Lévy CARMA models for shocks in mortality

Abstract

Recent literature on mortality modeling suggests to include in the dynamics of mortality rates the effects of time, age, the interaction of these two and a term for possible shocks. In this paper we investigate models that use Legendre polynomials for the inclusion of age and cohort effects. In order to capture the dynamics of the shock term it is suggested to consider continuous autoregressive moving average (CARMA) models due to their flexibility in reproducing different autoregressive profiles of the shock term. In order to validate the proposed model, different life tables are considered. In particular the male life tables for New Zealand, Taiwan and Japan are used for the presentation of in-sample fitting. Empirical analysis suggests that the inclusion of more flexible models such as higher-order CARMA(p,q) models leads to better in-sample fitting.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Notes

  1. 1.

    As observed in Loregian et al. (2012), in literature, several parametrizations for the Variance Gamma are available. In this paper, we use the parametrization considered by Madan and Seneta (1990), but we recall that it is always possible to switch from a parametrization to another through a proper parameter transformation.

  2. 2.

    The data on Human Mortality Database (2014) are collected yearly.

  3. 3.

    This is the same dataset used in Ahmadi and Gaillardetz (2015).

  4. 4.

    This is the null hypothesis.

  5. 5.

    This is the alternative hypothesis.

  6. 6.

    CARMA-BM(2,1) and CARMA-BM(2,0) are two nested models and therefore the LR-test is used.

References

  1. Ahmadi, S.S., Gaillardetz, P.: Modeling mortality and pricing life annuities with Lévy processes. Insur. Math. Econ. 64, 337–350 (2015)

    Article  Google Scholar 

  2. Antolin, P., Schich, S., Yermo, J.: The economic impact of protracted low interest rates on pension funds and insurance companies. OECD J. Financ. Mark. Trends 1, 1–20 (2011)

    Google Scholar 

  3. Ballotta, L., Haberman, S.: The fair valuation problem of guaranteed annuity options: the stochastic mortality environment case. Insur. Math. Econ. 38, 195–214 (2006)

    Article  Google Scholar 

  4. Barndorff-Nielsen, O.E.: Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 353(1674), 401–419 (1977)

    Article  Google Scholar 

  5. Barndorff-Nielsen, O.E.: Normal inverse gaussian distributions and stochastic volatility modelling. Scand. J. Stat. 24(1), 1–13 (1997)

    Article  Google Scholar 

  6. Brockwell, P.J.: Lévy-driven CARMA processes. Ann. Inst. Stat. Math. 53(1), 113–124 (2001)

    Article  Google Scholar 

  7. Brockwell, P.J.: Representations of continuous-time ARMA processes. J. Appl. Probab. 41(A), 375–382 (2004). https://doi.org/10.1017/S0021900200112422

    Article  Google Scholar 

  8. Brockwell, P.J., Marquardt, T.: Lévy-driven and fractionally integrated arma processes with continuous time parameter. Stat. Sin. 15(1), 477–494 (2005)

    Google Scholar 

  9. Brockwell, P.J., Davis, R., Yang, Y.: Estimation for non-negative Lévy-driven CARMA processes. J. Bus. Econ. Stat. 29(2), 250–259 (2011)

    Article  Google Scholar 

  10. Database, H.M.: University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany) (2014). www.mortality.org. Accessed 5 July 2018

  11. Eberlein, E., Raible, S.: Term structure models driven by general Lévy processes. Math. Finance 9(1), 31–53 (1999)

    Article  Google Scholar 

  12. Hitaj, A., Mercuri, L., Rroji, E.: Some Empirical Evidence on the Need of More Advanced Approaches in Mortality Modeling, pp. 425–430. Springer, Cham (2018)

    Google Scholar 

  13. Iacus, S.M., Mercuri, L.: Implementation of Lévy CARMA model in yuima package. Comput. Stat. 30(4), 1111–1141 (2015)

    Article  Google Scholar 

  14. Kendall, M., Stuart, A.: The Advanced Theory of Statistics. Distribution Theory, vol. 1, 4th edn. Griffin, London (1977)

    Google Scholar 

  15. Küchler, U., Tappe, S.: Tempered stable distributions and processes. Stoch. Process. Appl. 123(12), 4256–4293 (2013)

    Article  Google Scholar 

  16. Lee, D., Carter, L.: Modelling and forecasting U.S. mortality. J. Am. Stat. Assoc. 87, 659–671 (1992)

    Google Scholar 

  17. Loregian, A., Mercuri, L., Rroji, E.: Approximation of the variance gamma model with a finite mixture of normals. Stat. Probab. Lett. 82(2), 217–224 (2012)

    Article  Google Scholar 

  18. Madan, D.B., Seneta, E.: The variance gamma (v.g.) model for share market returns. J. Bus. 63, 511–524 (1990)

    Article  Google Scholar 

  19. McCullagh, P., Nelder, J.A.: Generalized Linear Models, vol. 37. CRC Press, Boca Raton (1989)

    Google Scholar 

  20. Oeppen, J., Vaupel, J.W.: Broken limits to life expectancy. Science 296(5570), 1029–1031 (2002)

    Article  Google Scholar 

  21. Poirot, J., Tankov, P.: Monte carlo option pricing for tempered stable (CGMY) processes. Asia Pac. Financ. Mark. 13(4), 327–344 (2006)

    Article  Google Scholar 

  22. Renshaw, A.E., Haberman, S., Hatzoupoulos, P.: The modelling of recent mortality trends in United Kingdom male assured lives. Br. Actuar. J. 2, 449–477 (1996)

    Article  Google Scholar 

  23. Rroji, E., Mercuri, L.: Mixed tempered stable distribution. Quant. Finance 15(9), 1559–1569 (2015)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Edit Rroji.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Proof of Proposition 1

Proof of Proposition 1

Proof

Starting from the explicit solution of the state-space process \(X_t\) in (7), we write the CARMA(p,q) model \(Y_t\) as follows:

$$\begin{aligned} Y_{t}=\mathbf {b}^{\intercal }e^{A\left( t-s\right) }X_{s}+\int _{s}^{t}\mathbf {b}^{\intercal }e^{A\left( t-u\right) }\mathbf {e}\text{ d }Z_{u},\ \ \forall t>s. \end{aligned}$$

As a first step, we observe that the quantity \(g\left( t-u\right) :=\mathbf {b}^{\intercal }e^{A\left( t-u\right) }\mathbf {e}\mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \) can be written as follows:

$$\begin{aligned} g\left( t-u\right) =\mathbf {b}^{\intercal }Re^{\Lambda (t-u)}R^{-1}\mathbf {e}\mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \end{aligned}$$
(20)

where \(\Lambda \) is a diagonal matrix whose elements are the eigenvalues of matrix A and R is the matrix of right eigenvectors of A, i.e.,

$$\begin{aligned} R= & {} \left[ \lambda _{j}^{i-1}\right] _{i,j=1}^{p}\\ \Lambda= & {} {\left\{ \begin{array}{ll} \lambda _{j} &{} j=i\\ 0 &{} j\ne i \end{array}\right. }\ i,j=1,\ldots ,p. \end{aligned}$$

As in Brockwell et al. (2011), we have:

$$\begin{aligned} R^{-1}\mathbf {e}= & {} \left[ \frac{1}{a^{\prime }\left( \lambda _{1}\right) },\frac{1}{a^{\prime }\left( \lambda _{2}\right) },\ldots ,\frac{1}{a^{\prime }\left( \lambda _{p}\right) }\right] ^{T} \\ Re^{\Lambda (t-u)}= & {} \left[ \begin{array}{cccc} e^{\lambda _{1}} &{}\quad e^{\lambda _{2}} &{}\quad \ldots &{}\quad e^{\lambda _{p}}\\ \lambda _{1}e^{\lambda _{1}} &{}\quad \lambda _{2}e^{\lambda _{2}} &{}\quad \ldots &{}\quad \lambda _{p}e^{\lambda _{p}}\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \lambda _{1}^{p-1}e^{\lambda _{1}} &{}\quad \lambda _{2}^{p-1}e^{\lambda _{2}} &{}\quad \ldots &{}\quad \lambda _{p}^{p-1}e^{\lambda _{p}} \end{array}\right] . \end{aligned}$$

The kernel \(g\left( t-u\right) \) in (20) can be written as:

$$\begin{aligned} g\left( t-u\right)= & {} \mathbf {b}^{\intercal }\left[ \begin{array}{cccc} e^{\lambda _{1}\left( t-u\right) } &{} e^{\lambda _{2}\left( t-u\right) } &{} \ldots &{} e^{\lambda _{p}\left( t-u\right) }\\ \lambda _{1}e^{\lambda _{1}\left( t-u\right) } &{}\quad \lambda _{2}e^{\lambda _{2}\left( t-u\right) } &{}\quad \ldots &{}\quad \lambda _{p}e^{\lambda _{p}\left( t-u\right) }\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \lambda _{1}^{p-1}e^{\lambda _{1}\left( t-u\right) } &{}\quad \lambda _{2}^{p-1}e^{\lambda _{2}\left( t-u\right) } &{}\quad \ldots &{}\quad \lambda _{p}^{p-1}e^{\lambda _{p}\left( t-u\right) } \end{array}\right] \left[ \begin{array}{c} \frac{1}{a^{\prime }\left( \lambda _{1}\right) }\\ \frac{1}{a^{\prime }\left( \lambda _{2}\right) }\\ \vdots \\ \frac{1}{a^{\prime }\left( \lambda _{p}\right) } \end{array}\right] \mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \\= & {} \mathbf {b}^{\intercal }\left[ \begin{array}{c} \underset{i=1}{\overset{p}{\sum }}\frac{1}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }\\ \underset{i=1}{\overset{p}{\sum }}\frac{\lambda _{i}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }\\ \vdots \\ \underset{i=1}{\overset{p}{\sum }}\frac{\lambda _{i}^{p-1}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) } \end{array}\right] \mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \\= & {} \left( \underset{i=1}{\overset{p}{\sum }}\frac{b_{0}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }+\underset{i=1}{\overset{p}{\sum }}\frac{b_{1}\lambda _{i}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }+\cdots +\underset{i=1}{\overset{p}{\sum }}\frac{b_{p-1}\lambda _{i}^{p-1}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }\right) \\&\times \mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \\= & {} \left( \underset{i=1}{\overset{p}{\sum }}\frac{b_{0}+b_{1}\lambda _{i}+\cdots +b_{p-1}\lambda _{i}^{p-1}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }\right) \mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \\= & {} \underset{i=1}{\overset{p}{\sum }}\left[ \frac{b\left( \lambda _{i}\right) }{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }\right] \mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) . \end{aligned}$$

The process \(Y_{t}\) in (5) becomes the result in (8). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hitaj, A., Mercuri, L. & Rroji, E. Lévy CARMA models for shocks in mortality. Decisions Econ Finan 42, 205–227 (2019). https://doi.org/10.1007/s10203-019-00248-9

Download citation

Keywords

  • Force of mortality
  • CARMA(p
  • q) model
  • Lévy process

JEL Classification

  • C02
  • C53
  • G22