Abstract
ABSTRACT This study aimed to set upper and lower bounds for the expected present value of whole life annuities and whole life insurance policies from incomplete mortality data, generalizing previous results on life expectancy. Since its inception, in the 17th century, actuarial science has been devoted to the study of annuities and insurance plans. Thus, setting intervals that provide an initial idea about the cost of these products using incomplete mortality data represents a theoretical contribution to the area and this may have major applications in markets lacking historical records or those having little reliability of mortality data, as well as in new markets still poorly explored. For both the continuous and discrete cases, upper and lower bounds were constructed for the expected present value of whole life annuities and whole life insurance policies, contracted by a person currently aged x, based on information about the expected present value of these respective financial products subscribed to by a person of age x + n and the probability that an individual of age x survives to at least age x + n. Through the bounds of a continuous annuity, in an environment where the instantaneous interest rate is equal to zero, the results shown also set bounds for the complete life expectancy, which implies that the contribution of this research generalizes previous results in the literature. It was also found that, for both annuities and insurance plans, the length of constructed intervals increases as the data gap size increases and it decreases as the survival curve becomes more rectangular. Illustratively, bounds for life expectancy at 40 and 60 years of age, for the 10 municipalities showing the highest life expectancy at birth in Brazil in 2010, were constructed by using data available in the Atlas of Human Development in Brazil.
Highlights
According to Pitacco, Denuit, Haberman and Olivieri (2009), actuarial science flourished in the mid-17th century, based on compound interest rate theory and probability theory, as well as observations on mortality
According to these authors, one of the first themes to be addressed by this new science was calculating the expected present value of annuities
This study aims to extend the results of Cohen (2011), by setting bounds for the expected present value of annuities and life insurance policies subscribed to by a person of age x, both in the continuous and discrete cases
Summary
According to Pitacco, Denuit, Haberman and Olivieri (2009), actuarial science flourished in the mid-17th century, based on compound interest rate theory and probability theory, as well as observations on mortality According to these authors, one of the first themes to be addressed by this new science was calculating the expected present value of annuities. The absence of complete mortality tables hinders the calculation of longevity measures, as in the case of the complete life expectancy In this way, Cohen (2011) established upper and lower bounds on life expectancy at a given age x, knowing only detailed mortality data from age x + n (and, life expectancy at age x + n), as well as the probability that a person aged x survives to at least age x + n. See Bowers, Gerber, Hickman, Jones and Nesbitt (1997)
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