Present Value of Annuities Under Random Rates of Interest By

Some attempts were made to evaluate the future value (FV) of the expected value and the variance for some classical cash flows (CF). The motivation stemmed from some recursive formulas. Similar recursive formulas do not hold for the case of the present value (PV) due to the lack of independence of some random variables. One can get some estimates for the PV using the results about the FV. We study the PV and we overcome the difficulty of independence by reversing the order of the CF. It turns out that we get similar recursive formulas for the PV as for the FV in the classical cash flows. It turns out to be similar to that used to evaluate the FV. Furthermore it makes it possible to study the PV of the classical CF directly, and it may suggest a method to study the PV of other CF as well