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 lump sums and annuities. In this section we will take a look at how to use the HP 20b 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 20b also has a key labeled CshFl to handle a series of uneven cash flows.

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

Period	Cash Flows
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 up the results. However, that is the hard way. Instead, we’ll use the cash flow key (CshFl). All we need to do is enter the cash flows exactly as shown in the table. First, we need to clear any previously entered cash flows from the list. To do this, press CshFl followed by SHIFT ← (Reset) INPUT. At this point you will be asked to confirm the deletion (Del.CF?). Press INPUT and then ON/CE to clear the list. You will now be back at the beginning of the list, which is labeled as CF(0).

Now, press 0 then INPUT to enter the time 0 cash flow. Next, you will be asked how many times this cash flow occurs in a row #CF(0). One time is the default, so you can simply press INPUT and move to cash flow 1. Continue entering the rest of the cash flows as follows: 100 INPUT INPUT, 200 INPUT INPUT, 300 INPUT INPUT, 400 INPUT INPUT, and finally 500 INPUT INPUT.

Finally, press the NPV button. You will be prompted for the interest rate, so enter 12 and then press INPUT. Now, press the ▼ key (down arrow) button to see that the present value is \$1,000.17922. In other words, you should be willing to pay up to $1,000.18, but not a penny more.

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 (still using a 12% interest rate). Fortunately, the HP 20b makes this much easier than most financial calculators. First enter the cash flows as shown above, if you haven’t already done so. Next, press NPV and enter 12 for the interest rate, and then press the down arrow button twice. You should see Net FV on the screen, and also the answer: \$1,762.65754. In other words, if you had invested the cash flows shown above at 12% per year, you would have ended up with about $1,762.66 after making the last investment (at period 5).

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 initial cost. 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 the cash flows, all that we need to do is to use the ▲ key (up arrow) to scroll back to cash flow 0. So, press CshFl to return to the list, and then scroll to cash flow 0 and enter 800 +/- INPUT. Now, press NPV, enter the discount rate if necessary, and then scroll down to the NPV. You’ll find that the NPV is \$200.17922. This means that the present value of the cash flows exceeds their cost by just over \$200. In other words, paying \$800 would be a great deal for you, and you should accept the investment.

Example 4.1 — Internal Rate of Return

Solving for the IRR is done exactly the same way, except that the discount rate is not necessary because that is the variable for which we are solving. This time, you’ll press IRR and immediately see that the IRR is 19.5382%. Note that the investment will earn about 19.50%, which substantially exceeds your 12% required return. So, just as with the NPV, we know that this is a good investment to purchase.

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 (including the 20b) 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 future value of each of the cash flows, starting from period 1 (leave out the initial outlay). Use the calculator’s Net FV function just like we did in Example 3, above. Use the reinvestment rate as your interest rate to find the future value.
2. Finally, using the TVM keys, 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 future value of the cash flows can be found as in Example 3. Press CshFl and change the amount of CF(0) to 0. The remaining cash flows should still be in the list. If not, then enter them as described above. Now press NPV, enter 10 for the interest rate, and then scroll down to Net FV. We find that the future value of the cash flows is $1,715.61.
2. Press ON/CE and then immediately press FV to enter this value into the FV button.
3. Next, we need to enter the rest of the data into the TVM keys: enter -800 into PV, 5 into N, and 0 into PMT.
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/YR) that equates these two numbers. Now press I/YR to find that the MIRR is 16.48385% 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.