Table 3-5. Fitting MAPE of the female for six countries (unit：%)
# of Principal Component
Country 1 2 3 4 5 6 7 LC
Austria 8.76 7.95 5.95 4.88 4.53 4.36 3.71 4.75
France 11.68 5.75 5.07 3.65 2.88 2.15 2.03 2.85
Ireland 16.05 9.12 7.50 7.20 6.12 5.43 4.85 7.06
Spain 9.19 7.93 7.02 4.12 3.59 2.81 2.54 4.75
Taiwan
9.47 7.52 5.78 5.34 4.14 3.74 3.33 4.46Total 10.90 7.73 6.46 5.27 4.54 4.00 3.58 5.11
We also calculate the AIC and BIC of the proposed method and compare with those of the LC model. However, since the proposed approach is for the combined data, it would be difficult to count the number of parameters in the calculation of AIC/BIC. Therefore, we plug into all parameters of the PCA to compute the AIC/BIC in Taiwan. Similarly, we also plug all parameters of the LC model used in six countries for the AIC/BIC. Table 3-5 lists the values of AIC and BIC, and it seems that fewer PC is better. Because the calculation of AIC/BIC is questionable, we will not use the results in Table 3-6 to choose PC’s.
Table 3-6. AIC and BIC of fitting results for Taiwan
# of Principal Component
Next, via the prediction MAPE, we compare the proposed method with the LC model. Time series methods (i.e., ARIMA model) are used to model the trend of future mortality. For the MAPE of predicting Taiwan mortality rates, the proposed method outperforms the LC model if the number of PC’s is two or more (Tables 3-7 & 3-8), with a reduction around 3% in MAPE. Similarly, on average, the proposed method also has smaller prediction MAPE if there are 3 or more PC’s. The proposed method has the larger reduction than the LC model for the Spain data (for both the male and female).Table 3-7. MAPE values of forecasting results for males of six countries (unit：%)
# of Principal Component
Country 1 2 3 4 5 6 7 LC
Austria 10.48 9.99 9.58 8.94 7.62 7.81 8.85 9.55 France 14.27 13.58 12.76 10.31 9.29 8.15 8.29 11.61 Ireland 23.25 21.82 21.12 19.02 20.09 21.22 21.29 22.72
Spain 10.57 9.79 9.77 8.62 8.75 8.66 9.13 14.04
Taiwan
22.26 15.10 15.70 15.13 14.75 14.90 14.86 18.13Total 15.73 13.93 13.71 12.54 12.23 12.22 12.40 14.72
Table 3-8. MAPE values of forecasting results for females of six countries (unit：%)
# of Principal Component
Based on the fitting and prediction errors, we found that the proposed method has smaller fitting and prediction errors than the LC model, if there are 5 or more PC’s. This result is true for Taiwan and for six countries combined (Austria, France, Ireland, Norway, Spain, and Taiwan).
Although the result might vary with different countries and periods of data, it seems that the proposed method provides a possible way for improving the mortality estimates of small populations.
4. Conclusions and Discussion
High volatility is always the most challenging problem in modeling mortality. For the countries with fewer populations, the mortality rates usually are more volatile due to small sample size. One way to reduce the volatility of mortality is to increase population size. The coherent model by Li and Lee (2005) can be treated as an example of increasing the population size, by adopting the information from a larger reference population which has similar mortality profile as the countries with fewer populations. In this study, we proposed another alternative for increasing population size by combining mortality data from other countries which have similar mortality experience as the country with small population.
The proposed method can be separated into two steps. The first step is using the cluster analysis method to select the countries sharing similar mortality experience. Next, apply the
acquire more reliable mortality estimates for the small population. The Taiwan data are used as a demonstration of the proposed method, and the countries from the Human Mortality Database are the reference groups. We found that the countries having similar mortality improvement as Taiwan are Austria, France, Ireland, Norway, and Spain. The data used are between 1970 and 2008, and they are separated into two periods: in-sample (or fitting period, 1970-1999) and out-sample (or testing data, 2000-2008). The proposed multi-PCA model produces significant smaller estimation and prediction errors than Lee-Carter model for the selected countries, provide that the number of principal components is 5 or more.
We shall continue evaluating whether the proposed method can also produce better mortality estimates and predictions for other countries with small population. In specific, we will try to check if the proposed method can work for the experienced data from Taiwan life insurance companies.
The experienced data of Taiwan life insurance have very few samples for the elderly group, and there is no guarantee that they share similar mortality experience as the whole population in Taiwan.
Reliable mortality estimates are crucial to pricing the annuity products, since Taiwan has been experiencing rapid population aging and there are no enough data to model mortality rates for the elderly.
Although the proposed approach has smaller estimation and prediction errors in the Taiwan case (comparing to the LC model), the process of applying the cluster analysis and principal component analysis still has room for improvement. For example, the current setting is to combine data from countries with similar mortality improvement as the target population at all ages. It is more likely to see that some set of countries share similar mortality experience at younger age groups, and another set of countries share similar mortality experience at older age groups. The intersection of two sets of countries might be empty or only contains very few countries, and this would not have significant increase in the population size.
On the other hand, the principal component analysis (PCA) is applied to the combined data, due to its ease of usage. However, there are a lot of data reduction methods, in addition to the PCA and singular value decomposition used in the LC model. For example, the function PCA receives
more attention in the recent years (Hyndman and Ullah, 2005; Hyndman et al., 2011). Also, like the coherent model by Li and Lee (2005), we can modify the mortality estimates of the small population from a larger reference population. In this study, we use the estimates from the larger reference population to replace the original estimates from the small population. Maybe we can adapt the idea similar to weighted average for modifying the small population estimates.
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