Annuity due vs Annuity immediate
Nathan Kamgang
Real-World Analogies
Think about how you pay for services in everyday life.
A post-paid phone plan is a perfect analogy for an annuity immediate. You make calls throughout the month, and only at the end of the period does the carrier tally up your usage and send you a bill. You consume first, pay later.
Conversely, a prepaid internet plan mirrors an annuity due. You load credit before the service begins. If you want internet access for the month, you pay upfront
Payment Schedule Over 6 Months ($1 per period)
| Time Point | Annuity Immediate (Post-paid Phone) | Annuity Due (Prepaid Internet) |
|---|---|---|
| $t = 0$ (Start of Month 1) | $0 | $1.00 |
| $t = 1$ (End of Month 1 / Start of Month 2) | $1.00 | $1.00 |
| $t = 2$ (End of Month 2 / Start of Month 3) | $1.00 | $1.00 |
| $t = 3$ (End of Month 3 / Start of Month 4) | $1.00 | $1.00 |
| $t = 4$ (End of Month 4 / Start of Month 5) | $1.00 | $1.00 |
| $t = 5$ (End of Month 5 / Start of Month 6) | $1.00 | $1.00 |
| $t = 6$ (End of Month 6) | $1.00 | $0 |
| Total paid | $6.00 | $6.00 |
General Cashflow
In the general case, an annuity is a sequence of equal, periodic payments made over $n$ periods. Depending on the timing convention, the cashflow looks like one of two patterns:
Annuity Immediate — payments at the end of each period:
$$\underbrace{0}_{t=0}, \underbrace{1}_{t=1}, \underbrace{1}_{t=2}, \; \ldots \;, \underbrace{1}_{t=n}$$Annuity Due — payments at the beginning of each period:
$$\underbrace{1}_{t=0}, \underbrace{1}_{t=1}, \underbrace{1}_{t=2}, \; \ldots \;, \underbrace{1}_{t=n-1}, \underbrace{0}_{t=n}$$Present Value and the Discount Factor
Motivation — the Phone Bill Example
Suppose you owe $1 at the end of month 1 on your post-paid phone plan. That dollar is worth less to you today than it will be at payment time, because money held now can earn interest. To find what that future payment is worth right now, we discount it.
Let $i$ be the effective interest rate per period. We define the discount factor as:
$$v = \frac{1}{1+i}$$A payment of $\$1$ made at time $t$ is worth $v^t$ dollars today. Intuitively: the further in the future the payment, the more times we divide by $(1+i)$, and the smaller its present value.
For your phone bill, the present value of each monthly payment of $\$1$ is:
| Payment at end of month $t$ | Present Value today |
|---|---|
| $t = 1$ | $v$ |
| $t = 2$ | $v^2$ |
| $t = 3$ | $v^3$ |
| $\vdots$ | $\vdots$ |
| $t = n$ | $v^n$ |
Present Value of an Annuity Immediate
Definition as a Sum
The present value of an annuity immediate of $\$1$ per period for $n$ periods, denoted $a_{\overline{n}|i}$ (or simply $a_{\overline{n}|}$), is the sum of all discounted payments:
$$a_{\overline{n}|} = v^1 + v^2 + v^3 + \cdots + v^n = \sum_{t=1}^{n} v^t$$Recognising the Geometric Series
This is a geometric series with:
- First term: $v$
- Common ratio: $v$
- Number of terms: $n$
Recall that the sum of a geometric series $S = a + ar + ar^2 + \cdots + ar^{n-1}$ is:
$$S = a \cdot \frac{1 - r^n}{1 - r}$$Applying this to our sum with first term $a = v$ and common ratio $r = v$:
$$a_{\overline{n}|} = v \cdot \frac{1 - v^n}{1 - v}$$Simplifying to the Closed Form
We now simplify $\dfrac{v}{1-v}$. Since $v = \dfrac{1}{1+i}$:
$$1 - v = 1 - \frac{1}{1+i} = \frac{(1+i) - 1}{1+i} = \frac{i}{1+i}$$Therefore:
$$\frac{v}{1-v} = \frac{\dfrac{1}{1+i}}{\dfrac{i}{1+i}} = \frac{1}{i}$$Substituting back:
$$\boxed{a_{\overline{n}|} = \frac{1 - v^n}{i}}$$This is the present value formula for an annuity immediate. It tells you the lump-sum equivalent today of $n$ future payments of $\$1$, given a periodic interest rate $i$.
Now if you try to find the present value of the annuity due, the process is pretty similar. Starting at $t=0$, we already have $\$1$ in the present, so it needs no discounting at all. For our prepaid internet example with 6 payments:
$$\ddot{a}_{\overline{6}|} = 1 + v + v^2 + v^3 + v^4 + v^5$$It is a geometric series with first term $1$ and common ratio $v$, so the sum is:
$$\ddot{a}_{\overline{n}|} = 1 \cdot \frac{1 - v^n}{1 - v}$$Substituting $1 - v = \dfrac{i}{1+i} = d$, where $d$ is the effective discount rate:
$$\boxed{\ddot{a}_{\overline{n}|} = \frac{1 - v^n}{d}}$$Here’s the interesting part. Let’s go back to our phone bill. Now shift every payment one period earlier, replacing $t$ by $t-1$:
$$ \begin{array}{lrrrrrrr} \textbf{Annuity Immediate:} & \$0 & \$1 & \$1 & \$1 & \$1 & \$1 & \$1 \\ \textbf{Shifted by } -1: & \$1 & \$1 & \$1 & \$1 & \$1 & \$1 & \$0 \\ \textbf{Annuity Due:} & \$1 & \$1 & \$1 & \$1 & \$1 & \$1 & \$0 \\ & t=0 & t=1 & t=2 & t=3 & t=4 & t=5 & t=6 \end{array} $$The shifted annuity immediate and the annuity due are identical row by row. An annuity due and an annuity immediate are the same cashflow, just shifted by one period.
Now, the present value of the annuity immediate over $n$ periods is:
$$a_{\overline{n}|} = \sum_{t=1}^{n} v^t = v + v^2 + \cdots + v^n = \frac{1 - v^n}{i}$$For our phone bill specifically, with 6 monthly payments of $1, this is:
$$a_{\overline{6}|} = v + v^2 + v^3 + v^4 + v^5 + v^6$$Now here’s the key insight. We just saw that shifting the annuity immediate one period earlier gives the annuity due. What does that shift mean algebraically? Each payment is discounted one fewer time, so each term loses one factor of $v$. In other words, we divide the whole sum by $v$, or equivalently multiply by $(1+i)$:
$$\frac{a_{\overline{6}|}}{v} = \frac{v + v^2 + v^3 + v^4 + v^5 + v^6}{v} = 1 + v + v^2 + v^3 + v^4 + v^5$$Compare this to the present value of the annuity due:
$$\ddot{a}_{\overline{6}|} = 1 + v + v^2 + v^3 + v^4 + v^5$$They are the same. In the general case, starting from $a_{\overline{n}|}$ and multiplying by $(1+i)$:
$$(1+i) \cdot a_{\overline{n}|} = (1+i)(v + v^2 + \cdots + v^n) = 1 + v + v^2 + \cdots + v^{n-1} = \sum_{t=0}^{n-1} v^t = \ddot{a}_{\overline{n}|}$$This gives birth to the fundamental relationship between the two:
$$\boxed{\ddot{a}_{\overline{n}|} = (1+i) \cdot a_{\overline{n}|}}$$Accumulated Value
$a_{\overline{n}|}$ brings all cashflows over $n$ periods back to the present $t = 0$.
Now to find the future value at time $t = n$, we multiply $a_{\overline{n}|}$ by the growth factor $(1+i)^n$. We denote this accumulated value $s_{\overline{n}|}$:
$$s_{\overline{n}|} = (1+i)^n \cdot a_{\overline{n}|} = (1+i)^n \cdot \frac{1 - v^n}{i} = \frac{(1+i)^n - 1}{i}$$since $(1+i)^n \cdot v^n = 1$. Just as $a_{\overline{n}|}$ is the lump-sum equivalent of all payments seen from the beginning, $s_{\overline{n}|}$ is their total worth seen from the end.
For the annuity due, the accumulated value is denoted $\ddot{s}_{\overline{n}|}$. Using the relation $\ddot{a}_{\overline{n}|} = (1+i) \cdot a_{\overline{n}|}$ and multiplying through by $(1+i)^n$:
$$\ddot{s}_{\overline{n}|} = (1+i)^n \cdot \ddot{a}_{\overline{n}|} = (1+i) \cdot s_{\overline{n}|} = \frac{(1+i)^n - 1}{d}$$The same one-period shift that connected $a_{\overline{n}|}$ to $\ddot{a}_{\overline{n}|}$ carries over cleanly to their accumulated counterparts.