Understand Finance - Finance Tutorials and Finance How To's

We’ve learned in previous tutorials that an annuity is simply a series of equal cash
flows that occur over some period of time. But what is a deferred annuity?
Simply put, it is just an annuity that doesn’t start until some point out in the future.
I’m sure everyone has seen an ad on television that promises, “If you buy now, you won’t
make payments until 2008.” The company is then dealing with a deferred annuity since your
equal payments won’t start until some time out in the future!

You’ve probably seen students studying deferred annuities with tears pouring down
their faces, wondering how their life will go on. Well, maybe not. The point is, deferred
annuities are very simple to value. It just takes one extra step. The first step is the
same–we’ll just find the value of the annuity at some point in time. Once we have the
value of the annuity somewhere on our timeline, we’ll just move that lump-sum anywhere on
the timeline that we want it. Let’s jump right into this one with a sample problem.

Suppose your construction company has been offered a job that will pay you $50,000 per
year from years 5 through 9. If your opportunity cost of funds is 9%, what is this job
worth to you today? Let’s take a look at this offer on a timeline.

On our timeline, we see the $50,000 payments we’ll receive, but they’re out in the
future. This is why it is called a deferred annuity. To find out what this is
worth to us today, as in time 0 on the timeline, we just have to do two easy steps.
First, we will find the value of the annuity. Then we can move that value back to time
zero as a lump-sum cash flow.

Texas Instruments BAII Plus

Step 1. Clear the calculator: Step 2. Annual compounding: Step 3. Set N = 5 Step 4. Set I/Y = 9% Step 5. Set PMT = 50,000 Step 6. Compute PV

Hewlett-Packard 10BII

Step 1. Clear the calculator: Step 2. Annual compounding: Step 3. Set N = 5 Step 4. Set I/YR = 9% Step 5. Set PMT = 50,000 Step 6. Compute PV

If your calculator gave you a value of $194,482.56 you’re halfway
there. Before we move on, lets clear up some of the parts that are somewhat confusing.
When dealing with annuities, just count the number of payments to find N. Here, we will
receive 5 payments so N=5. Also, notice we don’t have any future value. Now on with the
show…

The key to understanding deferred annuities is to realize where this value we computed
is on the timeline. It is in period 4. All you need to remember is that when we’re
dealing with an ordinary annuity, your calculator will give you the present value of the
annuity one period before the annuity starts. In this case, you can clearly see that the
annuity starts at year 5 on the timeline. Thus, the calculator gives us a present value
one period before, which is year 4.

Since we know what the annuity is worth in year 4, we can ignore the payments now. The
problem just got really easy! We now have a timeline like this:

All we need to do is set the $194,482 as our future value and solve for the present
value at time 0 to answer this problem!

Texas Instruments BAII Plus

Step 1. Clear the calculator: Step 2. Set N = 4 Step 3. Set I/Y = 9% Step 4. Set FV = 194,482 Step 5. Compute PV

Hewlett-Packard 10BII

Step 1. Clear the calculator: Step 2. Set N = 4 Step 3. Set I/YR = 9% Step 4. Set FV = 194,482 Step 5. Compute PV

Did you get $137,776? That is what our deferred annuity is worth at
time 0. Let’s quickly recap deferred annuities.

1. First find the value of the annuity anywhere on the timeline. Remember that if you
   solve for PV, it will be in the period before the annuity starts.
2. Move this value back to zero by setting it as your future value, setting payments to
   0, and solving for present value.
3. If you are asked to solve for the value of the annuity at some point out in the
   future, then just move the value at time 0 to that future point. For example, if we were
   asked what the annuity is worth in year 25, we could just move the value at time 0
   ($137,775) out to time 25 by setting it as our present value and solving for the future
   value.

That wasn’t too hard now was it? These deferred annuities take some practice though,
so keep trying. If you have any questions or comments, please leave them below.