Deferred Annuity

A deferred annuity ${}_{k|}a_{\overline{n}|}$ is an annuity-immediate (or annuity-due) whose first payment is delayed by a deferral period of $k$ periods.

${}_{k|}a_{\overline{n}|} = v^k \cdot a_{\overline{n}|}$

$= a_{\overline{k+n}|} - a_{\overline{k}|}$

${}_{k|}\ddot{a}_{\overline{n}|} = v^k \cdot \ddot{a}_{\overline{n}|}$

The difference form $a_{\overline{k+n}|} - a_{\overline{k}|}$ shows a deferred annuity as a full $(k+n)$ -payment annuity minus the payments withheld during the deferral.
$v = (1+i)^{-1}$ discounts the annuity-immediate value back over the $k$ -period deferral.
Common in retirement planning, where contributions accumulate during working years and payouts begin at a future date.

Present Value of a 3-Year Deferred Annuity An annuity pays $ $2{,}000$ at the end of each year for 5 years, with the first payment 4 years from now (i.e. deferred 3 years). The effective annual rate is $i = 5\%$ . Find the present value today.
Answer
 $\begin{align*} {}_{3|}a_{\overline{5}|} &= v^3 \cdot a_{\overline{5}|0.05} \\ &= (1.05)^{-3} \cdot \frac{1 - (1.05)^{-5}}{0.05} \\ &= 0.86384 \times 4.3295 \\ &\approx 3.7403 \end{align*}$ 
$PV = 2{,}000 \times 3.7403 \approx \$7{,}480.54$