1. Introduction

2. Mortality Models

3. Results and Discussion

3.1. Model Comparisons

3.1.1. Fitted ${\kappa }_t$ parameters

Firstly, we present the ${\kappa }_t$ parameters obtained from the fitted LC model for each country and proceed to compare these estimates with those derived from three distinct jump models: ${\kappa }_t$ with a permanent jump, ${\kappa }_t$ with a transitory jump, and ${\kappa }_t$ with a renewal process.

Figure 1 visually represents the comparison, highlighting that the ${\kappa }_t$ parameters obtained from the transitory jump models exhibit a closer alignment with the original LC parameters across all examined countries.

The utilisation of jump models is particularly insightful, as they contribute to the production of smoother ${\kappa }_t$ trajectories by explicitly incorporating jumps in the mortality time index. A smoother trajectory that closely mirrors the LC parameter is indicative of a superior fit.

As demonstrated in Figure 1, the permanent jump model consistently and notably underestimates the time-dependent parameters in Spain, while frequently overestimating ${\kappa }_t$ for Denmark, Sweden, Switzerland, and the UK in comparison to the LC parameter. In contrast, both transitory jump models yield more accurate fits. The renewal process utilising the exponential jump model is particularly noteworthy, as it generates trajectories that closely correspond to the LC parameter across all six countries.

It is crucial to acknowledge that, in the case of Japan, the limited number of jumps during the considered period complicates the applicability of any of the jump models. Consequently, none of the models appear to be ideal for this specific scenario. This observation highlights the intricate challenges inherent in mortality jump modelling, especially in situations where jump occurrences are sparse.

3.1.2. Bayes Information Criterion

The Bayes Information Criterion (BIC) serves as a means to compare models and discern which one better fits the data. A key aspect of BIC is its ability to compare models that are not necessarily nested.

The BIC for mortality models is defined as:

$$\mathrm{BIC}=l\left(\widehat{\varphi }\right)-\frac{1}{2}\upsilon logN$$

(7)

Here, $\varphi$ denotes the parameter vector, and $\widehat{\varphi }$ stands for its maximum likelihood estimate. In this context, $l\left(\widehat{\varphi }\right)$ represents the maximum likelihood, N is the number of observations, and $\upsilon$ is the number of parameters estimated from the mortality models [5].

To determine which jump effect better fits the mortality data, we initiate the comparison by examining the mortality time indices (for estimated ${\kappa }_t$s), parameter estimations, and BIC values obtained for those ${\kappa }_t$s in Table 2. Subsequently, we extend our comparison to encompass all jump-free and jump models, evaluating their overall BIC values (not limited to ${\kappa }_t$s alone) as presented in Table 3. It is noteworthy that a higher BIC value indicates a better fit.

Table 2 presents a comparative analysis between the Lee-Carter with random walk (LC - RW) model and three jump models, all fitted to the ${\kappa }_t$ parameters derived from the benchmark LC model and its derivatives. Among these models, the two transitory jump models exhibit superior fits, as evidenced by their lower BIC values, in contrast to both the LC-RW (except Japan for one of the models) and permanent jump model. It is essential to note that while the differences in BIC values may seem substantial, a more dependable basis for comparing the performance of the models is provided by the mean absolute percentage errors (MAPE), which is discussed in the following subsection. In Appendix A2, we utilised all available up-to-date data for six countries to observe changes in model parameters and in-sample fits for pre- and post-COVID periods. The results indicate that although including COVID data changed the parameters, the performance of the models did not change significantly.

On the other hand, Table 3 offers a more detailed comparison among three jump-free models—namely, LC-RW, RH, and CBD—and three jump models derived from the LC model. This comparison is facilitated by presenting the overall model BICs and MAPEs. Given the diverse structures of the models, the BIC values appear quite distinct. However, it is evident that, in general, and particularly with transitory jump models, the jump models consistently outperform the benchmark jump-free models.

3.1.3. Mean Absolute Percentage Error (MAPE)

The mean absolute percentage error (MAPE) stands out as a widely used metric for assessing prediction accuracy in mortality modelling. Its formulation is given by:

$$MAP{E}_i=\frac{1}{N_i}\sum _{x,t}\left|\frac{{\widehat{m}}_{x,i,t}-m_{x,i,t}}{m_{x,i,t}}\right|$$

In this equation, $N_i$ represents the number of observations in each population, calculated as the product of the number of ages and the number of years. Here, ${\widehat{m}}_{x,i,t}$ denotes the estimated number of deaths, while $m_{x,i,t}$ represents the observed number of deaths for a specific time (t), age (x), and population (i).

We employ MAPE to assess the in-sample prediction performance of the models. Initially, we consider all ages in Table 3 as model MAPEs. Subsequently, we present graphs for specific ages (20, 40, 60, and 80) to observe age-specific variations, focusing exclusively on jump models for different countries in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6 in Appendix. These graphs share a common y-scale, facilitating the comparison of different models and their effectiveness in capturing mortality deterioration across various countries.

Contrary to the BIC values, the overall MAPE values in Table 3 present a much closer scale when comparing jump-free and jump models. However, the LC-RH model stands out by producing relatively distinct and higher MAPEs compared to the others. Consistently lower MAPEs obtained from the jump models indicate better fits for the listed countries, with transitory jump models—particularly the renewal process model—exhibiting superior fits.

Furthermore, the MAPE results illustrated in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6 in Appendix indicate that better fits, characterised by lower MAPE values and reduced volatilities, are observed as ages increase for almost all countries and all three models. Notable differences emerge between the models, with the two transitory jump models consistently demonstrating superior fits and yielding the lowest MAPE values across all countries and ages, except for Japan.

The model incorporating an exponential jump with the renewal process consistently provides the lowest MAPE for all countries and ages, with the same exception—Japan, despite the volatilities observed in the graphs. Additionally, for Japan, the LC-RW model exhibits a slightly better fit as age advances, a detail not clearly depicted in the graphs. This observation aligns with the ${\kappa }_t$ parameter graphs in the preceding section for Japan, where the time series graph appears notably smoother, suggesting fewer significant jumps compared to other countries discussed in this paper. Essentially, this implies that when there are no jumps or only a few jumps, regardless of their frequency, jump-free models tend to outperform models with jumps.

3.1.4. Forecasts

Table 4 to Table 15 present forecasts for the COVID and post-COVID years, including quantiles and ranges derived from 100,000 simulations, along with the observed number of deaths and death rates (obtained by dividing the number of deaths by 2019 exposures) for age 75, serving as an example for six countries. The models, encompassing both jump-free and explicit jump models, were fitted up to 2019, the last pre-COVID year, and are compared in forecasting COVID deaths for the COVID and post-COVID years based on data availability from the HMD website.

In Table 4, only the CBD model forecasts COVID deaths for all three years from 2020 to 2023 for Denmark, with relatively high uncertainty compared to the other models. The LC-RW model captures the number of deaths for 2020 in the first quantile, along with the CBD and all three jump models, while the RH provides true forecasts for deaths at age 75 in the third quantile. Comparing to the number of deaths in the last pre-COVID year (2019) in Denmark, the jump models forecast the number of deaths for 2020 and 2022 in the first quantile with relatively small uncertainty given as the range in the final column. However, in Table 5, when we obtained the death rates by dividing the simulated number of deaths by the 2019 exposure, neither the LC-RW nor the jump models capture true rates. Jump models forecast higher death rates based on the 2019 exposures.

For Japan, in Table 6 all six models provide forecasts with varying degrees of uncertainty. RH, permanent jump, and transitory jump models offer smaller forecast intervals compared to the other two jump-free benchmark models. The transitory jump model with the renewal process presents a less accurate fit, consistent with its structure, which requires the distribution of inter-arrival times, a challenge when only two observations are available. It is important to note that the number of deaths in 2019 for age 75 is higher than the COVID and post-COVID years. Table 7 shows the death rates based on 2019 exposures, and only the CBD model and the transitory jump model with the renewal process seem to provide true forecasts at the expense of including high uncertainty.

In Table 8, only the jump-free models capture the high jump in the first COVID year (2020) in Spain with a huge range of forecasts. Jump models, although providing smaller forecast intervals, perform less accurately. On the other hand, Table 9indicates that the transitory jump model with renewal process manages to forecast the death rate for 2021 with much smaller uncertainty.

For Sweden in Table 16, except for the RH model, which only forecasts 2022 correctly, all other models forecast COVID deaths within the provided ranges for at least first two years (2020 and 2021). Jump models, particularly transitory jump models, offer more accurate forecasts with smaller ranges, although in Table 11, it causes the exclusion of the correct range for 2021 and 2022 when using the 2019 exposure.

Table 12 shows that the LC-RW and CBD models forecast the number of deaths for all three years from 2020 to 2022 with relatively large uncertainty compared to the transitory jump model with the renewal process for Switzerland. The permanent jump model provides the smallest ranges while forecasting the first year within the range. The performance of the models for forecasts for the death rates in Table 13 does not differ significantly from the number of deaths.

In the UK (Table 14), there is a notable jump in the number of deaths due to COVID compared to 2019, and except for the RH and permanent jump models, the other four models for the following two years manage to include the number of deaths in their forecast ranges, although the CBD model produces extremely large forecast intervals. In Table 15, two transitory jump models provide the correct death rates with more accuracy compared to the LC-RW and the CBD models, while the permanent jump model forecasts 2020 death rates within the range different from the number of death results.

When we consider younger ages, such as 60, none of the jump models accurately predicted the high number of deaths for Spain or the UK.

All six models perform differently for different countries. The CBD model successfully forecasts COVID deaths for a higher number of countries and years under examination. However, it is worth noting that the model produces wider intervals for the quantiles, which may not necessarily provide informative insights, as demonstrated in the cases of Spain, Sweden, and the UK.

4. Pricing of Mortality-Linked Securities

4.2. Change Measures via the Wang Transform

The Wang transform serves as a widely employed universal framework for pricing financial and insurance risks. For a given asset (or loss) variable X with a cumulative distribution function (CDF), $F\left(x\right)$, the Wang transform produces a risk-adjusted CDF, $F^{*}\left(x\right)$, given by the equation:

$$F^{*}\left(x\right)=\Phi [{\Phi }^{-1}\left(F\left(x\right)\right)-\lambda ]$$

Here, $\Phi$ represents the standard normal cumulative distribution, and $\lambda$ denotes the market price of risk. After obtaining the risk-adjusted distribution $F^{*}\left(x\right)$, the expectation of X under $F^{*}\left(x\right)$, denoted by $E^{*}\left(x\right)$, can be calculated. By discounting this value back to time zero using the risk-free interest rate, we can determine the fair value of the asset (or loss) X.

The payoff of the mortality instrument considered in this article is based on three different jump models. The first model is the permanent jump model. For comparison purposes, the Wang transform is applied to the same variables Z, Y, and N (or p for the permanent jump model) for all mortality models. Let ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ represent the market prices of risk associated with each of the variables Z, Y, and p. Then the dynamics of the mortality factor under the risk-adjusted measure Q becomes:

$$\left\{\begin{array}{c}{\kappa }_{t+1}^{*}={\kappa }_t^{*}+\mu -p^{*}m+\sigma Z_{t+1}^{*}\hfill \\ {\kappa }_{t+1}^{*}={\kappa }_t^{*}+\mu -p^{*}m+\sigma Z_{t+1}^{*}+Y_{t+1}^{*}\hfill \end{array}\right.$$

(8)

where $Z_t^{*}\sim N({\lambda }_1,1)$, $Y_t^{*}\sim N(m+{\lambda }_{2}s,s^2)$, $p^{*}=1-\Phi [{\Phi }^{-1}(1-p)-{\lambda }_3]$.

The second mortality model considered here is the transitory jump model, and after applying the Wang transform, the dynamics of the mortality factor under the risk-adjusted measure Q becomes:

$$\left\{\begin{array}{c}{\tilde{\kappa }}_{t+1}^{*}={\tilde{\kappa }}_t^{*}+\mu +\sigma Z_{t+1}\hfill \\ {\kappa }_{t+1}^{*}={\tilde{\kappa }}_{t+1}^{*}+Y_{t+1}^{*}{N}_{t+1}^{*}\hfill \end{array}\right.$$

(9)

where $Z_t^{*}\sim N({\lambda }_1,1)$, $Y_t^{*}\sim N(m+{\lambda }_{2}s,s^2)$, $N=\left\{\begin{array}{cc}1,\hfill & \mathrm{with}\mathrm{probability}{\mathrm{p}}^{*}\hfill \\ 0,\hfill & \mathrm{with}\mathrm{probability}1-{\mathrm{p}}^{*}\hfill \end{array}\right.$, and $p^{*}=1-\Phi [{\Phi }^{-1}(1-p)-{\lambda }_3]$ ([6]).

The last mortality model is the transitory mortality jump with the renewal process model, and we use the Wang transform on the variables Z, Y, and N. The risk-adjusted mortality factor becomes:

$${\kappa }_t={\kappa }_0+\left(\mu -\frac{1}{2}{\sigma }^2-{\delta }^{*}{\theta }^{*}\right)t+\sigma Z_t^{*}+\sum _{i=1}^{N_t^{*}}{Y}_i^{*}$$

(10)

where $Z_t^{*}\sim N({\lambda }_1,1)$, $Y_t^{*}$ is distributed exponentially with the parameter ${\eta }^{*}=\eta +{\lambda }_2$, and ${\theta }^{*}=1-\Phi [{\Phi }^{-1}(1-\theta )-{\lambda }_3]$.

Here, we apply the Wang transform to the expected jump frequency, which is different from other models. Since other mortality models with jumps assumed that mortality jumps could occur only once a year, they apply the Wang transform to the one-year jump probability. However, we assume that there could be more than one jump in a year; hence, we apply the Wang transform to the expected jump frequency.

We can derive the implied market price of risk $\lambda$ based on our hypothetical mortality bond. As mentioned before, our hypothetical bond is similar to the Swiss-Re bond. It has the same payment structure and issue price. However, our mortality index is a weighted average of Switzerland’s mortality rates. Age weights are obtained for all ages by dividing the exposures at each age by the total exposure. For the risk-free interest rate, Chen and Cox ([6]) used the US Treasury yield rates on December 30, 2003. Here, we use Switzerland’s 10-year government bond yield rate on December 30, 2003, as the risk-free interest rate 1. Moreover, we calculate the coupon payments by assuming they are paid annually, as we do not have quarterly data for the Switzerland government bond yield.

The estimation procedure for calculating the market price of mortality risk for the hypothetical bond issued in 2003, based on the Wang transform and following the approach of Chen and Cox ([6]), involves the following steps:

1. Simulate 10,000 times the future mortality time-series ${\kappa }_t$ for 2004-2006 based on the known 2003 mortality time series, using the three mortality jump models.

2. Apply the Wang transform to transition from the physical measure P to the risk-adjusted measure Q. Calculate the values of ${\kappa }_t^{*}$ on each path under Q using Equation 8, 9, 10, given initial values of the market prices of risk ${\lambda }_1,{\lambda }_2$, and ${\lambda }_3$.

3. Calculate the mortality rates for different age groups by the formula $m_{x,t}^{*}=exp(a_x+b_x{k}_t^{*})$ under Q. Compute the weighted average mortality index for each year using $M_t={\sum }_x{w}_x{m}_{x,t}$, where $w_x$ denotes the weight assigned to the age group x based on the year 2003.

4. Calculate the expected value of principal payment in every period T by the formula $E_T^{*}\left[\mathrm{payment}\right]=\$400,000,000\times [max(1-{\sum }_{t=2004}^{2006}{L}_t,0)]$. Calculate the coupon payment in every period based on the par spread plus 135 basis points obtained from the Swiss-Re bond.

5. Discount the coupon payments for each period and the principal repayment back to the beginning of the year 2004 using the risk-free rate. Equate the discounted expected payoff to the issue size of the Swiss-Re mortality bond ($400 million) to obtain the market prices of risk using the quasi-Newton method, as in Chen and Cox ([6]).

The estimated market prices of risk (${\lambda }_1,{\lambda }_2,{\lambda }_3$) for different jump models are presented in Table 16.

Under the permanent jump model, assuming that ${\lambda }_2={\lambda }_3=0$ (indicating that the risk associated with the jump process is diversifiable), the market price of mortality risk is 1.03. If ${\lambda }_1={\lambda }_3=0$, the market price of risk associated with the jump size is 1.15. The market price of risk associated with the jump frequency is 0.09 if ${\lambda }_1={\lambda }_2$.

Shifting to the model with transitory jump effects, the market prices of risk increase in each case. This increase can be explained by differences in the model setups and the volatility of the mortality index. Higher volatility leads to higher risk and, consequently, lower market prices of risk.

Under the transitory mortality jumps with the renewal process model, the market prices of risk are the highest in each case. This implies that the model with transitory jumps with the renewal process offers a higher risk premium to investors, making the bond with this model more attractive to investors. Although a high market price of risk may suggest high transaction costs, it could also be interpreted as the hypothetical bond with transitory mortality jumps with the renewal process overcompensating investors for taking on its mortality risk ([8]).

5. Conclusions

Abbreviations

References

1. Blake, D.; Cairns, A.J.G.; Dowd, K. Living with Mortality: Longevity Bonds and Other Mortality-Linked Securities. British Actuarial Journal 2006, 12, 153–197. [Google Scholar] [CrossRef]
2. Boruhns, N.; Denuit, M.; Vermunt, J.K. A Poisson Log-Bilinear Regression Approach to the Construction of Projected Lifetables. Insurance: Mathematics and Economics 2002, 31, 373–393. [Google Scholar] [CrossRef]
3. Cairns, A.J.G. The Common Cohort Effect Model for Cause of Death Data. This research forms part of the “Modelling Measurement and Management of Longevity and Morbidity Risk” research program.
4. Cairns, A.J.G.; Blake, D.; Dowd, K. A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. Journal of the Risk and Insurance 2006, 73, 687–718. [Google Scholar] [CrossRef]
5. Cairns, A.J.G.; Blake, D.; Dowd, K.; Coughlan, G.D.; Ong, A.; Balevich, I. A Quantitative Comparison of Stochastic Mortality Models Using Data From England and Wales and the United States. North American Actuarial Journal 2007, 13, 1–35. [Google Scholar] [CrossRef]
6. Chen, H.; Cox, S.H. Modeling Mortality with Jumps: Applications to Mortality Securitization. The Journal of the Risk and Insurance 2009, 3, 727–751. [Google Scholar] [CrossRef]
7. Chung, C.; Liu, L.M. Joint Estimation of Model Parameters and Outlier Effects in Time Series. Journal of the American Statistical Association 1993, 3, 187–211. [Google Scholar]
8. Cox, S.H.; Lin, Y.; Wang, S.S. Multivariate Exponential Tilting and Pricing Implications for Mortality Securitization. Journal of Risk and Insurance 2006, 73, 719–736. [Google Scholar] [CrossRef]
9. Cowley, A.; Cummins, J.D. Securitization of Life Insurance Assets. Journal of Risk and Insurance 2005, 72, 193–226. [Google Scholar] [CrossRef]
10. Currie, I.D.; Durban, M.; Eilers, P.H.C. Cummins J.D. Smoothing and forecasting mortality rates. Statistical Modelling 2004, 4, 279–298. [Google Scholar] [CrossRef]
11. Deng, Y.; Brocket, P.L.; MacMinn, R.D. Longevity/Mortality Risk Modeling and Securities Pricing. The Journal of Risk and Insurance 2012, 3, 697–721. [Google Scholar] [CrossRef]
12. Harrison, J.M.; Kreps, D.M. Martingales and Arbitrage in Multiperiod Security Markets. The Journal of Econometric Theory 1979, 20, 381–408. [Google Scholar] [CrossRef]
13. Human Mortality Database, www.mortality.org.
14. Li, R.D.; Carter, L.R. Modeling and Forecasting U.S. Mortality. Journal of the American Statistical Association 1992, 87, 659–671. [Google Scholar]
15. Liu, Y. Li, J.S.H. The Age Pattern of Transitory Mortality Jumps and Its Impact on the Pricing of Catastrophic Mortality Bonds. Insurance: Mathematics and Economics 2015, 64, 135–150. [Google Scholar]
16. Li, S.H.; Chan, W.S. Outlier Analysis and Mortality Forecasting: The United Kingdom and Scandinavian Countries. Scandinavian Actuarial Journal 2005, 3, 187–211. [Google Scholar] [CrossRef]
17. McShane, B.; Adrian, M.; Bradlow, E.T.; Fader, P.S. Count Models Based on Weibull Interarrival Times. Journal of Business and Economic Statistics 2008, 26, 369–378. [Google Scholar] [CrossRef]
18. Özen, S.; Şahin, Ş. Transitory Mortality Jump Modeling with Renewal Process and Its Impact on Pricing of Catastrophic Bonds. Journal of Computational and Applied Mathematics 2020, 376, 1–15. [Google Scholar] [CrossRef]
19. Özen, S.; Şahin, Ş. A Two-Population Mortality Model to Assess Longevity Basis Risk. Risks 2021, 9, 1–19. [Google Scholar] [CrossRef]
20. Regis, J.; Jevtic, P. Stochastic Mortality Models and Pandemic Shocks. In Pandemics: Insurance and Social Protection; Boado-Penas, M.C., Eisenberg, J., Şahin, Ş., Eds.; Springer Actuarial Series: Switzerland, 2021; pp. 61–73. [Google Scholar]
21. Renshaw, A.E.; Haberman, S. Lee-Carter Mortality Forecasting with Age-Specific Enhancement. Insurance: Mathematics and Economics, 2003; 33, 255–272. [Google Scholar]
22. Renshaw, A.E.; Haberman, S. A Cohort-Based Extension to the Lee-Carter Model for Mortality Reduction Factors. Insurance: Mathematics and Economics, 2006; 38, 556–570. [Google Scholar]
23. Villegas, A.M.; Millossovich, P.; Kaishev, V.K. StMoMo: Stochastic Mortality Modeling in R. Journal of Statistical Software 2018, 84, 1–38. [Google Scholar] [CrossRef]
24. Wang, S.S. A Universal Framework for Pricing Financial and Insurance Risks. ASTIN Bulletin 2002, 32, 213–234. [Google Scholar] [CrossRef]