Introduction to Actuarial and Financial Mathematical Methods Companionwebsite: http://booksite.elsevier.com/9780128037379/ Introduction to Actuarial and Financial Mathematical Methods S.J.GARRETT AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON NEW YORK (cid:129) OXFORD (cid:129) PARIS (cid:129) SAN DIEGO SAN FRANCISCO (cid:129) SINGAPORE (cid:129) SYDNEY (cid:129) TOKYO Academic Press is an imprint of Elsevier AcademicPressisanimprintofElsevier 125LondonWall,London,EC2Y5AS,UK 525BStreet,Suite1800,SanDiego,CA92101-4495,USA 225WymanStreet,Waltham,MA02451,USA TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK Copyright©2015ElsevierInc.Allrightsreserved. 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LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ForinformationonallAcademicPresspublications visitourwebsiteathttp://store.elsevier.com/ PrintedandboundintheUnitedStates ISBN:978-0-12-800156-1 Publisher:NikkiLevy AcquisitionEditor:ScottBentley EditorialProjectManager:SusanIkeda ProductionProjectManager:JasonMitchell Designer:MarkRogers To Yvette, foreverything past, present,and future. PREFACE Mathematics is a huge subject of fundamental importance to our individual lives and the collective progress we make in shaping the modern world. The importance of mathematicsisrecognizedinformaleducationsystemsaroundtheworldandindeedhow wechoosetoteachourownchildrenpriortotheirformalschooling.Forexample,from averyearlyagechildrenaretaughttheirnativelanguageinparalleltothefundamentals of working with numbers: children learn the names of objects and also how to count theseobjects. Despite claims to the contrary, most adults do have considerable mathematical knowledgeand anintuitiveunderstandingofnumbers.Irrespectiveoftheireducational choices and natural ability, most people can count and understand simple arithmetical operations.Forexample,mostknowhowtochecktheirreceiptatthesupermarket;that is, they understand the fundamental concepts of addition and subtraction, even if they prefertouseacalculatortoperformtheactualarithmetic.Asafurtherexample,givena distancetotravel,mostpeoplewouldintuitivelyknowhowtocalculateanapproximate timeof arrival froman estimateof their average speed. Given that mathematics is so engrained in our childhood and used in our everyday adult lives, any book on the practical use of mathematics must begin by drawing a line thatseparatesthematerialthatisassumedasprerequisiteandthatwhichthebookwishes to develop. The correct place to draw this line is difficult to determine and must, of course,depend ontheintendedaudienceofthebook.Thisparticularbook,as thetitle suggests,is intendedforpeoplewho ultimatelywishto studyand applymathematicsin the highly technical areas of actuarial science and finance. It is therefore assumed that the reader has a prior interest in mathematics that has manifest in some kind of formal mathematicalstudyto,say, thehigh-schoollevel at least.Itisatthelevel ofhigh-school mathematicsthat thelineis drawn for this book. It may be that high school was a long time ago and the mathematics learned there has since left you. For this reason I begin by softening what could be a sharp line. The first chapter on preliminary concepts summarizes some relevant mathematical terminologythatyouarelikelytohaveseenbefore.Chapters2–6thenproceedtodiscuss mathematicalconcepts and methodsthat you may also have been familiar with at some point, possibly at high school or maybe during the early months of an undergraduate program in a numerate subject. The material in Chapters 1–7 forms Part I which is intendedto give you thefoundationfor themoretechnicalPart II. A number of chapters close with a brief section on an example use of the ideas developed in that chapter within actuarial science. While this book does not aim to xi xii Preface covertopicsinactuarialscienceinanydetail,thesesectionsareincludedwhereatasteof atopiccanbegivenwithoutalsoprovidingasignificantamountofbackgroundmaterial. Notallchaptersincludesuchasectionandthisreflectseitherthattheapplicationwould be too esoteric or that sufficient “real-world” examples have already been included in themain materialof the chapter. Asyoumaybeaware,mathematicscanbeanextremelyformalandrigoroussubject. Whilesuchrigorisessentialforthedevelopmentofnewmathematicsanditsapplication to novel areas, there are many instances where formal rigor is a hindrance and a distractionfromtherealapplicationandpurposeofusingthemathematics.Ratherthan over complicating the descriptions with excessive technical considerations, the aim of thisbookistopresentaconciseaccountoftheapplicationofmathematicalmethodsthat mayberequiredwhenstudyingforactuarialexaminationsundertheInstituteandFaculty ofActuaries(IFoA)ortheSocietyofActuaries(SoA)intheUKandtheUSA,respectively. ThebookshouldalsobeofuseforthosestudyingundertheCFAInstitute,forexample, and many other professionalbodies related to finance professionals.The book does not give any formal proofs of the concepts used, although some attempt will be made to justifymanyof theideas. After studyingthisbook thereader shouldexpect to possessa well-stocked toolbox ofmathematicalconcepts,apracticalunderstandingofwhenand how to useeach tool, and an intuitiveunderstandingof why thetools work. The scope of the material discussed in this book has been heavily influenced by the statements of prerequisite knowledge for commencing studies with the IFoA and SoA. Certainly the book should be considered as covering all prerequisite material requiredforbeginningstudieswiththeIFoAandSoA.However,Ihavegonefurtherand includedsomeadditionaltopicsthat, inmyexperience,studentsfromdiverseacademic backgrounds have found useful to refresh during their early studies of actuarial science and financial mathematicsat thepostgraduatelevel. I am grateful for the many discussions regarding the content of this book with numerous students on the various actuarial and financial mathematics programs at the University of Leicester, particularly Marco De Virgillis. I would also like to thank Dr Jacqueline Butter who provided the additional perspective of someone who has gone throughanactuarialprogramfromabackgroundinphysicsandenteredindustryonthe other side. Writing a book is a lonely task and I would like thank my two sons, Adam and Matthew,andmywife,Yvette,forgivingmethetimeandspacetoimposethisloneliness onmyself.ThisbookisdedicatedtoYvettewhoisanunfailingsupporterofeverything I do. ProfessorStephenGarrett Leicestershire,UK Spring 2015 CHAPTER 1 Mathematical Language Contents 1.1 CommonMathematicalNotation 3 1.1.1 Numbersystems 3 1.1.2 Mathematicalsymbols 6 1.2 MoreAdvancedNotation 8 1.2.1 Setnotation 8 1.2.2 Intervalnotation 12 1.2.3 Quantifiersandstatements 13 1.3 AlgebraicExpressions 14 1.3.1 Equationsandidentities 14 1.3.2 Anintroductiontomathematicsonyourcomputer 17 1.3.3 Inequalities 18 1.4 Questions 20 Prerequisiteknowledge Learningobjectives • “School”mathematics • Define, recognize, anduse • useofacalculator • numbersystems • algebraicmanipulation • mathematicalnotationincludingsetnotation • analyticalsolutionofsimple • bracketnotation polynomialexpressions • quantifiers • FamiliaritywithbasicuseofExcel • equations,identities,andinequalities Inthischapter,westateandillustratetheuseofcommonmathematicalnotationthatwill beusedwithoutfurthercommentthroughoutthisbook.Itisassumedthatmuchofthis sectionwillhavebeenfamiliartoyouatsomepointofyoureducationandisincludedas anaide-mémoire.Ofcourse,giventhatthebookwillexploremanyareasoftheapplication ofmathematics,thematerialpresentedheremaywellprovetobeincomplete.Itshould thereforebeconsideredasanillustrationofthelevelofmathematicsthatwillbeassumed as prerequisite,rather than adefinitivelist. 1.1 COMMONMATHEMATICALNOTATION 1.1.1 Numbersystems We begin by summarizing the types of numbers that exist. As this book in concerned with the practical application of mathematics, it should be unsurprising that the set of IntroductiontoActuarialandFinancialMathematicalMethods ©2015ElsevierInc. Allrightsreserved. 3 4 IntroductiontoActuarialandFinancialMathematicalMethods −∞ ∞ 5 5.2 5.6767 6 7 Figure1.1 Therealnumberline. realnumbersformsthebuildingblocksofmost(butnotquiteall,seeChapter8)ofwhat we will study. A real number is a value that represents a position along a continuous number line. For example, numbers 5 and 6 have clear positions on the number line in Figure 1.1 and soarereal numbers.Thenumber5.2also has apositiononthenumberline, afifth between 5 and 6. Going further we see that 5.6767 is also on the line. In fact, we can keepgoingand,withasharpenoughpencil,markanumberwithanynumberofdecimal places onthenumberline.Withthisintuitiveunderstanding,itshouldbeclear thatthe set of all real numbers includes numbers to any number of decimal places and that we can also freely expand the number line without limit. As illustrated in Figure 1.1, real numberscanbepositiveornegative.Thesetofallrealnumbers,denotedR,istherefore seen as the fundamental collection of numbers that we might want to work with in real-world applications. Aswecaninprincipledefinearealnumberwithaninfinitenumberofdecimalplaces, there is in some sense an “infinity of infinities” of real numbers. It should then be of no surprisethat the set R has many subsets, each with an infinitenumber of members. Such subsetsinclude • positivereal numbers,R+ • negativereal numbers,R− • integers,Z • naturalnumbers,N • rationalnumbers,Q • irrationalnumbers,J The meaning of the terms positive real numbers and negative real numbers should be clear, althoughnotethat 0 is technicallyneither.You mayhowever need toberemindedthat theintegersarethesubsetofrealnumbersthatare“whole.”Forexample,0,−10,and34 are integers,but−10.1 and 34.8 are not. The natural numbers are easily understood as the positive integers and zero.1 For example,57and−6arebothintegers,butonly57isanaturalnumber.Naturalnumbers 1Notethatthereissomedisagreementastowhetherzeroisanaturalnumber.Someauthorsclaimthatit doesnot belongtothe setofnaturalnumbers, insteadisa memberofanadditional setcalledthewhole numberswhicharethepositiveintegersandzero. MathematicalLanguage 5 areusefulforcountingandarethefirstnumbersystemweworkwithaschildren.Itwill proveuseful to defineN+ as the nonzeronaturalnumbers. In addition to the sets of whole and natural numbers, a rational number is any real numberthat can be expressedas thefractionof two integers.Itshouldbe clear that the set of integers are also rational numbers,for example, 32= 32/1 and −7 = −7/1, but so arenumberslike 45/2 and −98,736/345,298. In contrast, irrational numbers are those which cannot be represented as a fraction of twointegers.Irrationalnumber√sarenumberswhichhaveaninfinitenumberofdecimal places, for example, π, e, and 2. Irrational numbers cannot therefore be integers or naturalnumbers. The relationship between the different sets of real numbers is summarized in Figure 1.2. From this it is clear that the “sum” of the sets of rational and irrational numbers form the broader set of real numbers. The set of rational numbers can be furthersubdividedintointegersandnonintegers;thesetofintegerscontainsthenatural numbers. EXAMPLE1.1 Wherewould0appearintheVenndiagramofFigure1.2? Solution According to the definitions given here, zero is a real number, a rational number, an integer,andanaturalnumber.ItwillthensitinsideofthecircleindicatedbyN.However, otherauthorsclaimthatitisnotanaturalnumberandsositsinsideofthecircleindicated byZbutoutsideofN. R Q J Z N Figure1.2 Venndiagramoftherealnumbersystems.