Fuzzy Sets and Systems 188 (2012) 27 – 44 www.elsevier.com/locate/fss Using fuzzy random variables in life annuities pricing Jorge de Andrés-Sánchez a , ∗ , Laura González-Vila Puchades b a Department of Business Administration, Faculty of Economics and Business Studies, Rovira i Virgili University, Av. de la Universitat 1, 43204 Reus, Spain b Department of Economic, Financial and Actuarial Mathematics, Faculty of Economics and Business,University of Barcelona, Av. Diagonal 696, 08034 Barcelona, Spain Received 15 September 2010; received in revised form 26 May 2011; accepted 26 May 2011 Available online 12 June 2011 Abstract This paper develops life annuity pricing with stochastic representation of mortality and fuzzy quantification of interest rates. We show that modelling the present value of annuities with fuzzy random variables allows quantifying their expected price and risk resulting from the uncertainty sources considered. So, we firstly describe fuzzy random variables and define some associated measures: the mathematical expectation, the variance, distribution function and quantiles. Secondly, we show several ways to estimate the discount rates to price annuities. Subsequently, the present value of life annuities is modelled with fuzzy random variables. We finally show how an actuary can quantify the price and the risk of a portfolio of annuities when their present value is given by means of fuzzy random variables. © 2011 Elsevier B.V. All rights reserved. Keywords: Economics; Finance; Life annuities; Fuzzy numbers; Fuzzy random variables 1. Introduction Pricing life annuities has to model the uncertainty of demographic events and financial variables. In this paper, as is done in standard life insurance mathematics (see, e.g., in [12]), we will consider for demographic phenomena its stochastic uncertainty. So, we will obtain the corresponding probabilities from life tables. Recent research in actuarial science has focused on formalizing the uncertainty related to the economic parameters by means of random variables and stochastic processes. The most important of those parameters is, undoubtedly, the discount rates used to price contracts. Therefore, since the early 70s the actuarial literature has extensively developed life annuities pricing with random interest rates, see, e.g., [2,4,26]. In the actuarial field, fuzzy sets theory (FST) has been used to model problems that require a great deal of actuarial subjective judgement and problems for which the information available is scarce or vague. Complete surveys of FST applications in actuarial science can be found in [25,29]. One of the applications of FST in actuarial science is to price insurance contracts with fuzzy interest rates. In this respect, [1,3,21] are particularly noteworthy in a life-insurance ∗ Corresponding author. Tel.: +34 977759833; fax: +34 977759810. E-mail addresses: jorge.deandres@urv.net (J. de Andrés-Sánchez), lgonzalezv@ub.edu (L. González-Vila Puchades). 0165-0114/$-see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.05.024