Arithmetic Progression

Study guide: Exam FM-2

In the context of annuities, an arithmetic progression is a sequence of payments that changes by a fixed amount $Q$ each period: starting from a first payment $P$ , the payments are $P,\ P+Q,\ P+2Q,\ \ldots,\ P+(n-1)Q$ . Paid at the end of each of $n$ periods at the effective rate $i$ per period, they have present value

$PV = P\,a_{\overline{n}|} + Q\,\frac{a_{\overline{n}|} - nv^n}{i}$

The first term values a level payment of $P$ every period; the second values what is added on top of it, $0, Q, 2Q, \ldots, (n-1)Q$ . $Q > 0$ gives an increasing annuity, $Q < 0$ a decreasing one.
Equivalently, write each payment $P + (t-1)Q$ as $(P-Q) + Qt$ : a level annuity of $P - Q$ plus $Q$ times the increasing annuity $1, 2, \ldots, n$ :

$PV = (P-Q)\,a_{\overline{n}|} + Q\,(Ia)_{\overline{n}|}$

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

The first increase arrives with the second payment, so a level annuity of $P$ plus $Q\,(Ia)_{\overline{n}|}$ overstates the value by $Q\,a_{\overline{n}|}$ .
The accumulated value at time $n$ is $P\,s_{\overline{n}|} + Q\,(s_{\overline{n}|} - n)/i$ . With $P, Q > 0$ and no end date, the perpetuity-immediate is worth $P/i + Q/i^2$ .
Paid at the start of each period instead, the present value is $P\,\ddot{a}_{\overline{n}|} + Q\,(a_{\overline{n}|} - nv^n)/d$ .

Payments Rising from 100 to 500 Payments of $100, 200, 300, 400, 500$ are made at the end of years 1–5. Find the PV at $i=6\%$ .
Answer $P = 100$ , $Q = 100$ , $n = 5$ , with $a_{\overline{5}|} = 4.212364$ and $5v^5 = 3.736291$ :
 $\begin{align*} PV &= 100\,a_{\overline{5}|} + 100\,\frac{a_{\overline{5}|} - 5v^5}{0.06} \\ &= 100(4.212364) + 100\,\frac{0.476073}{0.06} \\ &= 421.24 + 793.45 \\ &= 1{,}214.69 \end{align*}$ 
Here $P - Q = 0$ , so this is exactly $100\,(Ia)_{\overline{5}|} = 100(12.146912)$ .

A Lease That Rises Each Year A lease pays $1,000 at the end of the first year, and each later payment is $50 more than the one before, for 10 payments in all. Find its present value at $i = 5\%$ .
Answer $P = 1{,}000$ , $Q = 50$ , with $a_{\overline{10}|} = 7.721735$ and $10v^{10} = 6.139133$ :
 $\begin{align*} PV &= 1{,}000\,a_{\overline{10}|} + 50\,\frac{a_{\overline{10}|} - 10v^{10}}{0.05} \\ &= 7{,}721.73 + 50\,\frac{1.582602}{0.05} \\ &= 7{,}721.73 + 1{,}582.60 \\ &= 9{,}304.34 \end{align*}$ 
Check with the other split: $950\,a_{\overline{10}|} + 50\,(Ia)_{\overline{10}|} = 7{,}335.65 + 50(39.373783) = 9{,}304.34$ . Taking $1{,}000\,a_{\overline{10}|} + 50\,(Ia)_{\overline{10}|}$ instead gives $9{,}690.42$ — too much by $50\,a_{\overline{10}|} = 386.09$ .