE will not produce the same
results as the overall M AE and M AP E due to the square root. The overall RM SE
CHAPTER 5. NUMERICAL ILLUSTRATIONS
here is defined as
1
RM SE =
M
t0 +m+M
X
32
RM SEt .
t=t0 +m+1
In this project, the starting age x0 is selected from 40 to 50; thus, the mortality
rates for eleven age groups are forecasted for comparisons among models. For each
starting age x0 = 40, ..., 50, fifteen cohort sequences qxc 0 ,1961,35 , . . . , qxc 0 ,1975,35 are
projected, and the overall M AE, M AP E and RM SE are obtained. The three
types of projection errors plotted against the starting age x0 for the USA, the UK
and Japan are displayed in Figures 5.1, 5.2 and 5.3, respectively. The average levels
over the eleven starting ages are also shown in the corresponding figure.
As observed from Figure 5.1, the LLHT-LR model produces the least average
errors except RM SE for the USA females where the Lee-Carter model wins. As
the starting age gets older, the trends of M AE and RM SE for all four models
move up, whereas the M AP E does not show increasing trends; that is, low RM SE
and M AE does not imply small M AP E. Also, the overall shapes of three types
of errors between both genders look similar. The magnitudes of M AE and RM SE
for the males are generally higher than those for the females due to larger male
mortality rates; however, there is no much difference in the size of M AP E between
both genders since a higher absolute error in the numerator of M AP E is usually
offset by a larger real mortality rate in denominator.
Figure 5.2 shows the error curves for the UK males and females. Though the
LLHT-LR model does not outperform the other three for most of age groups, it
still produces the lowest average errors over the eleven age groups because of its
big win in the first age group [40, 74] (all the other three models have quite high
projection errors except the LLHT-LR model). Same situations as those for the USA
population, the patterns of three types of errors and the size of M AP E between
both genders of the UK are similar.
Observed from Figure 5.3, which displays the error curves for the Japan population, the two variations of the LLHT models are not performing as good as the other
countries. The Lee-Carter model is the best projection model for M AE and RM SE;
however, the LLHT-LR and LLHT-RW models still have satisfactory performance
CHAPTER 5. NUMERICAL ILLUSTRATIONS
33
Figure 5.1: Projected errors against x0 for the USA population
0.0020
USA M
0.0017
0.0020
0.0013
MAE
0.0025
0.0015
0.0005
0.0003
0.0000
0.0000
0.0045
0.0040
0.0038
0.0033
0.0030
0.0027
RMSE
0.0007
0.0023
0.0020
0.0015
0.0013
0.0008
0.0007
0.0000
0.0000
0.1200
0.1200
0.1000
0.1000
0.0800
0.0800
0.0600
0.0600
0.0400
0.0400
0.0200
0.0200
0.0000
0.0000
LC
CBD
LR
RW
USA F
0.0010
0.0010
MAPE
MAPE
RMSE
MAE
0.0030
LC
CBD
LR
RW
CHAPTER 5. NUMERICAL ILLUSTRATIONS
34
Figure 5.2: Projected errors against x0 for the UK population
0.0035
UK M
0.0029
0.0047
0.0023
MAE
0.0058
0.0035
0.0012
0.0006
0.0000
0.0000
0.0090
0.0050
0.0075
0.0042
0.0060
0.0033
RMSE
0.0012
0.0045
0.0025
0.0030
0.0017
0.0015
0.0008
0.0000
0.0000
0.3500
0.3500
0.2917
0.2917
0.2333
0.2333
0.1750
0.1750
0.1167
0.1167
0.0583
0.0583
0.0000
0.0000
LC
CBD
LR
RW
UK F
0.0017
0.0023
MAPE
MAPE
RMSE
MAE
0.0070
LC
CBD
LR
RW
CHAPTER 5. NUMERICAL ILLUSTRATIONS
35
Figure 5.3: Projected errors against x0 for the JAP population
0.0020
JAP M
0.0017
0.0023
0.0013
MAE
0.0029
0.0017
0.0006
0.0003
0.0000
0.0000
0.0060
0.0035
0.0050
0.0029
0.0040
0.0023
RMSE
0.0007
0.0030
0.0017
0.0020
0.0012
0.0010
0.0006
0.0000
0.0000
0.1400
0.1200
0.1167
0.1000
0.0933
0.0800
0.0700
0.0600
0.0467
0.0400
0.0233
0.0200
0.0000
0.0000
LC
CBD
LR
RW
JAP F
0.0010
0.0012
MAPE
MAPE
RMSE
MAE
0.0035
LC
CBD
LR
RW
CHAPTER 5. NUMERICAL ILLUSTRATIONS
36
in younger age groups. For the M AP E, the Lee-Carter model becomes the worst
for the males; if we combine both cases of males and females, the LLHT-LR and
LLHT-RW models outperform the Lee-Carter and CBD ones.
Table 5.3 gives the average projection errors, M AE, RM SE and M AP E, over
11 age groups. From Table 5.3, it can be observed that the two LLHT-based models perform quite well in all three error statistics. In fact, the LLHT-LR model
dominates the others for M AE and M AP E for both genders of the USA and the
UK. Regarding RM SE, both of the LLHT-LR and LLHT-RW models have relatively poorer performance than the Lee-Carter one. However, the two variations of
the LLHT model are ranked the first and the second on the average over the six
combinations of country and gender.
Table 5.3: Overall average projection errors over 11 age groups
error
MAE
RMSE
MAPE
method
LC
CBD
LR
RW
LC
CBD
LR
RW
LC
CBD
LR
RW
USA M
18.93
13.97
13.97
14.27
24.87
17.69
18.10
18.31
8.47
6.28
6.15
6.34
USA F
8.98
10.44
8.14
8.42
12.13
17.13
13.42
13.44
7.22
6.52
5.42
5.69
UK M
23.79
24.51
18.58
21.34
33.51
36.71
27.43
30.67
10.37
10.22
7.24
9.13
UK F
13.62
10.83
10.23
11.89
19.29
14.82
14.94
16.90
10.38
8.54
7.36
9.17
JAP M
17.57
18.81
19.76
19.63
23.82
27.98
30.51
30.27
9.78
9.10
9.30
9.13
JAP F
7.57
10.74
8.21
8.24
10.78
17.85
13.98
13.98
8.27
8.98
6.65
6.63
Average
15.08
14.88
13.15
13.97
20.73
22.03
19.73
20.59
9.08
8.27
7.02
7.68
Note that RM SE and M AE are scaled to (×10−4 ) and M AP E is a percentage.
5.2
Stochastic mortality projection
In Chapter 4, approaches to obtaining confidence intervals on the projected death
probabilities are explained in detail for all four models. To compare and evaluate
these models, 95% confidence intervals are constructed for the USA, the UK and
CHAPTER 5. NUMERICAL ILLUSTRATIONS
37
Japan populations. Here the middle age group [45, 79] of the latest projected cohort
c
is selected for comparisons, which is the cohort mortality sequence q45,1975,35
. Figures
c
5.4, 5.5 and 5.6 show the confidence intervals on q̂45,1975,35
along with the actual one
c
q45,1975,35
for both genders of the three countries.
For the USA shown in Figure 5.4, the curves of the projected mortality sequence
and its confidence interval for this age span produced by the Lee-Carter model
are least smooth and do not display the same trends as that of the real mortality
sequence, whereas those by the two LLHT-based models are most smooth. Besides,
the confidence intervals constructed from these four models cover more real female
mortality rates than real male ones. For the females, the Lee-Carter model covers
the fewest real mortality rates, but the LLHT-RW model covers all real ones. For
the males, the lower bound of the confidence interval by the LLHT-LR model is
close to the real sequence and that by the LLHT-RW model even overlaps the real
mortality rates up to age 69; the Lee-Carter model still performs the worst for the
males.
For the UK population in Figure 5.5, all the models overestimate the mortality
rates, and comparisons among models have results similar to those for the USA
population. The projected mortality rates and its associated confidence intervals derived from the Lee-Carter model remain unsmooth and are quite apart from
the real mortality rates for older ages. The CBD model produces the narrowest
confidence interval for the females, which fails to contain any real mortality rates.
Again, the confidence interval by the LLHT-RW model covers all real female mortality rates, and the lower bound of the confidence interval almost overlaps the real
male mortality rates up to age 69.
Observations from Figure 5.6 for Japan show the same smoothness as those for
the USA and the UK, and the LLHT-RW model still outperforms the other three.
It is worthy of mention that the confidence intervals by the CBD model fan out at
age 75 due to large standard deviations.
In summary, the LLHT-RW model has the best performance for this age group
due to its confidence interval being able to cover the most real mortality rates for
the three countries.
CHAPTER 5. NUMERICAL ILLUSTRATIONS
38
0.04
0.03
mortality
0.00
0.00
0.01
0.02
0.04
0.02
mortality
0.06
0.05
c
Figure 5.4: 95% confidence intervals on q45,1975,35
for the USA population
45
50
55
60
65
70
75
80
45
50
55
60
age
65
70
75
80
75
80
75
80
75
80
age
(b) LC-Female
0.04
0.03
mortality
0.00
0.00
0.01
0.02
0.04
0.02
mortality
0.06
0.05
(a) LC-Male
45
50
55
60
65
70
75
80
45
50
55
60
age
65
70
age
(d) CBD-Female
0.04
0.03
mortality
0.00
0.00
0.01
0.02
0.04
0.02
mortality
0.06
0.05
(c) CBD-Male
45
50
55
60
65
70
75
80
45
50
55
60
age
70
(f) LR-Female
lower
real
predicted
upper
0.03
mortality
0.00
0.00
0.01
0.02
0.02
0.04
0.04
0.06
0.05
(e) LR-Male
mortality
65
age
45
50
55
60
65
70
age
(g) RW-Male
75
80
45
50
55
60
65
70
age
(h) RW-Female
CHAPTER 5. NUMERICAL ILLUSTRATIONS
39
0.00
0.00
0.02
0.02
0.04
mortality
0.06
0.04
mortality
0.08
0.06
0.10
0.12
0.08
c
Figure 5.5: 95% confidence intervals on q45,1975,35
for the UK population
45
50
55
60
65
70
75
80
45
50
55
60
age
0.12
80
75
80
75
80
75
80
mortality
0.00
0.02
0.02
0.04
0.06
0.10
0.08
0.06
0.04
mortality
75
(b) LC-Female
0.00
45
50
55
60
65
70
75
80
45
50
55
60
age
65
70
age
(d) CBD-Female
0.04
0.00
0.00
0.02
0.02
0.04
0.06
mortality
0.08
0.06
0.10
0.12
0.08
(c) CBD-Male
mortality
70
0.08
(a) LC-Male
45
50
55
60
65
70
75
80
45
50
55
60
age
65
70
age
(f) LR-Female
lower
real
predicted
upper
0.04
0.00
0.00
0.02
0.02
0.04
0.06
mortality
0.08
0.06
0.10
0.08
0.12
(e) LR-Male
mortality
65
age
45
50
55
60
65
70
age
(g) RW-Male
75
80
45
50
55
60
65
70
age
(h) RW-Female
CHAPTER 5. NUMERICAL ILLUSTRATIONS
40
0.015
mortality
0.00
0.000
0.01
0.005
0.010
0.03
0.02
mortality
0.04
0.020
0.05
0.06
0.025
c
Figure 5.6: 95% confidence intervals on q45,1975,35
for the JAP population
45
50
55
60
65
70
75
80
45
50
55
60
age
0.06
80
75
80
75
80
75
80
mortality
0.000
0.01
0.005
0.010
0.015
0.020
0.05
0.04
0.03
0.02
mortality
75
(b) LC-Female
0.00
45
50
55
60
65
70
75
80
45
50
55
60
age
65
70
age
(d) CBD-Female
0.015
mortality
0.03
0.00
0.000
0.01
0.005
0.02
0.010
0.04
0.020
0.05
0.06
0.025
(c) CBD-Male
mortality
70
0.025
(a) LC-Male
45
50
55
60
65
70
75
80
45
50
55
60
age
65
70
age
(f) LR-Female
lower
real
predicted
upper
0.015
mortality
0.03
0.00
0.000
0.01
0.005
0.02
0.010
0.04
0.020
0.05
0.06
0.025
(e) LR-Male
mortality
65
age
45
50
55
60
65
70
age
(g) RW-Male
75
80
45
50
55
60
65
70
age
(h) RW-Female
CHAPTER 5. NUMERICAL ILLUSTRATIONS
5.3
41
Errors in pricing
Besides comparing the accuracy of mortality projections for all models, a simple application in pricing of life insurance helps to understand the impact of the projected
mortality rates on premiums of life insurance.
Recall that term insurance pays $1 death benefit at the end of the year of the insured’s death. The total of the discounted death benefits, i.e, the net single premium
(NSP) at issue for the insured aged x is given by
A1x:n =
n
X
k−1 px
· qx+k−1 · v k ,
k=1
where v = 1/(1 + i) is the discount factor and i is the interest rate.
An n-year temporary annuity-due is an annuity with $1 annual payment up to
n years at the beginning of each year as long as the annuitant survives. Its NSP is
given by
äx:n =
n−1
X
k px
· vk .
k=0
1 , can be
Consider an n-year term life policy; its level premium, denoted by Px:n
obtained by dividing A1x:n by äx:n , which is
1
Px:n
=
A1x:n
.
äx:n
c
, they both start from qx,t , but the former
For mortality sequences qx,t,n and qx,t,n
ends at qx+n−1,t and the latter ends at qx+n−1,t+n−1 . In this project, the cohort morc
tality sequence qx,t,n
for an individual aged x in year t is used for pricing A1x:n , äx:n
1 . The reason of using q c
and Px:n
x,t,n is that it conforms to the mortality path of the
cohort for those who buy the term life insurance at age x in year t.
The populations of the USA, the UK and Japan are selected for pricing. Both the
c
c
real and projected mortality sequences for 15 consecutive cohorts, qx,1961,35
, . . . , qx,1975,35
,
1
1
1 , A1
are used to calculate Px:n
x:n and äx:n . Thus, for each Xx:n , X = {A , ä, P },
there will be two values computed with both the predicted and real mortality rates.
1
Figures 5.7, 5.8 and 5.9 show the average relative errors on A1x:n , äx:n and Px:n
against the starting years 1961, . . . , 1975 for males and females of the USA, the
CHAPTER 5. NUMERICAL ILLUSTRATIONS
42
UK and Japan, respectively. Here the relative error is the ratio of the premium
calculated by the predicted cohort mortality sequence to that computed by the real
sequence less one. The average is taken over the 11 relative errors of premium for 11
age spans [x, x+n−1], x = 40, . . . , 50. Since k px = px ×· · ·×px+n−1 , overestimating
one px+i−1 for some i and underestimating the other px+j−1 for some j can offset
each other. As a result, a poorer forecast on a cohort mortality sequence does not
necessarily lead to a higher relative error on premium.
Observed from Figures 5.7 and 5.8, the overall trends of the relative errors of
A1x:n , äx:n
1
and Px:n
are increasing, decreasing and increasing, respectively. These
plots are similar among all models but they are further apart from zero over time,
consistent with that all models tend to perform worse for a longer term.
As shown in Figure 5.7 (a), the LLHT-LR model gives the average relative
error curve of A1x:n closest to zero for the males for all projection years and for
the females for later ones, which indicates that this model yields the minimum
difference in A1x:n based on the predicted and actual mortality rates for these years.
Regarding the case of äx:n , both the LLHT-LR and Lee-Carter models have quite
1
good performances compared with the other two. Furthermore, Px:n
is mainly
1
determined by A1x:n ; thus, the patterns of curves of both A1x:n and Px:n
look very
similar. Another important observation is that the relative errors on A1x:n for all
models and all projection years are positive for the males, indicating that all models
are overestimating mortality rates for the USA males.
For the UK population in Figure 5.8, the overall trends of curves are similar to
those for the USA in Figure 5.7. However, the LLHT-RW model produces results
1
closest to zero in early forecast
with the average relative errors on A1x:n and Px:n
years. Also, mortality rates are overestimated for most cases and underestimated
for some models in early years.
Opposite trends are shown in Figure 5.9 for the Japan population; A1x:n , äx:n and
1
Px:n
have decreasing, increasing and decreasing trends, respectively. Also negative
1 reveal that the projected cohort mortality sequences
relative errors on A1x:n and Px:n
for all 15 cohorts are underestimated. For the Japan males and females, the curves
1
of relative errors on A1x:n and Px:n
are closest to zero for the Lee-Carter model.
CHAPTER 5. NUMERICAL ILLUSTRATIONS
43
Overall, using cohort mortality sequence for pricing can capture mortality improvement or deterioration over years. The classical Lee-Carter and CBD models
are adjusted to provide ways of forecasting the future mortality rates for cohorts.
Compared with the famous Lee-Carter and CBD models, the two variations of the
1
based on the
LLHT model give satisfactory results of pricing A1x:n , äx:n and Px:n
mortality data for the three countries.
CHAPTER 5. NUMERICAL ILLUSTRATIONS
44
0.00
0.00
0.01
0.02
0.02
0.04
0.03
0.04
0.06
1
Figure 5.7: Average relative errors on A1x:n , äx:n and Px:n
for the USA population
1962
1966
1974
1962
1966
1970
year
year
A1x:n -Male
A1x:n -Female
(b)
1974
-0.015
-0.006
-0.010
-0.004
-0.005
-0.002
0.000
0.000
(a)
1970
1962
1966
1970
1974
1962
1966
year
1970
(d) äx:n -Female
0.00
0.02
0.04
0.06
0.00 0.01 0.02 0.03 0.04 0.05
0.08
(c) äx:n -Male
1962
1966
1974
year
1970
year
(e) Px:n -Male
1974
LC
CBD
LR
RW
1962
1966
1970
year
(f) Px:n -Female
1974
CHAPTER 5. NUMERICAL ILLUSTRATIONS
45
0.00
0.00
0.04
0.04
0.08
0.08
0.12
1
Figure 5.8: Average relative errors on A1x:n , äx:n and Px:n
for the UK population
1962
1966
1974
1962
1966
1970
year
year
A1x:n -Male
A1x:n -Female
(b)
1974
-0.015
-0.008
-0.010
-0.004
-0.005
0.000
0.000
(a)
1970
1962
1966
1970
1974
1962
1966
year
1970
1974
year
(d) äx:n -Female
0.08
LC
CBD
LR
RW
0.08
0.04
0.00
0.00
0.04
0.12
0.12
(c) äx:n -Male
1962
1966
1970
year
(e) Px:n -Male
1974
1962
1966
1970
year
(f) Px:n -Female
1974
CHAPTER 5. NUMERICAL ILLUSTRATIONS
46
-0.08
-0.12
-0.06
-0.08
-0.04
-0.04
-0.02
0.00
0.00
1 for the Japan population
Figure 5.9: Average relative errors on A1x:n , äx:n and Px:n
1962
1966
1974
1962
1966
1970
year
year
A1x:n -Male
A1x:n -Female
(b)
1974
1962
1966
1970
1974
-0.001
0.000
0.005
0.001
0.010
0.003
0.015
0.020
0.005
(a)
1970
1962
1966
year
1970
1974
year
(d) äx:n -Female
-0.15
-0.08
-0.10
-0.06
-0.04
-0.05
-0.02
0.00
0.00
(c) äx:n -Male
1962
1966
1970
year
(e) Px:n -Male
1974
LC
CBD
LR
RW
1962
1966
1970
year
(f) Px:n -Female
1974
Chapter 6
Conclusion
In this project, four mortality projection models are discussed and compared; among
all of them, the Lee-Carter and the CBD models have been used as benchmarks for
mortality forecast, while the LLHT-LR and LLHT-RW models are simple to implement and also have good performances in terms of accuracy of mortality forecast
for some data sets.
Ever since the LHT model was proposed as a method of forecasting mortality
rates by Tsai and Jiang (2010), several extensions or modifications have been made
on it. Besides the originally proposed arithmetic and geometric methods of mortality
projection, two new methods of projecting future mortality rates for cohorts are
proposed in this project. Extraction of mortality data are also adjusted from qx,t,n
c
to qx,t,n
for cohorts on the Lee-Carter and CBD models for comparisons. Mortality
data for three well-developed countries, the USA, the UK and Japan, are selected
to implement the models and compare the accuracy of mortality projection. Three
statistics, M AE, RM SE and M AP E, measuring the deviance between the real and
forecasted mortality rates are proposed, along with numerical values which show
that the LLHT-LR model is overall the best based on the given mortality data for
cohorts. Compared with the Lee-Carter and CBD models, the LLHT-based models
are simpler to understand and implement, and outperform the Lee-Carter and CBD
models in terms of accuracy of mortality projection for the given mortality data set.
The main disadvantage of fitting and forecasting cohort mortality rates is that
mortality data for a large number of years are needed. For example, to forecast the
47
CHAPTER 6. CONCLUSION
48
mortality rates of n-year age span for M cohorts with the parameters estimated from
those for m cohorts, we need mortality data for (m + M + n − 1) years for the LeeCarter and CBD models and (m + M + n) years for the LLHT-LR and LLHT-RW
models where M is included for comparing the real and forecasted mortality rates.
Instead of forecasting mortality rates for a cohort, mortality rates for a year can be
predicted with traditional mortality projection models. As a comparison, to predict
mortality rates of n-year age span for M years with the parameters estimated from
those for m years, only (m + M ) years of mortality data are needed for the classical
Lee-Carter and CBD models, and (m + M + 1) years for the regular LLHT-based
models. This is a restriction on projection for cohort mortality sequences with a
long age span since the recording of mortality rates data has not started until last
century. Thus, predicting cohort mortality rates with less data reliance is worthy of
further research.
Bibliography
[1] J.M. Alho. Discussions on Ronald Lee’s ‘The Lee-Carter method for forecasting
mortality, with various extensions and applications’. North American Actuarial
Journal, 4(1):91–93, 2000.
[2] H. Booth, J. Maindonald, and L. Smith. Applying Lee-Carter under conditions
of variable mortality decline. Population Studies, 56(3):325–336, 2002.
[3] H. Booth and L. Tickle. Mortality modelling and forecasting: A review of
methods. Annals of Actuarial Science, 3:3–43, 2008.
[4] N.L. Bowers, H.U. Gerber, J.C. Hickman, D.A. Jones, and C.J. Nesbitt. Actuarial Mathematics. 2nd ed. Schaumburg, Illinois: The Society of Actuaries,
1997.
[5] N. Brouhns, M. Denuit, and J.K. Vermunt. A poisson log-bilinear regression
approach to the construction of projected life tables. Insurance: Mathematics
and Economics, 31:373–393, 2002.
[6] A.J.G. Cairns, B. Blake, and K. Dowd. A two-factor model for stochastic
mortality with parameter uncertainty: Therory and calibration. Journal of
Risk and Insurance, 2:79–113, 2006.
[7] A.J.G. Cairns, B. Blake, K. Dowd, G. Coughlan, and M. Khallaf-Allah. Mortality density forecasts: An analysis of six stochastic mortality models. Heriot-Watt
University and Pensions Institute discussion paper PI-0801, 2008.
[8] A.J.G. Cairns, B. Blake, K. Dowd, G.D. Coughlan, D. Epstein, A. Ong, and
I. Balevich. A quantatitative comparison of stochastic mortality models using
date from england & wales and the united states. Heriot-Watt University and
Pensions Institute discussion paper PI-0701, 2007.
[9] H. Chen and S.H. Cox. Modeling mortality with jumps: Applications to mortality securitization. Journal of Risk and Insurance, 76(3):727–751, 2009.
[10] Y. Deng, P.L. Brockett, and R.D. MacMinn. Longevity/mortality risk modelling and security pricing. The Journal of Risk and Insurance, 79:697–721,
2012.
49
BIBLIOGRAPHY
50
[11] B. Gompertz. On the nature of the function expressive of the law of human
mortality, and on a new mode of determining the value of life contingencies.
Philosophical transactions of the Royal Society of London, 115:513–583, 1825.
[12] D. Hainaut. Multidimensional Lee-Carter model with switching mortality processes. Insurance: Mathematics and Economics, 50(2):236–246, 2011.
[13] R.D. Lee and L.R. Carter. Modeling and forecasting US mortality. Journal of
the American Statistical Association, 87:659–675, 1992.
[14] S.H. Li and W.S. Chan. Outlier analysis and mortality forecasting: the united kingdom and scandinavian countries. Scandinavian Actuarial Journal,
2005(3):187–211, 2005.
[15] A.E. Renshaw and S. Haberman. Lee-Carter mortality forecasting: a parallel
generalized linear modelling approach for england and wales mortality projections. Journal of the Royal Statistical Society: Series C (Applied Statistics),
52(1):119–137, 2003.
[16] A.E. Renshaw and S. Haberman. Lee-Carter mortality forecasting with agespecific enhancement. Insurance: Mathematics and Economics, 33:255–272,
2003.
[17] A.E. Renshaw and S. Haberman. A cohort-based extension to the Lee-Carter
model for mortality reduction factors. Insurance: Mathematics and Economics,
38:556–570, 2006.
[18] E. Tabeau, A. van den Berg Jeths, and C. Heathcote. Forecasting mortality in
developed countries: Insights from a statistical, demographic and epidemiological perspective, volume 9. Springer, 2001.
[19] C.C.L. Tsai. A linear logrithm hazard transform approach to modeling mortality rates. working paper, 2012.
[20] C.C.L. Tsai and L. Jiang. Actuarial applications of the linear hazard transform
in mortality fitting and prediction. 14th International Congress on Insurance:
Mathematics and Economics, Canada, Abstract available online at http://atlasconferences.com/cgi-bin/abstract/cbak-47, 2010.
[21] M. Yu. Modeling mortality rates with the linear logarithm hazard transform
approaches. 2013.