Annuities

Study guides: Exam FM-2, Exam FAM, Exam ALTAM

An annuity is a series of payments made at equal intervals of time. On Exam FM an annuity is an annuity-certain: its payments are certain to be made for the whole term — the syllabus's "non-contingent payments" — unlike a contingent (life) annuity, which pays only while a person survives. Its value on any date is the sum of the values of its payments on that date — for the present value at the effective rate $i$ per payment period:

$PV = \sum_{t} C_t\, v^{t}$

$C_t$ is the payment at time $t$ and $v = 1/(1+i)$ the discount factor. Valued at a date inside the term, payments before the date are accumulated and payments on or after it discounted — the current value.
Timing: an annuity-immediate pays at the end of each period, an annuity-due at the beginning; a perpetuity never stops; payments can also be payable m-thly or payable continuously.
Amount: a level payment annuity pays the same amount every period; non-level annuities vary — in arithmetic or geometric progression, decreasing, or in no pattern at all.

For $n$ level payments of 1, SOA's notation is $a_{\overline{n}|}$ and $s_{\overline{n}|}$ for the present and accumulated value of the annuity-immediate, and $\ddot{a}_{\overline{n}|}$ and $\ddot{s}_{\overline{n}|}$ for the annuity-due:

$a_{\overline{n}|} = \frac{1-v^n}{i}$

$\ddot{a}_{\overline{n}|} = (1+i)\,a_{\overline{n}|}$

$s_{\overline{n}|} = (1+i)^n\,a_{\overline{n}|}$

Lease Payments in Arrears or in Advance An equipment lease pays $ $2{,}000$ a year for 10 years. At an effective annual rate of $5\%$ , find the present value if the payments are made (a) at the end of each year, (b) at the start of each year.
Answer
 $\begin{align*} a_{\overline{10}|} &= \frac{1-1.05^{-10}}{0.05} \\ &= 7.72173 \\ \text{(a)}\quad 2{,}000\,a_{\overline{10}|} &= 15{,}443.47 \\ \text{(b)}\quad 2{,}000\,\ddot{a}_{\overline{10}|} &= 2{,}000(1.05)(7.72173) \\ &= 16{,}215.64 \end{align*}$ 
Paying in advance moves every payment one year earlier, so it is worth $5\%$ more: $ $772.17$ more here.

Value of an Annuity Partway Through Its Term A structured settlement pays $ $1{,}000$ at the end of each year for 8 years. At $i = 4\%$ , find its value at time 3, just after the third payment.
Answer The three payments already made are accumulated and the five still to come are discounted:
 $\begin{align*} CV_3 &= 1{,}000\,s_{\overline{3}|} + 1{,}000\,a_{\overline{5}|} \\ &= 1{,}000(3.12160) + 1{,}000(4.45182) \\ &= 7{,}573.42 \end{align*}$ 
Check: $1{,}000\,a_{\overline{8}|}(1.04)^3 = 1{,}000(6.73274)(1.124864) = 7{,}573.42$ — the value at time 0 moved forward three years.