BEA140 Quantitative Methods

1. BEA140 Quantitative Methods Lecture 1 By Leon Jiang - Summer Semester 2009 Leon Jiang, University of Tasmania

2. Introduction to the unit • Unit Coordinator – Mr. Steve Thollars • Contact: Please see the unit outline! • Lecturer/Tutor – Leon Jiang • Contact: lljiang@utas.edu.au • Work Phone: 03 6226 7685 • Consultation: 12:00 – 13:00 during days of teaching • Office: Annex Commerce A-103 Leon Jiang, University of Tasmania

3. Structure of this unit – 2 sections by 4 modules • Section One • Module 1 – Mathematics of Finance • Section Two • Module 2, 3, 4 • Quantitative Methods – a large part involving statistics Leon Jiang, University of Tasmania

4. Section One MOF Leon Jiang, University of Tasmania

5. Objectives of Learning Mathematics of Finance (MOF) • Interests – knowledge and calculations • Timeline diagram • Compounding rates – Annuity • Applications and typical examples Leon Jiang, University of Tasmania

6. Quantitative Methods – statistics • Why QM? • Standards – all degrees have a compulsory QM unit. • Professional requirements • A powerful scientific analysis instrument • As part of your intellectual fulfilment Leon Jiang, University of Tasmania

7. Three stages of QMing… • Data Collection and Presentation • - Module 2 • Data Analysis • - Module 3 • Estimation and Inference from the prepared data • - Module 4 Leon Jiang, University of Tasmania

8. Prerequisite of doing this unit! • TCE Mathematics Applied, Mathematics Stage 2 (or equivalent) or be enrolled in UPP090 Bridging Maths or BEA109 Introduction to Quantitative Methods. • Prior Knowledge/Skills • Arithmetic Skills • Basic Algebra Skills • Basic Calculator Operation Leon Jiang, University of Tasmania

9. Prescribed Text(s) • There is no prescribed text for this unit. Leon Jiang, University of Tasmania

10. Other Requirements • Calculators/Skills • Please make sure you have one proper calculator as prescribed in the unit outline and know how to well operate it. • Microsoft Excel Skills • Certain computational skills will be shown in this unit. Leon Jiang, University of Tasmania

11. Now, lets get down to our business today! Leon Jiang, University of Tasmania

12. ∑ (Sigma)- Summation Notation • The Greek letter Σ (a capital sigma) is used to designate summation. Leon Jiang, University of Tasmania

13. For example, suppose you are controlling the inventory level for a grocer’s; 4 bags of apples are on-shelf, each of which holds different number of apples respectively. Bag 1 will be referred to as X 1 , Bag 2 as X 2 , and so on. The numbers are shown below: • The way to use the summation sign to indicate the sum of all four X's is: Leon Jiang, University of Tasmania

14. Other examples… • There are lots of other forms and make sure you are familiarized with those often used! Leon Jiang, University of Tasmania

15. * Explaining the formula! • Often an abbreviated form of the summation notation is used. For example, ΣX means to sum all the values of X. • When only a subset of the values of X are to be summed then the full version is required. Leon Jiang, University of Tasmania

16. * When only a subset of the values of X are to be summed ! • This would be read as the sum of X with i going from 2 to N-1. Leon Jiang, University of Tasmania

17. * Example! • X – sample size • X1=7; X2=6; X3=5; X4=8 * Result? Leon Jiang, University of Tasmania

18. * Squared! • Some formulae require that each number be squared before the numbers are summed. Result: 72 + 62 + 52 + 82 = 174. Leon Jiang, University of Tasmania

19. * Two different formulae! • ΣX2 Vs. (ΣX)2 • Square first then sum! • Sum first then square! Leon Jiang, University of Tasmania

20. * Example! • ΣX2 72 + 62 + 52 + 82 = 174. • (ΣX)2 • (7 + 6 + 5 + 8)2= 262 = 676 Leon Jiang, University of Tasmania

21. * Σ(X + Y) = ΣX + ΣY *Basic Theorems Leon Jiang, University of Tasmania

22. * Example! Leon Jiang, University of Tasmania

23. * Σ(X + Y) = ΣX + ΣY • Σ(X + Y) = ΣX + ΣY • Σ(X + Y) = 11 + 5 + 5 = 21 • ΣX = 3 + 2 + 4 = 9 • ΣY = 8 + 3 + 1 = 12 • ΣX + ΣY = 9 + 12 = 21 Leon Jiang, University of Tasmania

24. *ΣaX = aΣX Leon Jiang, University of Tasmania

25. * ΣaX = aΣX • ΣaX = aΣX (“a” is a constant)For an example, let a = 2.ΣaX = (2)(3) + (2)(2) + (2)(4) = 18 • a ΣX = (2)(9) = 18 Leon Jiang, University of Tasmania

26. Part 2. Introduction to Mathematics of Finance Leon Jiang, University of Tasmania

27. A Question … • What is finance? Leon Jiang, University of Tasmania

28. Questions • Would you rather have $1000 now or a $1000 in a year’s time? • Would you rather have $1000 now or a $1000 one year ago? • Would you rather have $1000 in a year’s time or $966 now? Leon Jiang, University of Tasmania

29. * Aim of This Section! • · You will work with various forms of interest rates and the basic compound interest functions in order to value cash flows. Leon Jiang, University of Tasmania

30. What is interest? • Interest is a fee paid on borrowed assets. • By far the most common form in which these assets are lent is money, but other assets may be lent to the borrower, such as shares, consumer goods through hire purchase, major assets such as aircraft, and even entire factories in finance lease arrangements. In each case the interest is calculated upon the value of the assets in the same manner as upon money. Leon Jiang, University of Tasmania

31. * Concepts! • Principal: the money borrowed or lent out. • Interest: the cost for using the principal for a certain period of time. Leon Jiang, University of Tasmania

32. * Simple Interest! • When money is borrowed, interest is charged for the use of that money for a certain period of time. • When the money is paid back, the principal (amount of money that was borrowed) and the interest is paid back. • The amount to interest depends on the interest rate, the amount of money borrowed (principal) and the length of time that the money is borrowed. Leon Jiang, University of Tasmania

33. *Interest = Principal * Rate * TimeI=PRT • The formula for finding simple interest is: Interest = Principal * Rate * Time. • If $100 was borrowed for 2 years at a 10% interest rate, the interest would be $100*10%*2 = $20. • The total amount at due time would be $100+$20=$120. Leon Jiang, University of Tasmania

34. * Compound Interest - Making interest on interest! • Compound: The ability of an asset to generate earnings that are then reinvested and generate their own earnings. Leon Jiang, University of Tasmania

35. * A small example! • $10,000 put in bank deposit for three years. • Annual compound rate: 4% * At the end of the first year: 10,000+10,000*4%=$10,400 * At the end of the second year: 10,400+10,400*4%=$10,816 * At the end of the third year: 10,816+10,816*4%=11,248 Leon Jiang, University of Tasmania

36. Compound Interest * At the end of the 1st year: 10,000+10,000*4%=$10,400 * At the end of the 2nd year: 10,400+10,400*4%=$10,816 * At the end of the 3rd year: 10,816+10,816*4%=11,248 Simple Interest *At the end of the 3rd year 10,000+10,000*4%*3=11,200 * Comparison between Compound Interest and Simple Interest! Leon Jiang, University of Tasmania

37. * A set of notation! • P or PV - Principal or present value • F or FV - Compounded or future value • n – the number of interest periods • Jm – annual interest rate; • m: the number of times the interest is to be compounded per year. • i - the interest rate exercised in each compounding period. • i= Jm/m Leon Jiang, University of Tasmania

38. * Example! • We invest $10,000 in a bank with an interest rate (each compounding period) of 10% for n periods. • Here: P or PV=$10,000 • i = 10% • n Leon Jiang, University of Tasmania

39. At the end of period 1. 10,000+10,000*10%=11,000 At the end of period 2. 11,000+11,000*10%=12,100 At the end of period 3. 12,100+12,100*10%=13,310 P+P*i P+P*i=P(1+i) (P+P*i)+ (P+P*i)*i or P(1+i)+i*P(1+i) =P(1+i) 2 P(1+i) 2 +i*P(1+i) 2 = P(1+i) 3 * Compounding starts! Leon Jiang, University of Tasmania

40. Future Value • In general, after n periods, the initial sum P will have compounded to : FV=PV(1+i)n Leon Jiang, University of Tasmania

41. * Present Value - PV Leon Jiang, University of Tasmania

42. * PV- Present Value! • The amount today that a sum of money in the future is worth, given a specified rate of return. • This sounds a bit confusing, but it really isn't. An investment that earns 10% per year and can be redeemed for $1000 in 5 years would have a present value of $620. In other words, $620 today is worth $1000 in 5 years. Leon Jiang, University of Tasmania

43. * Example! • An investment that earns 10% per year and can be redeemed for $1000 in 5 years. • How much money we need to invest now? • FV=$1000 • i=10% (annually compounding rate) • n=5 Leon Jiang, University of Tasmania

44. * Example! • PV=FV*(1+i) -n • PV=$1000*(1+10%) -5 • PV=$620 This means that to have $1000 at a compound interest rate of 10%, one needs to invest $620 for 5 years. Leon Jiang, University of Tasmania

45. Interest rate vs. discount rate • The two concepts are in fact the same. • The only difference is ‘interest rate’ mostly for future values while ‘discount rate’ for present values. Leon Jiang, University of Tasmania

46. * Comparing FV and PV! Leon Jiang, University of Tasmania

47. Solutions for “n” and “i”! Leon Jiang, University of Tasmania

48. * Finding n and i! • n – the number of periods or times a compound interest rate is exercised! • i – compound interest rate • Where are they? • By manipulating PV or FV equation! Leon Jiang, University of Tasmania

49. * Equivalent Interest Rate! * Effective Rates! Leon Jiang, University of Tasmania

50. * Equivalent Rates! • An interest rate to be applied m times a year can be converted to another interest rate exercised p times a year. • To be equivalent, the two different interest rates must produce a same compound value by taking same amount of time and starting from a same sum of money. • An interest rate of 16% p.a. will not necessarily result in a greater future value than an interest rate of 15%. Leon Jiang, University of Tasmania