Continuous Compounding and Annualization


 Gilbert Reeves
 3 years ago
 Views:
Transcription
1 Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem 6 1 Intoduction In ou discussion of highway maintenance we will discuss optimal maintenance policies, in which we choose the best inteval to esuface the oads. This involves maintenance costs spead out ove time and we know, in pinciple, how to handle them: we convet eveything to pesent value. Howeve, if we want to attack the optimality poblem using calculus, then the discetetime methods developed in (eg) CRP 763 won't quite do. The eason is that if we conside, say, a 30yea policy and then think of extending it one yea, we get a jump discontinuity in the esult, and this makes calculus effectively unusable. The solution is to conside time as vaying continuously, and this note discusses the analytics of continuoustime discounting. A elated poblem is that we 1
2 want to conside a single epesentative yea of the maintenance policy, and this equies that we annualize the elevant pesent value that is, convet it to an equivalent onceayea quantity. A nal poblem is that the paticula sequence of maintenance costs we shall be concened with is special: it consists of costs which ae incued peiodically, once evey somany yeas (that is, the yeas in which we actually pefom the maintenance). The nal section consides this special poblem. 2 Continuous Compounding Suppose you deposit P.0/ today (at time 0) in a bank. At the end of 1 yea the bank pays you inteest, calculating it at an annual ate of. Then at the end of one yea, you P.0/ has become P.0/.1 C / Now suppose that the bank pays you inteest twice a yea, a sixmonth intevals. If is the annual ate, then it will compute each payment based on a ate of =2; and thee will be two payments. So at the end of yea 1 P.0/ will become h S.1; 2/ D P.0/ 1 C i 1 C 2 2 D P.0/ 1 C 2 2 whee in S.1; 2/ the st agument (1) indicates that this is at the end of 1 yea, and the second (2) indicates that compounding is done twice a yea. What about thee times a yea? The effective inteest ate is =3; and thee ae thee compounding peiods, so at the end of the yea P.0/ becomes S.1; 3/ D P.0/ 1 C 1 C 1 C D P.0/ 1 C 3 : 3 We can see whee this is going. If payments ae compounded k times a yea, then at the end of 1 yea P.0/ becomes S.1; k/ D P.0/ 1 C k : k 2
3 If you leave you money on deposit fo t yeas with compounding k times a yea thee will be kt sepaate compoundings, so at the end of t yeas you will have Note that S.t; k/ is a peiodt quantity. S.t; k/ D P.0/ 1 C kt k Now suppose that we allow the numbe of annual compoundings k to become lage and lage. In the limit we aive at continuous compounding in effect, the bank is making you inteest payments all the time. Of couse, the effective ate pe payment the analog of =k gets smalle and smalle; but the numbe of payments gets lage and lage. What will P.0/ become the end of t yeas? To study this we need to look at what happens as the numbe of annual compoundings k tends to innity, that is: S.t; 1/ D lim 1 P.0/ C k!1 k D P.0/ lim 1 C k!1 k If we wite x D =k then we can wite the limit expession as lim.1 C x/1=x t k!1 (since.1 C x/ 1=x t D.1 C x/ t=x D.1 C x/ t=.=k/ D.1 C x/ kt /: Since x D =k then k! 1 means that x! 0: Note that the oute exponent doesn't depend on k, so we can wite this as t lim.1 C x/1=x x!0 Fom the Binomial Theoem the quantity (1 C x/ 1=x tends to e.d 2:718 3 : : : / as x tends to zeo, and we have lim x!0.1 C x/1=x t D e t : In othe wods, with continuous compounding, P.0/ will gow in t yeas to S.t; 1/ D P.0/e t kt kt 3
4 3 Pesent Value with Continuous Compounding Suppose that someone offes you S.t/ to be eceived in yea t. To nd the pesent value of this we need to nd a quantity P.0/ such that you will be indiffeent between eceiving P.0/ now and S.t/ in yea t. If you maket oppotunities ae given by a banking system which compounds continuously at annual ate ; then at the end of t yeas you P.0/ will compound to P.0/e t In ode fo you to be indiffeent between S.t/ in peiod t note that since we ae now assuming continuous compounding, this is the same as what we wote as S.t; 1/ in the last section and eceiving P.0/ now and leaving it on deposit until peiod t we must have S.t/ D P.0/e t o, solving fo the yea0 quantity, the pesent value: P.0/ D S.t/e t : In othe wods, with continuous compounding, the pesent value of S.t/ eceived in yea t is S.t/e t Fo efeence, hee is a table compaing the pesent value of $1 eceived at vaious times t and at vaious inteest (discount) ates, using continuous and discete time compounding; t e t 1.1C/ t :03 10 :7408 : :5488 : :4066 :4120 :05 10 :6065 : :3679 : :2231 :2314 :08 10 :4492 : :2019 : :0967 :0994 As we can see, the esults ae geneally quite close. 4
5 4 Annualization Conside a poject which has a single upfont (time0) cost and a benets extending though time. One way to evaluate the poject is to convet the steam of benets to its pesent value; then we can diectly compae the two. But we sometimes want to think about what happens at each yea of the poject, and in this case we need to compae the annual benet to some potion of the cost. To handle this, it is logical to think about evesing the idea of pesent value: that is, to constuct a steam of costs which is equivalent to the oiginal (time0) cost. Since thee ae an innite numbe of ways to do this, we shall also equie that each of the costs in the constucted steam be the same: in othe wods we constuct an annuity which is equivalent to the oiginal time0 cost. This pocess is known as annualization: it convets a single quantity to an equivalent annuity. Suppose you invest P.O/ now in some (public) poject that is pojected to last T yeas. We seek an (annual) annuity amount A that is equivalent in pesent value to P.0/ now. With continuous compounding, we will need to add up the pesent value at each possible time between 0 and T: With continuous time, this means that the pesent value of ou T peiod annuity is an integal: A Z tdt td0 e t dt D A e T 1 D A 1 e T To nd the annuity amount A which is equivalent to a poject expenditue of P.0/ now, we must solve P.0/ D A 1 e T fo A; and the solution is A D P.0/ 1 e T D P.0/ 1 e T In othe wods, incuing a poject cost of P.0/ now is equivalent to incuing a cost of P.0/ 1 e T 5
6 in each of yeas 0 to T: This is the annualized cost ove a T yea poject life. If the poject lasts foeve, we need to see what happens as T! 1: By inspection, the tem e T tends to zeo and the annualized cost ove an innite poject lifetime is theefoe: P.0/: 5 A Special Poblem In ou discussion of oad maintenance we will conside a special situation: a maintenance policy incus a cost of C evey T yeas stating in yea T, and this patten continues indenitely. We want to nd the annualized cost of the policy. We do this in two steps: st, we nd the pesent value of the steam, and then we annualize that pesent value. Fist, what is the pesent value of the steam? The st cost comes in yea T I its pesent value is Ce T : The second comes in yea 2T I its pesent value is Ce 2T : So we'e looking at a seies of pesent value tems like Ce T C Ce 2T C Ce 3T C : : : D C.e T C e 2T C e 3T C : : : / whee the seies extends foeve. We handle this as follows: we st compute the value of the seies when it extends out fo J tems, and then we take the limit as J tends to innity. Ignoing the constant C fo the moment, ou Jtem seies is U D e T C e 2T C e 3T C C e JT We now use a tick vey much like the one we use when summing a geometic seies, except that this time we multiply each tem by e T : The esult is Subtacting, we obtain Ue T D e 2T C e 3T C C e JT C e.jc1/t : U Ue T D U.1 e T / D e T e.jc1/t (since eveything in between cancels out), o U D e T e.jc1/t 1 e T : 6
7 Now, what happens as J tends to innity? The second tem in the numeato vanishes (tends to zeo), and we have U D e T 1 e T : We can make this look a bit neate by multiplying top and bottom by e T esult is 1 U D e T 1 : the and ou conclusion is that the pesent value of in innite steam of costs C incued evey T yeas is: C e T 1 Finally we annualize this pesent value. Since the steam of costs continues indenitely, the annualization is the one shown at the end of the last section, and we see that the annualized pesent value is C e T 1 : 7
AMB111F Financial Maths Notes
AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed
More informationValuation of Floating Rate Bonds 1
Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationCONCEPT OF TIME AND VALUE OFMONEY. Simple and Compound interest
CONCEPT OF TIME AND VALUE OFMONEY Simple and Compound inteest What is the futue value of shs 10,000 invested today to ean an inteest of 12% pe annum inteest payable fo 10 yeas and is compounded; a. Annually
More informationFI3300 Corporate Finance
Leaning Objectives FI00 Copoate Finance Sping Semeste 2010 D. Isabel Tkatch Assistant Pofesso of Finance Calculate the PV and FV in multipeiod multicf timevalueofmoney poblems: Geneal case Pepetuity
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationBasic Financial Mathematics
Financial Engineeing and Computations Basic Financial Mathematics Dai, TianShy Outline Time Value of Money Annuities Amotization Yields Bonds Time Value of Money PV + n = FV (1 + FV: futue value = PV
More informationQuestions for Review. By buying bonds This period you save s, next period you get s(1+r)
MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the twopeiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume
More informationThe LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.
Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationGESTÃO FINANCEIRA II PROBLEM SET 1  SOLUTIONS
GESTÃO FINANCEIRA II PROBLEM SET 1  SOLUTIONS (FROM BERK AND DEMARZO S CORPORATE FINANCE ) LICENCIATURA UNDERGRADUATE COURSE 1 ST SEMESTER 20102011 Chapte 1 The Copoation 113. What is the diffeence
More informationIlona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
More informationThe Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationYIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE
YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE Septembe 1999 Quoted Rate Teasuy Bills [Called Banke's Discount Rate] d = [ P 1  P 1 P 0 ] * 360 [ N ] d = Bankes discount yield P 1 = face
More informationNontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More informationWeek 34: Permutations and Combinations
Week 34: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More information9.5 Amortization. Objectives
9.5 Aotization Objectives 1. Calculate the payent to pay off an aotized loan. 2. Constuct an aotization schedule. 3. Find the pesent value of an annuity. 4. Calculate the unpaid balance on a loan. Congatulations!
More informationThe Time Value of Money
he ime Value of Money Inteest Rates and Futue Value Inteest ates ae a facto in the valuation of vitually all financial instuments. While all money maket ates () ae quoted on an annual basis (PR nnual Pecentage
More informationFirstmark Credit Union Commercial Loan Department
Fistmak Cedit Union Commecial Loan Depatment Thank you fo consideing Fistmak Cedit Union as a tusted souce to meet the needs of you business. Fistmak Cedit Union offes a wide aay of business loans and
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationOn Some Functions Involving the lcm and gcd of Integer Tuples
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 91100. On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact:
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194114 Online ISSN: 213953 Pint ISSN: 2138423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationLoyalty Rewards and Gift Card Programs: Basic Actuarial Estimation Techniques
Loyalty Rewads and Gift Cad Pogams: Basic Actuaial Estimation Techniques Tim A. Gault, ACAS, MAAA, Len Llaguno, FCAS, MAAA and Matin Ménad, FCAS, MAAA Abstact In this pape we establish an actuaial famewok
More informationON THE (Q, R) POLICY IN PRODUCTIONINVENTORY SYSTEMS
ON THE R POLICY IN PRODUCTIONINVENTORY SYSTEMS Saifallah Benjaafa and JoonSeok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poductioninventoy
More informationCHAPTER 10 Aggregate Demand I
CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationRisk Sensitive Portfolio Management With CoxIngersollRoss Interest Rates: the HJB Equation
Risk Sensitive Potfolio Management With CoxIngesollRoss Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance the availability
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance
More informationLife Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109
Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationReal Estate Equity Derivatives
Real Estate Equity Deivatives Geltne Mille 2 nd Edition Chapte 26 Section 26.3 Real Estate Deivatives (Index Retun Swaps) Real Estate Equity Deivatives A deivative is an asset whose value depends completely
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More informationExam #1 Review Answers
xam #1 Review Answes 1. Given the following pobability distibution, calculate the expected etun, vaiance and standad deviation fo Secuity J. State Pob (R) 1 0.2 10% 2 0.6 15 3 0.2 20 xpected etun = 0.2*10%
More informationSymmetric polynomials and partitions Eugene Mukhin
Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation
More information30 H. N. CHIU 1. INTRODUCTION. Recherche opérationnelle/operations Research
RAIRO Rech. Opé. (vol. 33, n 1, 1999, pp. 2945) A GOOD APPROXIMATION OF THE INVENTORY LEVEL IN A(Q ) PERISHABLE INVENTORY SYSTEM (*) by Huan Neng CHIU ( 1 ) Communicated by Shunji OSAKI Abstact. This
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationDefinitions and terminology
I love the Case & Fai textbook but it is out of date with how monetay policy woks today. Please use this handout to supplement the chapte on monetay policy. The textbook assumes that the Fedeal Reseve
More informationReduced Pattern Training Based on Task Decomposition Using Pattern Distributor
> PNN05P762 < Reduced Patten Taining Based on Task Decomposition Using Patten Distibuto ShengUei Guan, Chunyu Bao, and TseNgee Neo Abstact Task Decomposition with Patten Distibuto (PD) is a new task
More informationCONCEPTUAL FRAMEWORK FOR DEVELOPING AND VERIFICATION OF ATTRIBUTION MODELS. ARITHMETIC ATTRIBUTION MODELS
CONCEPUAL FAMEOK FO DEVELOPING AND VEIFICAION OF AIBUION MODELS. AIHMEIC AIBUION MODELS Yui K. Shestopaloff, is Diecto of eseach & Deelopment at SegmentSoft Inc. He is a Docto of Sciences and has a Ph.D.
More informationThe Supply of Loanable Funds: A Comment on the Misconception and Its Implications
JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently FieldsHat
More informationTheory and practise of the gindex
Theoy and pactise of the gindex by L. Egghe (*), Univesiteit Hasselt (UHasselt), Campus Diepenbeek, Agoalaan, B3590 Diepenbeek, Belgium Univesiteit Antwepen (UA), Campus Die Eiken, Univesiteitsplein,
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationSeshadri constants and surfaces of minimal degree
Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationFinancial Derivatives for Computer Network Capacity Markets with QualityofService Guarantees
Financial Deivatives fo Compute Netwok Capacity Makets with QualityofSevice Guaantees Pette Pettesson pp@kth.se Febuay 2003 SICS Technical Repot T2003:03 Keywods Netwoking and Intenet Achitectue. Abstact
More informationSELFINDUCTANCE AND INDUCTORS
MISN0144 SELFINDUCTANCE AND INDUCTORS SELFINDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. SelfInductance L.........................................
More informationHour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and
Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationMULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION
MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe and subsolution method with
More informationChapte 3 Is Gavitation A Results Of Asymmetic Coulomb Chage Inteactions? Jounal of Undegaduate Reseach èjurè Univesity of Utah è1992è, Vol. 3, No. 1, pp. 56í61. Jeæey F. Gold Depatment of Physics, Depatment
More informationPAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII  SPETO  1995. pod patronatem. Summary
PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8  TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC
More informationInaugural  Dissertation
Inaugual  Dissetation zu Elangung de Doktowüde de NatuwissenschaftlichMathematischen Gesamtfakultät de Rupecht  Kals  Univesität Heidelbeg vogelegt von DiplomMathematike Makus Fische aus Belin Datum
More informationHow Much Should a Firm Borrow. Effect of tax shields. Capital Structure Theory. Capital Structure & Corporate Taxes
How Much Should a Fim Boow Chapte 19 Capital Stuctue & Copoate Taxes Financial Risk  Risk to shaeholdes esulting fom the use of debt. Financial Leveage  Incease in the vaiability of shaeholde etuns that
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationIgnorance is not bliss when it comes to knowing credit score
NET GAIN Scoing points fo you financial futue AS SEEN IN USA TODAY SEPTEMBER 28, 2004 Ignoance is not bliss when it comes to knowing cedit scoe By Sanda Block USA TODAY Fom Alabama comes eassuing news
More informationPersonal Saving Rate (S Households /Y) SAVING AND INVESTMENT. Federal Surplus or Deficit () Total Private Saving Rate (S Private /Y) 12/18/2009
1 Pesonal Saving Rate (S Households /Y) 2 SAVING AND INVESTMENT 16.0 14.0 12.0 10.0 80 8.0 6.0 4.0 2.0 0.02.04.0 1959 1961 1967 1969 1975 1977 1983 1985 1991 1993 1999 2001 2007 2009 Pivate Saving Rate
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationLesson 7 Gauss s Law and Electric Fields
Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual
More informationApplication for Admission GENEVA COLLEGE
Application fo Admission GENEVA COLLEGE 3 2 0 0 C o l l e g e A v e n u e, B e a v e F a l l s, P A 1 5 0 1 0 Application Instuctions When to apply You may apply fo admission any time afte you junio yea
More informationChannel selection in ecommerce age: A strategic analysis of coop advertising models
Jounal of Industial Engineeing and Management JIEM, 013 6(1):89103 Online ISSN: 0130953 Pint ISSN: 013843 http://dx.doi.og/10.396/jiem.664 Channel selection in ecommece age: A stategic analysis of
More information9:6.4 Sample Questions/Requests for Managing Underwriter Candidates
9:6.4 INITIAL PUBLIC OFFERINGS 9:6.4 Sample Questions/Requests fo Managing Undewite Candidates Recent IPO Expeience Please povide a list of all completed o withdawn IPOs in which you fim has paticipated
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationSaving and Investing for Early Retirement: A Theoretical Analysis
Saving and Investing fo Ealy Retiement: A Theoetical Analysis Emmanuel Fahi MIT Stavos Panageas Whaton Fist Vesion: Mach, 23 This Vesion: Januay, 25 E. Fahi: MIT Depatment of Economics, 5 Memoial Dive,
More informationSolutions to Problems: Chapter 7
Solution to Poblem: Chapte 7 P71. P72. P73. P74. Authoized and available hae LG 2; Baic a. Maximum hae available fo ale Authoized hae 2,000,000 Le: Shae outtanding 1,400,000 Available hae 600,000 b.
More information9.4 Annuities. Objectives. 1. Calculate the future value of an ordinary annuity. 2. Perform calculations regarding sinking funds.
9.4 Annuities Objectives 1. Calculate the futue value of an odinay annuity. 2. Pefo calculations egading sinking funds. Soewhee ove the ainbow... skies ae blue,... and the deas that you dae to dea...eally
More informationA Capacitated Commodity Trading Model with Market Power
A Capacitated Commodity Tading Model with Maket Powe Victo MatínezdeAlbéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu
More informationA framework for the selection of enterprise resource planning (ERP) system based on fuzzy decision making methods
A famewok fo the selection of entepise esouce planning (ERP) system based on fuzzy decision making methods Omid Golshan Tafti M.s student in Industial Management, Univesity of Yazd Omidgolshan87@yahoo.com
More informationExperimentation under Uninsurable Idiosyncratic Risk: An Application to Entrepreneurial Survival
Expeimentation unde Uninsuable Idiosyncatic Risk: An Application to Entepeneuial Suvival Jianjun Miao and Neng Wang May 28, 2007 Abstact We popose an analytically tactable continuoustime model of expeimentation
More information2  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1
 ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation
More informationDefinitions. Optimization of online direct marketing efforts. Test 1: Two Email campaigns. Raw Results. Xavier Drèze André Bonfrer. Lucid.
Definitions Optimization of online diect maketing effots Xavie Dèze Andé Bonfe Lucid Easily undestood; intelligible. Mentally sound; sane o ational. Tanslucent o tanspaent. Limpid Chaacteized by tanspaent
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationFinance Practice Problems
Iteest Fiace Pactice Poblems Iteest is the cost of boowig moey. A iteest ate is the cost stated as a pecet of the amout boowed pe peiod of time, usually oe yea. The pevailig maket ate is composed of: 1.
More informationGravitational Mechanics of the MarsPhobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning
Gavitational Mechanics of the MasPhobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationCHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL
CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The
More informationThe impact of migration on the provision. of UK public services (SRG.10.039.4) Final Report. December 2011
The impact of migation on the povision of UK public sevices (SRG.10.039.4) Final Repot Decembe 2011 The obustness The obustness of the analysis of the is analysis the esponsibility is the esponsibility
More informationHow do investments in heat pumps affect household energy consumption?
Discussion Papes Statistics Noway Reseach depatment No. 737 Apil 203 Bente Halvosen and Bodil Meethe Lasen How do investments in heat pumps affect household enegy consumption? Discussion Papes No. 737,
More informationCHAT PreCalculus Section 10.7. Polar Coordinates
CHAT PeCalculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to
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 isketu tadeoff ad time value of
More informationProblem Set # 9 Solutions
Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new highspeed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 14 Intoduction: Pehaps one of the most unusual
More informationAnalytical Proof of Newton's Force Laws
Analytical Poof of Newton s Foce Laws Page 1 1 Intouction Analytical Poof of Newton's Foce Laws Many stuents intuitively assume that Newton's inetial an gavitational foce laws, F = ma an Mm F = G, ae tue
More informationAn Analysis of Manufacturer Benefits under Vendor Managed Systems
An Analysis of Manufactue Benefits unde Vendo Managed Systems Seçil Savaşaneil Depatment of Industial Engineeing, Middle East Technical Univesity, 06531, Ankaa, TURKEY secil@ie.metu.edu.t Nesim Ekip 1
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More informationTrading Volume and Serial Correlation in Stock Returns in Pakistan. Abstract
Tading Volume and Seial Coelation in Stock Retuns in Pakistan Khalid Mustafa Assistant Pofesso Depatment of Economics, Univesity of Kaachi email: khalidku@yahoo.com and Mohammed Nishat Pofesso and Chaiman,
More informationData Center Demand Response: Avoiding the Coincident Peak via Workload Shifting and Local Generation
(213) 1 28 Data Cente Demand Response: Avoiding the Coincident Peak via Wokload Shifting and Local Geneation Zhenhua Liu 1, Adam Wieman 1, Yuan Chen 2, Benjamin Razon 1, Niangjun Chen 1 1 Califonia Institute
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More informationSecure SmartcardBased Fingerprint Authentication
Secue SmatcadBased Fingepint Authentication [full vesion] T. Chales Clancy Compute Science Univesity of Mayland, College Pak tcc@umd.edu Nega Kiyavash, Dennis J. Lin Electical and Compute Engineeing Univesity
More informationHEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING
U.P.B. Sci. Bull., Seies C, Vol. 77, Iss. 2, 2015 ISSN 22863540 HEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING Roxana MARCU 1, Dan POPESCU 2, Iulian DANILĂ 3 A high numbe of infomation systems ae available
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More information