Perpetuityimmediate

Perpetuityimmediate refers to a perpetuity, a stream of equal payments that continues forever, with payments occurring at the end of each period. In actuarial and financial mathematics, this term distinguishes the timing of cash flows from other perpetuities or annuities. The payments are level and the horizon is infinite.

If the interest rate per period is i, the present value of a perpetuity immediate that pays 1 unit at the end of every period is 1/i. More generally, for a perpetuity immediate paying P units per period, the present value is P/i. This result follows from summing a geometric series: PV = P/(1+i) + P/(1+i)^2 + ... = (P/(1+i)) / (1 - 1/(1+i)) = P/i.

Perpetuity immediate is often contrasted with perpetuity due, where payments occur at the beginning of each period. The present value of a perpetuity due paying P per period is P(1+i)/i, reflecting the extra period of time value for each payment. Both concepts assume an infinite time horizon and positive interest rate.

Applications of perpetuity immediate appear in theoretical valuations of perpetual securities, such as consoles and other instruments that provide fixed payments indefinitely. They serve as useful benchmarks in actuarial science and financial theory, illustrating the relationship between payments, discount rates, and infinite cash flows. In practical modeling, the assumption of i > 0 is essential; as i tends to 0, the present value grows without bound.