Future Value Continuous Compounding Ba Ii Plus

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Strictly speaking, an annuity is a series of equal cash flows, equally spaced in time. However, a graduated annuity (also called a growing annuity) is one in which the cash flows are not all the same, instead they are growing at a constant rate (any other series of cash flows is an uneven cash flow stream).

So, the two types of cash flows differ only in the growth rate of the cash flows. Annuity cash flows grow at 0% (i.e., they are constant), while graduated annuity cash flows grow at some nonzero rate. The image below shows an example:

Graduated annuities are found in many places including pensions that have built-in cost of living adjustments, lotteries such as PowerBall, and others. Any finite series of cash flows that are growing at a constant rate is a graduated (or, growing) annuity. In fact, the growth rate can be positive, negative, or zero so this is really just a generalization of a typical annuity (which would have a zero growth rate).

You might want to know how to calculate the present value of a graduated annuity if you have, for example, a legal settlement from a lawsuit or insurance company. These are often paid out in a structured settlement as a graduated annuity. You might wish to sell it to a third party and you should know how to determine its worth. This article will help with that.

We have already seen how to calculate the present value and future value of annuities. The Texas Instruments BAII Plus makes that easy because it has built-in functions that automatically handle annuities. However, there are no functions that can calculate the present value or future value of a growing stream of cash flows. Fortunately, we can make the PV function do the work for us by altering the interest rate that we use.

The Interest Rate Adjustment

The first thing to understand is that there are two opposing rates when dealing with graduated annuities: The growth rate and the discount rate. The growth rate makes the cash flows larger, but the discount rate makes them smaller. Therefore, the "net" interest rate that we will use must be a combination of these two rates.

Because the two rates work in opposition to each other, we can approximate the correct rate to use by simply using the difference between the discount rate and the growth rate. Intuitively, this should make sense. However, because the rates compound over time, the adjustment is a bit more complex. Specifically, the net rate can be calculated using the following formula:

\[{\rm{Net\, Rate\, for\, Graduated\, Annuity}} = \frac{{1 + i}}{{1 + g}} - 1\]

where i is the discount rate and g is the growth rate. Remember that compounding is a multiplicative process, not additive. Because of this we need to divide, not subtract, to net out the rates. When we adjust the rate using this formula, we can use the resulting rate in the PV function.

PV of a Graduated Annuity Due

A graduated annuity due is one where the first cash flow occurs today, that is at the beginning of a period. These are slightly easier to deal with than a regular graduated annuity, so we will deal with them first. Let's look at an example:

You are considering the purchase of an investment that will pay $100 immediately, and then 4 additional payments that grow at a rate of 5% per year to account for expected inflation. If your required return is 8% per year, what is the value of this investment?

Note that this is a graduated annuity due with a 5% growth rate and we will use an 8% discount rate. The image below shows the time line of the cash flows:

To find the present value, we usually use the PV key, but we can't use it in the normal way because of the growing payments. However, using the above formula, we can calculate the net rate that we need (multiply by 100 because that is what the calculator expects):

\[{\rm{Net\, Rate}} = \left[ {\frac{{1 + 0.08}}{{1 + 0.05}} - 1} \right] \times 100 = 2.857\% \]

Now, recall that the first payment is today, so we need to put the calculator into Begin mode. Press 2nd PMT then 2nd ENTER until you see BGN. Now press 2nd CPT and make sure that you see the word "BGN" in the upper middle of the screen. Enter the data from the problem as follows: N = 5, I/Y = 2.857 (it is best to calculate this as above and then enter it rather than typing it directly), PMT = 100, and FV = 0.

Finally, solve for PV and you will get -472.98 (the negative value simply means that this is a cash outflow).

To recap the steps, here is how to find the present value of a graduated annuity due on the BAII Plus:

1. Place the calculator into Begin mode
2. Enter N and I/Y, being sure to use the net rate for the interest rate
3. Enter the first payment amount into PMT
4. Solve for the PV

PV of a Graduated Regular Annuity

We now change the cash flows to a graduated regular annuity (cash flows at the end of the period). It should be obvious the present value will be somewhat less than above because the cash flows are received one period later. The time line below is similar to the previous one, but notice that the cash flows have been shifted one period forward. We are still using a 5% annual growth rate and an 8% discount rate.

There are a couple of ways that we can do this calculation, but we will see the more intuitive method first.

Intuitive Method: Compare the two timelines above, and note that all we have done is to shift the cash flows by one period. This means that the present values must differ by a factor equal to 1 + the discount rate. Since we already know the present value of the graduated annuity due, all that we need to do is to adjust it as follows:

\[{\rm{PV\, Graduated\, Regular\, Annuity}} = \frac{{{\rm{PV\, Graduated\, Annuity\, Due}}}}{{1 + i}} = \frac{{472.98}}{{1.08}} = 437.94\]

Of course, if we already know the value of the regular graduated annuity, then we could simply "grow" it by the same factor to obtain the value of the graduated annuity due.

Using TVM Keys: First, place the calculator into End mode by pressing 2nd PMT then 2nd ENTER until you see END. Now press 2nd CPT to quit. In this case you will not see any indication of the mode on the screen (just make sure that it does not say BGN). Now, enter the data exactly as before: N = 5, I/Y = 2.857 (again, it is best to calculate this as above and then enter it rather than typing it directly), PMT = 100, and FV = 0.

Next, press CPT PV to solve for the present value. You will get -459.84, but this is not the final answer. The last step is to divide this result by 1 plus the growth rate (5%) as shown in the equation below:

\[{\rm{PV\, Graduated\, Regular\, Annuity}} = \frac{{459.84}}{{1 + g}} = \frac{{459.84}}{{1.05}} = 437.94\]

To recap the steps, here is how to find the present value of a graduated regular annuity on the HP 12C:

1. Place the calculator into End mode
2. Enter N and I/Y, being sure to use the net rate for the interest rate
3. Enter the first payment amount into PMT
4. Solve for the "PV"
5. Get the actual PV by dividing the result from step 4 by 1+ g

Also, note that you can verify the results from above by treating the cash flows as an uneven cash flow stream. This method is more work, and it isn't as practical if you have a lot of cash flows.

Future Value of a Graduated Annuities

Once one understands how to calculate the present value of a graduated annuity, then finding its future value is very easy. Simply find the present value and then calculate the future value of that number. The only thing to remember is that the future value of an annuity due is defined to be one per after the last cash flow. In the examples from above, the future value will be in period 5, regardless of whether it is an annuity due or a regular annuity. The same applies to normal (all cash flows equal) annuities.

For the graduated annuity due, recall that we found that the present value was 472.98. Therefore, to get the future value we simple enter the following: N = 5, I/Y = 8 (note that we use the discount rate, not the net rate), PV = -472.98, and PMT = 0. Now solve for FV and you will get 694.97.

For the graduated regular annuity, recall that we found that the present value was 437.94. Therefore, to get the future value we simple enter the following: N = 5, I/Y = 8 (note that we use the discount rate, not the net rate), PV = -437.94, and PMT = 0. Now solve for FV and you will get 643.49.

Note that your answers could be off by a small amount if you simply copied the numbers and re-entered them. It is always best practice to calculate the numbers and enter them directly from the calculator's memory as this avoids rounding errors.

I hope that you have found this tutorial about graduated annuities to be useful. If you have any further questions or comments, please contact me.

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Source: http://www.tvmcalcs.com/calculators/apps/ti-baii-plus-graduated-annuities

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