FINANCIAL COMPUTATIONS George A. Jahn Chairman, Dept. of Mathematics Palm Beach Community College Palm Beach Gardens Location http://www.pbcc.edu/faculty/jahng/ The TI-83 Plus and TI-84 Plus have a wonderful FINANCE application. Their TVM Solver (TVM denotes Time, Value, Money) allows you to investigate interest rates, future value, present value, payments, and number of payments. Other features allow for the investigation of annuities, amortization schedules, internal rate of return, nominal and effective interest rates, and days between dates. This paper will illustrate most of these features: BEST INTEREST RATE QUESTION: Which of the following gives the best interest rate for a savings account? a. 5.120% annual rate, compounded monthly. b. 5.116% annual rate, compounded daily. c. 5.115% annual rate, compounded continuously. FACTS: The nominal rate of a loan or savings account is the advertised rate, such as those listed above. This is also called the annual percentage rate (APR), or simply the annual rate. The effective rate is the equivalent percentage rate of the loan or savings account if it were compounded once a year. This is also called the annual percentage yield (APY). Sometimes it may be called the annual equivalent yield, or simply the annual yield. PROCESS: Converting the advertised nominal rates to effective rates allows you to compare rates which are compounded at different frequencies. To convert a nominal rate to an effective rate, press The format for the Eff command is Eff (nominal rate, number of compounding periods). Note: Since the TI-83/84 does not have a way of entering infinity, compounding continuously 12 must be entered as a very large number such as 10. To enter 10 12 press y `. SOLUTION: The highest effective rate comes from 5.116% compounded daily, so this is the best rate for a savings account. Had this been a loan instead of a savings account, the best rate would have been 5.120% compounded monthly. 1
PRESENT AND FUTURE VALUE QUESTION: On the day you were born, your grandmother put $50,000 in a trust fund for you with the stipulation that you could not touch the money until the account was worth at least $1,000,000. If interest is compounded quarterly at a fixed 8% rate, how old will you be before you can spend Granny s money? FACTS: Present value refers to the initial value, i.e., the value at the time when t = 0. For a loan, this is the initial amount of the loan; and for a savings account, this is the initial deposit. Future value refers to the value of the loan, savings account, etc., at some time in the future. PROCESS: Since we will henceforth be dealing with money, set the mode of the calculator to Float 2 (z~~~ yz) so that all decimals will be rounded to 2 places. Press to enter the TMV Solver. (TMV = time-value-ofmoney.) N is the number of payments. Leave it at whatever value is there, or set it to 0. This will be explained soon. Set I% to 8, the interest rate for Granny s account. Note: I% denotes the annual percentage rate and is entered as the actual percentage rate, not the decimal equivalent of the percentage rate. Set PV, the present value, equal to 50000. Caution: In the TVM Solver, outflow of money is negative and inflow is positive. Since Granny is sending her money out to the bank, this value must be negative. Set PMT, amount of payment, equal to zero since Granny will not be making any future deposits, and set FV, future value, equal to 1,000,000. Note: Future value, in this situation, is positive since it represents a future inflow of money. Set P/Y, payments per year, to 4 since this account is compounded quarterly. This will automatically set C/Y, compound periods per year, to 4. Leave the last line highlighted at END, indicating that payments are made at the end of each payment period. Even though Granny will not be making any payments, the interest will be compounded at the end of the payment period. Move the cursor to N and press to solve for N. 2
SOLUTION: N represents the number of payments made to the account, which in this case is the same as the number of compounding periods. Since N has a decimal part, the account will not be worth exactly $1,000,000 at the end of a compounding period. It is not until N= 152 that the account will reach, and exceed, the $1,000,000 mark. And since interest was compounded four times a year, you cannot spend Granny s money until you are N 4 = 38 years old.. ALTERNATE To get N in years, change P/Y to 1 and press ENTER. METHOD: Note that this automatically changes C/Y to 1, so you must reset it to 4 compounding periods per year. Move the cursor to N and press to find N in years. Since N is greater than 37.75 (37 years plus 3 more compounding periods) the account doesn t hit the $1,000,000 mark until you are 38 years old. QUESTION: Your retirement plan will allow you to retire as a millionaire in 20 years. Assuming a 3% inflation rate, how much will that $1,000,000 be worth in 20 years? PROCESS: In the TVM Solver, make the entries for N, I%, PMT, FV, P/Y, and C/Y which are shown at the right. (Why do we make these entries? What happens if FV is negative?) Then place the cursor on PV and press. SOLUTION: We see that 20 years from now, $1,000,000 will be worth what $553,675.75 is worth today. So in today s terms, you will be a half-a-millionaire! EXERCISE: You have $250,000 in your retirement plan which is earning approximately 9% APR compounded monthly. In 20 years you wish to be a millionaire by today s standards. The previous problem tells you that a million dollars by today s standards is approximately 2 million dollars 20 years from now. Assuming no changes in inflation and investment rates, how much should be invest monthly to achieve this goal? 3
LOANS QUESTION: You are buying your first house. You get a mortgage of $100,000 at 7.0% which is to be paid back over a period of 30 years. What are your monthly payments? PROCESS: In the TVM Solver, make the entries for N, I%, PV, FV, P/Y, and C/Y which are shown at the right. (Why do we make these entries? Why is PV positive?) Then place the cursor on PMT and press. SOLUTION: Your monthly payments will be $665.30. QUESTIONS: How much of your first payment goes toward paying off the balance (principal) of the loan? What is your balance after 20 years? How much interest do you pay during the life of the loan? PROCESS: Prn and Int ( and ) will calculate cumulative principal and interest. Int (n,m), for example, will give the total interest paid from the n th through the m th payment periods. bal ( n) ( fi) gives the balance at the n th payment period. To use these features, the appropriate values must first be entered in the TVM Solver. Then you must exit the TVM Solver using yz, and then re-enter the FINANCE application to select the appropriate command. SOLUTION: Only $81.97 of the first payment of $665.30 goes toward paying off your house. After 20 years you have a balance of $57,301.30, and after the loan has been paid off, you have paid $139,510.98 for the privilege of having this $100,000 mortgage. QUESTION: If you doubled your monthly payments, how long would it take you to pay off the loan and how much interest would you have paid over the life of the loan? While on the Home screen, you can enter the value of a TVM variable by pressing then keying in the number to the left of the variable you want to enter. ~ and 4
CREDIT CARDS QUESTION: Kim, a college senior, has a balance of $2151 on a credit card having a fixed APR of 16.9% and a $1700 credit limit. Each month her balance remains over the $1700 credit limit, she gets charges a $29 over limit fee. If she stops using the card and pays off the debt at the rate of $60 a month, when, if ever, will she be out of debt? SOLUTION: First find how long it takes Kim to get her balance at or below the $1700 credit limit. Note that during this period, $29 of her $60 monthly payment goes toward the over limit fee. So her payments on her credit card debt are only $31 a month. Entering the data in the TVM Solver, moving the cursor to N, and pressing, reveals that approximately 164.55 payments are needed to get to the $1700 credit limit. So Kim will no longer have to pay the over limit fee after 165 payments. Changing N in the TVM Solver to 165 and solving for FV, shows that Kim will then owe $1696.86. From then on, all of Kim s $60 monthly payments go towards paying off the remaining debt of $1696.86. To find how much longer it takes Kim to pay off the debt, enter the new values in the TVM Solver and solve for N. N = 36.32 indicates that Kim will be making 36 payments of $60.00, and a 37 th payment of less than $60.00. So Kim made a total of 201 monthly payments. That s 16 years and 9 months! And worse than that, Kim would have paid a little less than $12,060 to get rid of that $2151 credit card debt!! EXERCISE: How much of Kim s first $60 payment went towards paying off her debt? How long would it take Kim to pay off the debt at $80 a month? 5
INTERNAL RATE OF RETURN QUESTION: Your daughter is entering her freshman year at a college that requires each student to have a laptop computer. The college bookstore sells the appropriate kind of laptop, loaded with all necessary software, for $2000. Your research shows this to be a good deal, but with all the other college expenses you are encountering, you cannot afford to buy the laptop outright. The bookstore will lease the laptop to you for $600 a year for four years. At the end of the fourth year you have the option to buy the laptop for an additional $300. Your local bank will give you a four year loan at 12% APR. Which is the better deal? FACTS: The internal rate of return (irr) is the yearly rate that you earn on an investment plan. In this scenario, it is the yearly rate, APR, you would pay if the above leasing plan was converted to a loan. PROCESS: To compare the two, use the irr command to convert the bookstore offer to an APR. The format for irr ( ) is irr (initial cash flow, cash flow list, cash flow frequency list). In the given scenario, the initial cash flow is the $2000 cost of the laptop offered by the bookstore. This value is positive because the bookstore has given you the equivalent of $2000 in the form of the laptop. The cash flow list is the $600 you will pay for the first 3 years, and the $900 you pay in the last year, assuming you choose to purchase the laptop. These two values are negative since it is money out of your pocket. The frequency list is 3 and 1 since you will pay $600 for 3 years and $900 in the last year. Lists must be enclosed in braces (above the parentheses keys on the calculator). SOLUTION: The 12% bank loan is the better deal, assuming you wish to eventually own the laptop and software. DAYS BETWEEN DATES QUESTION: How many days until Christmas? PROCESS: dbd ( ) calculates the number of days between two dates, provided those dates are between the years 1950 and 2049. The dates are entered in the format MM.DDYY. SOLUTION: Assuming today is March 13, 2004, you have plenty of time to get your shopping done! 6