Annuity Immediate

Study guide: Exam FM-2

An annuity-immediate is a series of n level payments of 1 made at the end of each period. The present value, valued one period before the first payment, is a_{\overline{n}|}, and the accumulated value at the time of the last payment is s_{\overline{n}|}.

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

= \frac{1-(1+i)^{-n}}{i}

s_{\overline{n}|} = \frac{(1+i)^n - 1}{i}

• The two are related by s_{\overline{n}|} = (1+i)^n \cdot a_{\overline{n}|}.
• For a level payment of P per period, multiply through by P.
• The annuity-immediate is the standard building block for loan amortisation, bond pricing, and insurance reserve calculations.

Present Value of a 5-Year Annuity-Immediate An annuity pays $1{,}200 at the end of each year for 5 years. The effective annual interest rate is i = 6\%. Find the present value.

Answer

\begin{align*}
a_{\overline{5}|0.06} &= \frac{1-(1.06)^{-5}}{0.06} \\
&= \frac{1-0.747258}{0.06} \\
&= 4.212364 \\
PV &= 1{,}200 \times 4.212364 \\
&= \$5{,}054.84
\end{align*}