Nonlevel Annuities - Arithmetic Progression¶

TmVal’s Annuity class can also handle annuities with increasing arithmetic progression. This means that if an annuity makes an initial payment $P$ , the next payment is $P+Q$ , and the payment after that is $P+2Q$ , and so on. The present value of such an annuity is:

$(I_{P,Q}\ax{}){\angln i} = P\ax{\angln i} + \frac{Q}{i}(\ax{\angln i} - nv^n).$

The accumulated value is:

$(I_{P,Q}\sx{}){\angln i} = P\sx{\angln i} + \frac{Q}{i}(\sx{\angln i} - n).$

The Annuity class can also handle the companion formulas for the annuity-due case:

$(I_{P,Q}\sx**{}){\angln i} = P\sx**{\angln i} + \frac{Q}{d}(\sx{\angln i} - n),$

and

$(I_{P,Q}\ax**{}){\angln i} = P\ax**{\angln i} + \frac{Q}{d}(\ax{\angln i} - n).$

For special cases $(I\sx{}){\angln i}$ , $(I\ax{}){\angln i}$ , $(D\ax{}){\angln i}$ , $(I\sx**{}){\angln i}$ , $(D\ax**{}){\angln i}$ , see the Notation Guide.

Examples¶

Suppose we have a 10-year annuity with an initial end-of-year payment of 100, and subsequent end-of-year payments increasing by 100 for each of the next 9 years. If the interest is 5% compounded annually, what is the present value?

We can solve this problem by setting the aprog argument of the Annuity class to aprog=100:

In [1]: from tmval import Annuity

In [2]: ann = Annuity(
   ...:     gr=.05,
   ...:     amount=100,
   ...:     n=10,
   ...:     aprog=100
   ...: )
   ...: 

In [3]: print(ann.pv())
3937.378280472917

Now, suppose instead that the annuity makes beginning-of-year payments. What is the accumulated value?

In [4]: ann2 = Annuity(
   ...:     gr=.05,
   ...:     amount=100,
   ...:     n=10,
   ...:     aprog=100,
   ...: )
   ...: 

In [5]: print(ann2.sv())
6413.57432465254

The special cases mentioned earlier can be achieved by simply modifying the amount and aprog argument to be equal to the corresponding special case values. For more information on what these symbols mean and how to derive them, refer to a text on interest theory (some can be found in the References section).