Making Money With Algebra

Transcription

1 Making Money With Algebra Floyd Vest References Counte, S.D. and C. deboor Elementary Numerical Analysis. 3rd ed., New York: McGraw-Hill. Haeussler, Ernest F., and Richard S. Paul Introductory Mathematical Analysis for Business, Economics, and Life and Social Sciences. Englewood Cliffs, NJ: Prentice Hall. Hunt, Richard A Calculus with Analytic Geometry. New York: Harper and Row.

2 1 Many individuals make money by collecting interest on money which they loan or invest. Banks and other financial institutions also make a profit by charging interest on loans. If, like millions of people, you were working in one of these financial institutions, there would be several mathematics of finance formulas which determine the interest that is charged. You are probably familiar with some of them. (For discussion of these formulas, see the Pull-Out Sections in past issues of Consortium Numbers 30, 29, 27, 20 and 16.) One of these formulas is the Compound Interest Formula. Example If you were to loan a friend $100 principal for four months at 15% per year compounded monthly, and he was to pay you back the $100 principal plus interest, then he should pay you an amount which would be S = 100( /12) 4 where n = 4 months and i = 0.15/12. You may want to check these substituted values against the general formula. Calculating S with a typical scientific calculator, a code would be = y x 4 X 100 = (Underlined numerals appear as answers in the display.) Rounding to the nearest cent gives S = $ Of course the interest you earned from the loan is I = S P = = $5.09. Do"You Try It" #1. Compound Interest Formula: S = P(1 + i) n where P = principal (for example the amount of a loan), S = sum (for example the amount paid back including principal and interest), n = number of periods (for example number of months of a loan) = (number of years) X (number of periods per year), and i = interest rate per compounding period = (annual interest rate ) (number of compounding periods per year). The companion interest formula is I = S P where I = total interest (for example interest charged on a loan). You can relate these variables to the following time line: P S n - 1 n periods See the example in the sidebar. The Pull-Out Section of Consortium No. 20 has a derivation (explanation) of this formula.

3 2 Now let us use our algebra skills. Notice that the compound interest formula is solved for S. We say S is expressed explicitly (separated by itself). There are times when we need to solve this equation for the other three variables. Assume that the employees in your bank need to do this but they don't know algebra as well as you do. To assist them do "You Try Its" #2 #4. Present Value of Annuity Using Algebra, Symbolic Solutions Another formula which determines the interest a bank earns is called the present value formula for an ordinary annuity. It is $ 1 (1+i ) n A = R where A = present i value (for example, the amount of a loan), R = rent (for example, the amount of each of several equal payments), n = number of payments or periods = (number of periods per year) X (number of years), and i = interest rate per period = (annual interest rate) (number of periods per year). In case of a loan, the first loan payment is made one period after the beginning of the loan. Please relate these variables to the time-line. A R R R R R n 1 n In order to pay off the loan, the borrower can make n payments of R dollars each so that a bank receives interest I = nr A. The Pull-Out Section in Consortium No. 29 has a derivation of this formula.

4 3 HiMAP Pull-Out Section: Spring 1990 For an example, let us assume that our bank makes a $12,995 loan on an automobile at 12% per year compounded monthly for four years with payments to be made monthly. We need to calculate the amount R of each payment and the interest I that the bank would earn on the loan. Substituting into the formula, we have 12,955 = R 1 ( ) ) ( You can check the present value formula and the meaning of the variables to see if this is correct. Consider two notes: You can do the calculations correctly with codes different from this one and even do some steps mentally. Also, why not ask your local bank how they would round R? Would they use $ or $341.16? We can use a scientific calculator to calculate R. Starting inside the parentheses, a typical code would be: = y x (12 x 4) +/ =+/ x 12 = 2nd 1/x x 12,955 = This gives R = $ rounded to the nearest cent. See the note in the sidebar. We should not forget that the idea is to make money (interest) on the loan. Using the formula for interest, we get I = 48(341.15) 12,955 = $3, For an example of continuous compound interest, let us assume that our bank loans $100,000 to another bank by contracting for a CD (Certificate of Deposit) for 3 years paying to our bank at the end of the three years principal plus interest earned at 9% compounded continuously. How much interest do we earn? Substituting we have S = 100,000e (0.09)(3). Calculating with a scientific calculator with the code 0.09 X 3 = inv 1nx X = gives S = $130, The interest our bank earns is I = 130, ,000 = $30, This is not small potatoes. You have heard that it takes money to make money. Do "You Try It" #6. Now let us solve this formula for the present value of an ordinary annuity for the variables n and R. Remember that we do this to assist the employees in the bank. Do "You Try It" #5. Continuous Compound Interest A bank will sometimes offer what is called Continuous Compound Interest. The interest is not compounded annually, monthly, or daily, but more frequently. It is compounded continuously. The continuous compound interest formula is S = Pe it where P = principal (amount of the loan, for example), i = annual interest rate compounded continuously, t = number of years, and S = sum (for example, the amount paid back including principal and interest). As you know, e is the natural number where e The companion interest formula is I = S P. See the sidebar for an example.

5 Using Algebra and Calculators, Numerical Solutions 4 Now is the time for our bank to make some money with the three formulas. Find numerical solutions for the following problems. Label all numbers and answers carefully. Remember that when working in a bank, someone may need to read your notes. We will tell you which formulas to use for the first few problems. You figure this out for the other problems. It helps to draw a time-line for each problem and to study the difference between the formulas. Do "You Try Its" #7 #19. Summary You have demonstrated how a knowledge of algebra and calculators can be used to perform banking calculations in the area of Symbolic Solutions solving equations with letters, and Numerical Solutions solving equations with numbers. You Try It #1 This $5.09 interest may not seem like much money, but if you loaned $1000 on the above terms to 1000 customers in a year, how much would you make in interest? You may want to work this problem two ways. You Try It #2 Solve the compound interest formula for P and write the meaning of each variable. You Try It #3 Similarly, give the symbolic solutions for i and n. For n, you may need to review logarithms. You Try It #4 Since one of your employees is not very good with a scientific calculator, use the letters in the compound interest formula to write a general calculator code which solves this last formula for n. Use the numerical values in the above example to check your code. You Try It #5 Solve the present value formula for an annuity for R. State the value of each variable. Similarly, express n explicitly. You Try It #6 Help the workers in your bank by giving symbolic solutions for the other three variables in the continuous compound interest formula. Organize your work neatly for the sake of your employees. Write the meaning of each variable. You Try It #7 Our bank is to collect a $10, payoff of a loan, which was made at 10% per year compounded monthly for three years. How much did we loan? How much interest should we earn? Use the compound interest formula.

6 5 HiMAP Pull-Out Section: Spring 1990 You Try It #8 A loan of $10,000 was made at 9% per year compounded monthly for several years. The debt at maturity is $14, What was the duration of the loan? How much interest was earned? Use the compound interest formula. You Try It #9 A car loan for $15,000 was made by our bank and is to be paid off by 48 monthly payments. The interest is 13% per year compounded monthly. How much are the payments? How much interest did the bank collect? Use the present value formula for an ordinary annuity. You Try It #10 A home improvement loan was made for $10,000 at 12% compounded monthly. The payments were $ per month. What was the number of payments? What was the duration of the loan? How much interest did the bank collect? Use the present formula for an ordinary annuity. You Try It #11 Our bank bought a $100,000 continuous compound interest CD from another bank. In two years the value at maturity will be $119, What was the annual interest rate? How much interest did the bank earn? You guessed it use the continuous compound interest formula. You Try It #12 A loan of $20,000 was made at an annual rate compounded quarterly for two years. The debt at maturity is $27, What was the quarterly rate of interest? What was the annual rate? How much interest was earned? You Try It #13 A church borrowed $50,000 to be paid with interest at 10% compounded annually in five years. How much will they owe? How much interest did the bank earn? You Try It #14 Again our bank bought a $90,000 CD at 9% compounded continuously and collected at maturity $98, What was the duration of the CD? How much interest did the bank earn? You Try It #15 A loan of $8, was made on a boat at 14% per year compounded monthly for four years. What monthly payments should be made? How much interest did the bank collect? You Try It #16 An old sage once told a young man that he would never have much money until he arranged things so that he made money while he was asleep. What did the old sage mean? Develop a carefully arranged mathematical calculation to dramatically illustrate this point. Just how much money can one earn while asleep? You Try It #17 Discuss the calculations in this Pull-Out with someone who works in a financial institution. Ask them how they do the calculations. Ask them how they think different customers do the calculations. Write a report on the different ideas expressed as well as your observations in this area. Compare the advantages and disadvantages of the methods used in this article and those reported. How do you think a computer does these calculations? You Try It #18 Use the equations in this Pull-Out section to verify and understand by mathematical calculations the numbers and terms in an advertisement in a newspaper for such an item as a car loan and discount on a new car, or an advertisement of a home mortgage for a new home in the real estate section. You Try It #19 Have you noticed that we did not ask you to solve the formula for the present value of an ordinary annuity for numerical solutions for i? There is a reason for this. You are not likely to know how. This problem is much more difficult than the ones we have been doing. Difficult problems like this one lead to new areas of mathematics. If you really want to know how to obtain these numerical solutions for i, you can find methods for numerical solutions in a calculus book (Newton's Method), or in a numerical analysis textbook. You can actually program an inexpensive programmable calculator (which you may own) to calculate i.

7 Answers for "You Try Its": 1 2 If the formula gives $5.095 on $100, you would make $50.95 on $1000. From 1000 customers you make I = 50.95(1000) = $50,950. P = S(1 + i) n, P = principal, S = sum, I = interest rate per compounding period, n = number of periods R R R R R payments 48 months The payments were R = $ per month. I = $ There were 60 monthly payments. I = $ i = n S P 1 n = log S P log (1 + i ) 1 11 The annual interest rate was 9% compounded continuously. I = $19, The quarterly interest rate was 4%. The annual rate was 16% compounded quarterly. I = $ A typical code: S P = log (1 + i) log =. gives n 13 They owed S = $80, I = $30, R = Ai log (1 Ai 1 (1 + i) n n = R ) log (1 + i) 14 The duration of the CD was 1 year. I = $ The monthly payments should be $ I = $ P = Se -it, i = 1 t ln S P, t = 1 i ln S P, 7 8 where S = sum, P = principal, i = annual interest rate compounded continuously, and t = time in years. The bank loaned P = $ and collected interest I = $ The duration of the loan was n = 4 years. The interest was I = $ Here is one answer. Yours might be better. The old sage meant that you can earn interest on your investments while asleep. If you just work for a salary and never invest, you are not likely to ever have much money. For most people, their wages are not high enough and there are not enough working hours in the day. But, if for example you invest $2000 on your twentieth birthday at 10% interest (tax-free) compounded annually, then you will have on your sixty-fifth birthday S = 2000( ) 45 = $145, Assuming that you were asleep 1/3 of the time, you made while asleep (145, )(1/3) = $47, How much would you have on your sixty-fifth birthday, if additionally you saved $2000 on each of the next birthdays up to and including the sixty-fifth?