ti 84 tutorial

TI-84 Plus Tutorial Part I

The TI-84 Plus is a fairly easy, but more difficult than most, to use financial

calculator which will serve you well in all finance courses. This tutorial will

demonstrate how to use the financial functions to handle time value of money

problems and make financial math easy. I will keep the examples rather

elementary, but understanding the basics is all that is necessary to learn the

calculator.

Initial SetupThere is one adjustment which needs to be made before using this calculator. By default the TI-

84 displays only two decimal places. This is not enough. Personally, I like to see five decimal

places, but you may prefer some other number. To change the display, press the MODE key, then

the down arrow key once (to the Float line). Next, use the right arrow key to highlight the 5 and

press Enter . Finally, press 2nd MODE to exit the menu. That's it, the calculator is ready to go.

This tutorial will make extensive use of the TVM Solver, but the TI 84 Plus offers additional

financial functions in the Finance menu.

If you have come here because you are experiencing a problem, you might check out the FAQ. If

you don't find the solution, please send me a note.

Example 1 - Future Value of Lump SumsWe'll begin with a very simple problem that will provide you with most of the skills to perform

financial math on the TI-84:

Suppose that you have $100 to invest for a period of 5 years at an interest rate of 10% per year.

How much will you have accumulated at the end of this time period?

In this problem, the $100 is the present value (PV), N is 5, and i is 10%. Before entering the data

you need to put the calculator into the TVM Solver mode. Press the Apps button, choose the

Finance menu (or press the 1 key), and then choose TVM Solver (or press the 1 key). Your

screen should now look like the one in the picture. Enter the data as shown in the table below.

N 5

I% 10

PV -100

PMT 0

FV 0

P/Y 1

C/Y 1

Now to find the future value simply scroll to the FV line and press Alpha Enter . The answer you

get should be 161.05.

A Couple of Notes

1. Every time value of money problem has either 4 or 5 variables (corresponding to the 5 basic

financial keys). Of these, you will always be given 3 or 4 and asked to solve for the other. In

this case, we have a 4-variable problem and were given 3 of them (N, I%, and PV) and had

to solve for the 4th (FV). To solve these problems you simply enter the variables that you

know on the appropriate lines and then scroll to the line for the variable you wish to solve

for. To get the answer press Alpha Enter . Be sure that any variables not in the problem are

set to 0, otherwise they will be included in the calculation.

2. The order in which the numbers are entered does not matter.

3. Always make sure that the P/Y (payments per year) and C/Y (compounding periods per

year) are set to 1. At least this is what I prefer. Since these are visible on the screen at all

times, it is not strictly necessary. If you can remember to change these to the appropriate

values for each problem (1 for annual compounding, 12 for monthly compounding, etc)

then you'll have no problems.

4. When we entered the interest rate, we input 10 rather than 0.10. This is because the

calculator automatically divides any number entered on the I% line by 100. Had you entered

0.10, the future value would have come out to 100.501 — obviously incorrect.

5. Notice that we entered the 100 in PV as a negative number. This was on purpose. Most

financial calculators (and spreadsheets) follow the Cash Flow Sign Convention. This is

simply a way of keeping the direction of the cash flow straight. Cash inflows are entered as

positive numbers and cash outflows are entered as negative numbers. In this problem, the

$100 was an investment (i.e., a cash outflow) and the future value of $161.05 would be a

cash inflow in five years. Had you entered the $100 as a positive number no harm would

have been done, but the answer would have been returned as a negative number. This would

be correct had you borrowed $100 today (cash inflow) and agreed to repay $161.05 (cash

outflow) in five years. Do not change the sign of a number using - (the "minus" key).

Instead, use (-) .

6. We can change any of the variables in this problem without needing to re-enter all of the

data. For example, suppose that we wanted to find out the future value if we left the money

invested for 10 years instead of 5. Simply enter 10 on the N line and solve for FV . You'll

find that the answer is 259.37.

Example 1.1 — Present Value of Lump SumsSolving for the present value of a lump sum is nearly identical to solving for the future value.

One important thing to remember is that the present value will always (unless the interest rate is

negative) be less than the future value. Keep that in mind because it can help you to spot

incorrect answers due to a wrong input. Let's try a new problem:

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 today as a lump sum to

achieve your goal?

In this case, we already know the future value ($100,000), the number of periods (18 years), and

the per period interest rate (8% per year). We want to find the present value. Go to the TVM

Solver and enter the data as follows: 18 into N , 8 into I% , and 100,000 into FV . Note that we

enter the $100,000 as a positive number because you will be withdrawing that amount in 18

years (it will be a cash inflow). Now move to PV and press ALPHA ENTER and you will see that

you need to invest $25,024.90 today in order to meet your goal. That is a lot of money to invest

all at once, but we'll see on the next page that you can lessen the pain by investing smaller

amounts each year.

Example 1.2 — Solving for the Number of PeriodsSometimes you know how much money you have now, and how much you need to have at an

undetermined future time period. If you know the interest rate, then we can solve for the amount

of time that it will take for the present value to grow to the future value by solving for N.

Suppose that you have $1,250 today and you would like to know how long it will take you double

your money to $2,500. Assume that you can earn 9% per year on your investment.

This is the classic type of problem that we can quickly approximate using the Rule of 72.

However, we can easily find the exact answer using the TI 84 Plus calculator. Enter 9 into I% , -

1250 into PV , and 2500 into FV . Now scroll up to N and press ALPHA ENTER and you will see

that it will take 8.04 years for your money to double.

One important thing to note is that you absolutely must enter your numbers according to the cash

flow sign convention. If you don't make either the PV or FV a negative number (and the other

one positive), then you will get ERR: DOMAIN on the screen instead of the answer. That is

because, if both numbers are positive, the calculator thinks that you are getting a benefit without

making any investment. If you get this error, just press 2 (Goto) to return to the TVM Solver and

then fix the problem by changing the sign of either PV or FV.

Example 1.3 — Solving for the Interest RateSolving for the interest rate is quite common. Maybe you have recently sold an investment and

would like to know what your compound average annual rate of return was. Or, perhaps you are

thinking of making an investment and you would like to know what rate of return you need to

earn to reach a certain future value. Let's return to our college savings problem from above, but

we'll change it slightly.



tuition, room and board, party supplies, etc. If you have $20,000 to invest today, what compound

average annual rate of return do you need to earn in order to reach your goal?

As before, we need to be careful when entering the PV and FV into the calculator. In this case,

you are going to invest $20,000 today (a cash outflow) and receive $100,000 in 18 years (a cash

inflow). Therefore, we will enter -20,000 into PV , and 100,000 into FV . Type 18 into N , and

then solve for I% to find that you need to earn an average of 9.35% per year. If you get ERR: NO

SIGN CHNG instead of an answer, it is because you didn't follow the cash flow sign convention.

Press 2 to return to the TVM Solver and fix the problem.

Note that in our original problem we assumed that you would earn 8% per year, and found that

you would need to invest about $25,000 to achieve your goal. In this case, though, we assumed

that you started with only $20,000. Therefore, in order to reach the same goal, you would need to

earn a higher interest rate.

When you have solved a problem, always be sure to give the answer a second look and be sure

that it seems likely to be correct. This requires that you understand the calculations that the

calculator is doing and the relationships between the variables. If you don't, you will quickly

learn that if you enter wrong numbers you will get wrong answers. Remember, the calculator

only knows what you tell it, it doesn't know what you really meant.

Please continue on to part II of this tutorial to learn about using the TI 84 Plus to solve problems

involving annuities and perpetuities.

TI-84 Plus Tutorial Part II

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 84 Plus 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 AnnuitiesSuppose 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. Enter the numbers onto the appropriate lines: 10 into N , 9

into I% , 1000 (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 AnnuitiesNow, suppose that you will be borrowing $1000 each year for 10 years at a rate of 9%, and then

paying back the loan immediate after receiving the last payment. How much would you have to

repay?

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 AmountWe 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:




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 PeriodsSolving 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 another 34.40 years.

Example 2.4 — Solving for the Interest RateSolving 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 DueIn 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 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 the right

arrow key 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:




rate of return of 8% per year, how much money would you need to invest at the beginning of

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 in 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 — PerpetuitiesOccasionally, 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 84 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. 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 = 1,000 ÷ 0.09

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 will 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.

TI-84 Plus Tutorial Part III

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 84 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 FlowsThis is where the TI-84 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 press APPS and 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, scroll down in the finance menu 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 FlowsNow 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 (you may have to do this twice to get back to the original 12% interest rate). 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 (use 2ND DEL to insert

numbers without overwriting). 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 APPS and 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 ReturnThe 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.

NPV(10,0,{100,200,300,400,500})

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

2. 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.

3. 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

ti 84 tutorial

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