**Annuity Calculations**

In the previous section we looked at the basic time value of money keys and how to use them to calculate present and future value of lump sums. In this section we will take a look at how to use the TI 83 to calculate the present and future values of regular annuities and annuities due.

A regular annuity is a series of equal cash flows occurring at equally spaced time periods. In a regular annuity, the first cash flow occurs at the end of the first period.

An annuity due is similar to a regular annuity, except that the first cash flow occurs immediately (at period 0).

**Example 2 — Present Value of Annuities**

Suppose that you are offered an investment that will pay you $1,000 per year for 10 years. If you can earn a rate of 9% per year on similar investments, how much should you be willing to pay for this annuity?

In this case we need to solve for the present value of this annuity since that is the amount that you would be willing to pay today. Launch the TVM Solver (`APPS` `1` `1`) and then enter the numbers onto the appropriate lines: `10` into N, `9` into I%, `1000` (a cash inflow) into PMT, and `0` for FV. Move to the PV line and press `ALPHA` `ENTER` to solve the problem. The answer is -6,417.6577. Again, this is negative because it represents the amount you would have to pay (cash outflow) today to purchase this annuity.

**Example 2.1 — Future Value of Annuities**

Now, suppose that you will be borrowing $1000 each year for 10 years at a rate of 9%, and then paying back the loan immediately after receiving the last payment. How much would you have to pay?

All we need to do is to put a `0` into PV to clear it out, and then solve for FV to find that the answer is -15,192.92972 (a cash outflow).

**Example 2.2 — Solving for the Payment Amount**

We often need to solve for annuity payments. For example, you might want to know how much a mortgage or auto loan payment will be. Or maybe you want to know how much you will need to save each year in order to reach a particular goal (saving for college or retirement perhaps). On the previous page, we looked at an example about saving for college. Let’s look at that problem again, but this time we’ll treat it as an annuity problem instead of a lump sum:

Suppose that you are planning to send your daughter to college in 18 years. Furthermore, assume that you have determined that you will need $100,000 at that time in order to pay for tuition, room and board, party supplies, etc. If you believe that you can earn an average annual rate of return of 8% per year, how much money would you need to invest at the end of each year to achieve your goal?

Recall that we previously determined that if you were to make a lump sum investment today, you would have to invest \$25,024.90. That is quite a chunk of change. In this case, saving for college will be easier because we are going to spread the investment over 18 years, rather than all at once. (Note that, for now, we are assuming that the first investment will be made one year from now. In other words, it is a regular annuity.)

Let’s enter the data: Type `18` into N, `8` into I%, and `100,000` into FV. Now, solve for PMT and you will find that you need to invest \$2,670.21 per year for the next 18 years to meet your goal of having \$100,000.

**Example 2.3 — Solving for the Number of Periods**

Solving for N answers the question, “*How long will it take…*” Let’s look at an example:

Imagine that you have just retired, and that you have a nest egg of $1,000,000. This is the amount that you will be drawing down for the rest of your life. If you expect to earn 6% per year on average and withdraw $70,000 per year, how long will it take to burn through your nest egg (in other words, for how long can you afford to live)? Assume that your first withdrawal will occur one year from today (End Mode).

Enter the data as follows: `6` into I%, `-1,000,000` into PV (negative because you are investing this amount), and `70,000` into PMT. Now, solve for N and you will see that you can make 33.40 withdrawals. Assuming that you can live for about a year on the last withdrawal, then you can afford to live for about 34.40 years.

**Example 2.4 — Solving for the Interest Rate**

Solving for I% works just like solving for any of the other variables. As has been mentioned numerous times in this tutorial, be sure to pay attention to the signs of the numbers that you enter into the TVM keys. Any time you are solving for N, I%, or PMT there is the potential for a wrong answer or error message if you don’t get the signs right. Let’s look at an example of solving for the interest rate:

Suppose that you are offered an investment that will cost $925 and will pay you interest of $80 per year for the next 20 years. Furthermore, at the end of the 20 years, the investment will pay $1,000. If you purchase this investment, what is your compound average annual rate of return?

Note that in this problem we have a present value (\$925), a future value (\$1,000), and an annuity payment (\$80 per year). As mentioned above, you need to be especially careful to get the signs right. In this case, both the annuity payment and the future value will be cash inflows, so they should be entered as positive numbers. The present value is the cost of the investment, a cash outflow, so it should be entered as a negative number. If you were to make a mistake and, say, enter the payment as a negative number, then you will get the wrong answer. On the other hand, if you were to enter all three with the same sign, then you will get an error message,

Let’s enter the numbers: Type `20` into N, `-925` into PV, `80` into PMT, and `1000` into FV. Now, solve for I% and you will find that the investment will return an average of 8.81% per year. This particular problem is an example of solving for the yield to maturity (YTM) of a bond.

**Example 2.5 — Annuities Due**

In the examples above, we assumed that the first payment would be made at the end of the year, which is typical. However, what if you plan to make (or receive) the first payment today? This changes the cash flow from a regular annuity into an annuity due.

Normally, the calculator is working in End Mode. It assumes that cash flows occur at the end of the period. In this case, though, the payments occur at the beginning of the period. Therefore, we need to put the calculator into Begin Mode. To change to Begin Mode, scroll down to the bottom of the TVM Solver. You should see that END is currently highlighted. Now, press `⯈` to highlight BEGIN, and then press `ENTER`. Note that nothing will change about how you enter the numbers. The calculator will simply shift the cash flows for you. Obviously, you will get a different answer.

Let’s do the college savings problem again, but this time assuming that you start investing immediately:

Suppose that you are planning to send your daughter to college in 18 years. Furthermore, assume that you have determined that you will need $100,000 at that time in order to pay for tuition, room and board, party supplies, etc. If you believe that you can earn an average annual rate of return of 8% per year, how much money would you need to invest at thebeginningof each year (starting today) to achieve your goal?

As before, enter the data: `18` into N, `8` into I%, and `100,000` into FV. The only thing that has changed is that we are now treating this as an annuity due. So, once you have changed to Begin Mode, just solve for PMT. You will find that, if you make the first investment today, you only need to invest \$2,472.42. That is about \$200 per year less than if you make the first payment a year from now because of the extra time for your investments to compound.

Be sure to switch back to End Mode after solving the problem. Since you almost always want to be in End Mode, it is a good idea to get into the habit of switching back so that you don’t forget. Scroll down to the bottom of the TVM Solver, highlight END and press `ENTER`.

**Example 2.6 — Perpetuities**

Occasionally, we have to deal with annuities that pay forever (at least theoretically) instead of for a finite period of time. This type of cash flow is known as a perpetuity (perpetual annuity, sometimes called an infinite annuity). The problem is that the TI 83 Plus has no way to specify an infinite number of periods for N.

Calculating the present value of a perpetuity using a formula is easy enough: Just divide the payment per period by the interest rate per period.

$$PV=\frac{PMT}{i}$$

In our example, the payment is $1,000 per year and the interest rate is 9% annually. Therefore, if that was a perpetuity, the present value would be:

$$\$11,111.11 = \frac{1,000}{0.09}$$

#### Clever Hack!

If you can’t remember that formula, you can “trick” the calculator into getting the correct answer. The trick involves the fact that the present value of a cash flow far enough into the future (way into the future) is going to be approximately $0. Therefore, beyond some future point in time the cash flows no longer add anything to the present value. So, if we specify a suitably large number of payments, we can get a very close approximation (in the limit it will be exact) to a perpetuity.

Let’s try this with our perpetuity. Enter `500` into N (that should always be a large enough number of periods), `9` into I%, and `1000` into PMT. Now scroll to PV and press `ALPHA` `ENTER` and you will get $11,111.11 as your answer.

Please note that there is no such thing as the future value of a perpetuity because the cash flows never end (period infinity never arrives).

Please continue on to part III of this tutorial to learn about uneven cash flow streams, net present value, internal rate of return, and modified internal rate of return.

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