A Reexamination of the Joint Mortality Functions


 Melinda Kathlyn Wilcox
 2 years ago
 Views:
Transcription
1 Norh merican cuarial Journal Volume 6, Number 1, p (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali analsis involving muliple lives is easil one of he more complicaed aspecs in he heor of life coningencies. In his paper, we reinvesigae join morali funcions and in paricular, we eamine an asserion ha relaes he joinlife and lassurvivor random variables. This common asserion saes ha he sum of he lifeimes of he join life and he lassurvivor sauses is equal o he sum of he lifeimes of he single sauses. However, we show ha his asserion is no precisel correc. We herefore offer a modificaion o he definiions of he sauses so ha his common asserion holds. 1. Inroducion. The equaion relaing he joinlife and lassurvivor fuure lifeime random variables is given as (1) T ( ) T( ) = T() + T() +, wihou an independence assumpion. This asserion relies on cerain criical, e unsaed, assumpions. In Secion 4, we will presen a modified version of (1) wih proper assumpions. Since man formulae rel on his asserion, a number of basic relaionships in muliple life morali funcions urn ou o be wrong. For insance, 1
2 (2) p + p = p + p and (3) + = + are no valid wihou cerain independence assumpions. (ll necessar noaions and definiions are given in he following secion). Since hese relaions are used o price join life insurance producs, i is imporan o invesigae assumpions ha make hese relaions hold, and when hese assumpions do no hold, i is imporan o undersand he correc relaions. In Secion 3 eamples are given o illusrae ha hese equaliies do no hold. In Secion 4 he noaions are eamined in greaer deail, and valid equaliies are given. 2. Noaions and definiions. We follow closel he noaions in he ebook, Bowers, e al. (1997). We pu he cone in erms of spousal morali. Le X be he random variable represening female s age a deah, Y be he random variable represening male s age a deah, () is he saus denoing a personaged, ( ) is he joinlife saus ha survives as long as boh () and () survive, ( ) is he join lassurvivor saus ha eiss as long as a leas one of () and () is alive, and T(u) is he fuure lifeime for saus (u). 2
3 We define, for a saus (u), he condiional survival funcion p and he ne single u premium u for life insurance polic ha pas a he end of he ear he saus (u) fails. The annual ineres rae is denoed b i and 1 v =. 1 + i ( () > ) = P( X > + X ) p = P T >, ( () > ) = P( Y > + Y ) p = P T >, ( ( ) > ) = P( X > + Y > + X > Y ) p = P T and, >, ( ( ) > ) = P( X > + o r Y > + X > Y ) = P T, > p k+ 1, and u = v ( k pu k+ 1pu ), where u represens a saus,,, or. k 3. Eamples. The following are counereamples ha show equaliies (2) and (3) do no hold. Eample 1. ( p + p p + p ) Consider a pair of animals from a purel ficional species. Shorl afer birh he mae for life. probabili breakdown of heir ages a deah is given in he able below ( K m is he curae fuure lifeime of a newborn male animal, ec.): P( K f = 0) P( K f = 1) P( K f = 2) P( K m = 0) P( K m = 1) P( K m = 2)
4 Thus, for a newl maed pair, he probabili ha he male dies in is hird ear (K m = 2) while he female dies in is hird ear (K f = 2) is.20. Consider a pair, which have boh survived heir firs ear. Then we have he following: p 1:1 = P(boh survive a leas one more ear (given he survived hus far)) =.20/.50 = 2/5 p 1:1 = P(a leas one survives one or more ears (given he survived hus far)) =.40/.50 = 4/5 p 1 m = P(he male survives a leas one more ear (given i survived hus far)) =.40/.70 = 4/7 p 1 f = P(he female survive a leas one more ear (given i survived hus far)) =.40/.70 = 4/7 Clearl, , so p + p p + p. Eample 2. ( + ) For calculaing lassurvivor acuarial presen values, = + is commonl used. However, i urns ou ha his formula does no hold in general. Consider a copula model (e.g., Hougaard s copula wih Weibull marginals): Le he bivariae survival funcion (, ) P{ X > and Y > } = C( S ( ) S ( ) S () S 1, 2 =, where m j = > = σ j P{ X } ep for j = 1, 2, and j m j 4
5 1 α [ ] α ( ) α C u, v = ep ( lnu) + ( lnv). Noe ha C ( u, 1) = u and C (, v) = v bivariae survival funcion ( ) 1, hus he marginal survival funcions derived from he S, coincide wih S 1 () and (), respecivel. Le X be he "female age a deah" random variable, and le Y be he "male age a deah" random variable as defined earlier. We consider he model wih m 1 = , σ 1 = 8. 99, m 2 = 85.98, σ 2 =11. 24, α = (hese parameers come from using he maimum likelihood mehod on eperience daa for join annui conracs from an insurance f compan see Youn and Shemakin, 1999) and compue he ne single premiums 65, S 2 m 70, 65:70, and 65:70 ( 65:70 and 65:70 were compued using he bivariae survival funcion wih all he compuaions done on Mahemaica). Wih i =. 05, we have he following resuls: f 65 = m, 70 = , 65 :70 = = :70, and f m :70 = The epression ields an error of more han 10 %! The following able demonsraes he values of he raio f 65 + m 70 65:70 65:70 for various values of he associaion parameer α and he ineres rae i. Table 1. The raio f 65 + m 70 65:70 65:70 for various values of α and i. 5
6 Ineres rae 3% 6% 9% 12% ssociaion α ssociaion α =1 corresponds o independence beween male and female lives. Higher values of associaion and higher ineres raes bring abou a subsanial discrepanc beween he eac value and is approimaion : :70 f m Proposiions and nalsis. We presen a modified version of equali (1) T ( ) T( ) = T() + T() + ha correcl relaes he lifeimes of joinlife and lassurvivor sauses o he lifeimes of single sauses. Firs, le us closel eamine wh equali (2) p + p = p + p is no rue in general. We sar wih absrac probabili argumens, raher han morali argumens. The reason is ha man of us are so familiar wih he equali (2), i is worhwhile o sep back a lile. Le, B, C, and D be random evens. Then he condiional probabiliies of B and B given C D are relaed b he equaion 6
7 (a) P( B C D) + P( B C D) = P( C D ) + P(B C D ). Noe ha he same idenical condiion (ha C D be rue) is presen in each of he four probabiliies; o change some of hese condiions would creae a saemen which is no alwas rue. For eample, (b) P( B C D) + P( B C D) P( C ) + P(B D ) in he general case (alhough i ma be fairl close in man cases). We will now ranslae his resul ino acuarial erms. Le = {X > + }, B = {Y > + }, C = {X > }, and D = {Y > }. Then P( B C D) = p, P( B C D) = p, P( C ) = p, and P(B D ) = p, so inequali (b) shows us ha p + p p + p. We see ha he equali (2) does no hold because p makes no assumpions on he curren saus of Y and p likewise makes no assumpions on wheher X survives o age or no, while p and p require boh X> and Y> as condiions. The same is rue for he relaion beween T(), T(), T() and T( ). To define T(), one makes no assumpions on he curren saus of Y and, o define T(), one makes no assumpions on wheher X survives o age or no, bu o define T() and T( ), boh X> and Y> are required. In order o relae join life funcions wih single life funcions, one needs o consider single sauses in a join cone wih spousal informaion. 7
8 Noaion. We inroduce ( ) o denoe a saus of a personaged whose spouse is a personaged. Thus he fuure lifeime random variables for he sauses ( ) and ( ) are given b T( )=X, defined when boh X> and Y>, and T( )=Y, defined when boh X> and Y>. Now we can sae ha T()=min(X,Y), defined when boh X> and Y>, and equals min(t( ),T( )), and T( )=ma(x,y), defined when boh X> and Y>, and equals ma(t( ),T( ). The condiional survival funcions for he sauses ( ) and ( ) would become ( ( ) > ) = P( X > + X > Y ) p = P T, > and ( ( ) > k) = P( Y > + X > Y ) p = P T, >. We now can properl sae (4) p + p = p + p. We sae he following proposiion wihou an furher proof. Proposiion 1. (5) T()=min(T( ),T( )), (6) T ( ) (7) T()+ T ( ) =ma(t( ),T( )), and = T( )+T( ). 8
9 We wan o noe ha, while T( ), T( ), T(),and T ( ) are all defined on he common domain X>, Y>, T() and T() are no: in a join cone, T() is defined when X>, Y>0 and T() is defined when X>0, Y>. In such a cone, T() and T() do no have a common domain and canno be added as random variables. Thus, (7) ma be considered as a correcion of (1). Proposiion 2. There eis random variables X and Y, ha for some,, and (8) p + p p + p (9) + +. Proof. Saemen (8) is demonsraed b Eample 1. Saemen (9) is demonsraed b Eample 2. We noe ha epressing p and p in a join cone would be ( ) ( ) (10) = P T() > = P X > + X >, Y > 0 p ( ) ( ) (11) = P T() > = P Y > + Y >, X > 0 p. In he same vein, idenifing T() wih T( 0) would be mahemaicall correc, alhough is inerpreaion ma seem unnaural. In he ppendi, we eamine he difference beween T () and T( ) as he relae o life insurance premiums. furher reamen of insurance premiums wih spousal saus is 9
10 given in "Pricing Pracices for Join Las Survivor Insurance" b Youn and Shemakin, RCH If lives X and Y are indeed independen, he saemens (2) and (3) become rue. Unforunael, recen sudies show ha survivals of pairs of husbands and wives are no independen. (See nnui Valuaion wih Dependen Morali b Frees, e. al.) The Proposiion 2 is no as absrac or rivial as i migh seem. n assumpion of equali in (2) and (3) is he basis of man imporan relaionships in muliple life morali funcion heor. The Third Eaminaion of he Socie of cuaries ess knowledge of muliple life morali funcions (among oher opics) and hisoricall has made use of hese formulae. 5. Conclusion. Wha kind of independence assumpion is required o creae equali in (8)? The answer is, as he condiions in he epressions (10) and (11) indicae, he morali rae of he female or he male should no depend on wheher he have a surviving spouse or no, nor on he surviving spouse s age. This is generall assumed in pracice. Insurance companies do no classif insurers according o wheher one has a surviving spouse or no, nor o spouse s age. I is worh noing ha, according o a copula model as illusraed b Tables 24 in he ppendi, he life insurance premiums wih spousal saus classificaion are lower han hose wihou he classificaion. The percenage differences are higher for older spouses and for higher ineres raes. 10
11 References BOWERS, N.; GERBER, H.; HICKMN, J.; JONES, D.; and NESBITT, C. (1997) cuarial Mahemaics, Schaumburg, Ill.; Socie of cuaries. FREES, E.; CRRIERE, J.; and VLDEZ, E. (1996) nnui Valuaion wih Dependen Morali, Journal of Risk and Insurance, Vol. 63, YOUN, H.; and SHEMYKIN,. (1999) Saisical specs of Join Life Insurance Pricing, 1999 Proceedings of he Business and Economic Saisics Secion of he merican Saisical ssociaion, YOUN, H.; and SHEMYKIN,. (2000) Pricing Pracices for Join Las Survivor Insurance", cuarial Research Clearing House,
12 ppendi To illusrae an effec he difference beween T () and ( ) insurance, we compare insurance premiums based on T () and ( ) T has in pricing life T. k+ 1 We noe ha = P( T() > k) = P( X > + k X ), = v p p ) and k p > ( ( ) > k) = P( X > + k X > Y ) ( k k+ 1 k p = P T, >, and compue he insurance k+ 1 ( k k+ 1 premium wih spousal saus = v p p ). The following ables demonsrae he raio / beween he premium values for females age wih and wihou spousal classificaion, spousal ages allowed o var. Thus, denoes female s age and denoes her spouse s age. Compuaions were preformed according o Hougaard s copula model wih Weibull marginals m 1 = , σ 1 = 8.99, m 2 = , σ 2 =11. 24, and α = Ineres rae varies from 3% o 7%. Table 2. The raio / wih ineres rae 3%. Male ge Female ge Table 3. The raio / wih ineres rae 5%. Male ge Female ge
13 Table 4. The raio / wih ineres rae 7%. Male ge Female ge
Mortality Variance of the Present Value (PV) of Future Annuity Payments
Morali Variance of he Presen Value (PV) of Fuure Annui Pamens Frank Y. Kang, Ph.D. Research Anals a Frank Russell Compan Absrac The variance of he presen value of fuure annui pamens plas an imporan role
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationEconomics 140A Hypothesis Testing in Regression Models
Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationVector Autoregressions (VARs): Operational Perspectives
Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101115. Macroeconomericians
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationGoRA. For more information on genetics and on Rheumatoid Arthritis: Genetics of Rheumatoid Arthritis. Published work referred to in the results:
For more informaion on geneics and on Rheumaoid Arhriis: Published work referred o in he resuls: The geneics revoluion and he assaul on rheumaoid arhriis. A review by Michael Seldin, Crisopher Amos, Ryk
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More information4.2 Trigonometric Functions; The Unit Circle
4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationTask is a schedulable entity, i.e., a thread
RealTime Scheduling Sysem Model Task is a schedulable eniy, i.e., a hread Time consrains of periodic ask T:  s: saring poin  e: processing ime of T  d: deadline of T  p: period of T Periodic ask T
More informationResearch Question Is the average body temperature of healthy adults 98.6 F? Introduction to Hypothesis Testing. Statistical Hypothesis
Inroducion o Hypohesis Tesing Research Quesion Is he average body emperaure of healhy aduls 98.6 F? HT  1 HT  2 Scienific Mehod 1. Sae research hypoheses or quesions. µ = 98.6? 2. Gaher daa or evidence
More informationState Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University
Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween
More informationReturn Calculation of U.S. Treasury Constant Maturity Indices
Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationII.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal
Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.
More informationRisk Management of Policyholder Behavior in EquityLinked Life Insurance
Risk Managemen of Policholder Behavior in EquiLinked Life Insurance Anne MacKa, Maciej Augusniak, Carole Bernard and Mar R. Hard April 8, 2015 Absrac The financial guaranees embedded in variable annui
More informationRisk Management of Policyholder Behavior in EquityLinked Life Insurance
Risk Managemen of Policholder Behavior in EquiLinked Life Insurance Anne MacKa, Maciej Augusniak, Carole Bernard and Mar R. Hard April 9, 2015 Absrac The financial guaranees embedded in variable annui
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationAn empirical analysis about forecasting Tmall airconditioning sales using time series model Yan Xia
An empirical analysis abou forecasing Tmall aircondiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationWhy Do Real and Nominal. InventorySales Ratios Have Different Trends?
Why Do Real and Nominal InvenorySales Raios Have Differen Trends? By Valerie A. Ramey Professor of Economics Deparmen of Economics Universiy of California, San Diego and Research Associae Naional Bureau
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationTime Series Analysis Using SAS R Part I The Augmented DickeyFuller (ADF) Test
ABSTRACT Time Series Analysis Using SAS R Par I The Augmened DickeyFuller (ADF) Tes By Ismail E. Mohamed The purpose of his series of aricles is o discuss SAS programming echniques specifically designed
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
More informationA Mathematical Description of MOSFET Behavior
10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical
More informationSupplementary Appendix for Depression Babies: Do Macroeconomic Experiences Affect RiskTaking?
Supplemenary Appendix for Depression Babies: Do Macroeconomic Experiences Affec RiskTaking? Ulrike Malmendier UC Berkeley and NBER Sefan Nagel Sanford Universiy and NBER Sepember 2009 A. Deails on SCF
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationUsefulness of the Forward Curve in Forecasting Oil Prices
Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,
More informationThe Grantor Retained Annuity Trust (GRAT)
WEALTH ADVISORY Esae Planning Sraegies for closelyheld, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business
More informationImpact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences
S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough
More informationDensity Dependence. births are a decreasing function of density b(n) and deaths are an increasing function of density d(n).
FW 662 Densiydependen populaion models In he previous lecure we considered densiy independen populaion models ha assumed ha birh and deah raes were consan and no a funcion of populaion size. Longerm
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationHANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.
Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can
More informationCointegration Analysis of Exchange Rate in Foreign Exchange Market
Coinegraion Analysis of Exchange Rae in Foreign Exchange Marke Wang Jian, Wang Shuli School of Economics, Wuhan Universiy of Technology, P.R.China, 430074 Absrac: This paper educed ha he series of exchange
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationUSE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES
USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationChapter 6 Interest Rates and Bond Valuation
Chaper 6 Ineres Raes and Bond Valuaion Definiion and Descripion of Bonds Longerm debloosely, bonds wih a mauriy of one year or more Shorerm debless han a year o mauriy, also called unfunded deb Bondsricly
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationThe Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas
The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationInsurance: Mathematics and Economics. Tail bounds for the distribution of the deficit in the renewal risk model
Insurance: Mahemaics and Economics 43 (8 97 Conens liss available a ScienceDirec Insurance: Mahemaics and Economics journal homepage: www.elsevier.com/locae/ime Tail bounds for he disribuion of he defici
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.
The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationPREMIUM INDEXING IN LIFELONG HEALTH INSURANCE
Far Eas Journal of Mahemaical Sciences (FJMS 203 Pushpa Publishing House, Allahabad, India Published Online: Sepember 203 Available online a hp://pphm.com/ournals/fms.hm Special Volume 203, Par IV, Pages
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees
1 The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationACTUARIAL FUNCTIONS 1_05
ACTUARIAL FUNCTIONS _05 User Guide for MS Office 2007 or laer CONTENT Inroducion... 3 2 Insallaion procedure... 3 3 Demo Version and Acivaion... 5 4 Using formulas and synax... 7 5 Using he help... 6 Noaion...
More informationSPEC model selection algorithm for ARCH models: an options pricing evaluation framework
Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,
More informationThe Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies
1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany
More informationThis document is downloaded from DRNTU, Nanyang Technological University Library, Singapore.
This documen is downloaded from DRNTU, Nanyang Technological Universiy Library, Singapore. Tile A Bayesian mulivariae riskneural mehod for pricing reverse morgages Auhor(s) Kogure, Asuyuki; Li, Jackie;
More informationFifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance
Fifh Quaniaive Impac Sudy of Solvency II (QIS 5) Naional guidance on valuaion of echnical provisions for German SLT healh insurance Conens 1 Inroducion... 2 2 Calculaion of besesimae provisions... 3 2.1
More informationOptimal Life Insurance Purchase, Consumption and Investment
Opimal Life Insurance Purchase, Consumpion and Invesmen Jinchun Ye a, Sanley R. Pliska b, a Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA b Dep.
More informationChapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
More informationChapter 2 Kinematics in One Dimension
Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationForecasting Malaysian Gold Using. GARCH Model
Applied Mahemaical Sciences, Vol. 7, 2013, no. 58, 28792884 HIKARI Ld, www.mhikari.com Forecasing Malaysian Gold Using GARCH Model Pung Yean Ping 1, Nor Hamizah Miswan 2 and Maizah Hura Ahmad 3 Deparmen
More informationChapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )
Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y
More informationDopamine, dobutamine, digitalis, and diuretics during intraaortic balloon support
Dopamine, dobuamine, digialis, and diureics during inraaoric balloon suppor Sephen Slogoff, M.D. n his presenaion, should like o discuss some conceps of drug herapy for inraaoric balloon paiens. Figure
More informationJCER DISCUSSION PAPER No.136
JCER DISCUSSION PAPER No.136 Belief changes and expecaion heerogeneiy in buy and sellside professionals in he Japanese sock marke Ryuichi Yamamoo and Hideaki Hiraa February 2012 公 益 社 団 法 人 日 本 経 済 研
More informationSteps for D.C Analysis of MOSFET Circuits
10/22/2004 Seps for DC Analysis of MOSFET Circuis.doc 1/7 Seps for D.C Analysis of MOSFET Circuis To analyze MOSFET circui wih D.C. sources, we mus follow hese five seps: 1. ASSUME an operaing mode 2.
More information