Precificação de rendas para casais: uma abordagem via modelos de fragilidade

Abstract

In actuarial calculations for more than one individual, it is common to assume independence between residual lifetimes. However, for certain groups of individuals, such as couples, twins, and families, this assumption may not be valid, since lifetimes may be interconnected due to various causes; among them, the effects of common disasters, dependence due to lifestyle, and the broken heart syndrome, observed especially among couples. Consequently, when this reality is ignored, it may affect the assumptions of actuarial calculations, as independence is presumed. To verify this issue empirically, this study implements a dependence approach through a shared frailty model with a gamma distribution to assess whether there is dependence between the residual lifetimes of couples and to explore whether this relationship impacts the pricing of multi-life annuities. For the model application, a secondary dataset collected from historical records on the FamilySearch website was used. The results revealed a significant existence of dependence between residual lifetimes and the absence of a statistically significant difference in survival by sex. When considering frailty in survival, it is observed that survival under dependence tends to be higher compared to the common independence practice, and that this difference increases as ages rise. This also corresponds to the effect generated in annuities, indicating that the independent annuity value is undervalued. Thus, it is concluded that there is statistical evidence of dependence between the residual lifetimes of couples and that this dependence impacts annuity pricing, which becomes higher compared to standard market practice.