12 Arithmetic Progression Annuities
12.1 Annuities Increasing Once per Time Unit
Consider an annuity-immediate whose payments increase by 1 per unit of time. The annuity pays \(1\) at the end of the first unit of time, pays \(2\) at the end of the second, and so on until time \(n\) when the annuity makes the final payment of \(n\).
We use \((Ia)_{\overline{n|}}\) to denote the present value of this increasing annuity-immediate, and \[(Ia)_{\overline{n|}} = v + 2v^2 + 3v^3 + ... + (n-1)v^{n-1} + nv^n\] To obtain a convenient expression for \((Ia)_{\overline{n|}}\), we multiply the expression with \((1+i)\) and then subtract the original expression from both sides:
\[\begin{align} i(Ia)_{\overline{n|}} &= (1 + 2v + 3v^2 + ... nv^{n-1}) - (v + 2v^2 + ... + (n-1)v^{n-1} + nv^n) \\&= (1 + v + v^2 + ... + v^{n-1}) - nv^n \\ &= \ddot a_{\overline{n|}} - nv^n \\ \implies & (Ia)_{\overline{n|}} = \frac{\ddot a_{\overline{n|}} - nv^n}{i}\end{align}\]
The accumulated value of the increasing annuity-immediate at time \(n\) is denoted by \((Is)_{\overline{n|}}\), and it is found below by accumulating the present value for \(n\) years:
\[\begin{align}(Is)_{\overline{n|}} &= (1+i)^n (Ia)_{\overline{n|}} = (1+i)^n\frac{\ddot a_{\overline{n|}} - nv^n}{i} \\ \implies & (Is)_{\overline{n|}} = \frac{\ddot s_{\overline{n|}} - n}{i}\end{align} \]
The present value of the perpetuity-immediate can be calculated by letting \(n\) tends to infinity:
\[(Ia)_{\overline{\infty|}} = \lim_{n\to\infty}(Ia)_{\overline{n|}} = \lim_{n\to\infty} \bigg[\frac{1-v^n}{di} - \frac{nv^n}{i}\bigg] = \frac{1}{di}\times \lim_{n\to\infty}[1-v^n - \frac{nv^n}{d}] = \frac{1}{di}\] ### Increasing Annuity-Immediate
The present value at time 0, the accumulated value at time n of an increasing annuity-immediate and the present value of the perpetuity that pays \(1\) at time \(1\), \(2\) at time \(2\), and so on until a final payment of \(n\) is made at time \(n\) are:
\[(Ia)_{\overline{n|}} = \frac{\ddot a_{\overline{n|}} - nv^n}{i} \hskip4em (Is)_{\overline{n|}} = \frac{\ddot s_{\overline{n|}} - n}{i} \hskip4em (Ia)_{\overline{\infty|}} = \frac{1}{di}\] ### Increasing Annuity-Due
The present value at time 0 and the accumulated value at time n of an increasing annuity-due that pays \(1\) at time \(0\), \(2\) at time \(1\), and so on until a final payment of \(n\) is made at time \(n-1\) are:
\[(I\ddot a)_{\overline{n|}} = \frac{\ddot a_{\overline{n|}} - nv^n}{d} \hskip4em (I\ddot s)_{\overline{n|}} = \frac{\ddot s_{\overline{n|}} - n}{d}\hskip4em (I\ddot a)_{\overline{\infty|}} = \frac{1}{d^2}\]
12.1.1 Annuity-Immediate Increasing Once per Time Unit, Payable m-thly
The present value and accumulated value of an annuity that pays \(1/m\) at the end of each period in the first unit of time, \(2/m\) at the end of each period in the second unit of time, and so on, until the annuity pays \(n/m\) at the end of each period in the \(n\)-th unit of time are:
\[(Ia)_{\overline{n|}}^{(m)} = \frac{\ddot a_{\overline{n|}} - nv^n}{i^{(m)}} \hskip4em (Is)_{\overline{n|}}^{(m)} = \frac{\ddot s_{\overline{n|}} - n}{i^{(m)}} \hskip4em (Ia)^{(m)}_{\overline{\infty|}} = \frac{1}{d\times i^{(m)}}\]
12.1.2 Annuity-Due Increasing Once per Time Unit, Payable m-thly
The present value and accumulated value of an annuity that pays \(1/m\) at the beginning of each period in the first unit of time, \(2/m\) at the beginning of each period in the second unit of time, and so on, until the annuity pays \(n/m\) at the beginning of each period in the \(n\)-th unit of time are: \[(I\ddot a)_{\overline{n|}}^{(m)} = \frac{\ddot a_{\overline{n|}} - nv^n}{d^{(m)}} \hskip4em (I\ddot s)_{\overline{n|}}^{(m)} = \frac{\ddot s_{\overline{n|}} - n}{d^{(m)}}\hskip4em (I\ddot a)^{(m)}_{\overline{\infty|}} = \frac{1}{d\times d^{(m)}}\] ### Annuity Increasing Once per Time Unit, Payable Continuously
The present value and accumulated value of an annuity that pays continuously at a rate of \(1\) in the first unit of time, 2 in the second unit of time, and so until it pays at a rate of \(n\) per unit of time in the \(n\)-th unit of time are:
\[(I\bar a)_{\overline{n|}}^{(m)} = \frac{\ddot a_{\overline{n|}} - nv^n}{\delta} \hskip4em (I\bar s)_{\overline{n|}}^{(m)} = \frac{\ddot s_{\overline{n|}} - n}{\delta} \hskip4em (I\bar a)_{\overline{\infty|}} = \frac{1}{d\delta }\]
12.1.3 Comparison
Consider an annuity that pays at a rate of \(1\) in the first unit of time, \(2\) in the second unit of time, and so on until it pays at a rate of \(n\) in the \(n\)-th unit of time.
For each type of the annuity (annuity-immediate, annuity-due, annuity-immediate payable m-thly, annuity-due payable m-thly, annuity payable continuously), we have the formulae:
| Factor | Present Value | Accumulated Value | Perpetuity |
|---|---|---|---|
| \[(Ia)_{\overline{n\|}}\] | \[(Is)_{\overline{n\|}}\] | \[(Ia)_{\overline{\infty\|}}\] | |
| \[\ddot s_{\overline{n\|}} = \frac{i}{d}\] | \[(I\ddot a)_{\overline{n\|}} = \frac{i}{d}(Ia)_{\overline{n\|}}\] | \[(I\ddot s)_{\overline{n\|}} = \frac{i}{d}(Is)_{\overline{n\|}}\] | \[(I\ddot a)_{\overline{\infty\|}} = \frac{i}{d}(Ia)_{\overline{\infty\|}}\] |
| \[s^{(m)}_{\overline{n\|}} = \frac{i}{i^{(m)}}\] | \[(Ia)^{(m)}_{\overline{n\|}} = \frac{i}{i^{(m)}}(Ia)_{\overline{n\|}}\] | \[(Is)^{(m)}_{\overline{n\|}} = \frac{i}{i^{(m)}}(Is)_{\overline{n\|}}\] | \[(Ia)^{(m)}_{\overline{\infty\|}} = \frac{i}{i^{(m)}}(Ia)_{\overline{\infty\|}}\] |
| \[\ddot s_{\overline{n\|}} = \frac{i}{d^{(m)}}\] | \[(I\ddot a)^{(m)}_{\overline{n\|}} = \frac{i}{d^{(m)}}(Ia)_{\overline{n\|}}\] | \[(I\ddot s)^{(m)}_{\overline{n\|}} = \frac{i}{d^{(m)}}(Is)_{\overline{n\|}}\] | \[(I\ddot a)^{(m)}_{\overline{\infty\|}} = \frac{i}{d^{(m)}} (Ia)_{\overline{\infty\|}}\] |
| \[\bar s_{\overline{n\|}} = \frac{i}{\delta}\] | \[(I\bar a)^{(m)}_{\overline{n\|}} = \frac{i}{\delta}(Ia)_{\overline{n\|}}\] | \[(I\bar s)^{(m)}_{\overline{n\|}} = \frac{i}{\delta}(Is)_{\overline{n\|}}\] | \[(I\bar a)_{\overline{\infty\|}} = \frac{i}{\delta} (Ia)_{\overline{\infty\|}}\] |
12.2 Decreasing Annuities
Consider an annuity-immediate whose payments decrease by \(1\) per unit of time. The annuity pays \(n\) at the end of one unit of time. At the end of the second unit of time, it pays \((n-1)\). At the end of the third time unit, it pays \((n-2)\). This continues until time \(n\), when the annuity makes its final payment of \(1\).
Then the present value at time \(0\) and accumulated value at time \(n\) are: \[(Da)_{\overline{n|}} = \frac{n- a_{\overline{n|}}}{i} \hskip7em (Ds)_{\overline{n|}} = \frac{n(1+i)^n-s_{\overline{n|}}}{i}\] The present values and accumulated values for the other decreasing annuities are:
| Factor | Present Value | Accumulated Value |
|---|---|---|
| \[(Da)_{\overline{n\|}}\] | \[(Ds)_{\overline{n\|}}\] | |
| \[\ddot s_{\overline{n\|}} = \frac{i}{d}\] | \[(D\ddot a)_{\overline{n\|}} = \frac{i}{d}(Da)_{\overline{n\|}}\] | \[(D\ddot s)_{\overline{n\|}} = \frac{i}{d}(Ds)_{\overline{n\|}}\] |
| \[s^{(m)}_{\overline{n\|}} = \frac{i}{i^{(m)}}\] | \[(Da)^{(m)}_{\overline{n\|}} = \frac{i}{i^{(m)}}(Da)_{\overline{n\|}}\] | \[(Ds)^{(m)}_{\overline{n\|}} = \frac{i}{i^{(m)}}(Ds)_{\overline{n\|}}\] |
| \[\ddot s_{\overline{n\|}} = \frac{i}{d^{(m)}}\] | \[(D\ddot a)^{(m)}_{\overline{n\|}} = \frac{i}{d^{(m)}}(Da)_{\overline{n\|}}\] | \[(D\ddot s)^{(m)}_{\overline{n\|}} = \frac{i}{d^{(m)}}(Ds)_{\overline{n\|}}\] |
| \[\bar s_{\overline{n\|}} = \frac{i}{\delta}\] | \[(D\bar a)^{(m)}_{\overline{n\|}} = \frac{i}{\delta}(Da)_{\overline{n\|}}\] | \[(D\bar s)^{(m)}_{\overline{n\|}} = \frac{i}{\delta}(Ds)_{\overline{n\|}}\] |
12.3 Annuities Increasing More than Once per Time Unit
12.3.1 Annuity-Immediate Increasing m-thly, Payable m-thly
The present value at time 0, the accumulated value at time n of an increasing annuity-immediate and the present value of the perpetuity that pays \(1/m\) at time \(1/m\), \(2/m\) at time \(2/m\), and so on until a final payment of \(mn/m\) is made at time \(mn/m\) are:
\[(I^{(m)}a)_{\overline{n|}}^{(m)} = \frac{\ddot a^{(m)}_{\overline{n|}} - nv^n}{i^{(m)}} \hskip4em (I^{(m)}s)_{\overline{n|}}^{(m)} = \frac{\ddot s_{\overline{n|}} - n}{i^{(m)}} \hskip4em (I^{(m)}a)^{(m)}_{\overline{\infty|}} = \frac{1}{d^{(m)}\times i^{(m)}}\]
12.3.2 Annuity-Due Increasing m-thly, Payable m-thly
The present value at time 0, the accumulated value at time n of an increasing annuity-immediate and the present value of the perpetuity that pays \(1/m\) at time \(0\), \(2/m\) at time \(1/m\), and so on until a final payment of \(n\) is made at time \(n - (1/m)\) are:
\[(I^{(m)}\ddot a)_{\overline{n|}}^{(m)} = \frac{\ddot a^{(m)}_{\overline{n|}} - nv^n}{d^{(m)}} \hskip4em (I^{(m)}\ddot s)_{\overline{n|}}^{(m)} = \frac{\ddot s_{\overline{n|}} - n}{d^{(m)}} \hskip4em (I^{(m)}\ddot a)^{(m)}_{\overline{\infty|}} = \frac{1}{(d^{(m)})^2}\]
12.4 Calculator-Friendly Method
Consider an annuity whose payments increase by \(Q\) per unit of time. The annuity pays \(P\) at the end of the first unit of time. At the end of the second unit of time, it pays \(P+Q\). At the end of the third unit of time unit, it pays \(P+2Q\). This continues until time \(n\), when the annuity makes its final payment of \(P +(n-1)Q\).
The present value of the payments can be written as:
\[\begin{align} PV_0 &= Pv + (P+Q)v^2 + (P+2Q)v^3 + ... + (P+(n-1)Q)v^n \\ &= P \times a_{\overline{n|}} + Qv\times (Ia)_{\overline{n-1|}} \\ &= P\times a_{\overline{n|}} + Q \times \frac{a_{\overline{n-1|}} - (n-1)v^{n}}{i} \\ &= P\times a_{\overline{n|}} + Q\times \frac{a_{\overline{n|}} - nv^n}{i}\\ &=\bigg(P+\frac{Q}{i}\bigg)a_{\overline{n|}} - \frac{Qn}{i} v^n\end{align}\] To find the accumulated value at time \(n\), we would accumulate by a factor of \((1+i)^n\):
\[AV_n = PV_0\times (1+i)^n\] ### Formulae
Consider an annuity that pays at a rate of \(P\) in the first unit of time, \(P+2Q\) in the second unit of time, and so on until it pays at a rate of \(P+(n-1)Q\) in the \(n\)-th unit of time.
For each type of the annuity (annuity-immediate, annuity-due, annuity-immediate payable m-thly, annuity-due payable m-thly, annuity payable continuously), we have the formulae: \[\begin{align} PV_0\text{(annuity-immediate)} &= \bigg(P+\frac{Q}{i}\bigg)a_{\overline{n|}} - \frac{Qn}{i} v^n \\PV_0\text{(annuity-due)}&= \frac{i}{d} PV_0\text{(annuity-immediate)} \\ PV_0\text{(annuity-immediate payable m-thly)} &= \frac{i}{i^{(m)}}PV_0\text{(annuity-immediate)} \\ PV_0\text{(annuity-due payable m-thly)} &= \frac{i}{d^{(m)}}PV_0\text{(annuity-immediate)} \\ PV_0\text{(annuity payable continuously)} &= \frac{i}{\delta}PV_0\text{(annuity-immediate)}\end{align}\] ### Increasing Perpetuity-Due
The present value at time \(0\) of an increasing perpetuity-due that pays \(P_0\) at time \(0\), \(P_1\) at the time \(1\), \((P_1 +Q)\) at time \(2\), \((P_1 +3Q)\) at time \(3\), and so on is: \[PV_0 = P_0 + \frac{P_1}{i} + \frac{Q}{i^2}\]