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In the previous section we looked at the basic time worth of money secrets and how to use them to calculate current and future value of lump sums and continual annuities. In this area we will certainly take a look at just how to usage the TI 83 Plus to calculate the existing and also future values of unalso cash flow streams. We will certainly also check out how to calculate net present value (NPV), interior price of rerotate (IRR), and also the modified inner price of rerevolve (MIRR).

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Example 3 — Present Value of Unalso Cash Flows

This is wright here the TI 83 Plus is considerably more difficult than the majority of various other financial calculators. Its not as well bad one you obtain supplied to it, yet it is more tough than essential. Still, you usage what you"ve got, so allows plunge in. First, leave from the TVM Solver menu by pushing second MODE and also then go back to the Finance food selection.

To discover the present value of an uneven stream of cash flows, we have to usage the NPV attribute. This attribute is identified as:

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

Note that the Cash Flow Counts part is optional and we will certainly ignore it below. I will talk about it in the FAQ.

Suppose that you are readily available an investment which will pay the complying with cash flows at the end of each of the following five years:

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

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

We could resolve this problem by finding the present value of each of these cash flows individually and also then summing the outcomes. However before, that is the difficult method. Instead, we"ll use the NPV function. To begin, go to the Finance menu and also scroll down until you gain to the line that reads NPV(. Press Enter to choose that feature, and you will certainly view the beginning of the NPV attribute on your display screen. Now, finish the attribute as follows:


Press Go into to fix the feature and we find that the existing value is $1,000.17922. Keep in mind that you have the right to easily adjust the interemainder rate by pressing the 2nd Get in to retrieve the feature, and then utilizing the arrowhead secrets to modify it. For instance, to readjust the rate to 10%, push second Go into and also then usage the arrowhead secrets to relocate to the interemainder rate and push DEL to delete the 12, and then press second DEL (that"s the INS function) and also enter 10. Press Go into and you will certainly discover that the answer is currently $1,065.25883. Reset the interest rate to 12 prior to continuing.

Example 3.1 — Future Value of Uneven Cash Flows

Now suppose that we wanted to find the future worth of these cash flows rather of the present worth. Tbelow is no attribute to perform this so we should usage a tiny ingenuity. Realize that one method to discover the future value of any collection of cash flows is to first uncover the current worth. Next off, uncover the future worth of that existing worth and also you have actually your solution. In this situation, we"ve currently figured out that the current value is $1,000.17922, so we"ll recontact the NPV attribute by pushing second Get in. Now, include * 1.12 ^ 5 to the finish of the attribute, so that it now looks like:


Press Go into, and also you will certainly check out that the future value of these cash flows is $1,762.65754. Pretty simple, huh? Ok, at leastern its much easier than including up the future worths of each of the individual cash flows. It does require you to recognize the equation for the future worth of a lump sum, but you must recognize that anymeans.

Example 4 — Net Present out Value (NPV)

Calculating the net current worth (NPV) and/or inner rate of rerevolve (IRR) is virtually identical to finding the existing worth of an uneven cash circulation stream as we did in Example 3.

Suppose that you were available the investment in Example 3 at a cost of $800. What is the NPV? IRR?

To solve this difficulty we must not only tell the calculator around the yearly cash flows, however additionally the cost (formerly, we collection the cost to 0 because we just wanted the current value of the cash flows). Usually speaking, you"ll pay for an investment before you can get its benefits so the price (initial outlay) is sassist to take place at time period 0 (i.e., today). To discover the NPV recontact the NPV function and also modify it so that the initial outlay (previously 0) is -800. The attribute will certainly look favor this on screen:


Press Go into to acquire the solution and you"ll view that the NPV is $200.17922.

Example 4.1 — Internal Rate of Rerevolve

Solving for the IRR is done in a similar method, other than that we"ll use the IRR attribute.This feature is identified as:

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

For this problem, the attribute is:

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

Aacquire, note that the Cash Flow Counts component is optional and we will certainly overlook it right here, yet it is in the FAQ. To acquire the IRR attribute 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 till you watch IRR(. Get in the attribute as shown above and also then push Enter to obtain the answer (19.5382%).

Example 4.2 — Modified Internal Rate of Return

The IRR has been a well-known metric for evaluating investments for many kind of years — generally as a result of the simplicity through which it can be construed. However, the IRR suffers from a pair of significant flegislations. The most important flegislation is that it implicitly assumes that the cash flows will certainly be reinvested for the life of the task at a rate that equals the IRR. A great job may have an IRR that is significantly greater than any kind of reasonable reinvestment presumption. Because of this, the IRR can be misleadingly high at times.

The modified internal price of rerevolve (MIRR) solves this problem by using an explicit reinvestment price. Unfortunately, financial calculators don"t have actually an MIRR crucial prefer they have an IRR crucial. That indicates that we have to usage a tiny ingenuity to calculate the MIRR. Fortunately, it isn"t challenging. Here are the procedures in the algorithm that we will certainly use:

Calculate the complete present worth of each of the cash flows, beginning from period 1 (set the initial outlay to 0). Use the calculator"s NPV function simply like we did in Example 3, above. Use the reinvestment price as your discount rate to find the existing worth. Calculate the future worth as of the end of the task life of the present worth from action 1. The interemainder price that you will certainly usage to discover the future value is the reinvestment price. Finally, discover the discount rate that converts the initial expense of the investment through the future worth of the cash flows. This discount price is the MIRR, and also it have the right to be interpreted as the compound average annual rate of rerotate that you will earn on an investment if you reinvest the cash flows at the reinvestment rate.

Suppose that you were readily available the investment in Example 3 at a expense of $800. What is the MIRR if the reinvestment price is 10% per year?

Let"s go with our algorithm step-by-step:

The present value of the cash flows deserve to be discovered as in Example 3.


We find that the present worth is $1,065.26.

To discover the future worth of the cash flows, go to the TVM Solver and enter 5 into N, 10 right into I%, and also -1065.26 right into PV. Now resolve for the FV and also see that it is $1,715.61. At this allude our difficulty has actually been transformed into an $800 investment through a lump sum cash circulation of $1,715.61 at duration 5. The MIRR is the discount rate (I%) that equates these 2 numbers. Enter -800 into PV and also then resolve for I%. The MIRR is 16.48% per year.

Keep in mind that we have the right to actually integrate measures 1 and 2. Just as we did in Example 3, we have the right to calculate the future worth (utilizing our 10% reinvestment rate) as follows:


The future value is the exact same $1,715.61 that we discovered over. Now, go to the TVM Solver and also enter 5 into N, -800 into PV, and also 1715.61 right into FV. Solve for I% and also watch that the MIRR is 16.48% simply as prior to.

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

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Please proceed on to the next web page to learn exactly how to fix problems entailing non-yearly durations.