Arithmetic Increasing Annuity

Study guide: Exam FM-2

An arithmetic increasing annuity has payments 1, 2, 3, \ldots, n at the end of periods 1, 2, \ldots, n. Its present value is denoted (Ia)_{\overline{n}|}:

(Ia)_{\overline{n}|} = \frac{\ddot{a}_{\overline{n}|} - nv^n}{i}

• Here \ddot{a}_{\overline{n}|} is the annuity-due present value.
• The corresponding accumulated value is (Is)_{\overline{n}|} = (1+i)^n (Ia)_{\overline{n}|}.
• For the more general case with first payment P and constant increase Q per period (payments P,\, P+Q,\, P+2Q,\, \ldots,\, P+(n-1)Q), the present value is:

PV = P \cdot a_{\overline{n}|} + \frac{Q}{i}\!\left(a_{\overline{n}|} - nv^n\right)

Present Value of Payments 1, 2, 3, 4, 5 An annuity pays $100, $200, $300, $400 and $500 at the end of years 1 through 5. The effective annual rate is i = 6\%. Find the present value.

Answer Factor out 100: PV = 100\cdot(Ia)_{\overline{5}|0.06}.

\begin{align*}
v^5 &= (1.06)^{-5} \\
&= 0.747258 \\
\ddot{a}_{\overline{5}|} &= (1.06)\,\frac{1-0.747258}{0.06} \\
&= 4.465106 \\
(Ia)_{\overline{5}|} &= \frac{4.465106 - 5(0.747258)}{0.06} \\
&= \frac{4.465106 - 3.736291}{0.06} \\
&= \frac{0.728815}{0.06} \\
&= 12.1469 \\
PV &= 100 \times 12.1469 \\
&= \$1{,}214.69
\end{align*}