Actuarial Present Value and Variance for Changing Mortality and Stochastic Interest Rates

  • Bükre Yıldırım
  • A. Sevtap Selcuk-Kestel
  • N. Gülden Coşkun-Ergökmen
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 195)


Stochastic modeling of interest rates is expected to lead a better risk management in long-term investments due to the rapid changes and random fluctuations in the economies. Considering the fact that deterministic interest rate approach does not yield realistic future values, a country-specific stochastic model is aimed to fit the interest rates based on the United States Treasury Inflation Protected Securities (TIPS) at 10-year constant maturity by using time series techniques. Under the assumption that interest rate follows an ARMA(1, 1) model, the actuarial present value and its variance for a ten-year term life insurance policy are derived. Additionally, the stochastic mortality using Lee-Carter model for future mortality predictions is implemented to the U.S. Mortality tables over a period of 81 years. Based on these two stochastic patterns, the actuarial present value and the variance functions are calculated numerically for the years 2014 and forecasted for 2030. The accuracy of the proposed model is performed by assessing a comparative analysis with respect to a prespecified deterministic interest rate and mortality table.


Term-life insurance Stochastic interest rate Treasury securities ARMA(p, qLee-Carter model Actuarial present value Actuarial variance 


  1. 1.
    Boyle, P.P.: Rates of return as random variables. J. Risk Insur. 43(4), 693–713 (1976)CrossRefGoogle Scholar
  2. 2.
    Panjer, H.H., Bellhause, D.R.: Stochastic modeling of interest rates with applications to life contingencies. J. Risk Insur. 47, 91–110 (1980)CrossRefGoogle Scholar
  3. 3.
    Bellhouse, D.R., Panjer, H.: Stochastic modeling of interest rates with applications to life contingencies, Part II. J. Risk Insur. 48, 628–637 (1981)CrossRefGoogle Scholar
  4. 4.
    Giacotto, C.: Stochastic modeling of interest rates: actuarial versus equilibrium approach. J. Risk Insur. 53, 435–453 (1986)CrossRefGoogle Scholar
  5. 5.
    Dhaene, J.: Stochastic interest rates and autoregressive integrated moving average processes. ASTIN bull. 19(2), 131–138 (1989)Google Scholar
  6. 6.
    Frees, E.W.: Net premiums in stochastic life contingencies. Trans. Soc. Actuar. 40(1), 371–385 (1988)MathSciNetGoogle Scholar
  7. 7.
    Frees, E.W.: Stochastic life contingencies with solvency considerations. Trans. Soc. Actuar. 42, 91–148 (1990)Google Scholar
  8. 8.
    Parker, G.: Moments of the present value of a portfolio of policies. Scand. Actuar. J. 1, 53–67 (1994)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lai, S.-W., Frees, E.W.: Examining changes in reserves using stochastic interest models. J. Risk Insur. 6(3), 535–574 (1995)CrossRefGoogle Scholar
  10. 10.
    Zaks, A.: Annuities under random rates of interest. Ins. Math. Econ. 28, 1–11 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Debicka, J.: Moments of the cash value of future payment streams arising from life insurance contracts. Ins. Math. Econ. 33(3), 533–550 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Vasicek, O.: An equilibrium characterization of the term structure. J. Financ. Econ. 5, 177–188 (1977)CrossRefGoogle Scholar
  14. 14.
    Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Engle, R.F.: Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007 (1982)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31, 307–327 (1986)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nelson, D.B.: Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370 (1991)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ergökmen, N.G.: Stochastic modeling of random interest rates in life insurance. Unpublished M.Sc, Thesis, Middle East Technical University, Turkey (2001)Google Scholar
  19. 19.
    Market yield on U.S. Treasury securities.
  20. 20.
    Box, G.E.P., Jenkins, G.M.: Time Series Analysis, Forecasting and Control, 2nd edn. Holden-Day, San Francisco (1976)MATHGoogle Scholar
  21. 21.
    Said, D.E., Dickey, D.A.: Testing for unit roots in autoregressive moving average models of unknown order. Biometrika 71, 599–607 (1984)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lee, R.D., Carter, L.R.: Modeling and forecasting US mortality. J. Am. Stat. Assoc. 87(419), 659–671 (1992)MATHGoogle Scholar
  23. 23.

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Bükre Yıldırım
    • 1
  • A. Sevtap Selcuk-Kestel
    • 1
  • N. Gülden Coşkun-Ergökmen
    • 2
  1. 1.Middle East Technical University, Institute of Applied MathematicsAnkaraTurkey
  2. 2.Republic of Turkey, Prime Ministry, Undersecretariat TreasuryAnkaraTurkey

Personalised recommendations