Decreasing Annuity

Study guide: Exam FM-2

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

$(Da)_{\overline{n}|} = \frac{n - a_{\overline{n}|}}{i}$

The corresponding accumulated value is $(Ds)_{\overline{n}|} = (1+i)^n (Da)_{\overline{n}|}$ .
Paired with the increasing annuity, $(Ia)_{\overline{n}|} + (Da)_{\overline{n}|} = (n+1)\,a_{\overline{n}|}$ , since the two payment streams sum to $n+1$ in every period.
For a first payment $P$ decreasing by $Q$ 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 5, 4, 3, 2, 1 A decreasing annuity pays $ $500,\,\$400,\,\$300,\,\$200,\,\$100$ at the end of years 1 through 5. The effective annual rate is $i = 6\%$ . Find the present value.

Answer Factor out 100: $PV = 100\cdot(Da)_{\overline{5}|0.06}$ . $a_{\overline{5}|} = 4.2124$ $(Da)_{\overline{5}|} = \frac{5 - 4.2124}{0.06} = \frac{0.7876}{0.06} \approx 13.1273$ $PV = 100 \times 13.1273 \approx \$1{,}312.73$