Perpetuities¶

A perpetuity is a type of annuity that has an infinite number of payments. Perpetuities come in both immediate and due forms. For the former, the payments occur at the beginning of each period, whereas for the latter, they occur at the end of each period. A basic perpetuity (either immediate or due), is one that pays 1 for each period.

Like annuities, perpetuities have present value formulas that can be simplified to concise algebraic expressions. This fact can be proved via properties of infinite series. For a perpetuity-immediate:

$\ax{\angl{\infty} i} = \frac{1}{i}$

For a perpetuity-due:

$\ax**{\angl{\infty} i} = \frac{1}{d}.$

Unlike annuities, perpetuities do not have an accumulated value because the payments never end.

Examples¶

Suppose we have a perpetuity-immediate that pays 1 at the end of each year, and the annual effective interest rate is 5%. What is the present value of the annuity?

We can solve this problem by using TmVal’s Annuity class. In order to specify an infinite number of payments, we can set either the term or n argument to be infinite. We do so by importing Numpy and setting term=np.Inf or n=np.Inf:

In [1]: import numpy as np

In [2]: from tmval import Annuity, Rate

In [3]: ann = Annuity(
   ...:    term=np.Inf,
   ...:    gr=Rate(.05)
   ...: )
   ...: 

In [4]: ann2 = Annuity(
   ...:    n=np.Inf,
   ...:    gr=Rate(.05)
   ...: )
   ...: 

In [5]: print(ann.pv())
19.999999999999982

In [6]: print(ann2.pv())
19.999999999999982

For those who are unfamiliar with NumPy, NumPy is a scientific computing package that serves as the backbone of many other popular Python tools, such as Pandas (and hopefully someday, TmVal).