BAII Plus Professional Tutorial

8/10/2019 BAII Plus Professional Tutorial

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BAII Plus Professional TutorialPart I

The TI BAII Plus Professional is a fairly easy to use financial calculator that 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 SetupUnlike other financial calculators, the BAII Plus Professional comes from the factory set to assume

annual compounding (others default to monthly compounding which is less than optimal). That's

exactly what I have been wanting for years. Why? Well, the compounding assumption is hidden

from view and in my experience people tend to forget to set it to the correct assumption. Of course,most people don't recognize a wrong answer when they get one, so they blithely forge ahead. If you

ever wish to change the compounding assumption (which I don't recommend), press 2nd I/Y and

enter the number of periods per year (12 for monthly, 2 for semiannual, etc). Now press Enterand

then 2nd CPT (Quit) to return to a blank screen.

One adjustment is important. By default the BAII Plus Professional 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 2nd . , and, when prompted, enter the number of digits

you would like to see displayed. You may have to use the arrow keys to scroll through the list of

options until you see DEC = 2 (or whatever the current number is). Once you set the number of

decimal places, press Enterto lock in your choice. I would press 2nd. 5 Enter 2nd CPT (Quit) todisplay 5 decimal places. That's it, the calculator is ready to go.

If you don't find the answer that you are looking for, please check theFAQ.If it isn't there,

pleasedrop me a noteand I'll try to answer the question.

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

math on the BAII Plus Professional:

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), there are 5 periods (N), and the interest rate is10% (I/Y). Before entering the data you need to make sure that the financial registers (each key is

nothing more than a memory register, and you can recall the values with the RCL key) are

clear. Otherwise, you may find that numbers left over from previous problems will interfere with the

solution to this one. Press 2nd FV (CLR TVM) to clear the memory. Now all we need to do is enter

the numbers into the appropriate keys: 5 into N , 10 into I/Y , -100 into PV . Now to find the future

value simply press CPT (compute) and then the FV key. The answer you get should be 161.05.

A Couple of Notes1. 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. Inthis case, we have a 4-variable problem and were given 3 of them (N, I/Y, and PV) and had to

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

the appropriate keys and then press CPT and then the other key to get the answer.

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

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

automatically divides any number entered into the I/Y key by 100. Had you entered 0.10, the

future value would have come out to 100.501 obviously incorrect.

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

financial calculators (and spreadsheets) follow theCash Flow Sign Convention. This is simplya 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 - (the "minus" key). Instead, use the + |

- key.

5. 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 moneyinvested for 10 years instead of 5. Simply enter 10 into N and then press CPT 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
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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 of8% 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. Enter the data as follows:

18 into N , 8 into I/Y , 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

press CPT PV 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 thenext pagethat 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 BAII Plus calculator. Enter 9 into I/Y , -1250 into PV ,

and 2500 into FV . Now press CPT N 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 thecash

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 Error 5 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 CE\C to clear it 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 fromabove,but we'll change

it slightly.



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?
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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

press CPT I/Y to find that you need to earn an average of 9.35% per year. Again, if you get Error 5

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

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 youunderstand the calculations that the calculator is

doingand 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 topart IIof this tutorial to learn about using the BAII Plus Professional to solve

problems involving annuities and perpetuities.

BAII Plus Professional TutorialPart II

In theprevious sectionwe 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

BAII Plus to calculate the present and future values of regular annuities and annuities due.

Aregular annuityis 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.

Anannuity dueis similar to a regular annuity, except that the first cash flow occurs immediately (at

period 0).
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Example 2 Present Value of AnnuitiesSuppose that you are offered an investment which 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 youwould be willing to pay today. Press 2ndFV to clear the financial keys. Enter the numbers into the

appropriate keys: 10 into N , 9 into I/Y , and 1000 (a cash inflow) into PMT . Now press CPT PV to

solve for the present value. 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 press CPT 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

theprevious 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:



and board, party supplies, etc. If you believe that you can earn an average annual rate of return of8% 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 aregular annuity.)

Let's enter the data: Type 18 into N , 8 into I/Y , and 100,000 into FV . Now, press CPT 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
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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/Y , -1,000,000 into PV (negative because you are investing this

amount), and 70,000 into PMT . Now, press CPT 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/Y 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/Y, 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,

press CPTI/Y 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 anannuity 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 needto put the calculator into Begin Mode. To change to Begin Mode, press 2nd PMT . You should see

that it says END on the screen. Now, press 2nd ENTER to change that to BGN and finally

press 2nd CPT to exit from setting the calculation mode. The screen will now show BGN in the

upper-right corner. 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.
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Let's do the college savings problem again, but this time assuming that you start investing

immediately:



and board, party supplies, etc. If you believe that you can earn an average annual rate of return of8% per year, how much money would you need to invest at the beginningof each year (starting

today) to achieve your goal?

As before, enter the data: 18 into N , 8 into I/Y , 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 press CPTPMT . 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. Press 2nd PMT . You should see

that it says BGN on the screen. Now, press 2nd ENTER to change that to END and finallypress 2nd CPT to exit from setting the calculation mode. When in End Mode, the upper-right corner

of the screen will be blank.

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 aperpetuity(perpetual annuity, sometimes

called an infinite annuity). The problem is that the BAII Plus Professional has no way to specify an

infinite number of periods using the N key.

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/Y , and 1000 into PMT . Now press CPT PV 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 topart IIIof this tutorial to learn about uneven cash flow streams, net present

value, internal rate of return, and modified internal rate of return.
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BAII Plus Professional TutorialPart III

In theprevious sectionwe 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 theBAII Plus Professional 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 CashFlowsIn addition to the previously mentioned financial keys, the BAII Plus Professional 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
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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 (theprinciple 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 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, and make sure that the frequency is always at least 1 for each cash flow.

Now, press CF then 0 Enterdown arrow, 100 Enterdown arrow (twice), 200 Enterdown arrow

(twice), 300 Enterdown arrow (twice), 400 Enterdown arrow (twice), and finally 500 Enterdown

arrow (twice). Now, press the NPV key and enter 12 Enterdown arrow when prompted for the

interest rate. To get the present value of the cash flows, press the down arrow key and then CPT .

We find that the present value is $1,000.17922. Note that you can easily change the interest rate by

pressing the up arrow key to get back to that step.

Example 3.1 Future Value of Uneven CashFlowsNow suppose that we wanted to find the future value of these cash flows instead of the present

value. Unlike most other financial calculators, the BAII Plus Professional can do this easily. Since we

have already entered the cash flows, just press NPV and enter the interest rate if necessary. Now,

press down arrow twice to get to NFV (Net Future Value). Press CPT and you'll see that the future

value of these cash flows at 12% per year is $1,762.6575. Pretty easy, huh? (Ok, at least its easier

than adding up the future values of each of the individual cash flows.)

Note: At any time, you can return to cash flow mode by pressing CF . This will allow you to scroll

through the cash flows that you entered by using the arrow keys. You can change any of these cash

flows. However, if you are starting a completely new problem you should always

press 2nd CE/C (from within CF mode) to be sure that the cash flows from any previous problem are

cleared. Otherwise, you will very likely get a wrong answer.

Example 4 Net Present Value (NPV)Calculating thenet present value(NPV) and/orinternal 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).
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11/22

Solving Problems with Non-Annual Periods on theTI BAII Plus Professional

Many, perhaps most, time value of money problems in the real world involve other than annual time

periods. For example, most consumer loans (e.g., mortgages, car loans, credit cards, etc) requiremonthly payments. All of the examples in the previous pages have used annual time periods for

simplicity. On this page, I'll show you how easy it is to deal with non-annual problems.

General ConsiderationsThe first thing to understand is that all of the principles that you have learned to apply for annual

problems still apply for non-annual problems. In truth, nothing has changed at all. If you try to think in

terms of "periods" rather than years, you will be ahead of the game. A period can be any amount of

time. Most common would be daily, monthly, quarterly, semiannually, or annually. However, a time

period could be any imaginable amount of time (e.g., seven weeks, hourly, three days).The first, and most important, thing to think about when dealing with non-annual periods is the

number of periods in a year. The reason that this is so important is because you must be consistent

when entering data into the BAII Plus Professional. The numbers entered into

the N , I/Y and PMT keys must agree as to the length of the time periods being used. So, if you are

working on a monthly problem, then N should be the total number of months, I/Y should be the

monthly interest rate, and PMT should be the monthly annuity payment.

An Example

Very often in a problem, you are given annual numbers but then told that "payments are made on amonthly basis," or that "interest is compounded daily." In these cases, you must adjust the numbers

given in the problem. Let's look at an example:

You are considering the purchase of a new home for $250,000. Your banker has informed you that

they are willing to offer you a 30-year, fixed rate loan at 7% with monthly payments. If you borrow the

entire $250,000, what is the required monthly payment?


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Notice that we are told that the loan term is 30 years and the interest rate is 7% per year (that is

implied, not explicitly stated). So, you might be forgiven for expecting that a period is one year.

However, on further reading you see that the payments must be made every month. Therefore, the

length of a period is one month, and you must convert the variables to a monthly basis in order to

get the correct answer.

Since there are 12 months in a year, we calculate the total number of periods by multiplying 30 years

by 12 months per year. So, N is 360 months, not 30 years. Similarly, the interest rate is found by

dividing the 7% annual rate by 12 to get 0.5833% per month. Note that we do not make any

adjustments to the PV ($250,000) because it occurs at a single point in time, not repeatedly. The

same logic would apply if there was an FV in this problem. When you solve for the payment, the

calculator will automatically give you the monthly (per period to be exact) payment amount.

In this problem, then, we would solve for the payment amount by entering 360 in N , 0.5833 into I/Y ,

and 250,000 into PV . When you press CPT PMT you will find that the monthly payment is

$1,663.26.

One thing to be careful about is rounding. For example, when calculating the monthly interest rate,

you should do the calculation in the calculator and then immediately press the I/Y key. Do not do the

calculation and then write down the answer for later entry. If you do, you will be truncating the

interest rate to the number of decimal places that are shown on the screen, and your answer will

suffer from the rounding. The difference may not be more than a few pennies, but every penny

matters. Try sending your lender a payment that is consistently three cents less than required and

see what happens. It probably won't be long before you get a nasty letter.

Adjust First, Not After, Solving the Problem

You might be tempted to think that you could treat the problem as an annual one, and then adjustyour answer to be monthly. Don't do that! The math simply doesn't work that way. To prove it, let's

input annual numbers, and then convert the annual payment to monthly by dividing by 12. Enter 30

into N , 7 into I/Y , and 250,000 into PV . When you press CPT PMT , you will find that the annual

payment would be $20,146.60. However, you have to make monthly payments so if we divide that

by 12 we get a monthly payment of $1,678.88.

Do you see the problem? If you do the problem this way, you get an answer that is $15.63 too high

every month. So, when you make the adjustments matters. Always adjust your variables before

solving the problem. The reason for the difference is the compounding of interest. If you have read

through my tutorial on theMathematics of Time Value of Money,then you know that the more

frequently interest is compounded, the smaller the payment has to be in order to grow to a particular

future value.

Using the BAII Plus Professional Payments perYear Setting
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You may also be interested in my tutorial oncalculating bond yieldsusing the TI BAII Plus

Professional.

Bond Cash Flows

As noted above, a bond typically makes a series of semiannual interest payments and then, atmaturity, pays back the face value. Let's look at an example:

Draw a time line for a 3-year bond with a coupon rate of 8% per year paid semiannually. The bond

has a face value of $1,000.

The bond has three years until maturity and it pays interest semiannually, so the time line needs to

show six periods. The bond will pay 8% of the $1,000face valuein interest every year. However, the

annual interest is paid in two equal payments each year, so there will be sixcoupon paymentsof

$40 each. Finally, the $1,000 will be returned atmaturity(i.e., the end of period 6). Therefore, the

time line looks like the one below:

We will use this bond throughout the tutorial.

Bond Valuation on a Coupon DateWe will begin our example by assuming that today is either the issue date or acoupon payment

date.In either case, the next payment will occur in exactly six months. This will be important

because we are going to use the time value of money keys to find the present value of the cash

flows.

The value of any asset is the present value of its cash flows. Therefore, we need to know two things:

1. The size and timing of the cash flows.

2. The required rate of return (discount rate)that is appropriate given the riskiness of the cash

flows.

We have already identified the cash flows above. Take a look at the time line and see if you can

identify the two types of cash flows. Notice that the interest payments are a $40, six-periodregularannuity.The face value is a $1,000lump sumcash flow. Using theprinciple of value additivity,we

know that we can find the total present value by first calculating the present value of the interest

payments and then the present value of the face value. Adding those together gives us the total

present value of the bond.

We don't have to value the bond in two steps, however. The TVM keys on the BAII Plus Professional

can handle this calculation as we will see in the next example:
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Assuming that your required return for the bond is 9.5% per year, what is the most that you would be

willing to pay for this bond?

We can calculate the present value of the cash flows using the TVM keys. Enter the data: 6 into N ,

4.75 into I/Y (9.5/2 = 4.75), 40 into PMT , and 1,000 into FV . Now, press CPT PV and you will find

that the value of the bond is $961.63. (If you get $1,213.29 instead, then you have the calculator set

to assume monthly compounding. Please see theInitial Setupsection of the BAII Plus Professional

tutorial for how to correct this problem.)

Notice that the bond is currently selling at a discount (i.e., less than its face value). This discount

must eventually disappear as the bond approaches its maturity date. A bond selling at a premium to

its face value will slowly decline as maturity approaches. In the chart below, the blue line shows the

price of our example bond as time passes.

The red line shows how a bond that is trading at a premium will change in price over time. Both lines

assume that market interest rates stay constant. In either case, at maturity a bond will be worth

exactly its face value. Keep this in mind as it will be a key fact in the next section.

Bond Valuation In-between Coupon Dates, Part 1In the previous section we saw that it is very easy to find the value of a bond on a coupon payment

date. However, calculating the value of a bond in-between coupon payment dates is more complex.

As we'll see, the reason is that interest does not compound between payment dates. That meansthat you cannot get the correct answer by entering fractional periods (e.g., 5.5) into N .

Note that in this section we will deal with the generic idea of periods, and use the TVM keys. This

section is important so that you understand the process of valuing bonds between coupon payment

dates. In thenext section,we will use the BOND key and deal with exact dates.

Let's start by using the same bond, but we will now assume that 6 months have passed. That is,

today is now the end of period 1. What is the value of the bond at this point?
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The process so far is shown in the graphic below:

Now, to get the clean price (doesn't include accrued interest, this is the price that would be quoted

by a dealer) at period 0.5 we need to subtract the accrued interest.

Because interest accrues equally on each day of the payment period, we can calculate the accrued

interest by multiplying the total interest for the period by the fraction of the period that has elapsed:

Accrued Interest = Total Interest x Fraction of Period Elapsed

In this example, that works out to $20:

Accrued Interest = 40 x 0.5 = 20

Finally, to find the clean (quoted) price, we subtract the accrued interest from the dirty price:

Clean Price = Dirty Price - Accrued Interest

In this example, we get $964.20:

Clean Price = 984.20 - 20 = 964.20

The same procedure could be done for any fractional period. For example, if 2 months (out of 6)have elapsed, then the fraction is 1/3. So, the clean price of the bond would be $963.28. Prove that

for yourself to make sure that you understand the process.

Please note that you cannot get the correct answer by entering a fractional number into N . In this

case, if you simply entered 5.5 into N (because there are 5.5 periods remaining until maturity) you

would get an answer of $964.43. That is close, but it is not correct and it is not "close enough." The

reason that it won't work is because the formula used by the calculator assumes that the interest


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payments are an annuity. That is, the time between the cash flows must be exactly the same in

every case. Clearly, that isn't true when valuing a bond between coupon payment dates.

Bond Valuation In-between Coupon Dates, Part 2Unlike most financial calculators, the BAII Plus Professional can truly handle real world bond

valuation problems. That is because it has the BOND key, which allows you to enter exact dates and

to use the correct day-count basis (but only actual/actual or 30/360).

Let me recast our bond valuation problem slightly, so that it includes exact dates and other

information that we will need:

What is the value of a bond with an 8% coupon rate that last paid interest on 6/15/2007? Assume

that the settlement date for a trade made today is 9/15/2007, and that the bond will mature on

6/15/2010. The bond pays interest semiannually, and your required return is 9.5% per year. The

day-count basis is 30/360.

To get into the bond worksheet, press 2nd BOND (the 9 key). The first thing you are asked for is

thesettlement date(SDT). In the BAII Plus Professional, we can enter dates in either the mm.ddyy or

dd.mmyy formats. In this case, type 09.1507 ENTER . You should see the date change to 9-15-2007.

Press the down arrow button and type 8 ENTER for the coupon rate (CPN). Note that we entered the

annual coupon rate because the calculator will automatically adjust for the payment frequency.

Press the down arrow key again, and type 06.1510 ENTER for theredemption date(RDT).

Next, you will be asked for the redemption value (RV), which is the amount that you will receive at

maturity (or call date). This number must be entered as a percentage of the face value, so you will

type 100 ENTER . This indicates that you will be receiving 100% of the face value at maturity.

After pressing the down arrow key again, you will be prompted for theday-count basis.You can

choose either actual/actual (ACT) or 30/360 (360). If necessary, press 2nd ENTER to change the

day-count basis so that the screen shows 360.

Press the down arrow key again and you will be asked for the number of coupon payments per year.

You can choose either 1 per year (1/Y) or 2 per year (2/Y). This bond, as do most in the US, pays

interest semiannually so you should choose 2/Y. If necessary, you can change this setting by

pressing 2nd ENTER .

Next, we need to enter the required return (YLD). Type 9.5 and then press ENTER . Finally, to

calculate the value of the bond, press the down arrow button to get to PRI (price). Press CPT and

you will see that the price is 96.42.

Note that, as is the convention in the bond market, the price is given as a percentage of the face

value. In this case, the price is 96.42% of the $1,000 face value, or $964.20. This is the same price

that we found in the previous section.

One more thing: You can calculate the accrued interest using the BAII Plus Professional. After

calculating the price, press the down arrow key and you will see AI. It will show that the accrued

interest is 2.00% of the face value, or $20.00. Again, this is the same amount that we found in Part

1.
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Bond Price Quotes and Accrued InterestIt is important to understand that bond prices are quoted by dealers without the accrued interest. So,

if you get a quote of $950 to purchase a bond, then you will pay $950 plus however much interest

has accrued to the seller of the bond since the last coupon payment. That is, the invoiced price is the

quoted price plus accrued interest. There are three terms that you should understand:

Accrued Interest

Accrued interest is the interest that has been earned, but not yet been paid by the bond

issuer, since the last coupon payment. Note that interest accrues equally on every day

during the period. That is, it does not compound. So, halfway through the period, you will

have accrued exactly one-half of the period's interest payment. It works the same way for

any other fraction of a payment period.

Clean Price

The "clean price" is the price of the bond excluding the accrued interest. This is also known

as the quoted price.

Dirty Price

The "dirty price" is the total price of the bond, including accrued interest. This is the amount

that you would actually pay (or receive) if you purchase (or sell) the bond.

The dirty price is simply the clean price plus the accrued interest.

One final point: In the "real world" bond prices are quoted as a percentage of their face value, not in

dollars. So, if a bond dealer quoted the price of our example bond, they would say 96.443, not

964.43. This practice allows a bond price to be quoted without also having to state its face value,

and it makes price quotes comparable across different bonds regardless of their face value.

I hope that you have found this tutorial to be useful. Please continue on to the next pageto learn

about calculating the various bond return measures (current yield, yield to maturity, and yield to call).

Bond Yield Calculation on the BAII Plus

Professional CalculatorOne of the key variables in choosing any investment is the expected rate of return. We try to find

assets that have the best combination of risk and return. In this section we will see how to calculate

the rate of return on a bond investment. If you are comfortable using the TVM keys, then this will be

a simple task. If not, then you should first work through myTI BAII Plus Professionaltutorial.

The expected rate of return on a bond can be described using any (or all) of three measures:
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Current Yield

Yield to Maturity

Yield to Call

We will discuss each of these in turn below. In thebond valuationtutorial, we used an example bond

that we will use again here. The bond has a face value of $1,000, a coupon rate of 8% per year paid

semiannually, and three years to maturity. We found that the current value of the bond is $961.63.

For the sake of simplicity, we will assume that the current market price of the bond is the same as

the value. (You should be aware that intrinsic value and market price are two different, though

related, concepts.)

The Current YieldThe current yield is a measure of the income provided by the bond as a percentage of the current

price:

There is no built-in function to calculate the current yield, so you must use this formula. For the

example bond, the current yield is 8.32%:

Note that the current yield only takes into account the expected interest payments. It completely

ignores expected price changes (capital gains or losses). Therefore, it is a useful return measure

primarily for those who are most concerned with earning income from their portfolio. It is not a good

measure of return for those looking for capital gains. Furthermore, the current yield is a useless

statistic for zero-coupon bonds.

The Yield to Maturity, Part 1Unlike the current yield, the yield to maturity (YTM) measures both current income and expected

capital gains or losses. The YTM is the internal rate of return of the bond, so it measures theexpected compound average annual rate of return if the bond is purchased at the current market

price and is held to maturity.

In the case of our example bond, the current yield understates the total expected return for the bond.

As we saw in thebond valuation tutorial,bonds selling at a discount to their face value must

increase in price as the maturity date approaches. The YTM takes into account both the interest

income and this capital gain over the life of the bond.
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There is no formula that can be used to calculate the exact yield to maturity for a bond (except for

trivial cases). Instead, the calculation must be done on a trial-and-error basis. This can be tedious to

do by hand. Fortunately, the BAII Plus Professional has the time value of money keys, which can do

the calculation quite easily. Technically, you could also use the IRR function, but there is no need to

do that when the TVM keys are easier and will give the same answer.

To calculate the YTM, just enter the bond data into the TVM keys. We can find the YTM by solving

for I/Y . Enter 6 into N , -961.63 into PV , 40 into PMT , and 1,000 into FV . Now, press CPT I/Y and

you should find that the YTM is 4.75%.

But wait a minute! That just doesn't make any sense. We know that the bond carries a coupon rate

of 8% per year, and the bond is selling for less than its face value. Therefore, we know that the

YTM mustbe greater than 8% per year. You need to remember that the bond pays interest

semiannually, and we entered N as the number of semiannual periods (6) and PMT as the

semiannual payment amount (40). So, when you solve for I/Y the answer is a semiannual yield.

Since the YTM is always stated as an annual rate, we need to double this answer. In this case, then,

the YTM is 9.50% per year.So, always remember to adjust the answer you get for I/Y back to an annual YTM by multiplying by

the number of payment periods per year.

The Yield to Maturity, Part 2If you worked through the bond valuation tutorial, then you have already seen how we can use the

bond worksheet (BOND) to calculate the value of a bond between coupon payment dates. Rather

than present essentially the same material again, I'll just point you toBond Valuation In-between

Coupon Dates, Part 2.

The procedure for finding the yield to maturity in-between coupon payment dates is identical, exceptthat you need to enter the current market price of the bond for PRI and then solve for the yield

(YLD).

The Yield to CallMany bonds (but certainly not all), whether Treasury bonds, corporate bonds, or municipal bonds are

callable. That is, the issuer has the right to force the redemption of the bonds before they mature.

This is similar to the way that a homeowner might choose to refinance (call) a mortgage when

interest rates decline.

Given a choice of callable or otherwise equivalent non-callable bonds, investors would choose thenon-callable bonds because they offer more certainty and potentially higher returns if interest rates

decline. Therefore, bond issuers usually offer a sweetener, in the form of a call premium, to make

callable bonds more attractive to investors. A call premium is an extra amount in excess of the face

value that must be paid in the event that the bond is called.

The picture below is a screen shot (from theFINRA TRACEWeb site on 8/17/2007) of the detailed

information on a bond issued by Union Electric Company. Notice that the call schedule shows that
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the bond is callable once per year, and that the call premium declines as each call date passes

without a call. If the bond is called after 12/15/2015 then it will be called at its face value (no call

premium).

It should be obvious that if the bond is called then the investor's rate of return will be different than

the promised YTM. That is why we calculate the yield to call (YTC) for callable bonds.

The yield to call is identical, in concept, to the yield to maturity, except that we assume that the bond

will be called at the next call date, and we add the call premium to the face value. Let's return to our

example:

Assume that the bond may be called in one year with a call premium of 3% of the face value. What

is the YTC for the bond?

In this case, the bond has 2 periods before the next call date, so enter 2 into N . The current price is

the same as before, so enter -961.63 into PV . The payment hasn't changed, so enter 40 into PMT .

We need to add the call premium to the face value, so enter 1,030 into FV . Solve for I/Y and you

will find that the YTC is 7.58% per semiannual period. Remember that we must double this result, so

the yield to call on this bond is 15.17% per year.

Now, ask yourself which is more advantageous to the issuer: 1) Continuing to pay interest at a yield

of 9.50% per year; or 2) Call the bond and pay an annual rate of 15.17%. Obviously, it doesn't make

sense to expect that the bond will be called as of now since it is cheaper for the company to pay the

current interest rate.

I hope that you have found this tutorial to be helpful.

Documents

BAII Plus Professional Tutorial