Uneven Cash Flows

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 annuities. In this section we will take a look at how to use the BAII 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

In addition to the previously mentioned financial keys, the TI BAII Plus also has the CF (cash flow) key to handle a series of uneven cash flows. To exit from “cash flow mode” at any time, simply press 2nd CPT (quit).

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 (the principle of value additivity).  However, that is the hard way.  Instead, we’ll use the CF key.  All we need to do is enter the cash flows exactly as shown in the table.  Again, we must clear the cash flow registers first. In this case we need to press CF 2nd CE|C (note that pressing 2nd FV will have no effect on the cash flow registers). The calculator will prompt you to enter each cash flow and then the frequency with which it occurs.  For now, just accept the default frequency of 1 each time.  Now, press CF then 0 ENTER ↓, 100 ENTER ↓ ↓, 200 ENTER ↓ ↓, 300 ENTER ↓ ↓, 400 ENTER ↓ ↓, and finally 500 ENTER ↓ ↓. 

Now, press the NPV key and you will be prompted for the interest rate (the screen will show I = 0.00000). Type 12 ENTER and then press ↓ and you will see NPV = 0.00.  To get the present value of the cash flows, press CPT.  We find that the present value is $1,000.17922.  Note that you can easily change the interest rate by pressing the ↑ key to get back to that step and entering a different rate.

You can save the NPV in memory by pressing STO 1 (or any other number key) immediately after doing the calculation. In the next step, we will use this result and you can recall it from memory by pressing RCL 1. Doing this helps to avoid rounding errors, so it is always a good idea.

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 key 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.  Clear the financial keys (2nd FV) then enter -1000.17922 into the PV key.  N is 5 and I/Y is 12.  Now press CPT FV and you’ll see that the future value is \$1,762.65753.  Pretty easy, huh?  (OK, at least its easier than adding up the future values of each of the individual cash flows.)

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 above.

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). 

Since we have already entered all of the cash flows, we only need to change the initial outlay. Press CF to get back into cash flow mode, and then input -800 ENTER for CF0. Now, press NPV. Note that we need to supply a discount rate so the calculator will now prompt you for it. Input 12 for I when prompted, and then ENTER ↓ and CPT. You’ll find that the NPV is \$200.17922. 

Example 4.1 — Internal Rate of Return

Solving for the IRR is done exactly the same way as the NPV, except that the discount rate is not necessary.  This time, you’ll press IRR and then CPT, and you’ll find that the IRR is 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, most 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 CF0 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. Save this result to a memory location (STO 1).
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. Save this result to a memory location (STO 2).
3. Finally, find the discount rate that equates the initial cost of the investment with the future value of the cash flows using the TVM keys and the saved values from steps 1 and 2. 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. Clear the TVM keys and then enter the cash flows (remember that we are ignoring the cost of the investment at this point): press 2nd CE|C to clear the cash flow keys. Now, press CF then 0 ENTER ↓, 100 ENTER ↓ ↓, 200 ENTER ↓ ↓, 300 ENTER ↓ ↓, 400 ENTER ↓ ↓, and finally 500 ENTER ↓ ↓.  Now, press the NPV key and enter 10 ENTER ↓ when prompted for the interest rate.  To get the present value of the cash flows press CPT. We find that the present value is \$1,065.26.
2. To find the future value of the cash flows, enter -1,065.26 into PV, 5 into N, and 10 into I/Y. Now press CPT FV and see that the future value is \$1,715.61.
3. At this point our problem has been transformed into an \$800 investment with a lump sum cash flow of \$1,715.61 in period 5. The MIRR is the discount rate (I/Y) that equates these two numbers. Enter -800 into PV and then press CPT I/Y. The MIRR is 16.48% per year.

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.