Deferred Annuities¶

A deferred annuity is a type of annuity whose first payment begins more than one payment period later than its present valuation date. For example, we can purchase a deferred annuity-immediate today that makes annual payments beginning 5 years from now.

We denote a deferred annuity-immediate as $\ax[w|n]{}$ , where $w$ indicates that the first payment will occur $w + 1$ periods from the present valuation date. For example, the annuity in the preceding paragraph would be denoted $\ax[4|n]{}$ because the first payment occurs at time 5, which is equal to $w + 1$ , since in this case $w = 4$ .

Annuities-immediate, deferred annuities-due, and deferred annuities-immediate are related by the following expression:

$\ax{\angln} = \ax**[1|n]{} = \ax[0|n]{}$

Examples¶

Suppose we purchase an annuity-immediate deferred for 4 years. This annuity makes a payment of 1,000 each year for a term of five years, with the first payment beginning 5 years from now. If the annual effective interest rate is 5%, how much does the annuity cost?

We can solve this problem by using TmVal’s Annuity class, and specifying the deferral by setting the argument deferral=4.

In [1]: from tmval import Annuity, Rate

In [2]: ann = Annuity(
   ...:    amount=1000,
   ...:    n=5,
   ...:    gr=Rate(.05),
   ...:    deferral=4
   ...: )
   ...: 

In [3]: print(ann.pv())
3561.8711714816923

Now suppose we want to know, if we reinvest the payments at the 5% effective rate, how much will the investments grow to 20 years from now?

In [4]: print(ann.eq_val(20))
9450.704605312447