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MATH1510 Financial Mathematics I

MATH1510 Financial Mathematics IJitse NiesenUniversity of LeedsJanuary May 2012 Description of the moduleThis is the description of the module as it appears in the module to mathematical modelling of Financial and insurance markets withparticular emphasis on the time-value of money and interest rates. Introductionto simple Financial instruments. This module covers a major part of the Facultyand Institute of Actuaries CT1 syllabus ( Financial Mathematics , core technical).Learning outcomesOn completion of this module, students should be able to understand the timevalue of money and to calculate interest rates and discount factors. They shouldbe able to apply these concepts to the pricing of simple, fixed-income financialinstruments and the assessment of investment Interest rates.

ACTEX Publications, 2008. ISBN 978-1-56698-657-1. 2.The Faculty of Actuaries and Institute of Actuaries, Subject CT1: Finan-cial Mathematics, Core Technical. Core reading for the 2009 examinations. 3.Stephen G. Kellison, The Theory of Interest, 3rd ed., McGraw-Hill, 2009. ISBN 978-007-127627-6.

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Transcription of MATH1510 Financial Mathematics I

1 MATH1510 Financial Mathematics IJitse NiesenUniversity of LeedsJanuary May 2012 Description of the moduleThis is the description of the module as it appears in the module to mathematical modelling of Financial and insurance markets withparticular emphasis on the time-value of money and interest rates. Introductionto simple Financial instruments. This module covers a major part of the Facultyand Institute of Actuaries CT1 syllabus ( Financial Mathematics , core technical).Learning outcomesOn completion of this module, students should be able to understand the timevalue of money and to calculate interest rates and discount factors. They shouldbe able to apply these concepts to the pricing of simple, fixed-income financialinstruments and the assessment of investment Interest rates.

2 Simple interest rates. Present value of a single futurepayment. Discount factors. Effective and nominal interest rates. Real and money interest rates. Com-pound interest rates. Relation between the time periods for compoundinterest rates and the discount factor. Compound interest functions. Annuities and perpetuities. Loans. Introduction to fixed-income instruments. Generalized cashflow model. Net present value of a sequence of cashflows. Equation of value. Internalrate of return. Investment project appraisal. Examples of cashflow patterns and their present values. Elementary compound interest listThese lecture notes are based on the following books:1. Samuel A. Broverman, Mathematics of Investment and Credit, 4th ed.

3 ,ACTEX Publications, 2008. ISBN The Faculty of Actuaries and Institute of Actuaries, Subject CT1: Finan-cial Mathematics , Core Technical. Core reading for the 2009 Stephen G. Kellison, The Theory of Interest, 3rd ed., McGraw-Hill, John McCutcheon and William F. Scott, An Introduction to the Mathe-matics of Finance, Elsevier Butterworth-Heinemann, 1986. ISBN Petr Zima and Robert L. Brown, Mathematics of Finance, 2nd ed., Schaum sOutline Series, McGraw-Hill, 1996. ISBN syllabus for the MATH1510 module is based on Units 1 9 and Unit 11 ofbook 2. The remainder forms the basis of MATH2510 ( Financial Mathemat-ics II). The book 2 describes the first exam that you need to pass to become anaccredited actuary in the UK.

4 It is written in a concise and perhaps dry lecture notes are largely based on Book 4. Book 5 contains many exer-cises, but does not go quite as deep. Book 3 is written from a perspective, sothe terminology is slightly different, but it has some good explanations. Book 1is written by a professor from a background and is particularlygood in making connections to these books are useful for consolidating the course material. They allowyou to gain background knowledge and to try your hand at further , the lecture notes cover the entire syllabus of the for 2011/12 LecturerJitse (from outside: 0113 3435870)LecturesTuesdays 10:00 11:00 in Roger Stevens LT 20 Wednesdays 12:00 13:00 in Roger Stevens LT 25 Fridays 14:00 15:00 in Roger Stevens LT 17 Example classesMondays in weeks 3, 5, 7, 9 and 11,see your personal timetable for time and Abourashchi, Zhidi Du, James Fung, and hoursTuesdays.

5 (to be determined)or whenever you find the lecturer and he has workThere will be five sets of course work. Put your work inyour tutor s pigeon hole on Level 8 of School of Mathemat-ics. Due dates are Wednesday 1 February, 15 February,29 February, 14 March and 25 workOne mark (out of ten) will be deducted for every is allowed (even encouraged), copying the student handbook for exam will take place in the period 14 May 30 May; exact date and location to be course work counts for 15%, the exam for 85%.Lecture notesThese notes and supporting materials are available in theBlackboard 1 The time value of moneyInterestis the compensation one gets for lending a certain asset. For instance,suppose that you put some money on a bank account for a year.

6 Then, the bankcan do whatever it wants with that money for a year. To reward you for that,it pays you some asset being lent out is called thecapital. Usually, both the capital andthe interest is expressed in money. However, that is not necessary. For instance,a farmer may lend his tractor to a neighbour, and get 10% of the grain harvestedin return. In this course, the capital is always expressed in money, and in thatcase it is also called Simple interestInterest is the reward for lending the capital to somebody for a period of are various methods for computing the interest. As the name implies,simple interestis easy to understand, and that is the main reason why we talkabout it here.

7 The idea behind simple interest is that the amount of interestis the product of three quantities: the rate of interest, the principal, and theperiod of time. However, as we will see at the end of this section, simple interestsuffers from a major problem. For this reason, its use in practice is (Simple interest).The interest earned on a capitalClentover a periodnat a much interest do you get if you put 1000 pounds for twoyears in a savings acount that pays simple interest at a rate of 9% per annum?And if you leave it in the account for only half ar year? you leave it for two years, you get2 1000 = 180pounds in interest. If you leave it for only half a year, then you get12 1000 =45 this example shows, the rate of interest is usually quoted as a percentage;9% corresponds to a factor of Furthermore, you have to be careful thatthe rate of interest is quoted using the same time unit as the period.

8 In thisMATH15101example, the period is measured in years, and the interest rate is quoted perannum ( per annum is Latin for per year ). These are the units that are usedmost often. In Section we will consider other you put 1000 in a savings account paying simpleinterest at 9% per annum for one year. Then, you withdraw the money withinterest and put it for one year in another account paying simple interest at 9%.How much do you have in the end? the first year, you would earn 1 1000 = 90 pounds in interest, soyou have 1090 after one year. In the second year, you earn 1 1090 = in interest, so you have (= 1090 + ) at the end of the compare Examples and The first example shows that if youinvest 1000 for two years, the capital grows to 1180.

9 But the second exampleshows that you can get by switching accounts after a year. Even betteris to open a new account every inconsistency means that simple interest is not that often used in prac-tice. Instead, savings accounts in banks pay compound interest, which will beintroduced in the next section. Nevertheless, simple interest is sometimes used,especially in short-term (From the 2010 exam) How many days does it take for 1450 to accumu-late to 1500 under 4% simple interest?2. (From the sample exam) A bank charges simple interest at a rate of 7% a 90-day loan of 1500. Compute the Compound interestMost bank accounts usecompound interest. The idea behind compound interestis that in the second year, you should get interest on the interest you earned inthe first year.

10 In other words, the interest you earn in the first year is combinedwith the principal, and in the second year you earn interest on the happens with the example from the previous section, where the in-vestor put 1000 for two years in an account paying 9%, if we consider com-pound interest? In the first year, the investor would receive 90 interest (9%of 1000). This would be credited to his account, so he now has 1090. Inthe second year, he would get interest (9% of 1090) so that he endsup with ; this is the same number as we found before. The capital ismultiplied by every year: 1000 = 1090 and 1090 = generally, the interest over one year isiC, whereidenotes the interestrate andCthe capital at the beginning of the year.


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