Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL."

Transcription

1 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 of the late Professor Biyami Schwarz Abstract Some attempts were made to evaluate the future value (FV) of the expected value ad the variace for various cash flows (CF). The motivatio stemmed from some recursive formulas. This method does ot apply directly to the evaluatio of preset values (PV). Oe ca get some estimates for the PV usig results about the FV. We will preset a direct approach to evaluate the PV of both factors for some CF. It will tur to be similar to that used to evaluate the FV. Furthermore it maes it possible to study the PV of these CF directly, ad may suggest a method to study some other CF as well. Subj. class : IE50; IE5 Keywords : Radom Rates of Iterest; Idepedet variables; Expected value; Variace; Auities; future value; preset value.. Itroductio A auity is a sequece of paymets C,..., C made i years i, i =,...,. If the paymets are made i the begiig of each year we have a auity due, ad if the paymets are made at the ed of each year we have a auity i arrear. Assume that the iterest i the year i, is i, ad that these iterests for i =,..., are idepedet radom variables with : E(+j i ) = +j ad Var(+j i ) = s for all i, i =,...,. (.) It is our goal to study the expected value ad the variace, for the preset value (PV) ad for the future value (FV) of the auity. For a series of yearly paymets,let PV() deote the preset value at the begiig of the first year of the paymets, ad let FV() deote the future value at the ed of the th year. There are five classical cases : The first is that of a sigle ivestmet (d) at the begiig of the first year, ad (a) at the ed of the th year. I each of the others we have a auity due i case (id) or a auity i arrear i case (ia) for i =, 3, 4, 5. The paymets i the various cases, ad the values of the FV() of ay auity, as a auity certai with fixed yearly rate of iterest, are give below.the defiite fuctios that express the formulas of the symbols are well ow [ e.g. (MS86)]. Electroic copy available at:

2 The cases are : (d) C =, ad C i = 0 for i =,..., ; FV() = ( + j ) (a) C =, ad C i = 0 for i =,..., ; FV() = (d) C i = for i =,..., ; FV() = s (a) ; FV () = (3d) C i = i for i =,..., ; FV() = ( Is ) (3a) ; FV() = ( Is ) (4d) C i = -i + for i =,..., ; FV() = ( Ds ) (4a) ; FV() = ( Ds ) (5d) C i = (+r) i- for i =,..., ; FV() = ( Cs ) r (5a) ; FV() = ( Cs ) r Recall that the symbol for (5d) is evaluated as s for a rate of iterest f that satisfies (+f) (+r) = (+j),ad that for (5a) is evaluated as. Future Values. s s /(+r) for the rate of iterest f. For the FV of the auities due, let S deote the radom value of the FV of a auity due of paymets evaluated at the ed of years, the S = C ( + j ). The followig equality holds: S ( S + C ) ( +j + for =,..., - (.) The radom variables ( S + C ) ad ( + j + ) remai idepedet for all. Let : E( S ) =, E( S ) = m for =,..., (.) The, for =,..., the followig hold i cases, 3, 4, 5 : = FV() as give above (.3) ad, Var( S ) = m - (.4) I particular the detailed evaluatio of FV() is well ow [e.g.(ms86)], so we have ow formulas for E( S ) =, ad i particular for E( S Oe derives the regressio formulas : ) =. + = ( + C ) ( + j for =,..., (.5) ad m + = ( m + C + C ) [( +j) + s )] for =,..., (.6) that lead to the value Var( S ), via successive iteratios. Electroic copy available at:

3 It turs out that oe eed ot pass through m, ad oe eed ot use regressio to evaluate Var( S ). I fact it is possible to get a defiite formula for S ). Var( To verify this poit oe observes that i the case (d) it is well ow that for all, =,..., the followig hold : E( S ) = ( + ad S ) = [( + ) + ] - ( + Var( ad, i the other cases we have : m = ( m + C + C ) [( + ) + ] - [( + C ) ( + ] m = [( + ) + ] ( m - ) + + /( + If we set V() = Var( S )/[( + ) + ], ad () = / [( + ) + ] the : V(+) = V() + (+) /( + Set S = 0,ad Var( S ) = 0,ad add (.8) for =,..., to get the equality : 0 0 V() = [ /( + ] A detailed calculatio to evaluate () (.9) () for all cases, except the first oe, was made i (Z0). Ufortuately there are some miscalculatios there. These were tae care of i (BMW), ad the author is grateful for the correctios. For the FV of the auities i arrear, let S deote the radom value of the FV of a auity arrear of paymets evaluated at the ed of years, the S = C. A similar approach to the oe tae for the FV of auities due ca be tae for the FV of auities i arrear ad a similar treatmet will lead to similar results. For the auities i arrear we replace (.) with the correspodig followig equalities : S = S ( C + for =,..., - (.0) E( S ) =, E( S ) = r for =,..., (.) The radom variables S ad ( + + are idepedet ad C + is a costat for all. I the case (a) we will get : E( S ) = ( + ad Var( S ) = [( + ) + ] - - ( + () We get similar equatios for all, =,..., -, for the cases, 3, 4, 5 : + = ( + C + (.) r + = r [( + ) + ] + ( + C + + Var( S ) = r = ( r - )[( + ) + ] + As above we deote i a similar way for all, =,..., - : C (.3) (.4) W() = Var( S )/ [( + ) + ], () = / [( + ) + ] (.5) The, W(+) = W() + () /[( + ) + ] Observe that =, Var( S ) = 0, ad r = to obtai the equality : A detailed evaluatio of W() = { /[( + ) + ] above i the remars as to the evaluatio of (). () (.6) () results i a way similar to the oe metioed

4 3. Iterlude. Our ext aim is to ivestigate the PV case. The PV() of a auity is the value of paymets at the begiig of the first year. A direct approach similar to the oe we used for the FV case, seems impossible sice i the regressio formula for the PV() the radom variables that arise are ot liearly idepedet. I sectio 4 we will suggest a differet treatmet of the auities that will eable the direct approach. We will the explore the PV i a way similar to the oe used for the FV. The differet way to study the auities will avoid the liear depedece. At this poit we wish to cosider the value of the quotiet of the FV by the PV. We observe that for ay give auity, whether due or i arrear, i ay of the five cases, the quotiet FV()/PV() will satisfy the followig equalities : E[FV()/PV()] = ( + ad, Var[FV()/PV()] = [( + ) + ] - ( + ad these values may be used to get estimates for the PV(), usig the FV(). It is importat to observe that the above relatios apply to the case of auities that cosists of a series of aual paymets made at the begiig of the first years, ad where FV() is the value at the ed of the th year ad PV() is the value at the begiig of the first year. 4. Preset Values. Deote by i the yearly discout factor for the ( i + ) th year. It follows that : - i + -i+. Let us deote : E(- i ) = - ad Var(- i ) = for all i, i =,...,. ( 4.) I geeral the relatio - = ( + ) - does ot hold. The problem that arises, whe tryig to follow a lie of thought similar to the oe used for the FV i the secod sectio, is that the radom variables ivolved are o loger idepedet. To overcome this difficulty, we cosider the auity bacwards : we will cosider all the above cases with C, for =,...,, ad set D = C -+, i each of the five cases. The auity we will cosider is the sequece of paymets D,..., D. I particular, the PV() of that auity is the value i the begiig of the (-+) th year of the last yearly paymets, D,..., D. The paymets i the various cases, ad the values of the PV() of ay auity as a auity certai with fixed yearly rate of discout,are give below.the defiite fuctios that express the formulas of the symbols are well ow [ e.g. (MS86) ]. The cases are : (d) D =, ad D i = 0 for i =,..., ; PV() = ( d ) - (a) D =, ad D i = 0 for i =,..., ; PV() = ( - ) (d) D i = for i =,..., ; PV() = a (a) ; PV() = a

5 (3d) D i = -i+ for i =,..., ; PV() = ( Ia ) + ( ) a (3a) ; PV() = ( Ia ) + ( ) a (4d) D i = i for i =,..., ; PV() = ( Da ) (4a) ; PV () = ( Da ) (5d) D i = (+) -i for i =,..., ; PV() = (+) - ( Ca ) r (5a) ; PV() = (+) - ( Ca ) r The remars followig the list of cases i sectio, as for the evaluatio of the symbols i the 5 th case, apply i a similar way to the 5 th case above. let A deote the radom value of the PV() of a auity due, of a sequece of yearly paymets evaluated at the begiig of the first year, the A = D, ad we get : A ( - + D + for =,..., - (4.) A = with idepedet radom variables A ad ( - for all ad D + a costat. Let : E( A ) =, E( A ) = t for =,..., (4.3) The, for =,..., the followig hold i cases, 3, 4, 5 : = FV() as give above (4.4) ad, Var( A ) = t - (4.5) I particular, the detailed evaluatio of PV() is well ow [e.g.(ms86)], so we have ow formulas for E( A ) =, ad i particular for E( A Oe derives the regressio formulas : + = ( - +D + for =,..., ) =. (4.6) ad t + = t [( - ) + )] + D + + D for =,..., (4.7) that lead to the values of E( A ), ad Var( A ), via successive iteratios. It turs out that oe eed ot pass through t, ad oe eed ot use regressio to evaluate Var( A ). I fact it is possible to get a defiite formula for Var( A ). To verify this poit oe observes that i the case (d) it is well ow that : E( A ) = ( - - ad Var( A ) = [( - ) + ] - - ( - -) ad, i the other cases Oe ca derive the regressio formulas, for =,..., : t = [( - ) + ] ( t - ) + If we set M() = Var( A )/[( - ) + ], ad () = / [( - ) + ] the : M(+) = M() + () [( - ) + ] ad oticig that =, Var( A ) = 0, ad t = we obtai a similar equality :

6 M() = [ /[( - + ] A detailed calculatio to evaluate () (4.9) () for all cases, except the first oe, may be achieved alog the details as itroduced i (Z0) ad (BMW). A similar approach to the oe tae for the PV of auities due ca be tae for the PV of auities i arrear ad a similar treatmet will lead to similar results. Notice that it all starts by replacig (4.) with the correspodig equalities for this case : A = ( A + D + ) ( - + for =,..., - (4.0) E( A ) =, E( A ) = s for =,..., (4.) where A ad ( - + are idepedet radom variables ad D + is a costat for all. The, we get similar equatios for all, =,..., - : + = ( D + ) ( + (4.) s + = [( - ) + ] [s + D + + D ] (4.3) I the case (a) we will get : E( S ) = ( - ad Var( S ) = [( - ) + ] - ( - ad, i the other cases Oe ca derive the regressio formulas, for =,..., : Var( A ) = s = ( s - )[( - ) + ] + + /( ) (4.4) with otatios similar to the oes used for auities due, we get for =,..., - : N() = Var( A )/ [( - ) + ], () = / [( - ) + ] (4.5) The, N(+) = N() + (+)/(-) Set N(0) = 0,ad Var( A ) = 0,ad addig up for =,..., to get the equality : A detailed evaluatio of N() = [ /( - ) ] 0 () (4.6) () is similar to the oe used to evaluate (). Acowledgemet This research was supported by the Fud for the Promotio of Research at the Techhio Refereces (MS86) McCutcheo J. J., Scott W. F. 986 A Itroductio to the Mathematics of Fiace,Butterworth/Heiema, Lodo (Z0) Zas A. 00 Auities Uder Radom Rate of Iterest, IME 8, (00), - (BMW) Bureci K., Marciiu A., Wero A. Auities Uder Radom Rates of Iterest - revisited, IME 3, (003) (BD) Bedard D., Dufrese D. Pesio Fudig with Movig Average Rates of Retur, Scad. Act. J. (00) -7 (D) Dufrese D., Stability of pesio systems whe rates of retur are radom, IME 8,(989) 7-76 (DMW) Date, P., Mamo, R., Wag, I.C. Valuatio of cash flows uder radom rates of iterest: A liear algebraic approach IME (007) 4,

Abraham Zaks. Technion I.I.T. Haifa ISRAEL. and. University of Haifa, Haifa ISRAEL. Abstract

Abraham Zaks. Technion I.I.T. Haifa ISRAEL. and. University of Haifa, Haifa ISRAEL. Abstract Preset Value of Autes Uder Radom Rates of Iterest By Abraham Zas Techo I.I.T. Hafa ISRAEL ad Uversty of Hafa, Hafa ISRAEL Abstract Some attempts were made to evaluate the future value (FV) of the expected

More information

The Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract

The 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 information

Institute of Actuaries of India Subject CT1 Financial Mathematics

Institute 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 information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 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 information

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets

BENEFIT-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 information

I. Chi-squared Distributions

I. 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 information

Time Value of Money, NPV and IRR equation solving with the TI-86

Time Value of Money, NPV and IRR equation solving with the TI-86 Time Value of Moey NPV ad IRR Equatio Solvig with the TI-86 (may work with TI-85) (similar process works with TI-83, TI-83 Plus ad may work with TI-82) Time Value of Moey, NPV ad IRR equatio solvig with

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

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 information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.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 information

Terminology for Bonds and Loans

Terminology 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 information

I. Why is there a time value to money (TVM)?

I. 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 information

TO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC

TO: 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 information

Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014

Learning 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 information

Asymptotic Growth of Functions

Asymptotic 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 information

TO: Users of the ACTEX Review Seminar on DVD for SOA Exam FM/CAS Exam 2

TO: 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 information

where: 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

where: 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 information

Bond 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

Bond 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 information

Subject CT5 Contingencies Core Technical Syllabus

Subject 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 information

A probabilistic proof of a binomial identity

A 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 information

Present Values, Investment Returns and Discount Rates

Present 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 information

Incremental calculation of weighted mean and variance

Incremental 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 information

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Vladimir 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 information

Time Value of Money. First some technical stuff. HP10B II users

Time 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 information

Department of Computer Science, University of Otago

Department 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 information

Swaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps

Swaps: 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 information

Installment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate

Installment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate Iteratioal Coferece o Maagemet Sciece ad Maagemet Iovatio (MSMI 4) Istallmet Joit Life Isurace ctuarial Models with the Stochastic Iterest Rate Nia-Nia JI a,*, Yue LI, Dog-Hui WNG College of Sciece, Harbi

More information

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 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 information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A 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 information

CHAPTER 4: NET PRESENT VALUE

CHAPTER 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 information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In 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

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties 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 information

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Present 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 information

Sequences and Series

Sequences 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 information

Understanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions

Understanding 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 information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

Your 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:

Your 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 information

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 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 information

On Formula to Compute Primes. and the n th Prime

On Formula to Compute Primes. and the n th Prime Applied Mathematical cieces, Vol., 0, o., 35-35 O Formula to Compute Primes ad the th Prime Issam Kaddoura Lebaese Iteratioal Uiversity Faculty of Arts ad cieces, Lebao issam.kaddoura@liu.edu.lb amih Abdul-Nabi

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete 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 information

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

More information

Soving Recurrence Relations

Soving 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 information

Section 11.3: The Integral Test

Section 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 information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 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 information

Project Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments

Project Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments Project Deliverables CS 361, Lecture 28 Jared Saia Uiversity of New Mexico Each Group should tur i oe group project cosistig of: About 6-12 pages of text (ca be loger with appedix) 6-12 figures (please

More information

Data Analysis and Statistical Behaviors of Stock Market Fluctuations

Data Analysis and Statistical Behaviors of Stock Market Fluctuations 44 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 Data Aalysis ad Statistical Behaviors of Stock Market Fluctuatios Ju Wag Departmet of Mathematics, Beijig Jiaotog Uiversity, Beijig 00044, Chia Email:

More information

An Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function

An Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function A Efficiet Polyomial Approximatio of the Normal Distributio Fuctio & Its Iverse Fuctio Wisto A. Richards, 1 Robi Atoie, * 1 Asho Sahai, ad 3 M. Raghuadh Acharya 1 Departmet of Mathematics & Computer Sciece;

More information

Here 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.

Here 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 information

Amendments to employer debt Regulations

Amendments to employer debt Regulations March 2008 Pesios Legal Alert Amedmets to employer debt Regulatios The Govermet has at last issued Regulatios which will amed the law as to employer debts uder s75 Pesios Act 1995. The amedig Regulatios

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis 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 information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

Simple Annuities Present Value.

Simple Annuities Present Value. Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX-9850GB PLUS to efficietly compute values associated with preset value auities.

More information

Now here is the important step

Now here is the important step LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

More information

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51 Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

More information

Question 2: How is a loan amortized?

Question 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 information

Nr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996

Nr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996 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

More information

FI A CIAL MATHEMATICS

FI A CIAL MATHEMATICS CHAPTER 7 FI A CIAL MATHEMATICS Page Cotets 7.1 Compoud Value 117 7.2 Compoud Value of a Auity 118 7.3 Sikig Fuds 119 7.4 Preset Value 122 7.5 Preset Value of a Auity 122 7.6 Term Loas ad Amortizatio 123

More information

THE TIME VALUE OF MONEY

THE 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 information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case 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 information

THE ABRACADABRA PROBLEM

THE ABRACADABRA PROBLEM THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

More information

2 Time Value of Money

2 Time Value of Money 2 Time Value of Moey BASIC CONCEPTS AND FORMULAE 1. Time Value of Moey It meas moey has time value. A rupee today is more valuable tha a rupee a year hece. We use rate of iterest to express the time value

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL 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 information

A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length

A 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 information

INVESTMENT PERFORMANCE COUNCIL (IPC)

INVESTMENT 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 information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

Systems Design Project: Indoor Location of Wireless Devices

Systems Design Project: Indoor Location of Wireless Devices Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised

More information

A Guide to the Pricing Conventions of SFE Interest Rate Products

A 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 information

Approximating 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

Approximating 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 information

FM4 CREDIT AND BORROWING

FM4 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 information

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required. S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 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 information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

Lecture 2: Karger s Min Cut Algorithm

Lecture 2: Karger s Min Cut Algorithm priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.

More information

Floating Codes for Joint Information Storage in Write Asymmetric Memories

Floating Codes for Joint Information Storage in Write Asymmetric Memories Floatig Codes for Joit Iformatio Storage i Write Asymmetric Memories Axiao (Adrew Jiag Computer Sciece Departmet Texas A&M Uiversity College Statio, TX 77843-311 ajiag@cs.tamu.edu Vaske Bohossia Electrical

More information

The 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 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 information

Modified Line Search Method for Global Optimization

Modified 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 information

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 16804-0030 haupt@ieee.org Abstract:

More information

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,

NEW 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 information

1 Correlation and Regression Analysis

1 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 information

VALUATION OF FINANCIAL ASSETS

VALUATION OF FINANCIAL ASSETS P A R T T W O As a parter for Erst & Youg, a atioal accoutig ad cosultig firm, Do Erickso is i charge of the busiess valuatio practice for the firm s Southwest regio. Erickso s sigle job for the firm is

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

Elementary Theory of Russian Roulette

Elementary Theory of Russian Roulette Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

More information

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

More information

The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection

The 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 information

0.7 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

0.7 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

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

MMQ Problems Solutions with Calculators. Managerial Finance

MMQ Problems Solutions with Calculators. Managerial Finance MMQ Problems Solutios with Calculators Maagerial Fiace 2008 Adrew Hall. MMQ Solutios With Calculators. Page 1 MMQ 1: Suppose Newma s spi lads o the prize of $100 to be collected i exactly 2 years, but

More information

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011 15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes high-defiitio

More information

Automatic Tuning for FOREX Trading System Using Fuzzy Time Series

Automatic 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 information

NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE 11 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a -page formula sheet. Please tur over Mathematics/P DoE/November

More information

A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY

A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY J. Appl. Prob. 45, 060 070 2008 Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON

More information

CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION

CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION www.arpapress.com/volumes/vol8issue2/ijrras_8_2_04.pdf CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION Elsayed A. E. Habib Departmet of Statistics ad Mathematics, Faculty of Commerce, Beha

More information

Learning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV)

Learning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV) Leaig Objectives Chapte 2 Picig of Bods time value of moey Calculate the pice of a bod estimate the expected cash flows detemie the yield to discout Bod pice chages evesely with the yield 2-1 2-2 Leaig

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 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 information

THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK

THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for

More information

You are given that mortality follows the Illustrative Life Table with i 0.06 and that deaths are uniformly distributed between integral ages.

You are given that mortality follows the Illustrative Life Table with i 0.06 and that deaths are uniformly distributed between integral ages. 1. A 2 year edowmet isurace of 1 o (6) has level aual beefit premiums payable at the begiig of each year for 1 years. The death beefit is payable at the momet of death. You are give that mortality follows

More information