Annuities-Immediate¶

An annuity-immediate is a type of annuity in which the payments occur at the end of each payment period. We define a basic annuity-immediate as an annuity-immediate in which every payment equals 1.

Basic annuities-immediate are convenient because the formulas specifying their present and accumulated values can be reduced to simple, compact algebraic expressions. For example, the present value of a basic annuity-immediate that pays 1 for $n$ periods at compound effective interest rate $i$ per period is:

$\ax{\angln i} = v + v^2 + v^3 + \cdots + v^n = \frac{1 - v^n}{i}$

The accumulated value at time $n$ is:

$\sx{\angln i} = (1 + i)^{n-1} + (1+i)^{n-2} + \cdots + 1 = \frac{(1+i)^n - 1}{i}$

Examples¶

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

We can solve this problem by importing the Annuity class, TmVal’s general class that can be used to represent most types of annuities. We can specify the payment amount with the argument amount=1, the 5-year term as term=5, the 1-year payment interval as period=1, and the 5% interest rate as gr=Rate(.05).

We then use the method Annuity.pv() to get the present value:

In [1]: from tmval import Annuity, Rate

In [2]: ann = Annuity(
   ...:    amount=1,
   ...:    term=5,
   ...:    period=1,
   ...:    gr=Rate(.05)
   ...: )
   ...: 

In [3]: print(ann.pv())
4.329476670630819

Note that we can simplify the above answer and make it more consistent with actuarial notation by simply supplying the number of payments as n=5. Since the period defaults to 1 year, the amount defaults to 1, we only need to supply two arguments in the case of a basic annuity-immediate:

In [4]: ann = Annuity(
   ...:    n=5,
   ...:    gr=Rate(.05)
   ...: )
   ...: 

In [5]: print(ann.pv())
4.329476670630819

What is the accumulated value? We can use Annuity.sv() to find out:

In [6]: print(ann.sv())
5.525631250000002