FINANCIAL MATHEMATICS

Transcription

1 FINANCIAL MATHEMATICS Jimmy Thosago & Michele Demetriou CASIO DESCRIPTION OF CONTENT OF WORKSHOP: The aim of our workshop is to share ideas on financial mathematics through revision of basic simple and compound interest, not only in theory but through calculations too. However, the workshop is not limited to simple and compound interest. Nominal and effective interest, as well as continuous compound interest and depreciation have been included. Our purpose is to inspire teachers in newly expanded sections, such as financial mathematics, so that teachers will feel proficient in conveying their knowledge passionately. Calculator usage will play a vital role in the workshop, as we aim to bring teachers up to speed with the latest technological aids in teaching mathematics. The aim being not to replace mathematics, but rather to supplement it: conventional mathematics, new methods. MOTIVATION FOR RUNNING WORKSHOP: There are those who believe that the use of calculators in the classroom results in laziness in the learners, and others will argue that calculators are an irreplaceable learning tool in the classroom. We believe teachers wield the power. Careful planning and use of calculators during the lesson produces learners with a fine understanding of mathematics, yet a competence in using technological aids too. Calculators in classrooms mean more time is available to focus on higher order thinking mathematical concepts. Our aim is to empower teachers, through knowledge in mathematics and skills in technology. 1. Learning outcomes and assessment standards Use simple and Use simple and compound growth compound decay formulae formulae A = P (1 - ni) and A = P(1 - i) n to A = P(1 + ni) and A = solve problems, including straight (P + i) n to solve line depreciation and depreciation problems, including on a reducing balance. (link to LO interest, hire purchase, 2) inflation, Population growth and other real life problems (a) Calculate the value of n in the formula A = P(1 ± i) n. (b) Apply the knowledge of geometric series to solving annuity, bond repayment and sinking fund problems, with or without the use of the formulae: 81

2 Demonstrate an understanding of the implications of fluctuating foreign exchange rates (e.g. on the petrol price, imports, exports, overseas travel) Demonstrate an understanding of different periods of and compounding growth and decay (including effective compounding growth and decay and including effective nominal interest rates). F = P = x x n [( 1+ i) 1] i n [ 1 ( 1+ i) ] i Critically analyze investment and loan options and make informed decisions as to the best option(s) (including pyramid and micro-lenders schemes) 2. Workshop outcomes By the end of this workshop you will have Revised simple interest and compound interest on the Casio fx-82es in Natural Display Computation Mode as well as Table Mode. Defined interest and understand the different types of interest. Calculated nominal and effective interest rates Worked on timelines Calculated depreciation 3. Prior knowledge Know how to use the Casio fx-82es to find answers to financial calculations. Exercise 1: Fractions or shift (a) (b) (c) (d) S D and

3 Exercise 2: Powers X (a) 5 5 [3125] (b) [3971] Exercise 3: Roots (a) [13] (b) [ 18] (c) [5] Exercise 4: Compound Interest Shift mode 6 2 Set up fix 2 decimal places (a) (b) r A P = 1 9 = = 7693,12 A P = r 1 n ,87 = 48 1,25 1 = ,00 n 5 Shift mode 6 0 setup fix to nearest whole number 83

4 (c) A log n = P r log log = log 1 = 23 Shift mode 8 2 Setup normal Exercise 5: Memory Step 1: Store 5 in A 5 shift RCL (-) AC Step 2: Store 8 in B 8 shift RCL AC (a) A B [40] (b) A 2 - B [17] Shift 9 2 or 3 To clear memory or reset 4. Definitions and Formulae Interest - Money earned when money is borrowed or when money is invested. Simple interest - Interest calculated on principal amount each time. (e.g You invest R100 at 3% simple interest each year. You receive an additional R3 each year). 84

5 nr Formula A = P 1 A = Accumulated amount P = Principal/ original amount of money n = Number of years money invested/borrowed (years) r = interest rate (as percentage) Compound interest Interest calculated on interest. Investments yield better results when interest is compounded. (e.g You invest R100 at 3% compound interest, first year you receive R3, thereafter you receive 3% of yielded amount (R103), thereafter you receive more and more). Formula r A P = 1 n Continuously compounded interest Interest paid according to this formula is compounded continuously. E.g. You deposit R800 in a bank account that pays 12% compounded continuously. How much money will you have 5 years later? A o = R800 A(t ) = A o e rt t = 5 years A(5) = (800)e (0,12)(5) r = 12% = 0, 12 A(5) = 1 457,70 5. Simple Interest Calculations 1. R7000 is invested at 10% simple interest per annum for 4 years. Calculate the accumulated amount. 2. An amount of R2000 is invested at simple interest for 8 years. At the end of which the amount has grown to R5200. Calculate the rate of interest per annum. 85

6 3. (a) R12000 is invested at 15% simple interest per annum for 6 years. Calculate the amount accumulated at the end of each year. (b) Now draw the graph comparing the accumulated amount over the six years. (Hint: Use the ordered pairs on the Fx-82 ES to plot the graph). (c) What type of graph have you drawn? 6. Compound Interest Calculations 1. R12500 is deposited into a savings account. The interest is calculated at 15% compound interest per annum. How much money will be in the savings account after 4 years? 2. R5000 is loaned from the bank. After 5 years, the amount to be repaid is R7000. Calculate the rate of compound interest per annum. 3. (a) R2500 is invested at 16% compound interest per annum for 4 years. Calculate the accumulated amount after every year of investment. (b)draw a graph of accumulated amount over the 4 years. (Hint: Use the ordered pairs on the Fx-82 ES to plot the graph) (c) What type of a graph have you drawn? 86

7 7. Nominal and Effective Interest Rates Nominal and effective interest rate- Interest can be compounded more than just once a year. However, interest rates are usually quoted per annum. When this happens: You have to calculate number of times interest is compounded in a year. (n is multiplied by this number) You have to divide the annual interest rate by the number of times the principal is compounded. (r is divided by this number) 1. (a) R1000 is invested at 12% compound interest. Calculate the accumulated amount at the end of the year. (b) Calculate the accumulated amount if R1000 is invested for 1 year at 12% interest compounded monthly. (c) Now, using the Fx 82 ES, calculate the accumulated amount at the end of every month, if R1000 is invested for 1 year at 12% interest compounded monthly. (d) What is the nominal interest rate? (e) What is the quarterly interest rate? (f) Calculate the effective annual interest rate. 87

8 8. Timelines Timelines are used when investments involve a change in interest rate during the investment period, or when deposits or withdrawals are made on an account. Timelines summarize the information easily and effectively. 1. R is deposited into a savings account. Five years later R5 000 is deposited into the same account. The interest rate is 12% for the first three years, but increases to 15% thereafter. Find the value of the savings after 8 years. T1 T2 T3 T4 T5 T6 T7 T8 2. Jill has been saving for night classes since she was 16 years old. In the year she turns 22, she starts her 2 year night school course. Her first year s fees are R a) Calculate the value of her savings account before any withdrawals are made. Assume the interest rate is 13% compounded annually, and that the initial amount deposited into the account was R b) Calculate the value of Jill s savings account at the end of the first year of night classes. c) Assuming the class fees increase by 11,2% in the second year, calculate Jill s fees for year 2. 88

9 d) Does Jill have enough money saved to pay for both years of study? If so, what is the balance on her savings account? 9. Depreciation Depreciation is the loss in value of an asset through age. For example: vehicles, machinery and equipment reduce in value over a period of time as a result of usage and will eventually need to be replaced. Depreciation is most frequently calculated in one of two ways: A. Straight Line Depreciation Depreciation is the same each year. It is a percentage of the principal value and the asset is reduced to zero over a period of time. depreciati on For example nr = P 1 Catherine buys a cell phone for R The phone depreciates at 15% per annum on a straight line basis. Calculate the value of the phone after 3 years. B. Reducing Balance Depreciation Depreciation of the asset is based on the previous year s value. Every year the value by which an asset decreases is less. n r depreciation P = 1 Example: The company buys a vehicle for R Depreciation is calculated at 25% per annum on a reducing balance. Calculate the value of the vehicle at the end of each year, for 5 years. 89

10 REFERENCES: Laridon et al: Classroom Mathematics Grade 10 (2004). Heinneman Publishers, JHB, SA. Laridon et al: Classroom Mathematics Grade 11 (2006). Heinneman Publishers, JHB, SA. RL Finney and GB Thomas, Jr: Calculus (Addison- Wesley Publishing Company, Inc: Reading, Massachusetts: the second edition (1994). 90