Nonlevel Annuities - Geometric Progression¶

Annuities can have payments that increase geometrically. For example, an annuity might have payments that increase by 2% per year. If we have payments that increase by g% per year, we define the present value of an annuity-immediate with an initial payment $P$ as:

$P\left(\frac{1-\left(\frac{1 + g}{1 + i}\right)^n}{i-g}\right),$

where $i-g \neq 0$ , since this expression is undefined when the denominator is 0. If the payments increase at the rate of interest, we have:

$nP(1 + i)^{-1}.$

Examples¶

Suppose we have an annuity-immediate with end-of-year payments that pays 1 at the end of the first period, and then whose payments increase by 2% for each year for the next 4 years. If the interest rate is 5% compounded annually, what is its present value?

We can solve this problem by using TmVal’s Annuity class, and by providing the rate of payment increase to the argument gprog, which in this case is gprog=.02:

In [1]: from tmval import Annuity, Rate

In [2]: ann = Annuity(
   ...:    gr=Rate(.05),
   ...:    n=5,
   ...:    gprog=(.02)
   ...: )
   ...: 

In [3]: print(ann.pv())
4.497460026576225