Annuity Due

Study guide: Exam FM-2

An annuity-due is a series of n level payments of 1 made at the beginning of each period. Because each payment is received one period earlier than in an annuity-immediate, the annuity-due is worth exactly (1+i) times as much.

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

= \frac{1-v^n}{d}

• The accumulated value at the end of the last period (one period after the final payment) is \ddot{s}_{\overline{n}|}:

\ddot{s}_{\overline{n}|} = (1+i)\,s_{\overline{n}|}

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

• Here d = i/(1+i) is the effective annual Discount Rate.
• Annuities-due arise naturally when payments are made at the start of a period, such as lease payments, insurance premiums paid in advance, or tuition fees.

Present Value of a 4-Year Annuity-Due A lease requires payments of $800 at the beginning of each year for 4 years. The effective annual rate is i = 5\%. Find the present value.

Answer

\begin{align*}
a_{\overline{4}|0.05} &= \frac{1-(1.05)^{-4}}{0.05} \\
&= \frac{1-0.822702}{0.05} \\
&= 3.545951 \\
\ddot{a}_{\overline{4}|0.05} &= (1.05)(3.545951) \\
&= 3.723248 \\
PV &= 800 \times 3.723248 \\
&= \$2{,}978.60
\end{align*}