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This is from Professor David Baughman SECTION IV: ANNUITIES AND THE TI-83/84 GRAPHING CALCULATOR Calculations concerning annuities can be completed, with few exceptions, using the TI 83/84 Texas Instruments calculator. The following directions will facilitate calculations on the TI-83Plus calculator. The differences with the TI-83 and TI-84 are minimal. On the keyboard of the 83Plus press the blue “Apps” key. Next, in the resulting menu highlight (press) the item “Finance.” Number one on the resulting menu, which is already highlighted, is “TVM Solver.” Press the enter key. Most problems with annuities will be answered using “TVM- Solver.” The lines on this menu are listed as follows with explanations: N= total number of compounding/payment periods I%= Nominal interest rate(expressed as percent) PV= present value (amount of loan) PMT= size of periodic payment FV= future value of annuity;(amortization always equals 0) P/Y= payment periods per year C/Y= compounding periods per year PMT= End (Always set this key at “End”) Enter the information as follows: N=: the total number of compounding periods , i.e. the number of compounding periods in one year multiplied by the number of years. I%=: the stated nominal (yearly) interest rate expressed as a percent; e.g. 6% not 0.06. PV=: the amount that will be financed on the installment plan. This does not include things such as down-payment, default insurance, etc. PMT=: the periodic payment needed to either accrue a future value or to amortize a loan. In most amortization problems this is the value that will be computed at the final step. Hence when entering data skip this entry until the calculation is to be made (See below) FV=: the outstanding balance after the final payment; enter zero P/Y=: the number of payments to be made in one year. C/Y=: the number of compounding periods in one year For this course P/Y and C/Y will always be the same number. PMT=: the word “END” should be highlighted EXAMPLE I: An automobile is purchased on installment with $21500 financed. The loan is to be repaid in five years with interest at 6% compounded monthly. Determine the amount of the monthly payment: Using the pattern indicated above for the TI-83-84 Plus: N= 60 I%= 6 PV= -21500 PMT= (Skip over this value) FV= 0 P/Y=12 C/Y=12 With the data entered as above, move the cursor back to the fourth line (PMT). With this line highlighted first press the green “alpha” key, then press the enter key i.e the “alpha solve” function. The fourth line will read “PMT = 415.6552329.” To the nearest cent, the monthly payment will be $415.66. EXAMPLE II : Many other situations involving annuities can be solved using the “TVM SOLVER.” (The TVM stands for “Time, Value, Money.”) For example, suppose that a college fund is to be established by depositing $500 each quarter of the year for 12 years. The rate is 8% compounded quarterly, the first payment is entered three-months from now, find the future value immediately after the 48th payment. Enter, in order, the following amounts: N=48 I%=8 PV=0 PMT=-500 FV= (skip over this line) P/Y=4 C/Y=4 Next, move the cursor to the FV line and highlight. Enter “Alpha solve” as above. The highlighted FV line will read 39676.75964 representing a future value of $39676.76. SECTION V: AMORTIZATION SCHEDULE I The logic behind amortization (“to kill off”) of a loan is to find the monthly payment such that a portion will equal the interest owed on the outstanding balance of the loan while the remainder will be applied to the previous loan balance to determine the new outstanding balance immediately after the payment in question. The new balance will be the outstanding debt, and will be used to determine the amount of interest due for the following payment period. For successive payment periods, the interest will decrease while the principal, the amount by which the balance is reduced, will increase. An AMORTIZATION SCHEDULE is a table that displays the amount of the payment that is interest, the amount applied to the previous balance to reduce the debt, and the new outstanding balance. The columns of the table display from left to right the payment number, the payment amount, the principal applied to the previous outstanding balance and the new loan balance. Consider the first four rows of a schedule for Example I in the previous section: Period Payment Interest Principal Balance 0 $21500 1 $415.66 $107.50 $308.16 $21191.84 2 $415.66 $105.96 $309.70 $20882.14 3 $415.66 $104.41 $311.25 $2057.89 Row two begins with period zero, that is, the time when the agreement to finance $21500 is finalized. The only other entry in that row is in the Balance column and equals the amount that is initially financed, the present value of the loan. It is convenient to refer this as the outstanding balance. Interest in column three is found as the product of the periodic interest rate and the previous outstanding balance. Here the product of 0.005 and the previous outstanding balance. As is evident, line by line calculations can be tedious and subject to errors. Home mortgage loans are customarily thirty years in length, amounting to 360 payments. In the present financial milieu computer software such as EXCEL eliminates the drudgery. There are many uses for an amortization schedule. The total interest over one-year periods, which is most often tax-deductible can be determined. Also, the outstanding balance immediately after each payment is made can be readily observed (the Balance column). We will also show, in Section VI how this balance can be found with out an amortization schedule. Another tool in the financial applications of the TI 84. The total amount of interest for loans amortized over lengthy time periods is something we should all observe. The results are astounding. For thirty year mortgages the total of all payments often equals two to three times the amount of the original loan. For example, the finance charges for Example I in the previous section, financing $21500 for five years at 6% monthly amount to $3439 or approximately 16% of the amount financed. SECTION VI: OUTSTANDING BALANCE AND THE TI 83-84 PLUS Of the many and varied uses of the TI 83/84Plus calculator, two functions related to amortization schedules will be demonstrated. The problems addressed by these functions have been mentioned previously: (a) outstanding balance immediately after a specific payment, i.e. the present value of the loan at that point, and (b) calculation of interest paid over either the entire life of the loan or any part thereof.. (a) To determine the outstanding balance, the information concerning the amortization schedule, along with the calculation of the payment should be entered in the “TVM-Problem Solver. With this information in place return to the FINANCE menu of the calculator. Scroll down to the function “bal(“, number 9 on the TI83Plus, and enter. The screen will appear as “bal( “ with the cursor positioned to the right of the left parenthesis.. Type the payment number, right parenthesis, and then press enter. The figure given will be the outstanding balance immediately after the payment you have selected. Consider again Example I from Section V. To determine the outstanding balance, present value, immediately after payment number three, type “bal( 3)” and press enter. The value given is $20570.90 which is the same, considering round-off errors as the balance in row three of the Amortization Schedule in Section V. (b) To find the total interest paid, access the menu item under Finance appearing as “ Int ( “. Two entries are required corresponding to the total interest accrued beginning with the first of the two entries and ending with the second. Returning again to Example I, Section V, the total interest for 1 through 3 is found as Int(1,3). The amount of interest is $317.87.