Comparing present and accumulated value of annuities with different interest rates

Abstract

Assume we have $k$ immediate (due)-annuities with different interest rates. Let ${bf i}=(i_1,i_2,...,i_k)$ and ${bf i^*}=(i^*_1,i^*_2,...,i^*_k)$ be two vectors of interest rates such that ${bf i^*}$ is majorized by ${bf i}$. It's shown that sum of present and accumulated value of annuities-immediate with interest rate ${bf i}$ is grater than sum of present value of annuities-immediate with interest rate ${bf i^*}$. We also prove the similar results for annuities-due.