Loyalty Rewards and Gift Card Programs: Basic Actuarial Estimation Techniques


 Clinton Sanders
 3 years ago
 Views:
Transcription
1 Loyalty Rewads and Gift Cad Pogams: Basic Actuaial Estimation Techniques Tim A. Gault, ACAS, MAAA, Len Llaguno, FCAS, MAAA and Matin Ménad, FCAS, MAAA Abstact In this pape we establish an actuaial famewok fo loyalty ewads and gift cad pogams. Specifically, we pesent models to estimate edemption and beakage ates as well as to estimate cost and value fo use in both accued cost and defeed evenue accounting methodologies. In addition, we povide guidance on vaious issues and consideations that may be equied of an analyst when woking with loyalty ewads and gift cad pogams. Keywods: loyalty ewads, gift cads, edemption ate, beakage, liability estimation, cost estimation, accued costs, defeed evenue. INTRODUCTION The size and scope of loyalty ewad pogams has gown immensely ove the last seveal decades. Since the ise of ailine fequent flye pogams in the 980s, loyalty pogams in thei moden fom have become deeply intetwined within copoate maketing stategies. Fom the financial sevices industy with its ewadsbased cedit cads, to the hospitality sevices industy with hotel ewad pogams, to gift cads and othe coupons issued by common bick and mota industies such as food sevices and clothing etailes, to the fequent flye ailine miles pogams, ewad pogams can now be found almost eveywhee. While ewading fequent customes with peks, benefits, discounts o complimentay poduct has been a longstanding business pactice in maketing sphees, it has become eve moe impotant to othe aeas of business pactices within companies. In fact, membe loyalty and gift cad pogams have moved into uppe managements companywide puview as a coe component of band stategies and ae futhemoe now often an integal pat of copoate identities themselves. The elevation of impotance now equies pactitiones to stetch acoss the sometimes siloed pactices of maketing, finance, accounting, and infomation technology depatments within a company. Rewad pogams essentially consist of pomises made today to delive something tomoow, o next yea, o potentially neve. The natue of ewad pogams often bings with it significant challenges. Many ewad pogams stuctues ae built aound uncetain Casualty Actuaial Society EFoum, Summe 0
2 futue events: contingencies of how much, when, and if. Additionally, a pogam s tems and conditions can change as the pogam evolves, leading to mateial changes to the benefits that paticipating membes can obtain, o to the costs that sponsos will encounte. These uncetainties often obfuscate the value o costs that a sponso is pomising. Futhemoe, these uncetainties can often challenge one s ability to estimate the futue benefits and costs of a pogam in an accuate and substantive way. Fotunately, the amount of infomation collected and available to pogam povides pesents an exceptional oppotunity to tuly undestand the costs and evenue dives of thei pogams and to measue them in an accuate and timely manne. This lage amount of infomation can be used to design pogams that povide bette ewads to thei membes, maximize the value of the pogam to its sponso by geneating incemental evenue due to inceased membes loyalty, and help in poviding quantified feedback to management and othe financially inteested playes. It is ou hope that the tools pesented in this pape can povide guidance to an analyst (and an actuay!) as to how to think about some of the economic fundamentals of loyalty ewads and gift cads pogams, and to place a moe stuctued quantitative famewok aound undestanding and measuing thei impact on the companies that offe them.. OVERVIEW OF REWARD PROGRAMS. Pogam Basics The basic pemise of ewad pogams consists of membes puchasing goods o sevices in exchange fo a pomise, by the ewad pogam sponso, to povide additional futue goods, sevices, o value to the membe. One of the most impotant issues when attempting to undestand the wokings of a ewad pogam is to undestand the Tems & Conditions (T&C) that undelie the pogam. The T&C ae essentially laws of the pogam fom which all membes individual and aggegate behavios emege. The impotance of the T&C cannot be ovestated. Fo example, thee is geneally no equiement that membes actually claim the goods o sevices pomised to them and in many cases, T&Cs ae in place that make the pomises disappea though expiation and fofeitue ules. Theefoe, thee is no guaantee that the sponsos will eve be equied to make good on thei pomises. In fact, Casualty Actuaial Society EFoum, Summe 0
3 it is usually the case in ewad pogams that less than 00% of the ewads pomised will eve be claimed, o edeemed by the membes On a pogam sponso level, undestanding the potential fo ewad edemptions, as well as the incued cost when a ewad is edeemed, is a citical execise. The undelying costs of the pogam (o the quantification of the elative fai value being povided to membes) should be teated just as impotantly as the associated lift in evenues that is expected to be diven by the pogam. Togethe the two components dive pofitability, o lack theeof, fo the pogam s sponso. In fact, this undestanding can enhance decision making suounding the most pofitable membes and open up the potential to expand that pofitability. On the othe hand, high cost/low evenue centes o ineffective pomotional maketing campaigns can be phased out in a timely, costeffective, and custome peceptionsensitive manne. The uncetainties suounding the cost of the pomises made by the sponso, which ae themselves estimated based upon edemption ates and costs at edemptions, can lead to poo financial decision making and even pooe disclosue of the economic impacts that these pogams have on the sponso. The lack of guidance and established evaluation standads and methods, the uncetainties suounding the ultimate costs, as well as the fact that potential benefits on pomises may be immediate wheeas the associated costs can be defeed, sometimes into the fa distant futue, may have ceated an envionment fo some sponsos whee it is easie to addess the issue late athe than now. Due to the appaent challenges of undestanding how best to estimate and measue the uncetainties of both edemption fequency and edemption cost/value, it may sometimes appea to be a daunting task to estimate eithe. Howeve, actuaies and thei techniques ae uniquely pepaed to tackle these issues. By applying many commonly accepted actuaial appoaches, with appopiate modifications to addess the uniqueness of ewad pogams, obust estimates of both edemption ates and costs can be deived. In this pape we will geneically efe to the cuency of ewad pogams as points, the main benefactos of the pogams as membes, and the entities that ceate and manage the pogam on an ongoing capacity as sponsos. In addition, we will geneically efe to the value o cost of the awad simply as the cost, though the specific teminology that would Casualty Actuaial Society EFoum, Summe 0 3
4 be used would be dependent on the accounting standads unde which the pogam opeates. In pactice, thee is a geat divesity of names fo these things but fo claity and simplicity we will standadize them in this pape.. Total Cost in Rewad Pogams In its most basic fom, a ewad pogam s total cost can geneally be boken out into thee components: a cuency component, a edemption ate component, and a cost component. Points x Redemption Rate x Cost pe Point = Total Cost (.) This geneic equation will be used in a vaiety of applications. Geneally the fist item, the cuency, points, o miles, is a known value. In fact it is typically the only numbe known at the time that an analysis is pefomed. The edemption ate epesents the pecentage of points which ae expected to be utilized o edeemed by the pogam membes. The cost pe point epesents the economic value of each point given that such point will be edeemed. This fomula can be used in balance sheet contexts whee an analyst is inteested in valuing eithe the accued costs o the associated defeed evenue of a pogam. The fomula can also be used in income statement contexts whee the analyst is inteested in valuing eithe the incemental cost of an issued awad o the incemental defeed evenue at point of sale. When consideing the fomula above, it is impotant to maintain a common basis fo all thee components. Fo example, one should not apply a edemption ate expessed as a pecentage of issued points to an outstanding point balance. In the subsequent sections we will discuss the edemption ate and cost pe point components of the model in futhe detail. Casualty Actuaial Society EFoum, Summe 0 4
5 3. REDEMPTION RATE ESTIMATION APPROACHES One of the key components of nealy evey loyalty ewad o gift cad pogam is the edemption ate. The edemption ate is also fequently the single most challenging component to estimate. Developing a functional and pedictive edemption ate model can be an execise equiing significant time and effot. In many instances the degee of difficulty can be geatly inceased by data quality and availability issues o, on the opposite end of the spectum, ovewhelmingly lage quantities of data that ae difficult to manipulate and oganize. Redemption ates ae geneally expessed as a function of one of two diffeent bases; as the pecentage of the points that ae outstanding (points that have neithe been edeemed no fofeited) as of the valuation date o as a pecentage of the cumulative amount of points issued to date to pogam membes. As such it is impotant to keep in mind the basis on which edemption ates ae expessed. Thee ae specific qualities by which evey edemption ate must abide. Redemption ates, when expessed as a pecentage of cumulative points issued, must always be bounded by a minimum of zeo and by a maximum of unity. This can be intepeted to mean that thee can neve be moe point edemptions in the futue than the numbe of points issued to date o outstanding as of the evaluation date, and that thee can neve be negative edemptions, in aggegate. A situation whee histoical edemption ates ae below zeo o geate than unity would likely be due to data anomalies o exceptional situations elated to a pogam s T&C that need to be bette undestood and coected befoe moving fowad with the pojection of ultimate edemption ates. Beakage is fequently a facto of inteest. Beakage epesents the potion of points issued (o outstanding) that will neve be edeemed. Points that ae boken will eithe fofeit out of the pogam o sit domant until the pogam itself ceases to exist. The exact fate of the boken points is detemined by the T&C of the pogam. The beakage ate is, by definition, the complement of the edemption ate. Theefoe, unity less the edemption ate epesents the beakage ate. Because of the simple elationship between edemption and beakage ates, we will focus on the edemption ate heeafte with the knowledge that we can eadily convet the edemption ate into the beakage ate as needed. It should be noted Casualty Actuaial Society EFoum, Summe 0 5
6 that while out of the scope of this pape, an analyst may be equied to conside applicability of elevant laws elating to escheat popety and how these laws may potentially affect the pope teatment of beakage. The appoaches fo estimating an ultimate edemption ate fo loyalty ewad and gift cad pogams illustated in this pape povide an estimate of the ultimate edemption ate expessed as a pecentage of cumulative points issued as of the valuation date of the analysis. This edemption ate on issued points can be conveted to a edemption ate on outstanding points, if needed. In addition, it should be noted that thee ae altenative appoaches which may be moe appopiate given a pogam s stuctue, data availability, o othe easons that could be compaably easonable to the methods contained in this pape. 3. Point Issuance Peiod Method The Point Issuance Peiod method is built on the pemise that points can be tacked fom the peiod in which they wee eaned by membes until thei ultimate edemption o domancy/fofeitue, and that the lifecycle of a point fom olde issuance peiods can be applied to points issued in subsequent peiods. While it can be exceedingly difficult fo a pogam s sponso to tack individual points and to come up with meaningful pedictions of how, o even if, the points will be used, gouping points by issuance peiods can make the undelying pocess statistically moe pactical and povide accuate aggegate estimates. Constucting Point Redemption Tiangles The fist step to this method consists of constucting histoical point edemption tiangles. Redeemed points ae gouped by issuance peiod, and cumulative point edemptions associated with that issuance peiod (at multiple evenly spaced evaluations) ae obtained in ode to effectively tack how histoical edemptions ae elated to time since the oiginal issuance peiod. Constucting tiangles in this manne is analogous to constucting a cumulative loss development tiangle, but instead of using an accident peiod we use an issuance peiod. t In the tiangle below, R i epesents the cumulative numbe of edeemed points, out of the total points issued in issuance peiod i at time t. Casualty Actuaial Society EFoum, Summe 0 6
7 Issuance Evaluation Age Peiod 3 4 0X 3 R 0 X R 0 X R 0 X R 0X 3 R 0 X R 0 X R 0 X 0X3 R 0 X 3 R 0 X 3 0X4 R 0 X X We divide the cumulative point edemptions fo each issuance peiod by the espective numbe of points that wee issued in that peiod to geneate a cumulative edemption ate tiangle. In the tiangle shown below, t i epesents the cumulative numbe of edeemed points issued in peiod i at time t divided by the total numbe of points issued in that issuance peiod. It should be noted that the issued points in each issuance peiod ae effectively fozen so that the denominato acoss each ow is constant. Issuance Evaluation Age Peiod 3 4 0X 3 0 X 0 X 0 X 0X 3 0 X 0 X 0 X 0X3 0 X 3 0 X 3 0X4 0 X X Thee ae two pimay benefits to immediately conveting the edeemed points into edemption ates. Fist, this emoves the effect of changing volumes of issued points between issuance peiods and it also nomalizes the edemption activities between peiods making them moe easily compaable. Second, this appoach focuses diectly on edemption ates fom the outset of the analysis, which allows the analyst to immediately veify the bounday conditions so that edemption ates can neithe exceed unity (i.e., 00%) no be below 0%. Casualty Actuaial Society EFoum, Summe 0 7
8 Estimating Ultimate Redemption Rates Using the edemption ate tiangle that was developed in the pevious step, it should be immediately clea that one can apply standad actuaial pojection methods (such as the chain ladde appoach) to obtain estimates of the ultimate edemption ates by issuance peiod. Thee is geneally no significant diffeence in methodology between estimating ultimate edemption ates on issued points and estimating ultimate losses in an insuance application, though thee can be diffeent consideations that an analyst may need to contemplate (e.g., loyalty pogams may equie consideation of pomotions and expansion of enollment into new classes of membes instead of insuance consideations of claim handling stability and changes in undelying mix of coveages). Standad actuaial pojection techniques on tiangula data ae coveed in many othe souces of actuaial liteatue and as such we will not expand on that topic in this pape. At the end of the analysis one should have a completed tiangle as is shown below. Issuance Evaluation Age Peiod 3 4 Ult 0X 3 4 ult 0 X 0 X 0 X 0 X 0 X 0X 3 4 ult 0 X 0 X 0 X 0 X 0 X 0X3 3 4 ult 0 X 3 0 X 3 0 X 3 0 X 3 0 X 0X4 3 4 ult 0 X 4 0 X 4 0 X 4 0 X 4 0 X 3 4 The ultimate edemption ates by issuance yea can be used as is fo each individual issuance yea o, altenatively, a single volume weighted edemption ate on all issued points can be calculated if the analyst is focused on the oveall ultimate edemption ate ( URR ) fo all points issued by a loyalty ewad o gift cad pogam. We note that this appoach can be successfully applied to loyalty ewad o gift cad pogams that include a point expiation policy in thei T&C. In cases whee issued points only emain valid fo a fixed peiod afte issuance, an analyst can quickly obtain the actual URR fo each issuance peiod. Such an expiation policy can significantly facilitate the URR estimation fo moe ecent issuance peiods since the ultimate peiod is defined by the pogam sponso. Casualty Actuaial Society EFoum, Summe 0 8
9 Fo pogams without a point expiation policy, additional wok may be needed to obtain an estimated URR. Fo example, in many instances thee will be no histoical infomation available upon which to base futue expected point edemption activities beyond the most ecent evaluation date. Often, this is simply a esult of a ewad pogam not being sufficiently matue to have eached its point edemption ultimate in any histoical issuance peiod as of the evaluation date. Such an issue is compaable to detemining a tail facto in conventional loss development tiangles. In such instances an analyst may find that fitting a cuve that exhibits decay chaacteistics is the most appopiate method to apply. Obviously, multiple such cuves can be used to povide multiple pojections. In such instances, it is also ecommended that the analyst additionally apply a testing o anking appoach in ode to detemine which cuve might povide the best fit to histoical data. Fo a full numeical example of this method please efe to Appendix Aggegate Membe Join Peiod Method The Aggegate Membe Join Peiod method assumes that pogam membes cumulative edemption activity at any given time is elated to the time elapsed since the membes have joined the pogam. Membes ae typically combined into join peiod cohots so that points eaning o edemption activity ove the lifetime of the cohot can be elated to the age o matuity of the membes included in the cohot. Activities can be taced fom the date that membes fist enoll into the ewad pogam (join peiod) until thei ultimate lapse (i.e., fofeitue), depatue, o domancy. We will geneically efe to this as domancy, though the pogamspecific T&C will dictate if points actually do get fofeited out of membes accounts o not. In this method, the tiangle constuction includes membe join peiod cohot activity fo both domant and active membes. As a esult, any obseved changes in cumulative edemption activity between evaluation ages ae only attibutable to membes who emained active between evaluation peiods. Casualty Actuaial Society EFoum, Summe 0 9
10 Constucting Point Redemption and Points Issued Tiangles The fist step to the Membe Join Peiod method consists of constucting histoical point edemption tiangles. Redemption tiangles in this method use cumulative membe point edemptions at vaious matuities. Multiple evenly spaced evaluations of the cumulative edeemed points ae obtained so that one can effectively tack how edemptions ae elated to time passed since the oiginal membes join peiod. This tiangle constuction, simila to the Point Issue Peiod appoach descibed ealie, is also analogous to constucting a cumulative loss development tiangle, but instead of using an accident peiod appoach we use a join peiod appoach. t In the tiangle below, R j epesents the cumulative numbe of edeemed points, out of the cumulative points issued to membes joining in peiod j, at time t afte the join date. Join Evaluation Age Peiod 3 4 0X 3 R 0 X R 0 X R 0 X R 0X 3 R 0 X R 0 X R 0 X 0X3 R 0 X 3 R 0 X 3 0X4 R 0 X X In the tiangle below, I t j epesents the cumulative issued points associated with membes who joined in peiod j, at time t afte the join date. Join Evaluation Age Peiod 3 4 0X 3 I 0 X I 0 X I 0 X I 0X 3 I 0 X I 0 X I 0 X 0X3 I 0 X 3 I 0 X 3 0X4 I 0 X X By dividing the cumulative edeemed point tiangle by the cumulative issued point tiangle we obtain the tiangle shown below, which epesents the cumulative edemption Casualty Actuaial Society EFoum, Summe 0 0
11 ates ( ) fo membes, by join peiod. The cumulative edemption ates ae expessed as a t j pecentage of cumulative issued points. Unlike the Point Issuance Peiod method whee the points included in the denominato ae constant, the points included in this denominato continue to gow at each evaluation peiod, as long as at least one membe included in a join peiod cohot continues to be active in the pogam and eans moe points. Join Evaluation Age Peiod 3 4 0X 3 0 X 0 X 0 X 0X 3 0 X 0 X 0 X 0X3 0 X 3 0 X 3 0X4 0 X X Given this tiangle, an actuay can apply standad actuaial pojection methods to estimate the patten of futue estimated cumulative edemption ates at ultimate fo each join peiod. Pojected values coespond to the aeas within the boxed egion in the tiangle below. Join Evaluation Age Peiod 3 4 0X 3 0 X 0 X 0 X 0X 3 0 X 0 X 0 X 0X3 3 0 X 3 0 X 3 0 X 3 0X4 3 0 X 4 0 X 4 0 X X 4 0 X 4 0 X X 4 Teminal Redemption Peiod Consideations The Membe Join Peiod Method does not mathematically esolve itself to povide a clea cutoff whee the analyst can cease development. In fact, because of the cuvelike natue of the undelying cumulative data, mechanical development could pepetuate indefinitely with this method wee an analyst to poject out to infinity. Theefoe, it is necessay to establish a teminal peiod (o matuity) out to which the pojection should be pefomed. In geneal, thee is no eason that the teminal peiod used cannot vay by join peiod. Casualty Actuaial Society EFoum, Summe 0
12 An actuaial analyst should conside multiple factos befoe establishing a teminal edemption peiod fo the edemption ate pojection. Geneally, consideations include, but ae not limited to: Cuent pogam Tems & Conditions Expected futue changes in pogam Tems and Conditions Membe, o points, domancy pattens and tends Relative contibution of point activities associated with membes at each espective expected domancy peiod Additionally, an actuay should discuss the issue with the pogam sponso s management in ode to ensue a thoough undestanding of the pogam befoe implementing a specific matuity at which to end development. Lastly, it should be mentioned that in some instances, an analyst may want to avoid pojecting out to the estimated time of domancy fo the last active membe(s) in a join yea in the Membe Join Peiod method (i.e., the time at which all membes ae domant). The eason is that, wee one to do this, the edemption ate povided by the model could oveestimate the tue edemption ate since that estimated time would implicitly account fo points which would not yet have been eaned as of the time of the evaluation. This would be inconsistent with the natue of establishing liability estimates as of a detemined evaluation date fo the points outstanding as of that date. Fo a full numeical example of the Membe Join Peiod appoach please efe to Appendix Point Inventoy Method and Choice of Redemption Estimation Method It would be natual fo individuals to ty to daw compaisons between conventional inventoy systems and loyalty pogams. While such constuctions ae helpful in placing loyalty pogam opeations into a well established and undestood famewok of conventional inventoy systems, thee exists a notable diffeence between conventional inventoy and a loyalty pogam inventoy system. The pimay eason that the compaison is not pefect is due to the fact that tangible inventoy typically has a value that is geneally Casualty Actuaial Society EFoum, Summe 0
13 quantifiable via actual tansactional evidence at the time of acquisition o manufactue (i.e., the cost of puchasing o poducing an item included in the inventoy is known) wheeas the value of an issued and unedeemed point in a loyalty ewad pogam will not actually have a known cost until the date that that point is actually edeemed (if eve) sometime in the futue. Nevetheless, constucting an inventoy system that woks fo both financial epoting puposes and as a tool fo the analyst estimating the associated liability can still be a vey useful endeavo. Basic Oveview of Inventoy Systems Inventoy systems in loyalty pogams have simila stuctues to conventional inventoy systems. Below is a bief summay of the types.  Fist In, Fist Out: In this method, the oldest points owned by a membe ae the fist to get withdawn.  Last In, Fist Out: In this method, the newest points owned by a membe ae the fist to get withdawn. 3 Aveage Weighted Cost Method (a.k.a. Piggy Bank Method): In this method, the time at which a point is issued is ignoed and points go in and out of membes accounts iespective of when they wee issued (eithe because these dates ae intentionally disegaded o due to actual database constaints making them unavailable). As such, it is not possible to identify the exact issue time of any specific point and theefoe, it is neithe possible to identify the time of issuance fo any point that was edeemed o fofeited. In essence, evey point is completely impossible to distinguish fom evey othe point. Nevetheless, the aveage futue cost and aveage time of edemption can still be detemined. Geneally such a point inventoy system is constucted specifically to focus on membe point balances at any given time athe than to focus on the seies of tansactions that esult in a given balance. Inventoy Systems and Redemption Rate Estimation While thee is no specific ule as to the best edemption ate appoach to be used fo each inventoy system, o even which inventoy system should o should not be used, we believe that some methods moe natually accommodate the diffeent inventoy systems and make Casualty Actuaial Society EFoum, Summe 0 3
14 analyses moe tactable and moe easily explained. Fo example, the Point Issuance Peiod appoach geneally woks well unde a FIFO system. Howeve, the eviewing analyst may fequently be equied to conside issues which fall outside the scope of this pape befoe constucting o ecommending any specific inventoy system fo a given pogam. 3.4 Undestanding Redemption Rate Bases and Thei Application As noted peviously, edemption ates can be expessed in tems of eithe pecentage of outstanding points o pecentage of issued points. Both measuements ae potentially of inteest to an actuay and to a pogam sponso s management team. Up until this point, we have focused on estimating edemption ates stated on a points issued basis. Since thee is a quantifiable elationship between the two bases, one can geneally convet between the two as needed. Typically, edemption ates expessed as pecentage of issued points ae utilized in an income statement context, eithe fo defeed evenue o expense ecognition calculations as they occu though the accounting peiod. Convesely, edemption ates expessed as a pecentage of outstanding points ae typically used in a Balance Sheet context, eithe fo detemining unpaid liabilities o in estimating cumulative defeed evenue at the financial epoting date. Conveting Redemption Rate on Issued Points to Redemption Rate on Outstanding Points Fo the Point Issue Peiod method, the total edemption ate on outstanding points can be detemined using the following equation: OS, T i Ult T T I R I R / i i i i i (3.4.) The above equation can be intepeted as edemption ate on outstanding points fo issue peiod i is the poduct of the total ultimate edemption ate on issued points fo issue peiod i and the cumulative issued points less those points that have aleady been edeemed as of the evaluation date. This is then divided by the total outstanding points as of the evaluation date, which is itself equal to the total issued points less the total edeemed points. T epesents the evaluation date. Casualty Actuaial Society EFoum, Summe 0 4
15 In pogams that include fofeitue ules, the actuay must also subtact peviously fofeited points fom the denominato when expessing edemption ates as a pecentage of outstanding points. As the convesion fom issued to outstanding fo the Membe Join Peiod appoach is analogous to the method shown above, we have chosen not to show the equation. 3.5 Application to Gift Cads As peviously noted, the edemption ate appoaches descibed above can also be applied to the estimation of gift cad pogams edemption ates. The estimation method that is most appopiate is dependent on the natue of the pogam. Fo gift cad pogams whee cads ae typically not eused (i.e., additional value is neve o infequently added back to the cad afte initial issuance), the Point Issue Peiod method is pefeed. Fo gift cad pogams whee cad uses add value back to cads afte the initial cad issuance (i.e., cads can be eloaded ), the Aggegate Membe Join Peiod method is pefeed. 3.6 Consideations of the Intended Use of the Redemption Rate Estimate While it is geneally not the esponsibility of the actuay to detemine the appopiate use of the edemption ate in an accounting context, it is the esponsibility of the actuay to convey an appopiate undestanding of the natue of the edemption ate estimate to management. It should be kept in mind that the edemption ate estimate is exactly that, an estimate. In some cases, the detemination of a ange of easonable estimates aound the actuaial cental estimate povided to management may also be appopiate. The potential isk of undefunding the liability elated to the outstanding points (o unedeemed gift cads) may make management moe cautious when it comes to selecting the ultimate edemption ate to use in thei financial statements. As such, management may need to conside whethe the expected value o potentially a highe confidence level estimate (o a selection towad the high end of the ange of easonable estimates) is a moe appopiate estimate to use. Casualty Actuaial Society EFoum, Summe 0 5
16 Please efe to the Appendix fo futhe discussion on the potential accounting teatment of the methods. 4. COST OF REDEMPTION/VALUE OF DEFERRED REVENUE ESTIMATION APPROACHES In ode to fully undestand the economic natue of the tansactions elated to loyalty ewads o gift cad pogams, it is necessay to conside the costs incued by the plan sponso fo point edemptions (in an accued cost accounting appoach) o the value placed on the pomised futue edemptions (in a defeed evenue accounting appoach). We will geneically use the tems cost and value intechangeably heeafte, though the appopiate teminology will be detemined by the accounting appoach that the sponso uses fo financial epoting puposes. In some instances thee is little uncetainty suounding the value o cost of a point edemption as the point edemption oppotunities might be limited o piced in a fixed manne (i.e., a fixed numbe of points = a fixed amount of ewads). As such, no estimate of value is necessay. Fo example, in gift cad pogams the value of the tansaction is geneally aleady expessed in a cuency (i.e., the value that emains outstanding on the cad) and so the value to the cadholde is selfevident, egadless of when a edemption may eve occu. Howeve, in many ewad pogams, edemptions will occu in the futue and at a time when the value o cost of edemptions could be diffeent fom today and at values that ae not necessaily aleady expessed in an easily valuated fom. Since vaiations ove time in cost and value ae elatively common, it is impotant to conside how these change ove the duation of the expected edemptions. Costs can change fo a vaiety of easons: changes in T&C of the pogam, changes in edemption options available to membes, o even pice inflation of poviding loyalty ewads to membes at time of the edemption. Likewise, the actual value of ewads to the membes may also change ove time fo many of the same easons. To complicate mattes moe, many pogams offe multiple edemption options, many of which can vay, pehaps significantly, in cost o value fom each othe. A final complication elates to the detemination of the coect value of a point unde vaying accounting systems. Recent changes in intenational standads have intoduced the Casualty Actuaial Society EFoum, Summe 0 6
17 concept of "fai value" of a point to customes. This can be significantly diffeent fom the value that a pogam sponso believes to be easonable to use when estimating its outstanding liability, unde cuent US GAAP accounting standads. Since the objective of this pape is not to take a "deep dive" into the accounting wold, we will not discuss this issue any futhe. Howeve, an analyst should conside this issue and seek appopiate guidance when detemining the value of a point. Any fowad looking estimation of the potential cost of a point o of the value of a point equies a solid undestanding of the past, a thoough undestanding of expected futue changes, and a deep knowledge of the T&C of the pogam. The value of a point at time of issuance is a function of the value that the point will have at the time that it will actually be edeemed. In instances whee the value is constant ove time thee is no need to estimate that value (so long as the value is known today). When the value vaies, howeve, the value of a point at time of issuance is not likely to be the same as when that point is going to be edeemed. Unde these conditions, we can build a famewok that accommodates many potential scenaios of vaying values o costs. The basic pupose of the appoaches outlined in this pape is to detemine the expected cost o value of a point at the time of issuance in ode to include this vaiable in the cuent liability estimate. 4. Effectively Constant Cost/Value Pe Point Model Single Redemption Option This is the tivial example whee the value to the membe o cost to the company emains constant, o at least effectively constant, ove time. While, in this context, constant is elatively selfexplanatoy, effectively constant deseves moe explanation. When we efe to effectively constant, we efe to the fact that even though the cost o value of the ewad will change ove time, it is not expected to change between the issuance of the ewad pomise (i.e., the points) and the expected edemption of the points in etun fo that ewad. In such instances the value o cost of the pomised deliveable goods today, is the best indicato of the futue cost o value. Casualty Actuaial Society EFoum, Summe 0 7
18 4. Vaying Cost/Value Pe Point Model Single Redemption Option In many instances, the cost o value of a ewad could vay ove time in a manne that is easonably estimable. Examples of such situations ae plane tickets o hotel oom ewads, both of which ae impacted by elatively pedictable seasonal changes as well as geneal inflationay pessues. To incopoate changes in cost o value of points ove time into a pedictive famewok we can ceate a simple model. The model equies the following assumptions: ) A edemption patten, whee t is the pecentage of total point edemptions n t occuing in peiod t, and whee. 0. t ) An estimation of the costs o values that ovelap with the point edemption patten, t whee we define c as the cost o value of points edeemed at time t. With these two items an analyst can estimate the cuent aveage cost pe point as: 4.3 Multiple Redemption Options n t t c (4..) t This appoach essentially adds an exta level of complexity to the peceding method. This method includes a thid component, i.e., the utilization. This is stated in tems of the elative pecentages of all points that ae expected to be edeemed on each edemption option, in each futue peiod. This component eflects the fact that most ewads pogams offe multiple edemption options to thei membes. The objective is to captue the mix of futue point edemptions acoss a basket of goods that is available to membes. Once the utilization component has been defined, an analyst can apply this component to expected futue cost o value of each available awad type in each futue peiod to obtain the cuent weighted aveage cost pe point edeemed in each pospective peiod. In this way, the analyst can combine the estimated mix of edemptions with the espective costs associated at each expected time of edemption to obtain the total aveage cost o value pe point edeemed in the futue. Fo example, hotel pogams often allow thei membes to use thei eaned points to edeem fo hotels, ailine tickets and othe mechandise. Ailine pogams fequently allow Casualty Actuaial Society EFoum, Summe 0 8
19 thei membes to edeem fo fee flights and miscellaneous mechandise. In many instances, the cost o value of the multiple edemption options may vay significantly when viewed in a bypoint basis. While membes usually decide how to use thei points based on thei individual needs, that decision has a diect impact on the costs incued by a loyalty pogam sponso. Some ewad options can be significantly moe costly to a pogam than othes and theefoe it is cucial fo any pogam to have a good undestanding of its customes expected edemption behavio. The model equies the following assumptions; ) A edemption patten, whee t is defined as the pecentage of total point n t edemptions occuing in peiod t, and whee. 0. ) Estimation of costs o values fo each edemption options that ovelap with the point edemption patten, defined c t q as the cost o value of each edemption options at time t fo edemption option q. 3) Utilization pecentage, defined as edeemed at time t, fo edemption option q. k u.0 at each t, whee k is the total numbe of edemption options. q t q t t u q, which epesents the pecentage of total points t u q can vay ove time, howeve, With these thee items an analyst can estimate the total aveage cost pe point as: k n t t t cq uq (4.3.) t q 4.4 Additional Consideations in Cost/Value pe Point Models Redemption Patten The edemption patten can be estimated using eithe of the edemption ate methods descibed in Section 3. Altenatively, othe estimation appoaches not coveed in this pape may be used. Since edemption pattens can be expessed as eithe a pecentage of outstanding points o a pecentage of issued points, cae should be taken by the analyst to ensue that the appopiate patten is estimated and applied in a manne that is consistent with the intended pupose. Casualty Actuaial Society EFoum, Summe 0 9
20 Value o Cost at Time of Redemption t t x Thee is not necessaily any a pioi elationship between c and c, whee x is some time displacement fom t, though fequently the pogam T&C, business cycles, seasonal effects, and/o economic envionment will ceate some famewok into which to genealize futue costs. In addition, consideing expected futue inflation o pojected pice changes may be a easonable benchmak against which to detemine changes in the value o cost of futue ewad edemptions. Utilization Utilization is geneally expessed on a of the points expected to be edeemed basis. Theefoe it geneally ignoes futue points beakage. 5. ADDITIONAL GENERAL CONSIDERATIONS 5. Data Segmentation Just as with taditional actuaial analyses, data segmentation is vey impotant to conside in the analysis of any loyalty ewads o gift cad pogam. Utilizing well undestood data segments seves two oles. Fist, distotions can potentially occu when changes in the mix of business happen and appopiate segmentations can addess and coect fo these potential distotions. Second, it allows the actuay to dial in on smalle segments of the population and to bette identify the individual behavio of each segment. This knowledge, besides being of use to the actuaial analyst, can be incedibly useful to intenal paties such as a sponso s maketing, accounting o finance depatment, as well as with management epoting. Specifically, segmentation can help to undestand how things such as tageted mailings, pomotions, and pogam stuctue changes impact membes behavio, and can ultimately influence cost/benefit analyses of the activities. Identifying appopiate segmentations can be a significant task. This can be made even moe challenging when the segments ae fluid, such as in situations whee tansfes between segments ae possible (o fequent). Often, such tansfes ae obseved in hotel o ailine Casualty Actuaial Society EFoum, Summe 0 0
21 pogams whee membes can change membeship levels (upgade o downgade) due to thei ecent activities within thei pogam. Some potential segmentation citeions that ae often used ae: Membeship level/categoy, Poduct type, Aveage spend by membes, and/o Geogaphic location. This list is by no means intended to be compehensive but athe a selected numbe of options which may be consideed by the analyst. 5. Data Quality Many pogams have been in opeation fo seveal decades and, fo all intents and puposes, pedate the moden computing ea and compehensively managed database capabilities. As such, histoical data may not be complete o may simply not be available anymoe. Even in pogams that ae elatively young, the data may exhibit seious shotcomings o distotions. As a esult, thee may be limitations as to how the data can be povided to an analyst and, doubts may exist egading data integity. Given the impotance of data in actuaial analyses, it is impotant to make consideation of what is needed fo the analysis and compae that to what is actually available fom the pogam. In some cases, analytical decisions will be made based on data availability athe than theoetical optimization. In such instances, an analyst should conside and communicate to vested paties how data shotcomings may influence the estimated esults o incease the uncetainty aound the full undestanding of the pogam. 5.3 Changes in Pogam Tems & Conditions The Tems & Conditions of a pogam ae one of the single most impotant pats of a loyalty ewads pogam and they need to be well undestood befoe poceeding with an analysis of the estimated URR (o any othe component of such pogam fo that matte). In essence, the T&C ae the ules by which the membes and the pogam s sponso must abide (at least in theoy). It is impeative that the analyst gains a full undestanding of the T&C of any pogam that is unde eview. It is also impotant to undestand how stictly these ules ae actually applied by the pogam sponso. Casualty Actuaial Society EFoum, Summe 0
22 Changes in T&C can ceate lage vaiations in a pogam s cost stuctue, membes edemption behavio, membeship pofile, and moe. In some instances, changes may impact the fundamentals of a pogam to the extent that an analyst s ability to ely on histoical data to suppot a URR analysis may be limited, at least without including significant adjustments to the oiginal data. Fom an insuance point of view, changes in T&C can often be compaed to legislative changes that affect all insuance policies in foce (o even etoactively apply to all policies eve witten). These changes can fundamentally change the ules of the game to the extent that the past s emegence may povide only limited assistance in pedicting the futue. An actuaial analyst would likely apply some adjustment techniques to the histoical data pio to using it in an analysis. Simila adjustments can be made to histoical point accumulation o edemption activities. An analyst must be able to anticipate how a change (defined) can impact an analysis to avoid poducing biased URR esults. 5.4 Maketing As touched upon biefly above, maketing decisions (e.g., point pomotions) can intoduce lage shifts o spikes in membe behavio and theefoe can have an impact on actuaial analyses. In addition, it is not uncommon that these maketing campaigns will influence only potions of the membeship populations, wok in calenda yea manne (i.e., acoss entie diagonals when actuaial tiangles ae used) o have effects that wee vey diffeent fom the intended outcome. As such, an actuay should wok closely with a pogam s maketing depatment to undestand the upcoming plans o campaigns, if possible. Moe impotantly, the insights that can be gained fom quantitative analysis of the pogam can povide useful feedback to a company s maketing depatment as to the effectiveness (and costs) of vaious maketing pogams. Casualty Actuaial Society EFoum, Summe 0
23 In fact, the confluence of maketing and fundamental data analysis to moe deeply undestand costs and ewads is an aea that the authos believe to be a natual extension of the ideas contained in this pape. 5.5 Seasonal Effects Many pogams ae heavily impacted by seasonal effects. Fo example, ailine tickets typically tend to cost moe in summe months than in the fall o sping. Anothe example is that cedit cad companies typically issue significantly moe points in the holiday season due to the lage inceases in spending by membes. As such it is impotant to undestand how seasonal effects influence a ewad pogam fom both a membe pespective as well as fom the sponso s pespective. The good news fo an analyst is that it is likely that these effects ae consistent yea afte yea, which should help gain a pecise undestanding of thei timing and thei potential impact on calenda yea esults. This would also be helpful infomation when pefoming a patial yea analysis, with a ollfowad appoach to the upcoming yeaend evaluation date. As with any actuaial analysis elying on histoical data, data consistency though time is a key component of a loyalty ewads analysis. Casualty Actuaial Society EFoum, Summe 0 3
24 6. CONCLUSION The expansion in the univese of loyalty pogams has opened a new oppotunity fo actuaies to expand the application of thei taditional insuance pactice body of knowledge into anothe aea of expetise. The quantitative famewok developed by actuaies and the associated actuaial pojection methods ae exceptionally well suited to addess these nontaditional topics. While this pape focused on basic estimation techniques and thei application to loyalty ewads and gift cads pogams, we acknowledge that moe advanced techniques (including pedictive modeling methods) might also be successfully applied to the questions and poblems bought to us by these pogams. We puposely decided to exclude that discussion fom this pape in ode to maintain ou focus on the moe basic appoaches. It is always exciting to ventue into a new space and attempt to answe new questions. We hope that with this pape we will help the actuaial community continue its pogession and emain at the foefont of these new challenges. Casualty Actuaial Society EFoum, Summe 0 4
25 7. APPENDIX 7. Point Issue Peiod Appoach  Numeical Example Below we outline a simple case study example of how to obtain the estimated URR, expessed as a pecentage of total issued points, fo a hypothetical gift cad pogam. Step Undestand the pogam The pogam of inteest involves the issuance of point gift cads which ae chaged with a specified point value at time of puchase. The gift cad value can be edeemed by the cadholde fo goods at the issue s stoes as if the value on the cad wee a cash equivalent. Cadholdes cannot add additional value to the cad afte the oiginal time of issuance. The accounting standads unde which the epoting entity opeates allows fo the ecognition of the associated beakage evenue if the likelihood of nonedemption is pobable and the amount of beakage is easonably estimable. In this example, we assume that the cad issue has the capability to povide tansactional level infomation showing the time and amount of all tansactions well as the associated cad numbe fo each and evey histoical point edemption and issuance on a pe cad basis. Step Obtain Data The key data elements equied fo this appoach ae as follows: The total value of issued gift cads gouped by issuance peiod and the incemental edemptions ove time that coespond to the same issuance peiod This infomation is shown on Tables A and B of Appendix 7.. Step 3 Manipulate Data into Usable Fomat This appoach uses cumulative edemptions as a pecentage of the total issued value. As such we fist need to accumulate the incemental edemption tiangle. Table C in Appendix 7. contains the esult of this execise. In ou example the cumulative edemption pecentage Casualty Actuaial Society EFoum, Summe 0 5
26 coesponding to 0X at 36 months is calculated using the total incemental edemptions ( = 69) fo that issuance peiod as of the evaluation date. The next step is to divide the cumulative edeemed points fo each issuance peiod by the cumulative issued points fo each espective issuance peiod to obtain the cumulative edemption pecentages at each evaluation peiod. The esult of this is shown on Appendix 7., Table D. This table is the esult of dividing Table C by Table A. As an example, the 55.% on Table D is deived by dividing the 69 points edeemed at 36 months by the 5 points oiginally issued in that peiod. Step 4 Poject Ultimate Redemption Rate We can poject the ultimate edemption ate using one of many commonly accepted actuaial pojection methods. Fo this example, we have opted to use an exponential cuve fitted on motality basis edemptions fo ou ultimate pojection. The benefit of this method is that we can use the cuve to povide us with an estimate that extends beyond the oldest available data point (in this case actual data only extends to 48 months). The estimate of the tail potion is paticulaly impotant in this hypothetical example because we have assumed in this example that thee can be no fofeitues of value in this pogam. As such, edemptions can theoetically happen beyond ou latest data point, and pehaps significantly fathe. The fist step fo the exponential cuve fit is to convet ou cumulative edemption pecentages into incemental edemption pecentages. This can be seen on Appendix 7., Table E. We additionally ceate a tiangle of the cumulative amount that has not been edeemed at any given matuity (done by subtacting the cumulative edeemed pecentages fom 00.0%) The esult is shown in Appendix 7., Table F. We then calculate the motality ate by dividing the incemental pecentage edeemed in a given peiod by the cumulative unedeemed at the beginning of that peiod. Motality ates ae shown on Appendix 7., Table G and coesponds to Table E divided by Table F. We calculate the aveage motality at each matuity (fo example aveage motality ate at 4 months is 6.6% which is equal to [4.7% + 8.4% + 6.7%]/3). In this example we have chosen to fit an exponential decay Casualty Actuaial Society EFoum, Summe 0 6
27 function to the aveage motality ates, though numeous othe extapolation techniques could be used. Table H of Appendix 7. shows the esult of this execise..having estimated a motality cuve we can then poject out the ultimate edemption ate fo late matuities. Table I, on Appendix 7. shows the full pojection of ultimate edemption ates fo each issuance peiod. Fo example, the pojection of cumulative edemption pecentage of 53.8% fo 0X4 at 36 months of matuity is calculated as (00.0% %) x.% %. Having just estimated the ultimate edemption ate on issued gift cad value, we can easily convet this into the edemption ate on outstanding value, if needed (please see Appendix 7.4 fo an example of this convesion). 7. Aggegate Membe Join Peiod Appoach  Numeical Example Below we outline a simple case study example of how to obtain the estimated URR, expessed as a pecentage of total issued points, fo a hypothetical hotel loyalty pogam. Step Undestand the pogam This example pogam involves a hotel loyalty pogam whee membes ean points on evey puchase that they make at a paticipating popety. These eaned points can then be edeemed in the futue fo hotel ewads. All membes leave the pogam within thee yeas of thei oiginal date of enollment. Step Obtain Data The key data elements equied fo this appoach ae as follows: Cumulative issued and edeemed points, by join peiod at fixed inteval peiods  These ae shown on Appendix 7., Tables A and B, espectively. Step 3 Manipulate Data into Usable Fomat Taking the aw data elements, we can divide the cumulative edeemed points shown on Table B by the cumulative issued points shown on Table A. The cumulative edemption ate Casualty Actuaial Society EFoum, Summe 0 7
9:6.4 Sample Questions/Requests for Managing Underwriter Candidates
9:6.4 INITIAL PUBLIC OFFERINGS 9:6.4 Sample Questions/Requests fo Managing Undewite Candidates Recent IPO Expeience Please povide a list of all completed o withdawn IPOs in which you fim has paticipated
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More informationA framework for the selection of enterprise resource planning (ERP) system based on fuzzy decision making methods
A famewok fo the selection of entepise esouce planning (ERP) system based on fuzzy decision making methods Omid Golshan Tafti M.s student in Industial Management, Univesity of Yazd Omidgolshan87@yahoo.com
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationHow Much Should a Firm Borrow. Effect of tax shields. Capital Structure Theory. Capital Structure & Corporate Taxes
How Much Should a Fim Boow Chapte 19 Capital Stuctue & Copoate Taxes Financial Risk  Risk to shaeholdes esulting fom the use of debt. Financial Leveage  Incease in the vaiability of shaeholde etuns that
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194114 Online ISSN: 213953 Pint ISSN: 2138423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationIlona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More informationConverting knowledge Into Practice
Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationFirstmark Credit Union Commercial Loan Department
Fistmak Cedit Union Commecial Loan Depatment Thank you fo consideing Fistmak Cedit Union as a tusted souce to meet the needs of you business. Fistmak Cedit Union offes a wide aay of business loans and
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More informationDefine What Type of Trader Are you?
Define What Type of Tade Ae you? Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 1 Disclaime and Risk Wanings Tading any financial maket involves isk. The content of this
More informationExam #1 Review Answers
xam #1 Review Answes 1. Given the following pobability distibution, calculate the expected etun, vaiance and standad deviation fo Secuity J. State Pob (R) 1 0.2 10% 2 0.6 15 3 0.2 20 xpected etun = 0.2*10%
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance the availability
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationThe LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.
Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More informationValuation of Floating Rate Bonds 1
Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned
More informationOffice of Family Assistance. Evaluation Resource Guide for Responsible Fatherhood Programs
Office of Family Assistance Evaluation Resouce Guide fo Responsible Fathehood Pogams Contents Intoduction........................................................ 4 Backgound..........................................................
More informationDatabase Management Systems
Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012
More informationGESTÃO FINANCEIRA II PROBLEM SET 1  SOLUTIONS
GESTÃO FINANCEIRA II PROBLEM SET 1  SOLUTIONS (FROM BERK AND DEMARZO S CORPORATE FINANCE ) LICENCIATURA UNDERGRADUATE COURSE 1 ST SEMESTER 20102011 Chapte 1 The Copoation 113. What is the diffeence
More informationFixed Income Attribution: Introduction
18th & 19th Febuay 2015, Cental London Fixed Income Attibution: A compehensive undestanding of Fixed Income Attibution and the challenging data issues aound this topic Delegates attending this twoday
More informationIntroduction to Stock Valuation. Background
Intoduction to Stock Valuation (Text efeence: Chapte 5 (Sections 5.45.9)) Topics backgound dividend discount models paamete estimation gowth oppotunities piceeanings atios some final points AFM 271 
More informationBA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS
BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR John R. Gaham Adapted fom S. Viswanathan FUQUA SCHOOL OF BUSINESS DUKE UNIVERSITY 1 In this lectue we conside the effect of govenment
More informationQuestions for Review. By buying bonds This period you save s, next period you get s(1+r)
MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the twopeiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More informationChannel selection in ecommerce age: A strategic analysis of coop advertising models
Jounal of Industial Engineeing and Management JIEM, 013 6(1):89103 Online ISSN: 0130953 Pint ISSN: 013843 http://dx.doi.og/10.396/jiem.664 Channel selection in ecommece age: A stategic analysis of
More informationAn Analysis of Manufacturer Benefits under Vendor Managed Systems
An Analysis of Manufactue Benefits unde Vendo Managed Systems Seçil Savaşaneil Depatment of Industial Engineeing, Middle East Technical Univesity, 06531, Ankaa, TURKEY secil@ie.metu.edu.t Nesim Ekip 1
More informationON THE (Q, R) POLICY IN PRODUCTIONINVENTORY SYSTEMS
ON THE R POLICY IN PRODUCTIONINVENTORY SYSTEMS Saifallah Benjaafa and JoonSeok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poductioninventoy
More information883 Brochure A5 GENE ss vernis.indd 12
ess x a eu / u e a. p o.eu c e / :/ http EURAXESS Reseaches in Motion is the gateway to attactive eseach caees in Euope and to a pool of woldclass eseach talent. By suppoting the mobility of eseaches,
More informationElectricity transmission network optimization model of supply and demand the case in Taiwan electricity transmission system
Electicity tansmission netwok optimization model of supply and demand the case in Taiwan electicity tansmission system MiaoSheng Chen a ChienLiang Wang b,c, ShengChuan Wang d,e a Taichung Banch Gaduate
More information875 Grocery Products Purchase Order
875 Gocey Poducts Puchase Ode Functional Goup ID=OG Intoduction: This X12 Tansaction Set contains the fomat establishes the data contents of the Gocey Poducts Puchase Ode Tansaction Set (875) fo use within
More informationThe impact of migration on the provision. of UK public services (SRG.10.039.4) Final Report. December 2011
The impact of migation on the povision of UK public sevices (SRG.10.039.4) Final Repot Decembe 2011 The obustness The obustness of the analysis of the is analysis the esponsibility is the esponsibility
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationThe Predictive Power of Dividend Yields for Stock Returns: Risk Pricing or Mispricing?
The Pedictive Powe of Dividend Yields fo Stock Retuns: Risk Picing o Mispicing? Glenn Boyle Depatment of Economics and Finance Univesity of Cantebuy Yanhui Li Depatment of Economics and Finance Univesity
More informationHEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING
U.P.B. Sci. Bull., Seies C, Vol. 77, Iss. 2, 2015 ISSN 22863540 HEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING Roxana MARCU 1, Dan POPESCU 2, Iulian DANILĂ 3 A high numbe of infomation systems ae available
More informationEquity compensation plans New Income Statement impact on guidance Earnings Per Share Questions and answers
Investos/Analysts Confeence: Accounting Wokshop Agenda Equity compensation plans New Income Statement impact on guidance Eanings Pe Shae Questions and answes IAC03 / a / 1 1 Equity compensation plans The
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationIBM Research Smarter Transportation Analytics
IBM Reseach Smate Tanspotation Analytics Laua Wynte PhD, Senio Reseach Scientist, IBM Watson Reseach Cente lwynte@us.ibm.com INSTRUMENTED We now have the ability to measue, sense and see the exact condition
More informationFinancial Planning and Riskreturn profiles
Financial Planning and Risketun pofiles Stefan Gaf, Alexande Kling und Jochen Russ Pepint Seies: 201016 Fakultät fü Mathematik und Witschaftswissenschaften UNIERSITÄT ULM Financial Planning and Risketun
More informationLAL Update. Letter From the President. Dear LAL User:
LAL Update ASSOCIATES OF CAPE COD, INCORPORATED OCTOBER 00 VOLUME 0, NO. Lette Fom the Pesident Dea LAL Use: This Update will claify some of the statistics used with tubidimetic and chomogenic LAL tests.
More informationData Center Demand Response: Avoiding the Coincident Peak via Workload Shifting and Local Generation
(213) 1 28 Data Cente Demand Response: Avoiding the Coincident Peak via Wokload Shifting and Local Geneation Zhenhua Liu 1, Adam Wieman 1, Yuan Chen 2, Benjamin Razon 1, Niangjun Chen 1 1 Califonia Institute
More informationPromised LeadTime Contracts Under Asymmetric Information
OPERATIONS RESEARCH Vol. 56, No. 4, July August 28, pp. 898 915 issn 3364X eissn 15265463 8 564 898 infoms doi 1.1287/ope.18.514 28 INFORMS Pomised LeadTime Contacts Unde Asymmetic Infomation Holly
More informationIgnorance is not bliss when it comes to knowing credit score
NET GAIN Scoing points fo you financial futue AS SEEN IN USA TODAY SEPTEMBER 28, 2004 Ignoance is not bliss when it comes to knowing cedit scoe By Sanda Block USA TODAY Fom Alabama comes eassuing news
More informationYIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE
YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE Septembe 1999 Quoted Rate Teasuy Bills [Called Banke's Discount Rate] d = [ P 1  P 1 P 0 ] * 360 [ N ] d = Bankes discount yield P 1 = face
More informationNBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING?
NBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING? Daia Bunes David Neumak Michelle J. White Woking Pape 16932 http://www.nbe.og/papes/w16932
More informationFaithful Comptroller s Handbook
Faithful Comptolle s Handbook Faithful Comptolle s Handbook Selection of Faithful Comptolle The Laws govening the Fouth Degee povide that the faithful comptolle be elected, along with the othe offices
More informationTrading Volume and Serial Correlation in Stock Returns in Pakistan. Abstract
Tading Volume and Seial Coelation in Stock Retuns in Pakistan Khalid Mustafa Assistant Pofesso Depatment of Economics, Univesity of Kaachi email: khalidku@yahoo.com and Mohammed Nishat Pofesso and Chaiman,
More informationGravitational Mechanics of the MarsPhobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning
Gavitational Mechanics of the MasPhobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This
More informationCONCEPT OF TIME AND VALUE OFMONEY. Simple and Compound interest
CONCEPT OF TIME AND VALUE OFMONEY Simple and Compound inteest What is the futue value of shs 10,000 invested today to ean an inteest of 12% pe annum inteest payable fo 10 yeas and is compounded; a. Annually
More informationFI3300 Corporate Finance
Leaning Objectives FI00 Copoate Finance Sping Semeste 2010 D. Isabel Tkatch Assistant Pofesso of Finance Calculate the PV and FV in multipeiod multicf timevalueofmoney poblems: Geneal case Pepetuity
More informationReview Graph based Online Store Review Spammer Detection
Review Gaph based Online Stoe Review Spamme Detection Guan Wang, Sihong Xie, Bing Liu, Philip S. Yu Univesity of Illinois at Chicago Chicago, USA gwang26@uic.edu sxie6@uic.edu liub@uic.edu psyu@uic.edu
More informationMining Relatedness Graphs for Data Integration
Mining Relatedness Gaphs fo Data Integation Jeemy T. Engle (jtengle@indiana.edu) Ying Feng (yingfeng@indiana.edu) Robet L. Goldstone (goldsto@indiana.edu) Indiana Univesity Bloomington, IN. 47405 USA Abstact
More informationAn Infrastructure Cost Evaluation of Single and MultiAccess Networks with Heterogeneous Traffic Density
An Infastuctue Cost Evaluation of Single and MultiAccess Netwoks with Heteogeneous Taffic Density Andes Fuuskä and Magnus Almgen Wieless Access Netwoks Eicsson Reseach Kista, Sweden [andes.fuuska, magnus.almgen]@eicsson.com
More informationOffice Leasing Guide WHAT YOU NEED TO KNOW BEFORE YOU SIGN. Colliers International Office Leasing Guide P. 1
Office Leasing Guide WHAT YOU NEED TO KNOW BEFORE YOU SIGN Collies Intenational Office Leasing Guide P. 1 THE OFFICE LEASING GUIDE This stepbystep guide has been assembled to eflect Collies Intenational
More informationEvaluating the impact of Blade Server and Virtualization Software Technologies on the RIT Datacenter
Evaluating the impact of and Vitualization Softwae Technologies on the RIT Datacente Chistophe M Butle Vitual Infastuctue Administato Rocheste Institute of Technology s Datacente Contact: chis.butle@it.edu
More informationConfirmation of Booking
The Pesentes Rebecca Mogan Rebecca is a Taxation Consultant with the NTAA and has ove 15 yeas tax expeience. Rebecca holds a Bachelo of Ats and Law and a Mastes of Taxation. Rebecca has pesented a numbe
More informationSupply chain information sharing in a macro prediction market
Decision Suppot Systems 42 (2006) 944 958 www.elsevie.com/locate/dss Supply chain infomation shaing in a maco pediction maket Zhiling Guo a,, Fang Fang b, Andew B. Whinston c a Depatment of Infomation
More informationJapan s trading losses reach JPY20 trillion
IEEJ: Mach 2014. All Rights Reseved. Japan s tading losses each JPY20 tillion Enegy accounts fo moe than half of the tading losses YANAGISAWA Akia Senio Economist Enegy Demand, Supply and Foecast Goup
More informationReferral service and customer incentive in online retail supply Chain
Refeal sevice and custome incentive in online etail supply Chain Y. G. Chen 1, W. Y. Zhang, S. Q. Yang 3, Z. J. Wang 4 and S. F. Chen 5 1,,3,4 School of Infomation Zhejiang Univesity of Finance and Economics
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationManaging Card Compromises from the Issuer s Perspective
Managing Cad Compomises fom the Issue s Pespective Industy insides shae tips fo effectively managing mass compomises Numbe 60 June 2012 Lagescale payment cad compomises and account data beaches continue
More informationAn application of stochastic programming in solving capacity allocation and migration planning problem under uncertainty
An application of stochastic pogamming in solving capacity allocation and migation planning poblem unde uncetainty YinYann Chen * and HsiaoYao Fan Depatment of Industial Management, National Fomosa Univesity,
More informationChris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment
Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability
More informationChapter 11: Aggregate Demand II, Applying the ISLM Model Th LM t
Equilibium in the  model The cuve epesents equilibium in the goods maket. Chapte :, Applying the  Model Th t C ( T) I( ) G The cuve epesents money maket equilibium. M L(, ) The intesection detemines
More information867 Product Transfer and Resale Report
867 Poduct Tansfe and Resale Repot Functional Goup ID=PT Intoduction: This X12 Tansaction Set contains the fomat and establishes the data contents of the Poduct Tansfe and Resale Repot Tansaction Set (867)
More informationBasic Financial Mathematics
Financial Engineeing and Computations Basic Financial Mathematics Dai, TianShy Outline Time Value of Money Annuities Amotization Yields Bonds Time Value of Money PV + n = FV (1 + FV: futue value = PV
More informationCOMPLYING WITH THE DRUGFREE SCHOOLS AND CAMPUSES REGULATIONS
Highe Education Cente fo Alcohol and Othe Dug Abuse and Violence Pevention Education Development Cente, Inc. 55 Chapel Steet Newton, MA 024581060 COMPLYING WITH THE DRUGFREE SCHOOLS AND CAMPUSES REGULATIONS
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationFrom PLI s Treatise Initial Public Offerings: A Practical Guide to Going Public #19784 PREPACKAGED BANKRUPTCY AND PREARRANGED BANKRUPTCY PROCESS
Fom PLI s Teatise Initial Public Offeings: A Pactical Guide to Going Public #19784 16 PREPACKAGED BANKRUPTCY AND PREARRANGED BANKRUPTCY PROCESS Deyck Palme Jessica Fink Cadwalade, Wickesham & Taft LLP
More informationWasting the Percentage of Onhand Inventory in an Instantaneous Economic Order Quantity Model Due to Deterioration
Jounal of mathematics and compute Science 7 (2013) 154159 Wasting the Pecentage of Onhand Inventoy in an Instantaneous Economic Ode Quantity Model Due to Deteioation Aticle histoy: Received Mach 2013
More informationSIMULATION OF GAS TURBINES OPERATING IN OFFDESIGN CONDITION
SIMULAION OF GAS URBINES OPERAING IN OFFDESIGN CONDIION Analdo Walte: awalte@fem.unicamp.b Univesity of Campinas Dept. of Enegy DE/FEM/Unicamp P.O. Box 6122  ZIP code 13083970  Bazil Abstact. In many
More informationThe Capital Asset Pricing Model. Chapter 9
The Capital Asset Picing odel Chapte 9 Capital Asset Picing odel CAP centepiece of moden finance gives the elationship that should be obseved between isk and etun of an asset it allows fo the evaluation
More information30 H. N. CHIU 1. INTRODUCTION. Recherche opérationnelle/operations Research
RAIRO Rech. Opé. (vol. 33, n 1, 1999, pp. 2945) A GOOD APPROXIMATION OF THE INVENTORY LEVEL IN A(Q ) PERISHABLE INVENTORY SYSTEM (*) by Huan Neng CHIU ( 1 ) Communicated by Shunji OSAKI Abstact. This
More informationTowards Realizing a Low Cost and Highly Available Datacenter Power Infrastructure
Towads Realizing a Low Cost and Highly Available Datacente Powe Infastuctue Siam Govindan, Di Wang, Lydia Chen, Anand Sivasubamaniam, and Bhuvan Ugaonka The Pennsylvania State Univesity. IBM Reseach Zuich
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationA Capacitated Commodity Trading Model with Market Power
A Capacitated Commodity Tading Model with Maket Powe Victo MatínezdeAlbéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu
More informationLecture 8 Topic 5: Multiple Comparisons (means separation)
Lectue 8 Topic 5: Multiple Compaisons (means sepaation) ANOVA: H 0 : µ 1 = µ =... = µ t H 1 : The mean of at least one teatment goup is diffeent If thee ae moe than two teatments in the expeiment, futhe
More informationUniversal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool
Univesal Cycles 2011 Yu She Wial Gamma School fo Gils Depatment of Mathematical Sciences Univesity of Livepool Supeviso: Pofesso P. J. Giblin Contents 1 Intoduction 2 2 De Buijn sequences and Euleian Gaphs
More informationThe future challenges of Healthcare
The futue challenges of Healthcae D. Eich Hunzike CFO F. Hoffmann La Roche Ltd. JPMogan Healthcae Confeence San Fancisco, Januay 12, 2005 This pesentation contains cetain fowadlooking statements. These
More informationHow to create RAID 1 mirroring with a hard disk that already has data or an operating system on it
AnswesThatWok TM How to set up a RAID1 mio with a dive which aleady has Windows installed How to ceate RAID 1 mioing with a had disk that aleady has data o an opeating system on it Date Company PC / Seve
More informationSTABILITY ANALYSIS IN MILLING BASED ON OPERATIONAL MODAL DATA 1. INTRODUCTION
Jounal of Machine Engineeing, Vol. 11, No. 4, 211 Batosz POWALKA 1 Macin CHODZKO 1 Kzysztof JEMIELNIAK 2 milling, chatte, opeational modal analysis STABILITY ANALYSIS IN MILLING BASED ON OPERATIONAL MODAL
More informationAn Efficient Group Key Agreement Protocol for Ad hoc Networks
An Efficient Goup Key Ageement Potocol fo Ad hoc Netwoks Daniel Augot, Raghav haska, Valéie Issany and Daniele Sacchetti INRIA Rocquencout 78153 Le Chesnay Fance {Daniel.Augot, Raghav.haska, Valéie.Issany,
More informationGive me all I pay for Execution Guarantees in Electronic Commerce Payment Processes
Give me all I pay fo Execution Guaantees in Electonic Commece Payment Pocesses Heiko Schuldt Andei Popovici HansJög Schek Email: Database Reseach Goup Institute of Infomation Systems ETH Zentum, 8092
More information9.5 Amortization. Objectives
9.5 Aotization Objectives 1. Calculate the payent to pay off an aotized loan. 2. Constuct an aotization schedule. 3. Find the pesent value of an annuity. 4. Calculate the unpaid balance on a loan. Congatulations!
More informationAMB111F Financial Maths Notes
AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed
More informationDYNAMICS AND STRUCTURAL LOADING IN WIND TURBINES
DYNAMIS AND STRUTURAL LOADING IN WIND TURBINES M. Ragheb 12/30/2008 INTRODUTION The loading egimes to which wind tubines ae subject to ae extemely complex equiing special attention in thei design, opeation
More informationSimple Harmonic Motion
Simple Hamonic Motion Intoduction Simple hamonic motion occus when the net foce acting on an object is popotional to the object s displacement fom an equilibium position. When the object is at an equilibium
More informationThe Supply of Loanable Funds: A Comment on the Misconception and Its Implications
JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently FieldsHat
More informationDual channel closedloop supply chain coordination with a rewarddriven remanufacturing policy
Intenational Jounal of Poduction Reseach ISSN: 753 Pint 1366588X Online Jounal homepage: http://www.tandfonline.com/loi/tps Dual channel closedloop supply chain coodination with a ewaddiven emanufactuing
More informationThe Time Value of Money
he ime Value of Money Inteest Rates and Futue Value Inteest ates ae a facto in the valuation of vitually all financial instuments. While all money maket ates () ae quoted on an annual basis (PR nnual Pecentage
More informationAshfield Girls High School. Critical Incident Policy
Ashfield Gils High School A Specialist School fo ICT Citical Incident Policy Citical Incident Policy 2 Citical Incident Policy A Specialist School fo ICT Ashfield Gils High School CRITICAL INCIDENT POLICY
More informationControlling the Money Supply: Bond Purchases in the Open Market
Money Supply By the Bank of Canada and Inteest Rate Detemination Open Opeations and Monetay Tansmission Mechanism The Cental Bank conducts monetay policy Bank of Canada is Canada's cental bank supevises
More informationMETHODOLOGICAL APPROACH TO STRATEGIC PERFORMANCE OPTIMIZATION
ETHODOOGICA APPOACH TO STATEGIC PEFOANCE OPTIIZATION ao Hell * Stjepan Vidačić ** Željo Gaača *** eceived: 4. 07. 2009 Peliminay communication Accepted: 5. 0. 2009 UDC 65.02.4 This pape pesents a matix
More information