The earlier HECM models use static mortality tables and therefore fail to capture the dynamics of mortality over time. In addition, they do not model longevity risks. On the other hand, an unexpected mortality improvement will increase the life expectancy, and thereby increase both the term and the amount of the outstanding loan balance.
Recently a number of approaches have been developed for forecasting mortality. These methods are taken into account to describe the betterments in the mortality trend and to project survival tables. A recent paper, Cairns, et al. (2007), examined the empirical fits of eight different stochastic mortality models. Note that models M1 to M3 can be described as belonging to the family of generalized Lee-Carter (1992) models and models M5 to M8 can be described as members of the family of generalized CBD (2006) models.
Cairns, et al. (2008a), then examined the „goodness of fit‟ of the remaining six models by analyzing the statistical properties of their various residual series. Therefore, in this section we incorporate the classic Lee-Carter (1992) model in order to model the longevity and adverse mortality risks more accurately.
Stochastic mortality models either model the central mortality rate or the initial mortality rate (Coughlan, Epstein, Sinha, & Honig, 2007). The central mortality rate is defined as:
( 3-1)
where is the actual total number of deaths at time t , and is the population in age group x at time t. The initial mortality rate is the probability that a person aged x dies within the next year. The different mortality measures are linked by the following approximation:
( 3-2)
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3.1 The Lee-Carter Model
Lee and Carter (1992) base their model on the insight that age-specific death rates in the United States quite accurately follow a common exponential trend over the last decades, and propose the following parsimonious parameterization:
( 3-3)
where is the central death rate at age x in year t, coefficients describe the average shape of the age profile, and the coefficients describe the pattern of deviations from this age profile when the parameter varies. Note that all variables on the right side of the model are unobservable. Therefore, the Lee-Carter model cannot be fitted by the ordinary least square (OLS) approach. In addition, this model is obviously over-parameterized. Lee and Carter (1992) impose the following normalization conditions to obtain a unique solution:
and ( 3-4)
Then becomes the average value of over time, i.e.,
( 3-5)
where T is the length of the time series of mortality data. An important aspect of stochastic mortality models is the quality of the fit of the model to historical mortality data. We use the U.S. mortality rate data from the data of 1950 to 2006 and generate yearly mortality rate of age 60 and above to age 99, both male and female. The data can be obtained from human mortality database2.
2 Source: http://www.mortality.org/
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3.2 Fitting the Lee-Carter model
The estimation is completed in two steps according to Lee and Carter (1992). In the first step singular value decomposition (SVD) of the matrix is used to obtain estimates for , (x = 1, 2,…, ω), and (t = 1, 2,..., T). In the second step the time-series evolution of is recalculated based on the actual number of deaths in year t. Brouhns et al. (2002) described a fitting methodology for the Lee-Carter model based on a Poisson model.
The main advantage of this is that it accounts for heteroskedasticity of the mortality data for different ages. Therefore, the number of deaths is modeled using the Poisson model, implying:
( 3-6)
The parameter set ∅ is fitted with maximum likelihood estimation, where the log-likelihood function of model ( 3-6) is given by:
∅ ∅
∅ (3-7)
We used the R-code of the software package ''Lifemetrics'' as a basis for fitting (3-7).3 Therefore, we call the corresponding R function passing in vectors and arrays.
The fitting procedure will print in the R console the values of the log likelihood which is being maximized during the fitting process. The result of parameter estimates for the Lee-Carter model is plotted in Figure 3-1 and the fitted death rates for different sex and age group is plotted in Figure 3-2. The coefficients, as noted, are just the average values of the logs of the death rates. Not surprisingly, the male coefficients lie above the female at all ages, reflecting the fact that mortality was higher, on average, from 1950 to
3 Lifemetrics is an (open source) toolkit for measuring and managing longevity and mortality risk, designed by J.P. Morgan. www.lifemetrics.com
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2006. The coefficients describe the relative sensitivity of death rates to variation in the parameter. The male coefficients above the female before age 75, reflecting the fact that the male mortality improvement will increase than female before age 75. On the other hand, the female coefficients above the male after age 75. It can be seen that the younger the age, the greater its sensitivity to variation in the parameters. The exponential rate of change of an age group‟s mortality is proportional to the values:
. If declines linearly with time, then will be constant and each will decline at its own constant exponential rate.
Figure 3-1 Parameters Estimated by Lee-Carter Model
The next step is to model as a stochastic time series process. This is done using standard Box-Jenkins procedures. In most applications so far, is well-modeled as a random walk with drift:
( 3-8)
where d is a constant and is a normally distributed error term with zero mean.
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Figure 3-2 Fitted death rates for age group 60, 65, 70 and 75.
Figure 3-3 Fitted death rates for age group 80, 85, 90 and 95.
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