ad covexity Defie Macaulay duratio D Mod = r 1 = ( CF i i k (1 + r k) i ) (1.) (1 + r k) C = ( r ) = 1 ( CF i i(i + 1) (1 + r k) i+ k ) ( ( i k ) CF i
|
|
- Miranda Simpson
- 6 years ago
- Views:
Transcription
1 Fixed Icome Basics Cotets Duratio ad Covexity Bod Duratios ar Rate, Spot Rate, ad Forward Rate Flat Forward Iterpolatio Forward rice/yield, Carry, Roll-Dow Example Duratio ad Covexity For a series of cash flows {CF i }, the et preset value (NV) is a fuctio of aualized iterest rate r. Assume k-th periodic compoudig, the price or NV is give by Apply Taylor expasio (r) = V i (r) = CF i (1 + r k) i (r) = (r 0 ) + r r + 1 r ( r) By defiig = (r) (r 0 ), it ca be re-writte as = ( r 1 ) r + ( r ) ( r) letiaquat.wordpress. = D Mod r + 1 C( r) (1.1) where equatio (1.1) is referred to as Secod order approximatio. The modified duratio is defied as
2 ad covexity Defie Macaulay duratio D Mod = r 1 = ( CF i i k (1 + r k) i ) (1.) (1 + r k) C = ( r ) = 1 ( CF i i(i + 1) (1 + r k) i+ k ) ( ( i k ) CF i (1 + r k) D Mac = i ) V i = ( i V i k ) Notice that ( i k ) is the maturity of ith cash flow CF i. Therefore Macaulay duratio is a weighted average of maturities. From the defiitio of Macaulay duratio (1.), it is easy to see its relatioship with modified duratio, Dollar Duratio ad DV01 (V01) are defied as D Mac D Mod = 1 + r k D $ = r = D Mod DV01 = r ( r = 1bps) = D Mod ( r = 1bps) letiaquat.wordpress. This gives the liear (first order) approximatio of the price movemet due to 1bps jump of iterest rate. So far the yield curve is implicitly assumed to be flat. I reality, it is rarely the case. The parallel shift duratio (Fisher-Weil duratio) measures the effect of a small parallel shift o the spot yield curve. Cosider the market zero curve with term structure {r i }, where r i is the aualized zero rate of correspodig teor i. The I the formula, s is the parallel shift / spread. (s) = V i (s) = CF i (1 + r i + s k )i
3 D FS = 1 ( s ) s=0 = 1 CF i (1 + r i + s k )i i k (1 + r i + s k ) s=0 = 1 ( CF i i k (1 + r i k) i ) (1 + r i k) C FS = 1 ( s ) = 1 ( CF i i(i + 1) (1 + r s=0 i k) i+ k ) which equals the modified duratio oly if the zero curve is flat. I practice, they are differet but fairly close. I additio to the parallel shift, key rates duratio is itroduced to tackle o-parallel shifts. T.Ho (199) suggests usig key rates that correspod to the teors of o-the-ru treasuries (3M, 1Y, Y, 3Y, 5Y, 7Y, 10Y, 15Y, 0Y, 5Y, ad 30Y). Oe shifts oe key rate at a time while keeps the others uchaged. Key rate duratio ca be defied either o the spot rate curve or o the par rate curve (see below). Key rate duratio is computed as + Key Rate Duratio = 1bps 0 where ad + are bod price after a ±1bps chages o that duratio, respectively. Bod Duratios Let s cosider specific cash flows cotaied i a bod. Zero-Coupo Bod A zero-coupo bod, such as a T-bill, has oly oe paymet o the maturity date. The paymet equals its face value F. Cosider a T-bill of maturity T ad semi-aual compoudig, (=, k = ). The D Mac = V = ( V k = ) = T D Mod = T 1 + r C = T(T + 1 ) (1 + r ) Therefore the Macaulay duratio of a zero-coupo bod is its time to maturity. Fixed Coupo Bod letiaquat.wordpress.
4 A fixed coupo bod, such as a T-Note/T-Bod, has semi-aual coupo paymets ad the redemptio of the face value at the maturity. Deote coupo rate by c, the the semiaual coupo paymets equal F c. I the case of fixed coupo bod, the iterest rate r is also referred to as yield to maturity (YTM), the sigle flat rate that equates the NV with the quoted market dirty price. Floatig Rate Bod = F c F (1 + r + )i (1 + r ) D Mac = ( 1 F c (1 + r )i) (i ) + ( 1 F (1 + r D Mod = D Mac 1 + r )) (T) C = 1 ( F c i(i + 1) F T( + 1) (1 + r ) i+ ) + 4 (1 + r ) + Cosider a quarterly settled, paid i arrear floatig rate ote (FRN). After the floatig rate beig fixed at LIBOR rate, the holder of the FRN is expected to receive periodic iterest paymet o the ext paymet date whe the bod will is priced at par agai. Therefore its duratio equals to a zero-coupo bod matures o the ext paymet date. ar Rate, Spot Rate, ad Forward Rate letiaquat.wordpress. ar coupo rate of a fixed coupo bod is the coupo rate this bod should pay if it is priced at par. That is, the coupo rate c satisfies F = F c (1 + r + i )i F (1 + r ) The coupo rates of o-the-ru bods are close to the par coupo rates sice they are always issued with price ear par.
5 If the bod is priced at par, the par coupo rate should be equal to the yield to maturity. This rate is referred to as par rate. ar rate ca be calculated directly by solvig the equatio, or it ca be calculated from BS (basis-poit sesitivity) or VB (preset value of 1 bp). VB is defied as VB = c (1bps) = F (1 + r (1bps) = BS (1bps) )i the curret coupo rate should icrease/decrease ( F ) basis poits i order to reach the par coupo VB rate. If a bod is priced at par, par rate equals its yield-to-maturity. Therefore VB approximates V01 from 1bp (parallel) shift of YTM. Spot rate is the rate oe gets by holdig a zero coupo bod ow. Forward rate is the rate oe is expected to get by holdig a zero coupo bod i the future. The yield curves are referred to as spot (rate) curve/zero curve, (-moth) forward (rate) curve, ad par rate curve/par yield curve, respectively. There is aother oe called discout curve. Ay oe curve ca imply the other three (with the help of iterpolatio). Flat Forward Iterpolatio Curves are be built through bootstrap or global fittig (such as cubic splie). But o matter how oe builds curve, a iterpolatio method is ievitable. I this sectio we itroduce a popular iterpolatio method flat forward iterpolatio. Notatio is iherited from Brigo ad Mercurio (006). Specifically, deote zero coupo bod price at time t by (t, T). The the term structure o T at time 0 (that is, (0, T)) is the discout curve at time 0. The (cotiuously compouded) spot rate/zero curve is R(0, T), where R(0, T) is defied as e R(0,T)T (0, T) = 1 R(0, T)T = l(0, T) The istataeous forward rate f(0, T) is defied as letiaquat.wordpress. l(0, T) f(0, T) = T T l(0, T) = f(0, u)du 0
6 The flat forward iterpolatio assumes costat istataeous forward rate betwee ay two kots T 1 ad T. I the followig we show the iterpolatio procedure ad poit out that it is equivalet to the logliear iterpolatio o the discout curve. Give the zero rates o T 1 ad T, the aim is to get the zero rates o ay T [T 1, T ] via iterpolatio. Note that R(0, T 1 )T 1 = l(0, T 1 ) = 0 T 1 f(0, u)du T f(0, u)du = R(0, T )T R(0, T 1 )T 1 T 1 By assumig costat istataeous forward rate we have The for ay T [T 1, T ], R(0, T )T = l(0, T ) = f(0, u) f [T1,T ] where u [T 1, T ] f [T1,T ] = R(0, T )T R(0, T 1 )T 1 T T 1 R(0, T)T = R(0, T 1 )T 1 + f(0, u)du T T 1 = R(0, T 1 )T 1 + f [T1,T ](T T 1 ) = R(0, T 1 )T 1 + R(0, T )T R(0, T 1 )T 1 (T T T T 1 ) 1 = T T R(0, T T T 1 )T 1 + T T 1 R(0, T 1 T T )T 1 0 T f(0, u)du which idicates liear iterpolatio o R(0, T)T. By usig the equatio R(0, T)T = l(0, T), it ca be re-writte as letiaquat.wordpress. l(0, T) = T T T T 1 l(0, T 1 ) + T T 1 T T 1 l(0, T ) I sum, the flat forward iterpolatio o the zero curve is equivalet to the logliear (liear o the log) iterpolatio o the discout curve. This method is very stable, easy to implemet, ad provide meaigful results o both spot rates ad forward rates. Therefore it is widely used i the idustry.
7 Forward rice/yield, Carry, Roll-Dow This sectio is adapted from Sadr (009). Similar to the o-arbitrage argumet used for future price determiatio, the fair forward price of a bod is F dirty = Dirty + Fiacig cost FV(Coupo Icome) where the fiacig cost is called carry. Usually bods are fiaced i repo markets, ad the future value of ay coupo icome is supposed to be re-ivested at the repo rate r p. The, F Dirty (T Fwd ) = Dirty (1 + r p T Fwd t 360 ) C (1 + r p where T i are coupo paymet days, t is curret date, ad T Fwd is forward date. T Fwd T i ) 360 The price carry is the differece betwee the curret (spot) clea price ad the forward clea price: rice Carry = Clea F Clea (T Fwd ) The yield carry is the differece betwee the forward ad spot yields (to maturity): Yield Carry = Forward Yield Spot Yield Whe yield term structure is upward sloppig, short term fiacig (repo) rate is usually lower tha the yield of the bod. The borrowig-short-ivest-log strategy offers positive carry gais. Aother gai positively-sloped yield curve offers is the roll-dow retur, which is the capital gai caused by a fallig yield whe a bod is approachig maturity. Assumig that yield curve remais uchaged as time passes, letiaquat.wordpress. Moth Roll Dow = y(t) y(t Moths) Carry reflects the profit from lower fiacig cost. Roll-dow reflects time value of a bod (theta). Example The first part of accompayig C++ code shows relatioship betwee spot rate (curve), forward rate (curve), par rate (curve), ad discout rate (curve). The secod part of this curve shows carry ad rolldow retur. I use o-the-ru securities o
8 O 011-April-11, CB1 is quoted as BID ASK YTM CB1 is quoted at a discout from face value, i percetage, with ACT/360 day cout. Let Y d be the discout rate ad is the price, the 100 Y d = 360 t = F (1 Y d t 360 ) The bill is T+1 settled, or settled o It matures o Hece there are 3 days to maturity, or t = 3 days. Therefore the quoted bid/ask prices are BID = 100 ( % ) = ASK = 100 ( % ) = All the yields are based o the ask price. The bod equivalet yield (BEY) is the yield if oe buys the bill at the ask price ad holds it to maturity. It assumes simple compoudig BEY = 100 ASK 365 = 100 ASK 365 = = 0.01% ASK t ASK 3 The effective aual rate (EAR) aualizes simply-compouded BEY ad trasforms it ito compouded rate. The yield-to-maturity (YTM) equals EAR. O 011-April-11, CT is quoted as BID ASK YTM letiaquat.wordpress. Note ad bod prices are quoted i dollars plus 1/3 factios of a dollar. + stads for half a fractio or 1/64 of oe dollar. Therefore BID = = ASK = = Related Bloomberg commads: CB <Govt>, CT <Govt>, X1, CG I5, ad ALLQ.
9 Referece [1] Brigo, D. ad Mercurio, F (006). Iterest rate models: theory ad practice: with smile, iflatio, ad credit. Spriger Verlag. [] Fabozzi, F.J. (005). The hadbook of fixed icome securities, 7 th. McGraw-Hill. [3] Hull, J. (009). Optios, futures ad other derivatives. earso retice Hall. [4] Sadr, A. (009). Iterest rate swaps ad their derivatives: a practitioer's guide. Joh Wiley & Sos Ic. letiaquat.wordpress.
Chapter Six. Bond Prices 1/15/2018. Chapter 4, Part 2 Bonds, Bond Prices, Interest Rates and Holding Period Return.
Chapter Six Chapter 4, Part Bods, Bod Prices, Iterest Rates ad Holdig Period Retur Bod Prices 1. Zero-coupo or discout bod Promise a sigle paymet o a future date Example: Treasury bill. Coupo bod periodic
More information43. A 000 par value 5-year bod with 8.0% semiaual coupos was bought to yield 7.5% covertible semiaually. Determie the amout of premium amortized i the 6 th coupo paymet. (A).00 (B).08 (C).5 (D).5 (E).34
More informationCHAPTER 2 PRICING OF BONDS
CHAPTER 2 PRICING OF BONDS CHAPTER SUARY This chapter will focus o the time value of moey ad how to calculate the price of a bod. Whe pricig a bod it is ecessary to estimate the expected cash flows ad
More informationChapter Four Learning Objectives Valuing Monetary Payments Now and in the Future
Chapter Four Future Value, Preset Value, ad Iterest Rates Chapter 4 Learig Objectives Develop a uderstadig of 1. Time ad the value of paymets 2. Preset value versus future value 3. Nomial versus real iterest
More informationSubject CT1 Financial Mathematics Core Technical Syllabus
Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig
More informationChapter Four 1/15/2018. Learning Objectives. The Meaning of Interest Rates Future Value, Present Value, and Interest Rates Chapter 4, Part 1.
Chapter Four The Meaig of Iterest Rates Future Value, Preset Value, ad Iterest Rates Chapter 4, Part 1 Preview Develop uderstadig of exactly what the phrase iterest rates meas. I this chapter, we see that
More informationIntroduction to Financial Derivatives
550.444 Itroductio to Fiacial Derivatives Determiig Prices for Forwards ad Futures Week of October 1, 01 Where we are Last week: Itroductio to Iterest Rates, Future Value, Preset Value ad FRAs (Chapter
More informationDr. Maddah ENMG 602 Intro to Financial Eng g 01/18/10. Fixed-Income Securities (2) (Chapter 3, Luenberger)
Dr Maddah ENMG 60 Itro to Fiacial Eg g 0/8/0 Fixed-Icome Securities () (Chapter 3 Lueberger) Other yield measures Curret yield is the ratio of aual coupo paymet to price C CY = For callable bods yield
More informationFixed Income Securities
Prof. Stefao Mazzotta Keesaw State Uiversity Fixed Icome Securities Sample First Midterm Exam Last Name: First Name: Studet ID Number: Exam time is: 80 miutes. Total poits for this exam is: 400 poits Prelimiaries
More informationCourse FM Practice Exam 1 Solutions
Course FM Practice Exam 1 Solutios Solutio 1 D Sikig fud loa The aual service paymet to the leder is the aual effective iterest rate times the loa balace: SP X 0.075 To determie the aual sikig fud paymet,
More informationSTRAND: FINANCE. Unit 3 Loans and Mortgages TEXT. Contents. Section. 3.1 Annual Percentage Rate (APR) 3.2 APR for Repayment of Loans
CMM Subject Support Strad: FINANCE Uit 3 Loas ad Mortgages: Text m e p STRAND: FINANCE Uit 3 Loas ad Mortgages TEXT Cotets Sectio 3.1 Aual Percetage Rate (APR) 3.2 APR for Repaymet of Loas 3.3 Credit Purchases
More informationMark to Market Procedures (06, 2017)
Mark to Market Procedures (06, 207) Risk Maagemet Baco Sumitomo Mitsui Brasileiro S.A CONTENTS SCOPE 4 2 GUIDELINES 4 3 ORGANIZATION 5 4 QUOTES 5 4. Closig Quotes 5 4.2 Opeig Quotes 5 5 MARKET DATA 6 5.
More informationFixed Income Securities
Prof. Stefao Mazzotta Keesaw State Uiversity Fixed Icome Securities FIN4320. Fall 2006 Sample First Midterm Exam Last Name: First Name: Studet ID Number: Exam time is: 80 miutes. Total poits for this exam
More information1 The Power of Compounding
1 The Power of Compoudig 1.1 Simple vs Compoud Iterest You deposit $1,000 i a bak that pays 5% iterest each year. At the ed of the year you will have eared $50. The bak seds you a check for $50 dollars.
More information1 Savings Plans and Investments
4C Lesso Usig ad Uderstadig Mathematics 6 1 Savigs las ad Ivestmets 1.1 The Savigs la Formula Lets put a $100 ito a accout at the ed of the moth. At the ed of the moth for 5 more moths, you deposit $100
More informationChapter 4: Time Value of Money
FIN 301 Class Notes Chapter 4: Time Value of Moey The cocept of Time Value of Moey: A amout of moey received today is worth more tha the same dollar value received a year from ow. Why? Do you prefer a
More informationBond Valuation. Structure of fixed income securities. Coupon Bonds. The U.S. government issues bonds
Structure of fixed icome securities Bod Valuatio The Structure of fixed icome securities Price & ield to maturit (tm) Term structure of iterest rates Treasur STRIPS No-arbitrage pricig of coupo bods A
More informationCalculation of the Annual Equivalent Rate (AER)
Appedix to Code of Coduct for the Advertisig of Iterest Bearig Accouts. (31/1/0) Calculatio of the Aual Equivalet Rate (AER) a) The most geeral case of the calculatio is the rate of iterest which, if applied
More informationENGINEERING ECONOMICS
ENGINEERING ECONOMICS Ref. Grat, Ireso & Leaveworth, "Priciples of Egieerig Ecoomy'','- Roald Press, 6th ed., New York, 1976. INTRODUCTION Choice Amogst Alteratives 1) Why do it at all? 2) Why do it ow?
More informationCourse FM/2 Practice Exam 1 Solutions
Course FM/2 Practice Exam 1 Solutios Solutio 1 D Sikig fud loa The aual service paymet to the leder is the aual effective iterest rate times the loa balace: SP X 0.075 To determie the aual sikig fud paymet,
More informationClass Sessions 2, 3, and 4: The Time Value of Money
Class Sessios 2, 3, ad 4: The Time Value of Moey Associated Readig: Text Chapter 3 ad your calculator s maual. Summary Moey is a promise by a Bak to pay to the Bearer o demad a sum of well, moey! Oe risk
More informationModels of Asset Pricing
APPENDIX 1 TO CHAPTER 4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
More informationModels of Asset Pricing
APPENDIX 1 TO CHAPTER4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
More information2. The Time Value of Money
2. The Time Value of Moey Problem 4 Suppose you deposit $100 i the bak today ad it ears iterest at a rate of 10% compouded aually. How much will be i the accout 50 years from today? I this case, $100 ivested
More information1 + r. k=1. (1 + r) k = A r 1
Perpetual auity pays a fixed sum periodically forever. Suppose a amout A is paid at the ed of each period, ad suppose the per-period iterest rate is r. The the preset value of the perpetual auity is A
More informationSection 3.3 Exercises Part A Simplify the following. 1. (3m 2 ) 5 2. x 7 x 11
123 Sectio 3.3 Exercises Part A Simplify the followig. 1. (3m 2 ) 5 2. x 7 x 11 3. f 12 4. t 8 t 5 f 5 5. 3-4 6. 3x 7 4x 7. 3z 5 12z 3 8. 17 0 9. (g 8 ) -2 10. 14d 3 21d 7 11. (2m 2 5 g 8 ) 7 12. 5x 2
More information0.07. i PV Qa Q Q i n. Chapter 3, Section 2
Chapter 3, Sectio 2 1. (S13HW) Calculate the preset value for a auity that pays 500 at the ed of each year for 20 years. You are give that the aual iterest rate is 7%. 20 1 v 1 1.07 PV Qa Q 500 5297.01
More informationMATH : EXAM 2 REVIEW. A = P 1 + AP R ) ny
MATH 1030-008: EXAM 2 REVIEW Origially, I was havig you all memorize the basic compoud iterest formula. I ow wat you to memorize the geeral compoud iterest formula. This formula, whe = 1, is the same as
More informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS This set of sample questios icludes those published o the iterest theory topic for use with previous versios of this examiatio.
More informationMS-E2114 Investment Science Exercise 2/2016, Solutions
MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205 Perpetual auity pays a xed sum periodically forever. Suppose a amout A is paid at the ed of each period, ad suppose the per-period iterest rate
More informationBinomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge
Biomial Model Stock Price Dyamics The value of a optio at maturity depeds o the price of the uderlyig stock at maturity. The value of the optio today depeds o the expected value of the optio at maturity
More informationModels of Asset Pricing
4 Appedix 1 to Chapter Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
More informationChapter 11 Appendices: Review of Topics from Foundations in Finance and Tables
Chapter 11 Appedices: Review of Topics from Foudatios i Fiace ad Tables A: INTRODUCTION The expressio Time is moey certaily applies i fiace. People ad istitutios are impatiet; they wat moey ow ad are geerally
More informationClass Notes for Managerial Finance
Class Notes for Maagerial Fiace These otes are a compilatio from:. Class Notes Supplemet to Moder Corporate Fiace Theory ad Practice by Doald R. Chambers ad Nelso J. Lacy. I gratefully ackowledge the permissio
More information1 Basic Growth Models
UCLA Aderso MGMT37B: Fudametals i Fiace Fall 015) Week #1 rofessor Eduardo Schwartz November 9, 015 Hadout writte by Sheje Hshieh 1 Basic Growth Models 1.1 Cotiuous Compoudig roof: lim 1 + i m = expi)
More informationSolutions to Interest Theory Sample Questions
to Iterest Theory Sample Questios Solutio 1 C Chapter 4, Iterest Rate Coversio After 7.5 years, the value of each accout is the same: 7.5 7.5 0.04 1001 100e 1.336 e l(1.336) 7.5 0.0396 7.5 Solutio E Chapter
More informationAppendix 1 to Chapter 5
Appedix 1 to Chapter 5 Models of Asset Pricig I Chapter 4, we saw that the retur o a asset (such as a bod) measures how much we gai from holdig that asset. Whe we make a decisio to buy a asset, we are
More informationChapter 3. Compound interest
Chapter 3 Compoud iterest 1 Simple iterest ad compoud amout formula Formula for compoud amout iterest is: S P ( 1 Where : S: the amout at compoud iterest P: the pricipal i: the rate per coversio period
More informationof Asset Pricing R e = expected return
Appedix 1 to Chapter 5 Models of Asset Pricig EXPECTED RETURN I Chapter 4, we saw that the retur o a asset (such as a bod) measures how much we gai from holdig that asset. Whe we make a decisio to buy
More informationFINANCIAL MATHEMATICS
CHAPTER 7 FINANCIAL MATHEMATICS Page Cotets 7.1 Compoud Value 116 7.2 Compoud Value of a Auity 117 7.3 Sikig Fuds 118 7.4 Preset Value 121 7.5 Preset Value of a Auity 121 7.6 Term Loas ad Amortizatio 122
More informationWe learned: $100 cash today is preferred over $100 a year from now
Recap from Last Week Time Value of Moey We leared: $ cash today is preferred over $ a year from ow there is time value of moey i the form of willigess of baks, busiesses, ad people to pay iterest for its
More informationWhen you click on Unit V in your course, you will see a TO DO LIST to assist you in starting your course.
UNIT V STUDY GUIDE Percet Notatio Course Learig Outcomes for Uit V Upo completio of this uit, studets should be able to: 1. Write three kids of otatio for a percet. 2. Covert betwee percet otatio ad decimal
More informationAPPLICATION OF GEOMETRIC SEQUENCES AND SERIES: COMPOUND INTEREST AND ANNUITIES
APPLICATION OF GEOMETRIC SEQUENCES AND SERIES: COMPOUND INTEREST AND ANNUITIES Example: Brado s Problem Brado, who is ow sixtee, would like to be a poker champio some day. At the age of twety-oe, he would
More informationof Asset Pricing APPENDIX 1 TO CHAPTER EXPECTED RETURN APPLICATION Expected Return
APPENDIX 1 TO CHAPTER 5 Models of Asset Pricig I Chapter 4, we saw that the retur o a asset (such as a bod) measures how much we gai from holdig that asset. Whe we make a decisio to buy a asset, we are
More informationSIMPLE INTEREST, COMPOUND INTEREST INCLUDING ANNUITY
Chapter SIMPLE INTEREST, COMPOUND INTEREST INCLUDING ANNUITY 006 November. 8,000 becomes 0,000 i two years at simple iterest. The amout that will become 6,875 i years at the same rate of iterest is:,850
More informationFinancial Analysis. Lecture 4 (4/12/2017)
Fiacial Aalysis Lecture 4 (4/12/217) Fiacial Aalysis Evaluates maagemet alteratives based o fiacial profitability; Evaluates the opportuity costs of alteratives; Cash flows of costs ad reveues; The timig
More informationDate: Practice Test 6: Compound Interest
: Compoud Iterest K: C: A: T: PART A: Multiple Choice Questios Istructios: Circle the Eglish letter of the best aswer. Circle oe ad ONLY oe aswer. Kowledge/Thikig: 1. Which formula is ot related to compoud
More informationACTUARIAL RESEARCH CLEARING HOUSE 1990 VOL. 2 INTEREST, AMORTIZATION AND SIMPLICITY. by Thomas M. Zavist, A.S.A.
ACTUARIAL RESEARCH CLEARING HOUSE 1990 VOL. INTEREST, AMORTIZATION AND SIMPLICITY by Thomas M. Zavist, A.S.A. 37 Iterest m Amortizatio ad Simplicity Cosider simple iterest for a momet. Suppose you have
More information7 Swaps. Overview. I have friends in overalls whose friendship I would not swap for the favor of the kings of the world. Thomas A.
7 Swaps I have frieds i overalls whose friedship I would ot swap for the favor of the kigs of the world. Thomas A. Ediso Overview Mechaics of iterest rate swaps Day cout issues (Cofirmatios skip) The comparative-advatage
More informationHopscotch and Explicit difference method for solving Black-Scholes PDE
Mälardale iversity Fiacial Egieerig Program Aalytical Fiace Semiar Report Hopscotch ad Explicit differece method for solvig Blac-Scholes PDE Istructor: Ja Röma Team members: A Gog HaiLog Zhao Hog Cui 0
More informationThe self-assessment will test the following six major areas, relevant to studies in the Real Estate Division's credit-based courses:
Math Self-Assessmet This self-assessmet tool has bee created to assist studets review their ow math kowledge ad idetify areas where they may require more assistace. We hope that studets will complete this
More informationSubject CT5 Contingencies Core Technical. Syllabus. for the 2011 Examinations. The Faculty of Actuaries and Institute of Actuaries.
Subject CT5 Cotigecies Core Techical Syllabus for the 2011 Examiatios 1 Jue 2010 The Faculty of Actuaries ad Istitute of Actuaries Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical
More informationThe Time Value of Money in Financial Management
The Time Value of Moey i Fiacial Maagemet Muteau Irea Ovidius Uiversity of Costata irea.muteau@yahoo.com Bacula Mariaa Traia Theoretical High School, Costata baculamariaa@yahoo.com Abstract The Time Value
More informationLecture 16 Investment, Time, and Risk (Basic issues in Finance)
Lecture 16 Ivestmet, Time, ad Risk (Basic issues i Fiace) 1. Itertemporal Ivestmet Decisios: The Importace o Time ad Discoutig 1) Time as oe o the most importat actors aectig irm s ivestmet decisios: A
More information1 Random Variables and Key Statistics
Review of Statistics 1 Radom Variables ad Key Statistics Radom Variable: A radom variable is a variable that takes o differet umerical values from a sample space determied by chace (probability distributio,
More informationMath of Finance Math 111: College Algebra Academic Systems
Math of Fiace Math 111: College Algebra Academic Systems Writte By Bria Hoga Mathematics Istructor Highlie Commuity College Edited ad Revised by Dusty Wilso Mathematics Istructor Highlie Commuity College
More informationCAPITAL PROJECT SCREENING AND SELECTION
CAPITAL PROJECT SCREEIG AD SELECTIO Before studyig the three measures of ivestmet attractiveess, we will review a simple method that is commoly used to scree capital ivestmets. Oe of the primary cocers
More informationUsing Math to Understand Our World Project 5 Building Up Savings And Debt
Usig Math to Uderstad Our World Project 5 Buildig Up Savigs Ad Debt Note: You will have to had i aswers to all umbered questios i the Project Descriptio See the What to Had I sheet for additioal materials
More informationAnnual compounding, revisited
Sectio 1.: No-aual compouded iterest MATH 105: Cotemporary Mathematics Uiversity of Louisville August 2, 2017 Compoudig geeralized 2 / 15 Aual compoudig, revisited The idea behid aual compoudig is that
More information2013/4/9. Topics Covered. Principles of Corporate Finance. Time Value of Money. Time Value of Money. Future Value
3/4/9 Priciples of orporate Fiace By Zhag Xiaorog : How to alculate s Topics overed ad Future Value Net NPV Rule ad IRR Rule Opportuity ost of apital Valuig Log-Lived Assets PV alculatio Short uts ompoud
More informationChapter 5: Sequences and Series
Chapter 5: Sequeces ad Series 1. Sequeces 2. Arithmetic ad Geometric Sequeces 3. Summatio Notatio 4. Arithmetic Series 5. Geometric Series 6. Mortgage Paymets LESSON 1 SEQUENCES I Commo Core Algebra I,
More informationInstitute of Actuaries of India Subject CT5 General Insurance, Life and Health Contingencies
Istitute of Actuaries of Idia Subject CT5 Geeral Isurace, Life ad Health Cotigecies For 2017 Examiatios Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which
More informationOnline appendices from Counterparty Risk and Credit Value Adjustment a continuing challenge for global financial markets by Jon Gregory
Olie appedices from Couterparty Risk ad Credit Value Adjustmet a APPENDIX 8A: Formulas for EE, PFE ad EPE for a ormal distributio Cosider a ormal distributio with mea (expected future value) ad stadard
More informationOverlapping Generations
Eco. 53a all 996 C. Sims. troductio Overlappig Geeratios We wat to study how asset markets allow idividuals, motivated by the eed to provide icome for their retiremet years, to fiace capital accumulatio
More informationWhere a business has two competing investment opportunities the one with the higher NPV should be selected.
Where a busiess has two competig ivestmet opportuities the oe with the higher should be selected. Logically the value of a busiess should be the sum of all of the projects which it has i operatio at the
More informationContents List of Files with Examples
Paos Kostati Power ad Eergy Systems Egieerig Ecoomics Itroductio ad Istructios Cotets List of Files with Examples Frequetly used MS-Excel fuctios Add-Is developed by the Author Istallatio Istructio of
More informationEVEN NUMBERED EXERCISES IN CHAPTER 4
Joh Riley 7 July EVEN NUMBERED EXERCISES IN CHAPTER 4 SECTION 4 Exercise 4-: Cost Fuctio of a Cobb-Douglas firm What is the cost fuctio of a firm with a Cobb-Douglas productio fuctio? Rather tha miimie
More information1 Estimating sensitivities
Copyright c 27 by Karl Sigma 1 Estimatig sesitivities Whe estimatig the Greeks, such as the, the geeral problem ivolves a radom variable Y = Y (α) (such as a discouted payoff) that depeds o a parameter
More informationMonopoly vs. Competition in Light of Extraction Norms. Abstract
Moopoly vs. Competitio i Light of Extractio Norms By Arkadi Koziashvili, Shmuel Nitza ad Yossef Tobol Abstract This ote demostrates that whether the market is competitive or moopolistic eed ot be the result
More informationFirst determine the payments under the payment system
Corporate Fiace February 5, 2008 Problem Set # -- ANSWERS Klick. You wi a judgmet agaist a defedat worth $20,000,000. Uder state law, the defedat has the right to pay such a judgmet out over a 20 year
More informationChapter 5 Time Value of Money
Chapter 5 Time Value of Moey 1. Suppose you deposit $100 i a bak that pays 10% iterest per year. How much will you have i the bak oe year later? 2. Suppose you deposit $100 i a bak that pays 10% per year.
More informationNPTEL DEPARTMENT OF INDUSTRIAL AND MANAGEMENT ENGINEERING IIT KANPUR QUANTITATIVE FINANCE END-TERM EXAMINATION (2015 JULY-AUG ONLINE COURSE)
NPTEL DEPARTMENT OF INDUSTRIAL AND MANAGEMENT ENGINEERING IIT KANPUR QUANTITATIVE FINANCE END-TERM EXAMINATION (2015 JULY-AUG ONLINE COURSE) READ THE INSTRUCTIONS VERY CAREFULLY 1) Time duratio is 2 hours
More informationMafatlal Centre, 10th Floor, Nariman Point, Mumbai CIN: U65991MH1996PTC Tel.: Fax:
Mafatlal Cetre, 10th Floor, Narima Poit, Mumbai - 400 021 CIN: U65991MH1996PTC100444 Tel.: 91-22 66578000 Fax: 91-22 66578181 www.dspblackrock.com Jauary 8, 2018 Dear Uit Holder, Sub: Chage i Fudametal
More informationSingle-Payment Factors (P/F, F/P) Single-Payment Factors (P/F, F/P) Single-Payment Factors (P/F, F/P)
Sigle-Paymet Factors (P/F, F/P) Example: Ivest $1000 for 3 years at 5% iterest. F =? i =.05 $1000 F 1 = 1000 + (1000)(.05) = 1000(1+.05) F 2 = F 1 + F 1 i = F 1 (1+ = 1000(1+.05)(1+.05) = 1000(1+.05) 2
More informationMonetary Economics: Problem Set #5 Solutions
Moetary Ecoomics oblem Set #5 Moetary Ecoomics: oblem Set #5 Solutios This problem set is marked out of 1 poits. The weight give to each part is idicated below. Please cotact me asap if you have ay questios.
More informationCAPITALIZATION (PREVENTION) OF PAYMENT PAYMENTS WITH PERIOD OF DIFFERENT MATURITY FROM THE PERIOD OF PAYMENTS
Iteratioal Joural of Ecoomics, Commerce ad Maagemet Uited Kigdom Vol. VI, Issue 9, September 2018 http://ijecm.co.uk/ ISSN 2348 0386 CAPITALIZATION (PREVENTION) OF PAYMENT PAYMENTS WITH PERIOD OF DIFFERENT
More information2. Find the annual percentage yield (APY), to the nearest hundredth of a %, for an account with an APR of 12% with daily compounding.
1. Suppose that you ivest $4,000 i a accout that ears iterest at a of 5%, compouded mothly, for 58 years. `Show the formula that you would use to determie the accumulated balace, ad determie the accumulated
More informationSeasonally adjusted prices for inflation-linked bonds
CUING EDGE. INFLAION Seasoally adjusted prices for iflatio-liked bods Iflatio-liked bod markets are used ever more frequetly by policy-makers, ecoomists ad commetators to assess the market s opiio about
More informationEstimating Proportions with Confidence
Aoucemets: Discussio today is review for midterm, o credit. You may atted more tha oe discussio sectio. Brig sheets of otes ad calculator to midterm. We will provide Scatro form. Homework: (Due Wed Chapter
More informationTwitter: @Owe134866 www.mathsfreeresourcelibrary.com Prior Kowledge Check 1) State whether each variable is qualitative or quatitative: a) Car colour Qualitative b) Miles travelled by a cyclist c) Favourite
More informationMaximum Empirical Likelihood Estimation (MELE)
Maximum Empirical Likelihood Estimatio (MELE Natha Smooha Abstract Estimatio of Stadard Liear Model - Maximum Empirical Likelihood Estimator: Combiatio of the idea of imum likelihood method of momets,
More informationUnderstanding Financial Management: A Practical Guide Problems and Answers
Udestadig Fiacial Maagemet: A Pactical Guide Poblems ad Aswes Chapte 4 Time Value of Moey Note: You ca use a fiacial calculato to check the aswes to each poblem. 4.2 Futue Value of a Peset Amout. If a
More informationThis article is part of a series providing
feature Bryce Millard ad Adrew Machi Characteristics of public sector workers SUMMARY This article presets aalysis of public sector employmet, ad makes comparisos with the private sector, usig data from
More informationNotes on Expected Revenue from Auctions
Notes o Epected Reveue from Auctios Professor Bergstrom These otes spell out some of the mathematical details about first ad secod price sealed bid auctios that were discussed i Thursday s lecture You
More informationThe Time Value of Money
Part 2 FOF12e_C03.qxd 8/13/04 3:39 PM Page 39 Valuatio 3 The Time Value of Moey Cotets Objectives The Iterest Rate After studyig Chapter 3, you should be able to: Simple Iterest Compoud Iterest Uderstad
More informationAY Term 2 Mock Examination
AY 206-7 Term 2 Mock Examiatio Date / Start Time Course Group Istructor 24 March 207 / 2 PM to 3:00 PM QF302 Ivestmet ad Fiacial Data Aalysis G Christopher Tig INSTRUCTIONS TO STUDENTS. This mock examiatio
More informationNon-Inferiority Logrank Tests
Chapter 706 No-Iferiority Lograk Tests Itroductio This module computes the sample size ad power for o-iferiority tests uder the assumptio of proportioal hazards. Accrual time ad follow-up time are icluded
More informationEXERCISE - BINOMIAL THEOREM
BINOMIAL THOEREM / EXERCISE - BINOMIAL THEOREM LEVEL I SUBJECTIVE QUESTIONS. Expad the followig expressios ad fid the umber of term i the expasio of the expressios. (a) (x + y) 99 (b) ( + a) 9 + ( a) 9
More informationKEY INFORMATION DOCUMENT CFD s Generic
KEY INFORMATION DOCUMENT CFD s Geeric KEY INFORMATION DOCUMENT - CFDs Geeric Purpose This documet provides you with key iformatio about this ivestmet product. It is ot marketig material ad it does ot costitute
More information2.6 Rational Functions and Their Graphs
.6 Ratioal Fuctios ad Their Graphs Sectio.6 Notes Page Ratioal Fuctio: a fuctio with a variable i the deoiator. To fid the y-itercept for a ratioal fuctio, put i a zero for. To fid the -itercept for a
More informationAn Introduction to Certificates of Deposit, Bonds, Yield to Maturity, Accrued Interest, and Duration
1 A Itroductio to Certificates of Deposit, Bods, Yield to Maturity, Accrued Iterest, ad Duratio Joh A. Guber Departet of Electrical ad Coputer Egieerig Uiversity of Wiscosi Madiso Abstract A brief itroductio
More informationThe Valuation of the Catastrophe Equity Puts with Jump Risks
The Valuatio of the Catastrophe Equity Puts with Jump Risks Shih-Kuei Li Natioal Uiversity of Kaohsiug Joit work with Chia-Chie Chag Outlie Catastrophe Isurace Products Literatures ad Motivatios Jump Risk
More informationr i = a i + b i f b i = Cov[r i, f] The only parameters to be estimated for this model are a i 's, b i 's, σe 2 i
The iformatio required by the mea-variace approach is substatial whe the umber of assets is large; there are mea values, variaces, ad )/2 covariaces - a total of 2 + )/2 parameters. Sigle-factor model:
More informationGuide to the Deutsche Börse EUROGOV Indices
Guide to the Deutsche Börse EUROGOV Idices Versio.2 November 20 Deutsche Börse AG Versio.2 Guide to the November 20 Deutsche Börse EUROGOV Idices Page 2 Geeral Iformatio I order to esure the highest quality
More informationOnline appendices from The xva Challenge by Jon Gregory. APPENDIX 10A: Exposure and swaption analogy.
APPENDIX 10A: Exposure ad swaptio aalogy. Sorese ad Bollier (1994), effectively calculate the CVA of a swap positio ad show this ca be writte as: CVA swap = LGD V swaptio (t; t i, T) PD(t i 1, t i ). i=1
More informationUnbiased estimators Estimators
19 Ubiased estimators I Chapter 17 we saw that a dataset ca be modeled as a realizatio of a radom sample from a probability distributio ad that quatities of iterest correspod to features of the model distributio.
More information0.1 Valuation Formula:
0. Valuatio Formula: 0.. Case of Geeral Trees: q = er S S S 3 S q = er S S 4 S 5 S 4 q 3 = er S 3 S 6 S 7 S 6 Therefore, f (3) = e r [q 3 f (7) + ( q 3 ) f (6)] f () = e r [q f (5) + ( q ) f (4)] = f ()
More informationAn Empirical Study of the Behaviour of the Sample Kurtosis in Samples from Symmetric Stable Distributions
A Empirical Study of the Behaviour of the Sample Kurtosis i Samples from Symmetric Stable Distributios J. Marti va Zyl Departmet of Actuarial Sciece ad Mathematical Statistics, Uiversity of the Free State,
More informationCD Appendix AC Index Numbers
CD Appedix AC Idex Numbers I Chapter 20, we preseted a variety of techiques for aalyzig ad forecastig time series. This appedix is devoted to the simpler task of developig descriptive measuremets of the
More informationIntroduction to Probability and Statistics Chapter 7
Itroductio to Probability ad Statistics Chapter 7 Ammar M. Sarha, asarha@mathstat.dal.ca Departmet of Mathematics ad Statistics, Dalhousie Uiversity Fall Semester 008 Chapter 7 Statistical Itervals Based
More information