## TI 83

#### Definitions & Other

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In this tutorial we will see how to create an amortization schedule for a fixed-rate loan using the TI 83, 83 Plus, or TI 84 Plus graphing calculators from Texas Instruments. One of the advantages of these calculators over other financial calculators is their ability to create tables of data. We will use this functionality to generate a complete amortization table. If you prefer to use a spreadsheet, which I do, please see my spreadsheet amortization tutorial.

Fully amortizing loans are quite common. Examples include home mortgages, car loans, etc. Typically, but not always, a fully amortizing loan is one that calls for equal payments (annuity) throughout the life of the loan. The loan balance is fully retired after the last payment is made. Each payment in this type of loan consists of interest and principal payments. It is the presence of the principal payment that slowly reduces the loan balance, eventually to $0. An amortization schedule is a table that shows each loan payment and a breakdown of the amount of interest and principal paid. Typically, it will also show the remaining balance after each payment has been made. ## Calculating Interest and Principal in a Single Payment Let's start by reviewing the basics with an example loan (if you already know the basics, you can skip right to Creating an Amortization Schedule): Imagine that you are about to take out a 30-year fixed-rate mortgage. The terms of the loan specify an initial principal balance (the amount borrowed) of$200,000 and an APR of 6.75%. Payments will be made monthly. What will be the monthly payment? How much of the first payment will be interest, and how much will be principal?

Our first priority is to calculate the monthly payment amount. We can do this most easily by using the TVM Solver. So, enter the TVM Solver by pressing the APPS button and then choose the first item in the Finance Menu.

Note that since we are making monthly payments, we will need to adjust the number of periods (N) and the interest rate (I%) to monthly values. This can be done within the TVM Solver itself. Enter the data exactly as shown below:

Field Entry
N 30*12
I% 6.75/12
PV 200000
PMT 0
FV 0
P/Y 1
C/Y 1

Note that when you use the arrow keys to move away from any of the items, the calculator will automatically perform any calculations in that row. So, when you type 30*12 for N and then press the down arrow key, the value for N will automatically be converted to 360. The same trick works for I%, which in this case will be converted to 0.5625%. This is a handy shortcut.

Now, to calculate the monthly payment, scroll down to the PMT row and then press ALPHAENTER. You should find that the monthly payment is $1,297.196. (Note that your actual mortgage payment would be higher because it would likely include insurance and property tax payments that would be funneled into an escrow account by the mortgage service company.) That answers our first question. So, we now need to separate that payment into its interest and principal components. We can do this using a couple of simple formulas (we will use some built-in functions in a moment): Monthly Interest Payment = Principal Balance x Monthly Interest Rate Monthly Principal Payment = Monthly Payment - Monthly Interest Payment Using these formulas, we can see that the interest component of the first payment would be: Interest in 1st Payment = 200,000 x 0.005625 =$1,125

and the principal payment is:

Principal in 1st Payment = 1,297.196 - 1,125 = $172.196 Note that the sum of the interest and principal is the amount of the total payment: 1,125 + 172.196 =$1,297.196

That is the case for every single payment over the life of the loan. However, as payments are made the principal balance will decline. This, in turn, means that the interest payment will be lower, and the principal payment will be higher (because the total payment amount is constant), for each successive payment.

## Using the Built-in Functions

We've now seen how the principal and interest components of each payment are calculated. However, you can use a couple of built-in functions to do the math for you. These functions also make it easier to calculate the principal and/or interest for any arbitrary payment. Before we can use these functions, you must enter the loan details into the TVM Solver as we did above.

The two functions from the Finance menu that we are going to use are the ΣInt (sum interest) and the ΣPrn (sum principal) functions. These functions calculate the total amount of interest or principal paid between any two payments. They are defined as:

ΣInt(Begin Period, End Period)

ΣPrn(Begin Period, End Period)

For example, we can calculate the total interest paid in the first year of the loan by using:

ΣInt(1,12) = $13,434.858 We could do the same thing using ΣPrn(1,12) and find that the total principal paid in the first year is$2,131.496. Again, it is important to note that these functions take the loan data directly from the TVM Solver.

Now, we can also use these functions to calculate the interest and principal for any payment by specifying that the beginning and ending periods are the same. So, we can calculate the amount of interest in the first payment as:

ΣInt(1,1) = $1,125 and the amount of the principal in the first payment as: ΣPrn(1,1) =$172.196

Those answers match exactly the ones that we calculated manually above. It should now be clear that if we want to calculate the amount of interest in the second payment, we would use ΣInt(2,2). You can do the same for any interest or principal payment that you wish to find.

One last function that we will need is the Bal function. It calculates the remaining balance at the end of any time period and is defined as:

bal(period number)

If we want to know the remaining principal balance after the first payment, we can calculate it with:

bal(1) = $199,827.804 You can verify that by subtracting the amount of principal paid in the first payment from the beginning balance: Principal Balance After 1st Payment = 200,000 - 172.196 =$199,827.804

## Creating an Amortization Schedule

As noted in the beginning, an amortization schedule is simply a listing of each payment and the breakdown of interest, principal, and remaining balance. For this loan, an amortization table for the first six months would look like this:

If you haven't been following along, you will first need to enter the loan data (as shown above) into the TVM Solver before you can create an amortization schedule.

The first thing that we want to do is to set up the table. As shown in the picture above, we are going to want four columns (Payment, Interest, Principal, and Balance). So, we need to define four Y variables (note that our X variable is the period number, and we'll set that up next). Press the Y= button (the furthest left key under the screen) to get into the variable definition screen. There are 10 variables that you can define (Y1, Y2,...Y0). We will use the first four to define our variables. Note that you will use the Finance Menu to get the functions, and the X,T,Θ,n key to type the X. Enter the data as shown below (you may want to review the function definitions if necessary):

Variable Entry
\Y1= tvm_Pmt
\Y2= ΣInt(X,X)
\Y3= ΣPrn(X,X)
\Y4= bal(X)

Now press 2ndWINDOW (that gets you TBLSET for Table Setup). Set TblStart to 1 and ΔTbl to 1. Those settings will start the table with X = 1 (first period) and then increase it by 1 for each row in the table.

To create the amortization schedule with the variables that we have defined, press 2ndGRAPH (Table). That will take you to the table screen where you can scroll through the table using the arrow keys. Note that as you scroll around, the highlighted variable and its exact value will be shown below the table. For example, if you scroll over to the fourth column (Y4) and the sixth row, you will see that the remaining balance after six payments is \$198,952.184. That matches our amortization schedule that is pictured above.

Note that at any time you can change the loan terms for the amortization table. Simply go back to the TVM Solver, change the loan terms, and recalculate the payment amount. Now, press 2ndGRAPH to return to the table and you will see an amortization schedule for the new loan terms.

One final note: Scrolling through the entire table is tedious. If you want to see, for example, the amortization starting at period 348 (the beginning of the last year of payments), then you can press 2ndWINDOW and set TblStart to 348. This will recalculate the amortization and take you to period 348 when you press 2ndGRAPH. You can then scroll up and down the table starting from that point.

To clear out the table completely, press Y= and go to each line and press the CLEAR button. That will delete the definition for those list items.

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