Document Type: Research Paper


1 Social and Business Research Laboratory, Department of Business Management, Rovira i Virgili University, Spain

2 Department of Mathematics for Economics, Finance and Actuarial Science, University of Barcelona, Spain


The concept of fuzzy random variable has been applied in several papers to model the present value of life insurance liabilities. It allows the fuzzy uncertainty of the interest rate and the probabilistic behaviour of mortality to be used throughout the valuation process without any loss of information. Using this framework, and considering a triangular interest rate, this paper develops closed expressions for the expected present value and its defuzzified value, the variance and the distribution function of several well-known actuarial liabilities structures, namely life insurances, endowments and life annuities.


[1] A. Alegre and M. Claramunt, Allocation of solvency cost in group of annuities: Actuarial
principles and cooperative game theory, Insurance: Mathematics and Economics, 17 (1995),
[2] J. Andres-Sanchez and L. Gonzalez-Vila Puchades, Using fuzzy random variables in life
annuities pricing, Fuzzy sets and Systems, 188 (2012), 27-44.
[3] J. Andres-Sanchez and L. Gonzalez-Vila Puchades, A fuzzy random variable approach to life
insurance pricing, In A. Gil-Lafuente; J. Gil-Lafuente and J.M. Merigo (Eds.), Studies in
Fuzziness and Soft Computing; Soft Computing in Management and Business Economics,
Springer-Verlag, Berlin/Heidelberg, (2012), 111-125.
[4] J. Andres-Sanchez and L. Gonzalez-Vila Puchades, Pricing endowments with soft computing,
Economic Computation and economic cybernetics studies research, 1 (2014), 124-142.
[5] J. Andres-Sanchez and A. Terce~no, Applications of Fuzzy Regression in Actuarial Analysis,
Journal of Risk and Insurance, 70 (2003), 665-699.
[6] J. J. Buckley, The fuzzy mathematics of nance, Fuzzy Sets and Systems, 21 (1987), 257-273.
[7] J. J. Buckley and Y. Qu, On using -cuts to evaluate fuzzy equations, Fuzzy Sets and Systems,
38 (1990), 309-312.
[8] L. M. Campos and A. Gonzalez, A subjective approach for ranking fuzzy numbers, Fuzzy
Sets and Systems, 29 (1989), 145-153.
[9] I. Couso, D. Dubois, S. Montes and L. Sanchez, On various de nitions of the variance of
a fuzzy random variable, 5th International Symposium on Imprecise Probabilities and Their
Applications, Prague, (2007), 135-144.
[10] J. D. Cummins and R. A. Derrig, Fuzzy nancial pricing of property-liability insurance,
North American Actuarial Journal, 1 (1997), 21-44.
[11] R. A. Derrig and K. Ostaszewski, Managing the tax liability of a property liability insurance
company, Journal of Risk and Insurance, 64 (1997), 695-711.
[12] Y. Feng, L. Hu and H. Shu, The variance and covariance of fuzzy random variables and their
application, Fuzzy Sets and Systems, 120 (2001), 487-497.
[13] H. U. Gerber, Life Insurance Mathematics, Springer-Verlag, Berlin/Heidelberg, 1995.
[14] S. Heilpern, The expected value of a fuzzy number, Fuzzy Sets and Systems, 47(1) (1992),

[15] R. Korner, On the variance of fuzzy random variables, Fuzzy Sets and Systems, 92 (1997),
[16] V. Kratschmer, A uni ed approach to fuzzy random variables, Fuzzy Sets and Systems, 123
(2001), 1-9.
[17] H. Kwakernaak, Fuzzy random variables I: de nitions and theorems, Information Sciences,
15 (1978), 1-29.
[18] J. Lemaire, Fuzzy insurance, Astin Bulletin, 20 (1990), 33-55.
[19] M. Li Calzi, Towards a general setting for the fuzzy mathematics of nance, Fuzzy Sets and
Systems, 35 (1990), 265-280.
[20] M. Lopez-Diaz and M. A. Gil, The -average value and the fuzzy expectation of a fuzzy
random variable, Fuzzy Sets and Systems, 99 (1998), 347-352.
[21] H. T. Nguyen, A note on the extension principle for fuzzy sets, Journal of Mathematical
Analysis and Applications, 64 (1978), 369-380.
[22] K. Ostaszewski, An Investigation Into Possible Applications of Fuzzy Sets Methods in Actu-
arial Science, Society of Actuaries, Schaumburg, 1993.
[23] E. Pitacco, Simulation in insurance, In: Goovaerts, M. De Vylder, F. Etienne and J. Haezendonck
(Eds.), Insurance and risk theory, Reidel, Dordretch, (1986), 37-77.
[24] M. L. Puri and D. A. Ralescu, Fuzzy random variables, Journal of Mathematical Analysis
and Applications, 114 (1986), 409-422.
[25] E. Roventa and T. Spircu, Averaging procedures in defuzzi cation processes, Fuzzy Sets and
Systems, 136 (2003), 375{385.
[26] A. Shapiro, Modeling future lifetime as a fuzzy random variable, Insurance: Mathematics
and Economics, 53 (2013), 864-870.
[27] R. Viertl and D. Hareter, Fuzzy information and stochastics, Iranian Journal of Fuzzy Systems,
1 (2004), 43-56.
[28] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
[29] C. Zhong and G. Zhou, The equivalence of two de nitions of fuzzy random variables, Proceedings
of the 2nd IFSA Congress (1987), Tokyo, 59-62,