## TI 83

#### Definitions & Other

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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 and regular annuities. In this section we will take a look at how to use the TI 83 Plus to calculate the present and future values of uneven cash flow streams. We will also see how to calculate net present value (NPV), internal rate of return (IRR), and the modified internal rate of return (MIRR).

## Example 3 — Present Value of Uneven Cash Flows

This is where the TI 83 Plus is considerably more difficult than most other financial calculators. Its not too bad one you get used to it, but it is more difficult than necessary. Still, you use what you've got, so lets plunge in. First, exit from the TVM Solver menu by pressing 2nd MODE and then return to the Finance menu.

To find the present value of an uneven stream of cash flows, we need to use the NPV function. This function is defined as:

NPV( Rate, Initial Outlay, {Cash Flows}, {Cash Flow Counts})

Note that the {Cash Flow Counts} part is optional and we will ignore it here. I will discuss it in the FAQ.

Suppose that you are offered an investment which will pay the following cash flows at the end of each of the next five years:

Period Cash Flow
0 0
1 100
2 200
3 300
4 400
5 500

How much would you be willing to pay for this investment if your required rate of return is 12% per year?

We could solve this problem by finding the present value of each of these cash flows individually and then summing the results. However, that is the hard way. Instead, we'll use the NPV function. To begin, go to the Finance menu and scroll down until you get to the line that reads NPV(. Press Enter to select that function, and you will see the beginning of the NPV function on your screen. Now, complete the function as follows:

NPV(12,0,{100,200,300,400,500})

Press Enter to solve the function and we find that the present value is $1,000.17922. Note that you can easily change the interest rate by pressing the 2nd Enter to retrieve the function, and then using the arrow keys to edit it. For example, to change the rate to 10%, press 2nd Enter and then use the arrow keys to move to the interest rate and press DEL to delete the 12, and then press 2nd DEL (that's the INS function) and enter 10. Press Enter and you will find that the answer is now$1,065.25883. Reset the interest rate to 12 before continuing.

## Example 3.1 — Future Value of Uneven Cash Flows

Now suppose that we wanted to find the future value of these cash flows instead of the present value. There is no function to do this so we need to use a little ingenuity. Realize that one way to find the future value of any set of cash flows is to first find the present value. Next, find the future value of that present value and you have your solution. In this case, we've already determined that the present value is $1,000.17922, so we'll recall the NPV function by pressing 2nd Enter. Now, add * 1.12 ^ 5 to the end of the function, so that it now looks like: NPV(12,0,{100,200,300,400,500})*1.12^5 Press Enter, and you will see that the future value of these cash flows is$1,762.65754. Pretty easy, huh?  Ok, at least its easier than adding up the future values of each of the individual cash flows. It does require you to know the equation for the future value of a lump sum, but you ought to know that anyway.

## Example 4 — Net Present Value (NPV)

Calculating the net present value (NPV) and/or internal rate of return (IRR) is virtually identical to finding the present value of an uneven cash flow stream as we did in Example 3.

Suppose that you were offered the investment in Example 3 at a cost of $800. What is the NPV? IRR? To solve this problem we must not only tell the calculator about the annual cash flows, but also the cost (previously, we set the cost to 0 because we just wanted the present value of the cash flows). Generally speaking, you'll pay for an investment before you can receive its benefits so the cost (initial outlay) is said to occur at time period 0 (i.e., today). To find the NPV recall the NPV function and edit it so that the initial outlay (previously 0) is -800. The function will look like this on screen: NPV(12,-800,{100,200,300,400,500}) Press Enter to get the solution and you'll see that the NPV is$200.17922.

## Example 4.1 — Internal Rate of Return

Solving for the IRR is done in a similar way, except that we'll use the IRR function.This function is defined as:

IRR( Initial Outlay, {Cash Flows}, {Cash Flow Counts})

For this problem, the function is:

IRR(-800, {100,200,300,400,500})

Again, note that the {Cash Flow Counts} part is optional and we will ignore it here, but it is in the FAQ. To get the IRR function on the screen, press 2nd X-1 (or APPS then Finance on the TI-83 Plus) to return to the finance menu, and scroll down until you see IRR(. Enter the function as shown above and then press Enter to get the answer (19.5382%).

## Example 4.2 — Modified Internal Rate of Return

The IRR has been a popular metric for evaluating investments for many years — primarily due to the simplicity with which it can be interpreted. However, the IRR suffers from a couple of serious flaws. The most important flaw is that it implicitly assumes that the cash flows will be reinvested for the life of the project at a rate that equals the IRR. A good project may have an IRR that is considerably greater than any reasonable reinvestment assumption. Therefore, the IRR can be misleadingly high at times.

The modified internal rate of return (MIRR) solves this problem by using an explicit reinvestment rate. Unfortunately, financial calculators don't have an MIRR key like they have an IRR key. That means that we have to use a little ingenuity to calculate the MIRR. Fortunately, it isn't difficult. Here are the steps in the algorithm that we will use:

1. Calculate the total present value of each of the cash flows, starting from period 1 (set the initial outlay to 0). Use the calculator's NPV function just like we did in Example 3, above. Use the reinvestment rate as your discount rate to find the present value.
2. Calculate the future value as of the end of the project life of the present value from step 1. The interest rate that you will use to find the future value is the reinvestment rate.
3. Finally, find the discount rate that equates the initial cost of the investment with the future value of the cash flows. This discount rate is the MIRR, and it can be interpreted as the compound average annual rate of return that you will earn on an investment if you reinvest the cash flows at the reinvestment rate.

Suppose that you were offered the investment in Example 3 at a cost of $800. What is the MIRR if the reinvestment rate is 10% per year? Let's go through our algorithm step-by-step: 1. The present value of the cash flows can be found as in Example 3. 2. NPV(10,0,{100,200,300,400,500}) We find that the present value is$1,065.26.

3. To find the future value of the cash flows, go to the TVM Solver and enter 5 into N, 10 into I%, and -1065.26 into PV. Now solve for the FV and see that it is $1,715.61. 4. At this point our problem has been transformed into an$800 investment with a lump sum cash flow of $1,715.61 at period 5. The MIRR is the discount rate (I%) that equates these two numbers. Enter -800 into PV and then solve for I%. The MIRR is 16.48% per year. Note that we can actually combine steps 1 and 2. Just as we did in Example 3, we can calculate the future value (using our 10% reinvestment rate) as follows: NPV(10,0,{100,200,300,400,500})*1.10^5 The future value is the same$1,715.61 that we found above. Now, go to the TVM Solver and enter 5 into N, -800 into PV, and 1715.61 into FV. Solve for I% and see that the MIRR is 16.48% just as before.

So, we have determined that our project is acceptable at a cost of \$800. It has a positive NPV, the IRR is greater than our 12% required return, and the MIRR is also greater than our 12% required return.

Please continue on to the next page to learn how to solve problems involving non-annual periods.