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

Size: px
Start display at page:

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

Transcription

1 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 law of mortality was itroduced by Gompertz i 1825 i [G]. I 1860 Makeham itroduced i [M] a modificatio to obtai aother law of mortality. Both these laws assume that the populatio uder cosideratio is stable. The two laws differ by a costat term i the force of mortality. The updated approach to the study of populatio presume that mortality chages over time. The differece stems from observig that the expectacy of life chages over the years. There were made various attempts to itroduce dyamic life tables, ad the Lee-Carter model has this advatage. The Lee-Carter model i [LC] describes better the mortality uder chages over time as was show recetly by ([MR]). We ited to study the differece arisig from a fixed chage i the force of mortality ad the Gompertz ad Makeham cases may serve to demostrate such a chage. The differece i the expectacy of life i both cases affects directly the correspodig life tables, ad cosequetly the auities (for life, term auitie ad deferred ), as well as the assuraces (whole life, term ad edowmet), ad the premiums ad reserves i the various cases. The chages i a stable life table model may serve to evaluate the chages that arise i terms of a sesitive aalysis of various assurace plas. Cosequetly there results a tool to cope with evaluatig the effect of chage i the force of mortality. Keywords : force of mortality ; Expectacy of life ; Auities, a,, Assuraces, A, A ; Premiums ad Reserves i various cases. 1 1 of 6 Electroic copy available at:

2 1. Auities We let G deote the "Gompertz case", M the "Makeham case" ad GM the "Gompertz ad Makeham cases". We follow the defiitios ad otatios of [Z1], [Z2], ad [Z3]. I G the force of mortality is give by, [, ad i M the force of mortality is give by, [, for some fixed. We may have 0, 0 0, i the last case both models coicide. Note that we may replace by where ad the results that will follow will be symmetric to the results that we obtai whe we replace for (e.g. for ad for ad so o ad for. Recall that the expectacy of a live aged i these models is ad respectively. Let be the force of iterest, ad let (µ,δ) be the PV of the cotiuous auity of 1 p.a. for a live aged ad we have (e.g.[z1]) (µ,δ+ε) ε, δ ad i particular we have the equalities (µ,δ+ ), δ, ad,,. As is well kow, 0 ad sice (e.g.[ Z]) we may deduce :. :, 0, Recall the approximatio (e.g. [Z1] ) : µ, δ ε µ, δ µ, δ where deotes the PV of the cotuuous auity of 1 p.a. for a live aged icreasig by 1 aually. The followig approximatio results :. : The approximatio, µ, 0 holds. The results for geeralizes to, evaluatig i the same table but i differet itersets :. :, δ ) (,δ+ ad a similar geeralizatio hold for the approximatio i the same table ad iterset :. :, δ (,δ), δ The result follows from, δ (,δ+ ) ad the approximatio (,δ+ ) (,δ), δ. It seems that oe ca derive appropriate approximatios for temporary auities a ad deferred auities, as well as for due auities ad i arrear auities. 2 of 6 Electroic copy available at:

3 2. Assuraces Let (µ,δ) be the PV of the cotiuous whole life issurace of 1 for a live aged. The result of comparrig, evaluatig i the same table but i differet itersets yields :. : (µε,δ) µ, δ ε ε (µ,δε). From the well kow equality (µ,δ)δ (µ,δ)=1 there follows the equality (µ,δε)= 1 δ ε (µ,δε). Recall that : (µε,δ)= 1 δ (µ ε, δ) ad we may derive the equality : (µε,δ) µ, δ ε ε (µ,δε) as claimed. Note that sice, δ, 1 we may coclude i a similar way that,,, that is or there follows the equality : A, A, a,. Recall from [Z1] thatwe have the approximatio µ, δ ε µ, δ µ, δ where µ, δ deotes the PV of the cotiuous whole life assurace of 1 for a live aged icreasig aually by 1. ad a similar geeralizatio hold for the approximatio i the same table ad iterset :. : µ, δ ε µ, δ µ, δ ε µ, δ ε. From the approximatio µ, δ ε µ, δ µ, δ (e.g. [Z1]) ad from propositio 3 the result follows as stated. I particular we may derive :. (,δ),δ)δ, δ. The equality,δ)δ (,δ)=1 implies,δ)1 δ (,δ) ad the equality (,δ)δ (,δ)=1 implies (,δ)=1 δ (,δ). Therefore (,δ)=1 δ (,δ ad by [Z1] we may derive the approximatio, δ 1 δ (,δ)δ, δ, ad sice,δ)1 δ (,δ) the approximatio (,δ),δ)δ, δ follows as stated. It seems that oe ca derive appropriate approximatios i a similar way for term assurace 1 A, edowmet assurace various assurace plas paid at the ed of the year of death.. A,ad deferred assurace, as well as for the 3 of 6

4 3. Premiums The premium for the whole life isurace may be derived from. This implies, 1, ad i particular, 1,. Furthermore we have the equality, 1,. Recall the approximatio :,,,, δ. Thus we may state the followig resultig equatio ad approximatio :. :,,..,,, δ, The result follows from the equality,, ad the approximatio,,, δ,. Note that startig with the equalities :, 1, Ad, 1,, it results that,, Ad i particular we may derive the equality : P, P,. It seems that oe ca derive appropriate approximatios i a similar way for premiums for the various assurace plas ad pesio schemes (cosidered as deferred auities).. 4. Reserves The reserve, i time for the whole life isurace for a live aged may be derived from Ad we obtai :, ad i a similar way for M µ, δ (µ,δ ad (µ,δ)= 1δ (µ,δ), 1,,, 1,, ad for G we have, 1,,. Recall that,, ad,, 4 of 6

5 Ad we may derive :. : Ad there results the approximatio :,,.,,,,, δ,, δ, I a similar way if we start with the equalities :, 1,, ad, 1,,, we have, V, holds., ad i particular the equality V, It seems that oe ca derive appropriate approximatios i a similar way for reserves i the various assurace plas ad pesio schemes.. 5. Approximatig higher derivatios Recall [Z2] for relatios of higher derivatives of with ad. A life table satisfies the weak decreasig assumptio e.g. whe we presume ati selectio, i which case the premium decreases whe the force of iterest icreases. I this case the iequality holds. Therefore we have ad we ca derive the iequality 1, that is 1 ad cosequetly :.. I particular followig [Z2] start with the equality ad use iductio, to obtai the equalities A x a x for all 0 where = that is :. 0, 0. The proof of this propositio is a direct cosequece of the observatio : of 6

6 Refereces [FL] Föllmer, H., Leukert, P., Quatile hedgig. Fiace Stoch. 3, [G ] Gompertz, B. " O the Nature of the Fuctio Expressive of the Law of Huma Mortality, ad o a New Mode of Determiig the Value of Life Cotigecies." Phil. Tras. of the Royal Soc. of Lodo, Vol. 115 (1825), pp [M] [LC] Makeham, W. M. "O the Law of Mortality ad the Costructio of Auity Tables." J. Ist. Act. ad Assur. Mag. 8, , Lee, R.D., Carter, L.R., Modellig ad forecastig U.S. mortality. J. Amer. Statist. Assoc. 87 (14), [MR] A.Melikov, Y.Romaiuk "Evaluatig the performace of Gompertz, Makeham ad Lee Carter mortality models for risk maagemet with uit-liked cotracts" IME Volume 39(3)2006, Pp [Z1] Zaks A., Auities uder chages i Life Tables ad chages i the Iterest, Pravartak J. Is. & Risk Ma.2009(8) [Z2] [Z3] Auities uder chages i Life Tables ad chages i the Iterest II, Pravartak J. Is. & Risk Ma.2009(9) Auities uder chages i Life Tables ad chages i the Iterest III, or Premium Icreases whe Iterest Drcreases, Pravartak J. Is. & Risk Ma.2009(10) of 6

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

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More 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

Module 4: Mathematical Induction

Module 4: Mathematical Induction Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

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

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

Chapter 7. V and 10. V (the modified premium reserve using the Full Preliminary Term. V (the modified premium reserves using the Full Preliminary

Chapter 7. V and 10. V (the modified premium reserve using the Full Preliminary Term. V (the modified premium reserves using the Full Preliminary Chapter 7 1. You are give that Mortality follows the Illustrative Life Table with i 6%. Assume that mortality is uiformly distributed betwee itegral ages. Calculate: a. Calculate 10 V for a whole life

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

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

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

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

Approximating the Sum of a Convergent Series

Approximating the Sum of a Convergent Series Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece

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

Exponential function: For a > 0, the exponential function with base a is defined by. f(x) = a x

Exponential function: For a > 0, the exponential function with base a is defined by. f(x) = a x MATH 11011 EXPONENTIAL FUNCTIONS KSU AND THEIR APPLICATIONS Defiitios: Expoetial fuctio: For a > 0, the expoetial fuctio with base a is defied by fx) = a x Horizotal asymptote: The lie y = c is a horizotal

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

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

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

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

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

D I S C U S S I O N P A P E R

D I S C U S S I O N P A P E R I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S ( I S B A UNIVERSITÉ CATHOLIQUE DE LOUVAIN D I S C U S S I O N P A P E R 2012/27 Worst-case

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

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

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

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

Research Article Sign Data Derivative Recovery

Research Article Sign Data Derivative Recovery Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

More information

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

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

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

Transient Behavior of Two-Machine Geometric Production Lines

Transient Behavior of Two-Machine Geometric Production Lines Trasiet Behavior of Two-Machie Geometric Productio Lies Semyo M. Meerkov Nahum Shimki Liag Zhag Departmet of Electrical Egieerig ad Computer Sciece Uiversity of Michiga, A Arbor, MI 489-222, USA (e-mail:

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

The Euler Totient, the Möbius and the Divisor Functions

The Euler Totient, the Möbius and the Divisor Functions The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

More information

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

More information

On The Comparison of Several Goodness of Fit Tests: With Application to Wind Speed Data

On The Comparison of Several Goodness of Fit Tests: With Application to Wind Speed Data Proceedigs of the 3rd WSEAS It Cof o RENEWABLE ENERGY SOURCES O The Compariso of Several Goodess of Fit Tests: With Applicatio to Wid Speed Data FAZNA ASHAHABUDDIN, KAMARULZAMAN IBRAHIM, AND ABDUL AZIZ

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

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

Efficient tree methods for pricing digital barrier options

Efficient tree methods for pricing digital barrier options Efficiet tree methods for pricig digital barrier optios arxiv:1401.900v [q-fi.cp] 7 Ja 014 Elisa Appolloi Sapieza Uiversità di Roma MEMOTEF elisa.appolloi@uiroma1.it Abstract Adrea igori Uiversità di Roma

More information

Irreducible polynomials with consecutive zero coefficients

Irreducible polynomials with consecutive zero coefficients Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

More information

NPTEL STRUCTURAL RELIABILITY

NPTEL STRUCTURAL RELIABILITY NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

More information

BOUNDS FOR THE PRICE OF A EUROPEAN-STYLE ASIAN OPTION IN A BINARY TREE MODEL

BOUNDS FOR THE PRICE OF A EUROPEAN-STYLE ASIAN OPTION IN A BINARY TREE MODEL BOUNDS FOR THE PRICE OF A EUROPEAN-STYLE ASIAN OPTION IN A BINARY TREE MODEL HUGUETTE REYNAERTS, MICHELE VANMAELE, JAN DHAENE ad GRISELDA DEELSTRA,1 Departmet of Applied Mathematics ad Computer Sciece,

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

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

Estimating the Mean and Variance of a Normal Distribution

Estimating the Mean and Variance of a Normal Distribution Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

More information

Evaluating Model for B2C E- commerce Enterprise Development Based on DEA

Evaluating Model for B2C E- commerce Enterprise Development Based on DEA , pp.180-184 http://dx.doi.org/10.14257/astl.2014.53.39 Evaluatig Model for B2C E- commerce Eterprise Developmet Based o DEA Weli Geg, Jig Ta Computer ad iformatio egieerig Istitute, Harbi Uiversity of

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

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

1 Itroductio Let A be a complex matrix ad let C (A) be its th compoud. It was show i [10, Formula (12)] that the imal row sum (of moduli) of elemets o

1 Itroductio Let A be a complex matrix ad let C (A) be its th compoud. It was show i [10, Formula (12)] that the imal row sum (of moduli) of elemets o Bouds o orms of compoud matrices ad o products of eigevalues Ludwig Elser Faultat fur Mathemati Uiversitat Bielefeld Postfach 100131 D-33615 Bielefeld Germay Daiel Hershowitz Departmet of Mathematics Techio

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

A unified pricing of variable annuity guarantees under the optimal stochastic control framework

A unified pricing of variable annuity guarantees under the optimal stochastic control framework A uified pricig of variable auity guaratees uder the optimal stochastic cotrol framework Pavel V. Shevcheko 1 ad Xiaoli Luo 2 arxiv:1605.00339v1 [q-fi.pr] 2 May 2016 Draft paper: 16 April 2016 1 CSIRO

More information

Model points and Tail-VaR in life insurance

Model points and Tail-VaR in life insurance Model poits ad Tail-VaR i life isurace Michel Deuit Istitute of Statistics, Biostatistics ad Actuarial Sciece Uiversité Catholique de Louvai (UCL) Louvai-la-Neuve, Belgium Julie Trufi Departmet of Mathematics

More information

Ekkehart Schlicht: Economic Surplus and Derived Demand

Ekkehart Schlicht: Economic Surplus and Derived Demand Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/

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

Basic Elements of Arithmetic Sequences and Series

Basic Elements of Arithmetic Sequences and Series MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC. SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol

More information

Factors of sums of powers of binomial coefficients

Factors of sums of powers of binomial coefficients ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

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

Class Meeting # 16: The Fourier Transform on R n

Class Meeting # 16: The Fourier Transform on R n MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

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

Gibbs Distribution in Quantum Statistics

Gibbs Distribution in Quantum Statistics Gibbs Distributio i Quatum Statistics Quatum Mechaics is much more complicated tha the Classical oe. To fully characterize a state of oe particle i Classical Mechaics we just eed to specify its radius

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

More 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

CDs Bought at a Bank verses CD s Bought from a Brokerage. Floyd Vest

CDs Bought at a Bank verses CD s Bought from a Brokerage. Floyd Vest CDs Bought at a Bak verses CD s Bought from a Brokerage Floyd Vest CDs bought at a bak. CD stads for Certificate of Deposit with the CD origiatig i a FDIC isured bak so that the CD is isured by the Uited

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

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

THE problem of fitting a circle to a collection of points

THE problem of fitting a circle to a collection of points IEEE TRANACTION ON INTRUMENTATION AND MEAUREMENT, VOL. XX, NO. Y, MONTH 000 A Few Methods for Fittig Circles to Data Dale Umbach, Kerry N. Joes Abstract Five methods are discussed to fit circles to data.

More information

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

More information

Sharp Nonasymptotic Bounds and Three Term Asymptotic Expansions for the Mean and the Median of a Gaussian Sample Maximum

Sharp Nonasymptotic Bounds and Three Term Asymptotic Expansions for the Mean and the Median of a Gaussian Sample Maximum Sharp Noasymptotic Bouds ad Three Term Asymptotic Expasios for the Mea ad the Media of a Gaussia Sample Maximum Airba DasGupta, Jorda Stoyaov August, 22 Abstract We are iterested i the sample maximum X

More information

An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process

An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process A example of o-queched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which

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

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

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

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More 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

Checklist. Assignment

Checklist. Assignment Checklist Part I Fid the simple iterest o a pricipal. Fid a compouded iterest o a pricipal. Part II Use the compoud iterest formula. Compare iterest growth rates. Cotiuous compoudig. (Math 1030) M 1030

More information

A Constant-Factor Approximation Algorithm for the Link Building Problem

A Constant-Factor Approximation Algorithm for the Link Building Problem A Costat-Factor Approximatio Algorithm for the Lik Buildig Problem Marti Olse 1, Aastasios Viglas 2, ad Ilia Zvedeiouk 2 1 Ceter for Iovatio ad Busiess Developmet, Istitute of Busiess ad Techology, Aarhus

More information

This chapter considers the effect of managerial compensation on the desired

This chapter considers the effect of managerial compensation on the desired Chapter 4 THE EFFECT OF MANAGERIAL COMPENSATION ON OPTIMAL PRODUCTION AND HEDGING WITH FORWARDS AND PUTS 4.1 INTRODUCTION This chapter cosiders the effect of maagerial compesatio o the desired productio

More information

Capacity Management for Contract Manufacturing

Capacity Management for Contract Manufacturing OPERATIONS RESEARCH Vol. 55, No. 2, March April 2007, pp. 367 377 iss 0030-364X eiss 526-5463 07 5502 0367 iforms doi 0.287/opre.060.0359 2007 INFORMS Capacity Maagemet for Cotract Maufacturig Diwakar

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

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

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

2.7 Sequences, Sequences of Sets

2.7 Sequences, Sequences of Sets 2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For

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

RISK TRANSFER FOR DESIGN-BUILD TEAMS

RISK TRANSFER FOR DESIGN-BUILD TEAMS WILLIS CONSTRUCTION PRACTICE I-BEAM Jauary 2010 www.willis.com RISK TRANSFER FOR DESIGN-BUILD TEAMS Desig-builD work is icreasig each quarter. cosequetly, we are fieldig more iquiries from cliets regardig

More information

Exploratory Data Analysis

Exploratory Data Analysis 1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios

More information

ADAPTIVE NETWORKS SAFETY CONTROL ON FUZZY LOGIC

ADAPTIVE NETWORKS SAFETY CONTROL ON FUZZY LOGIC 8 th Iteratioal Coferece o DEVELOPMENT AND APPLICATION SYSTEMS S u c e a v a, R o m a i a, M a y 25 27, 2 6 ADAPTIVE NETWORKS SAFETY CONTROL ON FUZZY LOGIC Vadim MUKHIN 1, Elea PAVLENKO 2 Natioal Techical

More information

Degree of Approximation of Continuous Functions by (E, q) (C, δ) Means

Degree of Approximation of Continuous Functions by (E, q) (C, δ) Means Ge. Math. Notes, Vol. 11, No. 2, August 2012, pp. 12-19 ISSN 2219-7184; Copyright ICSRS Publicatio, 2012 www.i-csrs.org Available free olie at http://www.gema.i Degree of Approximatio of Cotiuous Fuctios

More information

Present Value Tax Expenditure Estimate of Tax Assistance for Retirement Saving

Present Value Tax Expenditure Estimate of Tax Assistance for Retirement Saving Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig Tax Policy Brach Departmet of Fiace Jue 30, 1998 2 Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig This

More information

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

Extreme changes in prices of electricity futures

Extreme changes in prices of electricity futures Isurace Marets ad Compaies: Aalyses ad Actuarial Computatios, Volume 2, Issue, 20 Roald Huisma (The Netherlads), Mehtap Kilic (The Netherlads) Extreme chages i prices of electricity futures Abstract The

More information

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio

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

Chapter 14 Nonparametric Statistics

Chapter 14 Nonparametric Statistics Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

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

Standard Errors and Confidence Intervals

Standard Errors and Confidence Intervals Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5-year-old boys. If we assume

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

Iran. J. Chem. Chem. Eng. Vol. 26, No.1, 2007. Sensitivity Analysis of Water Flooding Optimization by Dynamic Optimization

Iran. J. Chem. Chem. Eng. Vol. 26, No.1, 2007. Sensitivity Analysis of Water Flooding Optimization by Dynamic Optimization Ira. J. Chem. Chem. Eg. Vol. 6, No., 007 Sesitivity Aalysis of Water Floodig Optimizatio by Dyamic Optimizatio Gharesheiklou, Ali Asghar* + ; Mousavi-Dehghai, Sayed Ali Research Istitute of Petroleum Idustry

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

9.8: THE POWER OF A TEST

9.8: THE POWER OF A TEST 9.8: The Power of a Test CD9-1 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based

More information

Example Consider the following set of data, showing the number of times a sample of 5 students check their per day:

Example Consider the following set of data, showing the number of times a sample of 5 students check their  per day: Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day

More information