Present Value of Annuity Due

by Obaidullah Jan, ACA, CFA

Annuity due is a stream of cash flows in which there is equal duration between different cash flows and each cash flow occurs at the start of each period. It is an annuity in which payments occur in advance instead of arrears.

Let’s say you take out a 3-year auto lease that require $10,000 as down payment and $400 as monthly lease payment to be made at the start of each month. Your total outflow at the start of 1st month 1 is $10,4000 (down payment of $10,000 plus first monthly instalment of $400) and your last cash flow occur at the start of 60th month.

Annuity due differs from ordinary annuity (or just annuity) in that periodic cash flows occur at the end of each period in an ordinary annuity.

Formula

The basic concept behind the present value of annuity due is the same as that of an (ordinary) annuity. You just need to discount cash flow to time zero.

One way to find present value is to manually discount each cash flow based on the following formula and sum all present values together:

$$ PV=\frac{{PMT}_n}{({1+i)}^n} $$

But it is tedious, you can use the following formula to directly calculate the present value:

$$ PV=PMT\times\frac{1\ -\ {(1+i)}^{-n}}{i}\times(1+i) $$

Where i is the relevant periodic interest rate, n is the total number of payments in the annuity due. If you look closely, one component of the right-hand side is the formula for present value of an (ordinary annuity). Let’s rewrite the above equation as follows:

$$ PV=PVA\times(1+i) $$

Where PVA stands for present value of annuity.

Because in an annuity due each payment occurs one period closer to time 0, each payment must be discounted over one less period as compared to an identical (ordinary) annuity. Multiplication of the present value of an (ordinary) annuity with (1 + i) has the effect of shifting the cash flows one period towards time 0.

Alternatively, you can use the following formula directly to calculate present value of an annuity due:

$$ PV=PMT+PMT\times\frac{1\ -\ {(1+i)}^{-(n-1)}}{i} $$

The above formula is intuitive. Since payment occurs at the start of each period, the first payment occurs at time 0 and it has no discounting effect hence the first PMT term on the right side of the above equation. The rest of the payments can be treated as an (ordinary) annuity with one less payment hence the -(n-1) term instead of just -n.

You can also use Excel PV function to calculate present value of an annuity due. PV function syntax is PV(rate, nper, pmt, [fv], [type]). Specifying 1 in [type] argument treats the cash flow stream as an annuity due.

Example

Your company has entered a lease requiring four annual payments of $10 million in advance. The lease rate is 10%

If the payments were required at the end of each year, the present value would be:

$$ PV\ (assuming\ ordinary\ annuity)\\=$10\ million\times\frac{1-{(1+10\%)}^{-4}}{10%}\\=$31.70\ million $$

To shift the present value one period closer to time 0, we need to multiply by (1+ 10%), which gets us $34.87 million.

We can also find present value by individually discounting each cash flow as follows:

$$ PV\ of\ lease\\=10M+\frac{10M}{{(1+10\%)}^1}+\frac{10M}{{(1+10\%)}^2}+\frac{10M}{{(1+10\%)}^3}\\=$34.87\ million $$

Now, let’s calculate the present value using the direct formula:

$$ PV\ of\ lease\\=10M+10M \times \frac{1-{(1+10\%)}^{-(4-1)}}{10\%}\\=$34.87\ million $$

If you have Excel handy, enter the following in any cell: PV(10%,4,-10000000,0,1).