**3.
Modeling
Simple
Tontine
Annuity
Schemes**

**3.3
Mortality
Model
Set
Up**

period. The resulting benefit payments are summarized in a distribution for different future times and ages.

**3.2 Taiwan Database **

To perform the simulation of mortality development and the operation of tontine annuity schemes, several assumptions in terms of data have been made.

Mortality model used in this paper is based on Taiwan’s male central death rates from 1970-2010 due to a limited number of years covered in the Human Mortality Database (HMD)- University of California, Berkeley. The HMD began in the year 2000 as a collaborative project involving research teams from the Department of Demography at the University of California, Berkley and the Max Planck Institute for Demographic Research (MPIDR) in Rostock, Germany. The HMD currently provides detailed population and mortality data for 37 countries or areas, which play an important role in mortality research.

Although our projections begin in year 2011, we assumed year 2050 to be time t = 0. For calibration purposes only age 60 to age 100 are considered, which is consistent with cohort’s initial age x=60 at time t = 0, and the schemes period here is limited to T=40 at the outset. These assumptions are made to prevent from calibration problems because the scarce number of observations at ages beyond age 100 leads to results of death rates anomaly.

**3.3 Mortality Model Set Up **

Mortality models can be divided into static models and dynamic models.

Static models only consider the relationship between mortality rates and age while dynamic models account for time-dependent changes in the state of the system. Static models cannot readily depict how the future mortality rate changes with time;

therefore, the need for applying a dynamic model to assess mortality clearly arises.

The method proposed in Lee and Carter (1992) has become one of the most well-known models, and it is applied in different countries around the world to forecast age-specific death rates. Lee and Carter developed their approach specifically for U.S.

mortality data from 1993-1987, which is further used to project future mortality in the United States for all ages. The model’s basic premise is that there is a linear

## ‧

relationship between the logarithm of age-specific death rates and two explanatory factors: the initial age interval and time. Wilmoth (1996) projects future mortality in Japan by using its own parameter estimation approach to tailor the Lee-Carter model to Japan mortality data.

Since 1992, a number of stochastic models have been developed to improve
the Lee-Carter model. Chen and Cox (2009) incorporated a jump process into the
original Lee-Carter model and estimated 𝑘_{!}with a Geometric Brownian Motion and
Compound Poisson Process. Renshaw, Haberman (2006) incorporated cohort effects
into the parametric structure of the basic Lee-Carter model for mortality projection.

Principal component analysis was also applied to add the second time factor. Cairns et al. (2006a, 2006b) estimated mortality rates with a yield rate model and further proposed a short-term mortality model, long-term mortality model, and market mortality model.

Although many other models were proposed to improve on the Lee-Carter model’s poor performance in projecting mortality rates at advanced ages, the model has been used as a benchmark for Census Bureau population forecasts in America (Lee and Miller, 2001) and Japan. For various reasons, the Lee-Carter model is used in this paper to generate simulations of mortality. First, the model has a relatively small number of parameters, and the parameters are fairly easy to interpret. Second, it attaches probabilistic confidence intervals to central mortality forecasts so that we can examine how light future mortality improvements turn out. Third, sample paths of future mortality can be generated via the stochastic components of the model. Thus, we use the Lee-Carter model to produce stochastic Taiwan mortality forecasts.

Lee-Carter (1992) specified a log-bilinear form for the central mortality rate, which is,

ln 𝑚_{!,!} = 𝛼_{!} + 𝛽_{!}𝑘_{!}+ 𝜀_{!,!}.

𝛼_{!}, 𝛽_{!}, 𝑘_{!} are the parameters to be estimated. Assuming calendar years of the
observed data are 𝑡 = 𝑡_{!}, 𝑡_{!}, … , 𝑡_{!} and set of ages are 𝑥 = 𝑥_{!}, 𝑥_{!}, … , 𝑥_{!} ,
interpretation of the parameters is as followed:

𝑚_{!,!} is the age-specific death rate for x interval and the year

## ‧

𝛼_{!} is the average age-specific mortality parameter; it reflects the general shape
of the mortality schedule

𝛽_{!} is a deviation in mortality due to changes in the 𝑘_{!} index. In indicates the
sensitivity of logarithm of the central mortality rate at age x to variations in the time
index 𝑘_{!}, that it, ^{! !" !}^{!,!}

!" = 𝛽_{!}(^{!!}^{!}

!").

𝜀_{!,!} is the random error.

To satisfy the constraints for parameter uniqueness, the estimates of 𝛽_{!} and 𝑘_{!}
are normalized so that they sum to one and zero, respectively.

𝜅_{!} = 0
higher-order terms are incorporated in the model as

𝑙𝑛𝑚_{!} 𝑡 = 𝛼_{!}+ 𝛽_{!}^{[!]}

!

!!!

𝜅_{!}^{[!]}

where 𝛾 is the rank of the matrix of {ln 𝑚_{!,!} − 𝛼_{!}}. 𝛽_{!}^{[!]}𝜅_{!}^{[!]} is the jth order term of
*the approximation (Pitacco et al. 2009). *

To reconcile the fitted and observed number of deaths, we re-estimate
parameter 𝜅_{!} with drift model. The dynamics of the estimated 𝜅_{!} are given by

## ‧

*and where d is the drift parameter. The conditional variance of the forecast is*

𝑉𝑎𝑟 𝜅_{!}_{!} 𝜅_{!}_{!}, 𝜅_{!}_{!}, … , 𝜅_{!}_{!} = 𝜅𝜎^{!}.

*Thus, the estimators of d and 𝜎*^{!} are given by the sample mean and variance of
the 𝜅_{!}− 𝜅_{!!!}’s, that is,
plausibility. Therefore, the last annuity payment is due at the beginning of time t = 40.

The initial benefit payment is considered to be $400 and the number of generated scenarios in the Monte Carlo simulation is set to N=10,000 throughout the subsequent analyses. Assumptions of all simulations are summarized as follows:

1. A starting benefit payment value of $400;

2. A time horizon of 40 years (so the oldest age included is 100);

3. Pool is consisted of a single cohort with initial number of insured (lx): 10,000;

4. Initial entry age of participants (x) is 60;

5. All fund members are males because the mortality data used for calibration of the model were Taiwanese males; and