Nr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996
|
|
- Barnaby Fisher
- 8 years ago
- Views:
Transcription
1 Nr 2 Iterpolatio of Discout Factors Heiz Cremers Willi Schwarz Mai 1996 Autore: Herausgeber: Prof Dr Heiz Cremers Quatitative Methode ud Spezielle Bakbetriebslehre Hochschule für Bakwirtschaft Dr Willi Schwarz Commerzbak AG, Frakfurt/M Risk Maagemet Hochschule für Bakwirtschaft Private Fachhochschule der BANKAKADEMIE Sterstraße Frakfurt/M Tel: 69 / Fax: 69 / hfb@mailpop-frakfurtcom
2 Abstract This paper deals with the problem of iterpolatio of discout factors betwee time buckets The problem occurs whe price ad iterest rate data of a market segmet are assiged to discrete time buckets A simple criterio is developed i order to idetify arbitrage-free robust iterpolatio methods Methods closely examied iclude liear, oetial ad weighted oetial iterpolatio Weighted oetial iterpolatio, a method still preferred by some baks ad also offered by commercial software vedors, creates several problems ad therefore makes simple oetial iterpolatio a more logical choice Liear iterpolatio provides a good approximatio of oetial iterpolatio for a sufficietly dese time grid 1 Itroductio Valuatio ad pricig of fiacial istrumets geerally requires kowledge of discout factors ad/or zero bod prices Fudametal to the calculatio of discout factors is detailed iformatio o iterest rates, as well as o prices of fixed icome securities i special market segmets (Bod-, FRA-, Swap-market) at preset time t The procedure for calculatig a discout structure df from this iformatio is as follows: Startig with market data we defie a discrete time structure t 1, t 2, K, t N ad calculate the implied discout factor df ( t, t ) for every time to maturity t, = 1,, N eg by usig a bootstrappig techique The calculatio of the preset value of a cash flow CF(t) occurrig at time t requires the coversio of the discrete structure df ( t, t1), df ( t, t2),, df ( t, t N ) ito a cotiuous discout curve t df ( t, t), t [ t, t N ] The complete set of empirical data is employed i order to derive the discrete discout structure, so that the secod step of the problem is reduced to a pure iterpolatio problem If the market data is icomplete the a iterpolatio problem may occur i the first step (eg this would be caused by a missig bod) I this paper we study several widely used iterpolatio methods thereby cofiig ourselves to the study of those iterpolatio problems which require the kowledge of oly two adjacet discout factors McCulloch [1] has developed splie iterpolatio techiques by usig the whole spectrum of market data Splie iterpolatio offers a higher degree of smoothess, which has its price i terms of precisio or eve arbitrage-freeess For a detailed discussio of this matter we 2
3 refer to Brecklig, Dal Dosso [2], [3] ad Shea [4] I a forthcomig paper we will ivestigate iterpolatio methods usig all available market iformatio Let us state the problem i more precise terms: Problem Let t deote the preset time, t 1, t 2, K, t N the desigated grid structure ad let df ( t, t1), df ( t, t2), K, df ( t, t N ) be the discout factors The problem is the valuatio of a give cash flow CF() t = (,) t 1, which pays a amout 1 at a time t with t t 1 < t < t, cosiderig oly the discout factors df 1 = df ( t, t 1) ad df = df ( t, t) (without ay restrictio we assume df 1 > df) Let the idex be fixed with { 1, K, N} t -1 t t 1 The problem ca be looked upo from two differet poits of view which are somehow dual to each other: Iterpolatio We calculate from discout factors df 1 ad df a iterpolated value df ( t, t ) = Ip( t, df 1, df) ad determie the preset value (PV = Preset Value) of the paymet (,) t 1 to be (1) PV (,) t 1 = 1 df ( t, t) Bucketig Two fuctios B1 = B1(, t df 1, df) ad B2 = B2(, t df 1, df) (bucketig fuctios) are to be determied i such a way that the cash flow ( t,) 1, which pays oe uit i t ca be replaced by the cash flows ( t 1, B1 ) ad ( t, B 2 ) (Buckets) The preset value of the paymet (,) t 1 the is calculated as (2) PV (,) t 1 = 1 B1 df ( t, t 1) + 1 B2 df ( t, t) Tyig together the two dual view poits, ie equatig (1) ad (2) we obtai (*) df ( t, t ) = B1 df ( t, t 1) + B2 df ( t, t) ; Therefore all bucketig methods ca be cosidered as special iterpolatio methods This formula ad coditios resultig from bucket hedgig will be the key poit i our aalysis Bucket hedgig has bee extesively studied by Turbull [5] 3
4 The paper is orgaized as follows First, we set some otatio ad state a o arbitrage coditio suited for our purpose I the secod part, commoly applied iterpolatio techiques such as liear, oetial ad weighted oetial iterpolatio are ivestigated i a qualitative maer Their impact o zero rate structures as well as o forward rate curves is discussed i coectio with some selected iterest rate scearios It ca be see that the weighted oetial iterpolatio already has remarkable drawbacks The fial sectio cotais the mai results of this paper A simple coditio described by a system of differetial equatios is imposed o equatio (*) Solutios to this system iclude the liear ad oetial iterpolatio method Iterestigly, these two solutios are related by the fact that liear iterpolatio is the first order term of the Taylor series asio of the oetial iterpolatio 2 Notatio A cotiuous fuctio df = Ip(, t df, df ), t t, t t 1 1, with boudary coditios (3) Ip( t 1, df 1, df) = df 1 ad Ip( t, df, df ) df 1 = is called iterpolatio fuctio Let df 1 > df for all { 1, K, N} A iterpolatio fuctio Ip is called arbitrage-free, if Ip is strictly decreasig i t, that meas (4) Ip( s1, df 1, df) > Ip( s2, df 1, df) for t 1 s1< s2 t Furthermore, we assume that the variables df are idepedet give the above restrictio Remark 1 No arbitrage is equivalet to the fact that all forward iterest rates r( t, s1, s2 ) with t 1 s1 < s2 t are positive t -1 s 1 s 2 t r t s s (,, ) 1 2 4
5 Proof: For the forward iterest rate r( t, s, s ) oe has 1 2 df ( t, s2) rt (, s1, s2) > < 1 Ip( s2, df1, df2) < Ip( s1, df1, df2) df ( t, s ) 1 Sice we are oly iterested i the relative distace of the time parameter t to the left boudary t -1, we will use the parameter λ istead of t where t t λ = λ() t = 1 t t 1 We deote by df λ the followig ressio df λ = Ip( λ, df 1, df ), λ [ 1, ] The the above boudary coditios ca be restated i terms of the ew parameter λ as: (5) Ip(, df 1, df) = df 1 ad Ip(, 1 df 1, df) = df 3 Examples of iterpolatio fuctios I the followig sectio we look at differet iterpolatio fuctios ad discuss their qualitative behaviour I aalyzig the zero rate curve ad the forward rate structure the followig three zero rate scearios are cosidered Maturity Sceario 1 Sceario 2 Sceario 3 1 yr 5, % 8,5 %, % 2 yrs 6,5 %, %, % 3 yrs,5 % 6, %, % 4 yrs 8,2 % 5,3 %, % 31 Liear Iterpolatio Liear iterpolatio is obtaied by assigig the relative distaces 1 λ ad λ as weights to the discout factors df 1 ad df,ie: 5
6 li (6) df λ = Ip ( λ, df 1, df ) = ( 1 λ) df 1 + λdf The boudary coditios (5) are easily verified The o arbitrage coditio follows from Ip λ li = df df 1 < Discout curve The resultig curve t dft = df λ( t ) is a cotiuous piecewise liear fuctio which is i geeral ot differetiable at ( t1, df1),( t2, df2 ),,( tn, df N) Zero rate curve If r t deotes the cotiuously compouded zero rate of the discout factor df t = df λ( t), the the iterpolated iterest rate r t is ressed as follows: r t r ( t t ) r ( t t ) df t t e t e = l( λ() ) = l(( 1 λ( )) λ( ) ) t t t t For the period [1 yr, 4 yrs] we obtai, usig a time iterval of legth =,1 yrs, ad give sceario 1 the followig zero rate curve Maturity[yrs] 4 Graph 1 Zero rate curve with ormal term structure (sceario 1) Similarly we obtai for a iverse term structure (sceario 2) a strictly decreasig zero rate curve with covex parts of the curve I case of a flat term structure (sceario 3), liear iterpolatio yields a fuctio which has slightly covex pieces 6
7 ,5 6,5 Graph 2 Zero rate curve with flat iterest rate structure (sceario 2) Forward rate curve Let df ( t, s1, s2 ) deote the forward discout factor ad r ( t, s1, s2 ) its oetial forward iterest rate for the time iterval s1, s2 t 1, t The discout factor, respectively the forward rate, ca be ressed by the followig formulas df ( t s s df s df df ( t, s, s ), 2) ( 1 λ( )) ( ) 1 2 = = λ 2 df ( t, s ) ( 1 λ( s )) df + λ( s ) df 1 l( df ( t, s, s )) r( t, s1, s2) = 1 2 s s For costat time itervals of legth = s s = yrs] the forward curve is as follows: 2 1 1, yrs ad time iterval [1 yr, Graph 3 Forward rate curve with ormal term structure (sceario 1)
8 Graph 4 Forward rate curve with iverse term structure (sceario 2) I both scearios (ormal term structure as well as iverse term structure) oe obtais icreasig forward rates withi the iterpolatio iterval; discotiuities appear at the boudary of the time itervals The discotiuities are due to the method of iterpolatio choose, which calculates discout factors as a average of adjacet discout factors I a flat term structure sceario (sceario 3), forward rates are ot oly icreasig but also show a periodic behaviour 8 6 Graph 5 Forward rate curve with flat term structure (sceario 3) 32 Expoetial Iterpolatio This form of iterpolatio is obtaied by assigig certai oets to the discout factors df 1, df : 1 λ λ () df λ = Ip ( λ, df 1, df ) = df df 1 The boudary coditios (5) are easily verified, the o arbitrage coditio (4) follows from Ip λ 1 λ λ 1 λ λ = (l df 1) df df + df (l df df df λ df df ) = (l l < )
9 Discout curve Sice 2 Ip = df 2 λ (l df l df 1) 2 > λ the oetial iterpolatio yields strogly covex pieces i the discout curve The discout curve t df df is a cotiuous fuctio, but i geeral ot t = λ( t ) differetiable at the poits t 1, t 2, K, t N Zero rate curve Let r t deote the cotiuously compouded zero rate of the discout factor df t = df λ( t) It is computed usig the liearly iterpolated value of the adjacet zero rates ad 1 λ () t λ() ( rt( t t )) = dfλ() t = df 1 df t = ( ( 1 λ)( t 1 t) r 1) ( λ( t t) r) t 1 t t t = ( 1 λ) r 1 + λ r ( t t ) t t t t r t t t t t r = ( 1 λ) λ r t t t t Give sceario 1, the zero rate curve appears as follows, oce agai by usig time itervals of =,1 yrs ad time periods [1 yr, 4 yrs]: Graph 6 Zero rate curve with ormal term structure (sceario 1) Give sceario 2, the zero rate curve decreases yieldig covex curve pieces Give a flat zero rate structure (sceario 3), the oetial iterpolatio maitais this property, which ca be derived as follows: If r 1 = r oe obtais 9
10 r t t 1 t t t t 1 t t t = ( 1 λ) r 1 + λ r = ( 1 λ) + λ t t t t t t t t ( t t)( t 1 t) + ( t t 1)( t t) = r = r for all t ( t t )( t t ) 1 r,5 6,5 Graph Zero rate curve with flat term structure (sceario 3) Forward rate curve Expoetial iterpolatio implies costat forward rates r( t, s1, s2 ) for time itervals s1, s2 of equal legth Let s 1 ad s 2 be such that t 1 s1 < s2 t The give λ1 = λ( s 1), λ2 = λ( s 2) ad a forward discout factor df ( t, s1, s2 ) it ca be rewritte as s s 1 λ df t s df 2 λ (, ) df 2 λ df 2 λ1 df df t s s t t (, 1, 2) = 2 = 1 = = df ( t, s1) 1 df 1df 1 df 2 1 λ λ λ λ df ie df ( t, s1, s2 ) ad r( t, s 1, s 2 ) as well oly deped o the distace s2 s1 For time itervals with a legth of = s2 s1 = 1, yrs ad time periods [1 yr, 4 yrs] we obtai the followig forward rate curve 2 1 1, 1
11 Graph 8 Forward rate curve with ormal term structure (sceario 1) Zero rates [%] Graph 9 Forward rate curve with iverse term structure (sceario 2),5 6,5 Graph 1 Forward rate curve with flat term structure (sceario 3) 33 Weighted Expoetial Iterpolatio This iterpolatio method is obtaied by assigig additioal time weights to the oets i (): weight 1 t 1 t (8) df = Ip (, t df, df ) = df df t α ()( λ()) α() t λ() t
12 t t 1 where: λ = λ() t = t t 1 ad αi = t t αi() t = t t weight Ip satisfies the boudary coditios (3), however the o arbitrage coditio (4) does ot hold Couterexample Let df 1 = 91, df 2 = 89, t =, t 1 = 1, t 2 = 2 ad t = 18 the df ( 1 8) = < 89 = df2 Accordig to Remark 1 i Sectio I, egative or zero forward rates caot be excluded by iterpolatio method (8) Remark 2 I order to obtai a valid ressio for the divisor t 1 t = t t = i formula (8) for the first time iterval t, t1 where = 1, we set: t1 = t + 1 day ad r 1 = overight-rate Discout curve The discout curve t df t is a cotiuous fuctio, but ot ecessarily differetiable at poits t 1, t 2, K, t N Term structure Let r t be the oetial iterest rate with discout factor df iterpolated rate r t is give by r = ( λ) r + λ r, t 1 1 i t = df λ( t), the the ie r t is obtaied by iterpolatig adjacet rates i a liear fashio The term structure as defied by the previous scearios yields the followig shape: Graph 11 Iterest rate curve with ormal term structure (sceario 1) 12
13 Similar graphs are obtaied for iverse (sceario 2) ad flat (sceario 3) term structures usig piecewise liear fuctios Forward curve For the forward discout factor df ( t, s1, s2 ) ad its associated weighted oetial forward rate r( t, s1, s2 ) for the time period s1, s2 t 1, t we have df ( t, s2) df ( t, s1, s2) = df ( t, s ) = df 1 ( s2 t)( t s2) ( s1 t)( t s1) 1 ( t 1 t)( t t 1) df ( s2 t)( s2 t 1) ( s1 t)( s1 t 1) ( t 1 t)( t t 1) l( df ( t, s, s )) r( t, s1, s2) = 1 2 s2 s1 For time itervals of equal legth = s2 s1 = 1, yrs ad time periods [1 yr, 4 yrs] we obtai the followig forward rates, give the aforemetioed scearios: Graph 12 Forward iterest rate curve with ormal term structure (sceario 1) Graph 13 Forward rate curve with iverse term structure (sceario 2) 13
14 ,5 6,5 Graph 14 Forward rate curve with flat term structure (sceario 3) 4 Results A large class of iterpolatio methods is obtaied by usig so called bucketig procedures As metioed i the itroductio,»buckets«for a cash flow ( t,) 1 where t t 1 < t < t are cofied to the time period t 1 ad t Two cotiuous fuctios B1 = B1(, t df 1, df) ad B2 = B2(, t df 1, df), t t 1, t with B 1 1 ad B 2 1 satisfyig the boudary coditios (9) t = t 1: B1( t 1, df 1, df) = 1 ad B2( t 1, df 1, df) = t = t : B1( t, df 1, df) = ad B2( t, df 1, df) = 1 are called bucketig fuctios or a bucketig procedure As metioed iitially, every bucketig procedure defies a iterpolatio method If B 1 ad B 2 are bucketig fuctios, the (1) Ip(, t df 1, df) = B1(, t df 1, df) df 1 + B2(, t df 1, df) df is the associated iterpolatio fuctio Give (9), the boudary coditios (3) are satisfied A bucketig procedure B 1, B 2 is called arbitrage-free, if the associated iterpolatio fuctio Ip is arbitrage-free, ie if Ip is strictly decreasig i t A sufficiet coditio is B1(, t df 1, df ) B2(, t df 1, df ) df 1 + df < t t provided B 1 ad B 2 are differetiable i t Boudary coditios ad the o arbitrage property of bucketig procedures have aalogue cocepts for the associated iterpolatio fuctio However, the cocept of robustess which is discussed below, seems to have o apparet similarities to iterpolatio Robustess is the essetial igrediet i derivig reasoable 14
15 iterpolatio/bucketig procedures Further, we assume that the fuctio Ip is cotiuously differetiable i the variables df 1 ad df A bucketig procedure is called robust, if B 1, B 2, ad its associated iterpolatio fuctio satisfy the followig system of partial differetial equatios Ip( t, df df (**) 1, ) = B1(, t df 1, df), df 1 Ip( t, df 1, df) = B2( t, df 1, df) for all t t 1, t df Iterpretatio The Taylor series of the associated iterpolatio fuctio satisfyig (**) is give for fixed t [ t 1, t] ad ( df, df ) by 1 Ip Ip(, t df 1, df ) = Ip(, t df 1, df ) + (, tdf 1, df )( df 1 df 1) df 1 Ip + (, tdf 1, df )( df df ) + R1 df = B1( t, df 1, df ) df 1 + B2( t, df 1, df ) df + R1 Cosequetly, small chages i discout factors df 1 ad df ( a small error term R 1 ) will result i ivariat bucketig fuctios B1( t, df 1, df ) ad B2( t, df 1, df) Therefore, a hedge based o bucketig does ot have to be adjusted for small chages i market factors The mai coclusio of the paper is Theorem: Let B 1, B 2 be as stated above, ad Ip the associated iterpolatio fuctio The li t t (a) B t df df t li t t (, 1 1, ) = 1 λ ( ) = ad B ( t, df df t 1 t t 2 1, ) = λ( ) = 1 t t 1 is a arbitrage-free solutio to the system (**) where λ deotes the relative distace of t to t 1 The associated iterpolatio fuctio is liear ad ressed by li Ip ( t, df 1, df ) = ( 1 λ( t )) df 1 + λ ( t ) df 15
16 λ() t df (b) B t df df t 1 (, 1, ) = ( 1 λ()) ad df 1 λ() t 1 df B t df df t 2 (, 1, ) = λ() df 1 is a arbitrage-free solutio to the system (**) where λ is as above The associated iterpolatio fuctio is as follows: Ip ( λ( t), df 1, df) = B ( t, df, df) df + B ( t, df, df) df λ( t ) λ( t ) = df df 1 (c) The two bucketig procedures are approximately the same which ca be see from the first term of the Taylor series asio of the oetial iterpolatio Let λ be betwee ad 1 ad ( df, df ) be fixed The 1 Ip Ip ( λ, df 1, df ) = Ip ( λ, df 1, df ) + ( df 1, df )( df 1 df 1) df 1 Ip + ( df 1, df )( df df ) + R1( df 1, df ) df λ λ = Ip ( λ, df 1, df ) + ( 1 λ)( df 1) ( df ) ( df 1 df 1) 1 λ λ 1 + λ ( df 1) ( df ) ( df df ) + R1( df 1, df ) λ 1 df df = ( 1 ) df 1 + λ λ λ df R + 1( df 1, df ) df 1 df 1 For small values of t t 1 oe has df R1( df 1, df ) ad 1, df 1 ad therefore, li Ip ( λ, df 1, df) ( 1 λ) df 1 + λ df = Ip ( λ, df 1, df) Remark 3 (1) The boudary coditios specified for our differetial equatios by o meas guaratee a uique solutio (2) The solutio i (b) ca be slightly geeralized, if λ( t ) is replaced by a strictly icreasig cotiuous fuctio with values betwee ad 1 16
17 (3) The weighted oetial iterpolatio does ot yield a robust bucketig procedure as weight Ip (, t df 1, df) df weight 1 Ip (, t df 1, df) df df = α 1( 1 λ) df df = αλ df α 1 αλ ( 1 λ) 1 produces the followig ressios for the bucketig fuctios B1 ad weight B2 ad 1()( t 1 ()) t 1 gew df B t df df t t 1 (, 1, ) = 1() ( 1 ()) α λ α λ df 1 () t () t 1 gew df B t df df t t 2 (, 1, ) = () () α λ α λ df 1 weight The weighted oetial iterpolatio method, although still ofte used, satisfies either the o arbitrage or the robustess coditio ad Refereces: [1] Mc Culloch, HJ: Measurig the Term Structure of Iterest Rates, Joural of Busiess, XLIV (Jauary 191), [2] Brecklig, J; L Dal Dasso: A No-parametric Approach to Term Structure Estimatio, i Hrsg G Bol, G Nakhaeizadeh, K-H Vollmer: Fiazmarktaweduge euroaler Netze ud ökoometrischer Verfahre, Physica Verlag Heidelberg 1994 [3] Brecklig, J; L Dal Dasso: Modellig of Term Structure Dyamics Usig Stochastic Processes, i Hrsg G Bol, G Nakhaeizadeh, K-H Vollmer: Fiazmarktaweduge euroaler Netze ud ökoometrischer Verfahre, Physica Verlag Heidelberg 1994 [4] Shea, GS: Pitfalls i Smoothig Iterest Rate Term Structure Data: Equilibrium Models ad Splie Approximatios, Joural of Fiacial ad Quatitative Aalysis, Vol 19 No 3 (September 1984), [5] Turbull, SM: Evaluatig ad Implemetig Bucket Hedgig 1
18 Arbeitsberichte der Hochschule für Bakwirtschaft Nr Autor/Titel Jahr 1 Moorma, Jürge 1995 Lea Reportig ud Führugsiformatiossysteme bei deutsche Fiazdiestleister 2 Cremers, Heiz; Schwarz, Willi 1996 Iterpolatio of Discout Factors Bestelladresse: Hochschule für Bakwirtschaft z H Frau Glatzer Sterstraße Frakfurt/M Tel: 69/ Fax: 69/
Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationBond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond
What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixed-icome security that typically pays periodic coupo paymets, ad a pricipal
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationTerminology for Bonds and Loans
³ ² ± Termiology for Bods ad Loas Pricipal give to borrower whe loa is made Simple loa: pricipal plus iterest repaid at oe date Fixed-paymet loa: series of (ofte equal) repaymets Bod is issued at some
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationA Guide to the Pricing Conventions of SFE Interest Rate Products
A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios
More informationNEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,
NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationPresent Values, Investment Returns and Discount Rates
Preset Values, Ivestmet Returs ad Discout Rates Dimitry Midli, ASA, MAAA, PhD Presidet CDI Advisors LLC dmidli@cdiadvisors.com May 2, 203 Copyright 20, CDI Advisors LLC The cocept of preset value lies
More informationSwaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps
Swaps: Costat maturity swaps (CMS) ad costat maturity reasury (CM) swaps A Costat Maturity Swap (CMS) swap is a swap where oe of the legs pays (respectively receives) a swap rate of a fixed maturity, while
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationLearning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014
1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the time-value
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationThis chapter considers the effect of managerial compensation on the desired
Chapter 4 THE EFFECT OF MANAGERIAL COMPENSATION ON OPTIMAL PRODUCTION AND HEDGING WITH FORWARDS AND PUTS 4.1 INTRODUCTION This chapter cosiders the effect of maagerial compesatio o the desired productio
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationTHE TIME VALUE OF MONEY
QRMC04 9/17/01 4:43 PM Page 51 CHAPTER FOUR THE TIME VALUE OF MONEY 4.1 INTRODUCTION AND FUTURE VALUE The perspective ad the orgaizatio of this chapter differs from that of chapters 2 ad 3 i that topics
More informationTime Value of Money. First some technical stuff. HP10B II users
Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationSubject CT5 Contingencies Core Technical Syllabus
Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value
More informationZ-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about
More informationFM4 CREDIT AND BORROWING
FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationTO: Users of the ACTEX Review Seminar on DVD for SOA Exam FM/CAS Exam 2
TO: Users of the ACTEX Review Semiar o DVD for SOA Exam FM/CAS Exam FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Exam FM (CAS
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationForecasting. Forecasting Application. Practical Forecasting. Chapter 7 OVERVIEW KEY CONCEPTS. Chapter 7. Chapter 7
Forecastig Chapter 7 Chapter 7 OVERVIEW Forecastig Applicatios Qualitative Aalysis Tred Aalysis ad Projectio Busiess Cycle Expoetial Smoothig Ecoometric Forecastig Judgig Forecast Reliability Choosig the
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationA Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 49-60 A Faster Clause-Shorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius
More informationCHAPTER 4: NET PRESENT VALUE
EMBA 807 Corporate Fiace Dr. Rodey Boehe CHAPTER 4: NET PRESENT VALUE (Assiged probles are, 2, 7, 8,, 6, 23, 25, 28, 29, 3, 33, 36, 4, 42, 46, 50, ad 52) The title of this chapter ay be Net Preset Value,
More informationI. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
More informationBENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationUnderstanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions
Udestadig Fiacial Maagemet: A Pactical Guide Guidelie Aswes to the Cocept Check Questios Chapte 4 The Time Value of Moey Cocept Check 4.. What is the meaig of the tems isk-etu tadeoff ad time value of
More informationChapter 6: Variance, the law of large numbers and the Monte-Carlo method
Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationResearch Article Sign Data Derivative Recovery
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationStatement of cash flows
6 Statemet of cash flows this chapter covers... I this chapter we study the statemet of cash flows, which liks profit from the statemet of profit or loss ad other comprehesive icome with chages i assets
More informationOption pricing. Elke Korn Ralf Korn 1. This publication is part of the book Mathematics and Economy, which is supported by the BertelsmannStiftung.
MaMaEuSch Maagemet Mathematics for Europea Schools http://www.mathematik.uikl.de/~mamaeusch/ Optio pricig Elke Kor Ralf Kor This publicatio is part of the book Mathematics ad Ecoomy, which is supported
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationTO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC
TO: Users of the ACTEX Review Semiar o DVD for SOA Eam MLC FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Eam M, Life Cotigecies
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationNon-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationQuestion 2: How is a loan amortized?
Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationhp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More informationChapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationThe Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract
The Gompertz Makeham couplig as a Dyamic Life Table By Abraham Zaks Techio I.I.T. Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 32000, Haifa, Israel Abstract A very famous
More informationI. Why is there a time value to money (TVM)?
Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationCHAPTER 11 Financial mathematics
CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationCapacity of Wireless Networks with Heterogeneous Traffic
Capacity of Wireless Networks with Heterogeeous Traffic Migyue Ji, Zheg Wag, Hamid R. Sadjadpour, J.J. Garcia-Lua-Aceves Departmet of Electrical Egieerig ad Computer Egieerig Uiversity of Califoria, Sata
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationOn the Capacity of Hybrid Wireless Networks
O the Capacity of Hybrid ireless Networks Beyua Liu,ZheLiu +,DoTowsley Departmet of Computer Sciece Uiversity of Massachusetts Amherst, MA 0002 + IBM T.J. atso Research Ceter P.O. Box 704 Yorktow Heights,
More informationODBC. Getting Started With Sage Timberline Office ODBC
ODBC Gettig Started With Sage Timberlie Office ODBC NOTICE This documet ad the Sage Timberlie Office software may be used oly i accordace with the accompayig Sage Timberlie Office Ed User Licese Agreemet.
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationBaan Service Master Data Management
Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :
More informationPre-Suit Collection Strategies
Pre-Suit Collectio Strategies Writte by Charles PT Phoeix How to Decide Whether to Pursue Collectio Calculatig the Value of Collectio As with ay busiess litigatio, all factors associated with the process
More informationHCL Dynamic Spiking Protocol
ELI LILLY AND COMPANY TIPPECANOE LABORATORIES LAFAYETTE, IN Revisio 2.0 TABLE OF CONTENTS REVISION HISTORY... 2. REVISION.0... 2.2 REVISION 2.0... 2 2 OVERVIEW... 3 3 DEFINITIONS... 5 4 EQUIPMENT... 7
More informationChapter XIV: Fundamentals of Probability and Statistics *
Objectives Chapter XIV: Fudametals o Probability ad Statistics * Preset udametal cocepts o probability ad statistics Review measures o cetral tedecy ad dispersio Aalyze methods ad applicatios o descriptive
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationTradigms of Astundithi and Toyota
Tradig the radomess - Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationProbabilistic Engineering Mechanics. Do Rosenblatt and Nataf isoprobabilistic transformations really differ?
Probabilistic Egieerig Mechaics 4 (009) 577 584 Cotets lists available at ScieceDirect Probabilistic Egieerig Mechaics joural homepage: wwwelseviercom/locate/probegmech Do Roseblatt ad Nataf isoprobabilistic
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationAutomatic Tuning for FOREX Trading System Using Fuzzy Time Series
utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which
More informationBond Mathematics & Valuation
Bod Mathematics & Valuatio Below is some legalese o the use of this documet. If you d like to avoid a headache, it basically asks you to use some commo sese. We have put some effort ito this, ad we wat
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success
More information