Perpetuities - Arithmetic Progression¶

TmVal’s Annuity class can also handle perpetuities with payments of increasing arithmetic progression:

$(I_{P,Q}\ax{}){\angl{\infty} i} = P\ax{\angl{\infty} i} + \frac{Q}{i}{\angl{\infty} i} = \frac{P}{i} + \frac{Q}{i^2}$

$(I_{P,Q}\ax**{}){\angl{\infty} i} = P\ax**{\angl{\infty} i} + \frac{Q}{i}{\angl{\infty} i} = \frac{P}{d} + \frac{Q}{id}$

Examples¶

Suppose we have a perpetuity-immediate with an initial end-of-year payment of 100. Subsequent end-of-year payments increase 100 each year forever. If the interest rate is 5% compounded annually, what’s the present value?

To solve this problem, we need the special value np.Inf from NumPy to specify an infinite term, passing term=np.Inf to TmVal’s Annuity class.

In [1]: import numpy as np

In [2]: from tmval import Annuity

In [3]: ann = Annuity(
   ...:    amount=100,
   ...:    gr=.05,
   ...:    term=np.Inf,
   ...:    aprog=100
   ...: )
   ...: 

In [4]: print(ann.pv())
41999.99999999993

What if we have a perpetuity-due instead?

In [5]: ann2 = Annuity(
   ...:    amount=100,
   ...:    gr=.05,
   ...:    term=np.Inf,
   ...:    aprog=100,
   ...:    imd='due'
   ...: )
   ...: 

In [6]: print(ann2.pv())
44099.99999999993