9  Annuity-Due

Consider an annuity with payments of \(1\) unit each, made at the beginning of every payment period for \(n\) periods. This kind of annuity is called an annuity-due (also called an annuity in advanced).

9.1 PV and AV of an Annuity Due

9.1.1 Present Value of an Annuity-Due

The symbol \(a_{\overline{n}|i}\) denotes the present value of an annuity-due of \(n\) payments with the rate of interest per payment period is \(i\), when the valuation point is right before the first payment. The formula for \(\ddot a_{\overline n|i}\) is:

\[\begin{align} \ddot a_{\overline n|i} &= 1+v+v^2 + ... + v^{n-1} \\ &=\sum_{t=0}^{n-1} v^t = \frac{1-v^n}{d} \end{align}\]

9.1.2 Accumulated Value of an Annuity-Due

The symbol \(\ddot s_{\overline{n}|i}\) denotes the accumulated value of an annuity-due of \(n\) payments with the rate of interest per payment period is \(i\), when the valuation point is one payment period after the last payment. The formula for \(\ddot s_{\overline n|i}\) is

\[\begin{align} \ddot s_{\overline n|i} &= (1+i)^{n} + (1+i)^{n-1} + ... + (1+i)\\ &= \sum_{t=1}^n (1+i)^t =\frac{(1+i)^n-1}{d}\end{align}\]

The cash flow of an annuity-due can be demonstrated through this time diagram:

Example: Calculate the present value of an annuity-immediate of amount $100 payable quarterly for 10 years at the annual rate of interest of 8% convertible quarterly. Also calculate its future value at the end of 10 years.

Solution: Note that the number of periods is \(4 \times 10 = 40\). The one-period rate of interest \(j = 8\%/4 = 2\%\). The present value of this annuity-immediate is \[100a_{\overline{40}|0.02} = 100\times \frac{1-(1.02)^{-40}}{0.02} = 2735.55\] And the future value of this annuity-immediate is \[100 s_{\overline{40}|0.02} = 100 \times a_{\overline{40}|0.02} \times (1.02)^{40 }= 6040.20\] ### Calculator Guide

Example: The annual effective interest rate is \(4\%\). An annuity-due makes payments of \(\$5\) at the beginning of each year for 8 years. Find the present value of the annuity-due.

Solution: Using the BA II, we can calculate the present value with the following keystrokes (with TVM mode is set to END):

8 [N] 4 [I/Y] 5[PMT] [CPT] [PV] -> PV = -33.66

Next, we multiply the present value of the annuity-immediate by 1.04 to find the present value of the annuity-due: [×] 1.04 [=] 35.01

Thus the present value of this annuity-due is 35.01.

Another method is that we set the TVM mode to BGN with the following keystroke: [2nd] [BGN] [2nd] [SET] [QUIT].

Then, entering 8 [N] 4 [I/Y] 5[PMT] [CPT] [PV] will yield the result of PV = -35.01.

9.2 Perpetuity-Due

A perpetuity-due is an annuity-due that pays out forever. The present value of a perpetuity-immediate that pays 1 at the end of each time unit forever is: \[\ddot a_{\overline{\infty|}i} = \lim_{n\to\infty} \ddot a_{\overline{n|}d} = \lim_{n\to\infty}\frac{1-v^n}{i} = \frac{1}{d}\] Another way to find the value of a perpetuity-due is to observe: \[\ddot a_{\overline{\infty|}i} = \frac{1}{d} =\frac{1+i}{i} = a_{\overline{\infty|}i} + 1\]

Example: The annual effective interest rate is \(6\%\). A perpetuity-due makes payments of \(\$5\) at the end of each year forever. Calculate the present value of the perpetuity-immediate.

Solution: The present value is \(PV_0 = 5 a_{\overline{\infty|}6\%} = \frac{5}{d} = \frac{5}{0.06/1.06}= 88.33\)