BY JOHN VUNDI MUSAU I56/61270/2013

Transcription

1 THE RELIABILITY OF CAPITAL ASSET PRICING MODEL ON VALUATION OF LISTED FIRMS AT THE NAIROBI SECURITIES EXCHANGE BY JOHN VUNDI MUSAU I56/61270/2013 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN ACTUARIAL SCIENCE AT THE UNIVERSITY OF NAIROBI NAIROBI, KENYA SUPERVISORS: DR PHILIP NGARE DR TOBIAS OLWENY NOVEMBER

2 DECLARATION Ths Research Project s my orgnal wor and has not been presented for an award of a degree n any other unversty or nsttute of learnng. Sgnature:. Date: John Vund Musau I56/61270/2013 Ths Research Project has been submtted for examnaton wth my approval as the Unversty supervsor. Sgnature:. Date:... Dr Phlp Ngare Lecturer, Department of Fnancal Economcs School of Mathematcs, Unversty of Narob

3 ACKNOWLEDGEMENT I wsh to than the Almghty God n whom I fnd my beng. I express my deep apprecaton to my dad for hs gentle prod, not to menton my lecturers, famly and frends who were nstrumental n pecng ths project together.

4 ABSTRACT One of the most mportant modern fnancal theores that has domnated captal marets and nfluenced the feld of fnance and busness for over the past fve decades s the Captal Asset Prcng Model. However, due to ts emprcal shortcomngs several researchers have attempted to mprove CAPM and ts assumptons by advancng varous extensons of the model. The objectve of ths study was to conduct emprcal tests of CAPM and ts extensons on the Narob Securtes Exchange and determne other applcatons of CAPM n nsurance. Usng decade-long data from forty seven lsted companes we undertae the weely study from January 2004 December We fnd that CAPM s not fully supported by the emprcal data. Alphas for a majorty of the NSE stocs are sgnfcantly dfferent from zero whereas all the betas were sgnfcant as expected. The Securty Maret Lne was lnear and a postve correlaton was observed between beta and returns but the SML ntercept was sgnfcantly dfferent from zero. The study was extended to predct ex-ante beta usng CAPM for year 2014 and compare wth the realzed ex-post beta durng the year. The test assumes mean reverson of returns and reveals hgh forecast errors whch negate CAPM as a true predctor of ex-ante betas. Zero-Beta CAPM ndcated a much lower correlaton and was forthwth rejected at the NSE. On the other hand, the Fama-French Three Factor Model reveals a hgher correlaton wth NSE but also faled crucal compatblty tests. The CAPM s therefore not a robust tool to predct the NSE and other model extensons need to be tested for conformty wth the Kenyan maret. Addtonal tests can help elmnate errors caused by unrealstc assumptons such as lac of wealth consumpton, foregn nvestment, varyng beta and rs prema. In case these extensons fal, then CAPM can be wholly rejected as a model of asset prcng at the NSE. Key Words: Captal Asset Prcng Model, Narob Securtes Exchange, Systematc rs, Non- Systematc rs v

5 TABLE OF CONTENTS DECLARATION... II ACKNOWLEDGEMENT... III ABSTRACT... IV TABLE OF CONTENTS... V CHAPTER 1: INTRODUCTION PROBLEM STATEMENT OBJECTIVES OF STUDY... 3 CHAPTER 2: LITERATURE REVIEW HISTORICAL ORIGINS EMPIRICAL TESTS OF THE CAPM... 8 CHAPTER 3: METHODOLOGY MARKOWITZ MEAN-VARIANCE PORTFOLIO THEORY MATHEMATICAL DERIVATION OF CAPITAL ASSET PRICING MODEL THE ZERO-BETA CAPITAL ASSET PRICING MODEL: THE FAMA-FRENCH THREE FACTOR MODEL THE CONDITIONAL CAPITAL ASSET PRICING MODEL THE ARBITRAGE PRICING THEORY THE INTERTEMPORAL CAPITAL ASSET PRICING MODEL THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL THE INTERNATIONAL CAPITAL ASSET PRICING MODEL THE PRODUCTION-BASED CAPITAL ASSET PRICING MODEL CHAPTER 4: APPLICATION OF CAPM IN THE INSURANCE INDUSTRY CHAPTER 5: DATA ANALYSIS & RESULTS NSE THROUGH THE YEARS TESTING CAPM ASSUMPTIONS TESTING CAPM THE SECURITY MARKET LINE PREDICTING EX ANTE BETA BASED ON CAPM TESTING THE ZERO-BETA CAPM FAMA FRENCH 3 FACTOR ANALYSIS SUMMARY OF RESULTS CHAPTER 6: CONCLUSION AND RECOMMENDATIONS CHAPTER 7: REFERENCES v

6 CHAPTER 1: INTRODUCTION Wllam Sharpe (1964) and John Lnter (1965) constructed the well-nown Captal Asset Prcng Model (CAPM) showng that the true measure of rs for stocs s beta. The model s wdespread popularty stems from ts ntutve appeal, computatonal ease and was an extenson of the earler wor of Harry Marowtz (1959) mean-varance portfolo model. A Nobel Prze was later awarded to Sharpe as well as Harry Marowtz and Merton Mller n 1990 for ther enormous contrbuton to fnancal theory. CAPM s a prcng theory that apples to a fnancal maret under the condton of sngle perod, perfect competton, and no conflct. The model tself s appled to obtan the lnear relatonshp between rs and return of a fnancal asset based on numerous restrcted assumptons. The assumptons made were that all nvestors focus on a sngle holdng perod, and they see to maxmze the expected utlty of ther termnal wealth by choosng among alternatve portfolos on the bass of each portfolo's expected return and standard devaton. All nvestors can borrow or lend an unlmted amount at a gven rs-free rate of nterest and there are no restrctons on short sales of any assets. All nvestors have dentcal estmated of the expected returns, varances, and covarance among all assets (that s, nvestors have homogeneous expectatons). All assets are perfectly dvsble and perfectly lqud (that s, maretable at the gong prce). There are no transacton costs and no taxes. All nvestors are prce taers (that s, all nvestors assume that ther own buyng and sellng actvty wll not affect stoc prces). The purpose of the assumptons related to fnancal asset maret s to obtan a clear representaton of the relatonshp between the expected return and the rs n the maret, so that the neffcency due to transacton costs, taxes, and the delay of nformaton s avoded n the applcaton of the model. Although the assumptons do not perfectly match the realstc stuaton, they smplfed the ssues to generate the prcng model of the fnancal asset. Under the restrcted assumptons, Wllam Sharpe and Lnter obtaned the lnear relatonshp between the expected return and the rs of the asset. The model of CAPM can be presented as: E [R ] = R f + β E [(R m ) R f ] R : The expected return of the stoc. R f : Rs free rate normally s the treasury bond rate wth 90-day maturty. R m : The expected return of the maret portfolo. β : β measures the systematc rs of stoc, or, t represents the reacton level of a fnancal asset s expected return towards the rs of t. Generally, t s defned as the volatlty of an asset n relaton to the volatlty of the benchmar that sad asset s beng compared to. The model of CAPM presents that the return of a rsy assets conssts of two parts, one s the return of rs free assets, represented by R f ; and the other part of the return of rsy assets s the return of rsy maret, represented by (R m ) R f. β represents the level of the rs of the fnancal maret, whch ndcates that a hgher rs s always accompaned wth a hgher return. 1

7 CAPM suggests that not all the rs of the assets would need a remedy of the rs; the one wth a remedy s systematc rs. Due to the reason that systematc rs cannot be reduced and elmnated through dversfcaton, an accompanyng return wth the rs can attract nvestors; n contrast, unsystematc rss can be dversfed; therefore, a remedy n order to attract nvestors s not needed. CAPM also ndcates that the best portfolo s maret portfolo due to the smallest rs t has among all, all the rsy asset nvestors would prefer maret portfolo. The approach of defnng the rs-return relatonshp of a sngle fnancal asset leads to the development of the securty maret lne (SML). As stated n Graph 1, Stoc Maret Lne ndcates the rs-return relatonshp of a sngle stoc under the condton of an effcent maret. When effcent maret apples, the relatonshp formula of a sngle stoc rs and ts return, the SML, can be obtaned. E (R m ) M SML r f 0 1 β Graph 1: Securty Maret Lne The SML represents the nvestment's opportunty cost after nvestng n a combnaton of the maret portfolo and the rs-free asset. The slope of the CAPM s equal to the maret rs premum and dsplays the levels of rs aganst the expected rate of return of the entre maret at a gven pont n tme. It s obvous that, based on the model of CAPM, for any sngle stocs, the expected return ncreases wth an ncrease of β, therefore, t s a postve lnear relaton between the value of β and the expected return of any sngle stocs. CAPM can be used n classfcaton of assets. Accordng to the model of CAPM, an effectve classfcaton of assets can be done through the result of CAPM applcaton. By usng the rs factor β n CAPM to classfy stocs, the classfcaton of stocs can avod rss and realze the returns for nvestors. As an example, when β>1, for nstance, β=2, when the maret portfolo return ncreased by 1%, such stocs would have an expected return ncrease 2%; when the maret portfolo return decreased by 1, however, such stocs would have an expected return decrease 2%. Therefore, t s obvous that the stocs n ths category carres a relatvely hgher rs than that wth a β=1. When β=1, such stocs wll have a volatlty same to the maret volatlty, and also they accurately reflect the maret portfolo prce change; when β<1, such stocs are more defensve compared to those dscussed above. Apparently, dfferent stocs carry dfferent characterstcs of return, based on whch, an effcent fnancal asset management can be conducted accordng to the nvestors rs preference and ther nvestment profle. Further, CAPM s used n prcng fnancal assets, provdng nvestment gudance to nvestors CAPM s a forecastng model for expected return of rsy fnancal assets based on a balanced 2

8 rs-return relatonshp; however, n the real lfe marets, returns of stocs are not balanced. Assumng that the calculated expected return s balanced, and a comparson can be conducted, so that the under-prced and overprced fnancal assets can be easly explored. Furthermore, based on the rule of nvestng n low, shortng n hgh, such an applcaton s gudng the nvestment behavour of nvestors. 1.1 PROBLEM STATEMENT Varous asset prcng models have been postulated by scholars over the years but the challenge of choosng relably proftable nvestments stll contnues to concern nvestors and fnancal nsttutons. How well do these economc theores fare n explanng asset prces and returns n actual marets and what other applcatons are derved from these theores. The current research n ths area seems to provde no conclusve support for the CAPM model. Several research studes have reled upon short NSE data perods and few stocs and have not studed the model extensons and ts varous applcatons. 1.2 OBJECTIVES OF STUDY 1. We carry out a mathematcal and emprcal study at how CAPM was derved, what dffcultes are faced n ts applcatons and what varous extensons were put forward to address ts shortcomngs. 2. We shall examne whether CAPM s applcable n the Kenyan bourse - Narob Securtes Exchange - and f t can be reasonably reled upon n explanng and predctng stoc prces to ad n captal budget decson mang. 3. In addton, we shall see to fnd out how CAPM has been appled by actuares n asset lablty management of nsurance frms and prcng of nsurance premums. Ths study conducts the applcaton and analyss of CAPM and ts extensons at the Narob Securtes Exchange over the decade commencng from year 2004 to year Chapter 1 manly provdes the bacground, objectves and ntroducton of the CAPM. Chapter 2 dscusses the development of asset prcng theores, lterature n exstence for the CAPM model and research undertaen n dfferent marets. Chapter 3 develops the mathematcal dervaton of CAPM, ts crtque and varous extensons. In Chapter 4, the applcatons of the CAPM model n the nsurance feld wll be dscussed. Chapter 5 studes the data methodology, sources, test results and analyss of the results for the standard CAPM, Zero-Beta CAPM and Fama-French Three Factor Models. Further, an ex-ante beta forecast for year 2014 wll be carred out usng CAPM predcton and results compared wth the actual beta realzed n the NSE. Chapter 6 wll provde the concluson of the study and related recommendatons. References used n the text shall be enumerated n the fnal secton. 3

9 CHAPTER 2: LITERATURE REVIEW The Captal Asset Prcng Model (CAPM) has been dscovered and dscussed snce early 1960 s through both theoretcal and emprcal aspects. Due to the long hstory of CAPM, the authors and researchers conducted numercal accomplshments related to the model tself and emprcal cases, whch s the man reason why CAPM has been developed nto many drectons as t s varous dependng on the applcaton crcumstances. Several authors have contrbuted to development of a model descrbng the prcng of captal assets under condton of maret equlbrum ncludng Eugene Fama, Mchael Jensen, John Lntner, John Long, Robert Merton, Myron Scholes, Wllam Sharpe, Jac Treynor and Fscher Blac, some of whose fndngs wll be dscussed n ths chapter of lterature revew. The CAPM was developed as a result of the meanvarance model whch fnds ts bass on Bernoull s utlty theory. 2.1 HISTORICAL ORIGINS Danel Bernoull (1738) wrote a remarable paper, orgnally n Latn whch was referenced wdely n the felds of mathematcs, logc and subsequently economcs. Ths paper was avalable n Englsh only after the 1950s. At the begnnng of the paper Bernoull states, that all mathematcans who had studed the measurement of rs agree, that: "Expected values are computed by multplyng each possble gan by the number of ways n whch t can occur, and then dvdng the sum of these products by the total number of cases where, n ths theory, the consderaton of cases whch are all of the same probablty s nssted upon." Before Bernoull turns to the famous Petersburg Paradox, he starts wth an observaton from the multple publcatons on the topc, all clamng that: "snce there s no reason to assume that of two persons encounterng dentcal rss, ether should expect to have hs desres more closely fulflled, the rss antcpated by each must be deemed equal n value." Ths s a formulaton of the prncple of suffcent reason. (Ths states that nothng s wthout a ground or reason why t s the way t s). He further tres to formulate the problem n general terms by strppng ndvdual characterstcs of the persons themselves and bannng them from the consderaton; only those matters should be weghed carefully that pertan to the terms of the rs. Bernoull deduces from hs concept of utlty a fundamental rule: If the utlty of each possble proft expectaton s multpled by the number of ways n whch t can occur, and we then dvde the sum of these products by the total number of possble cases, a mean utlty [moral expectaton] wll be obtaned, and the proft whch corresponds to ths utlty wll equal the value of the rs n queston. For Bernoull t becomes evdent that no vald measurement of the value of a rs can be obtaned wthout consderaton beng gven to ts utlty. However, t hardly seems plausble to mae any precse generalzatons snce the utlty of an tem may change wth crcumstances. The queston f ths nd of problem s accessble to mathematcs was answered by Bernoull wth: yes. He clams "Now t s hghly probable that any ncrease n wealth, no matter how nsgnfcant, wll always result n an ncrease n utlty whch s nversely proportonal to the quantty of goods already possessed." 4

10 Ths s the frst publshed exposton of the Prncple of Decreasng Margnal utlty. Ths prncple, later wdely accepted n the theory of economc behavour, states that margnal utlty (the extra utlty obtaned from consumng a good) decreases as the quantty consumed ncreases; n other words, that each addtonal good consumed s less satsfyng than the prevous one. Bernoull s concept of decreasng margnal utlty became central to economcs, notably n the wors of Jevons, Menger, Walras and Marshall. Manstream economcs orgnates from Jevons and Menger s margnal utlty and Walras and Marshall s equlbrum approach. Menger and Jevns each managed to ground a theory of economc exchange on the fact that people are ratonal enough to choose whch of two goods wll provde them wth the greatest beneft. Bernoull also ntroduced the concept of maxmzaton of expected utlty. However, despte endorsement by Laplace and others, Bernoull s approach had lttle mpact on the economcs of decson mang under rs untl the formal development of the expected utlty theory by Savage (1954) and Von Neuman and Oscar Morgenstern (1944, 1947) n ther boo Theory of Games and Economc Behavour. The von Neumann-Morgenstern expected utlty model s not wthout ts lmtatons. One lmtaton s that t treats uncertanty as objectve rs that s, as a seres of con flps where the probabltes are objectvely nown. Savage s (1954) approach to choce under uncertanty, whch rather than assumng the exstence of objectve probabltes attached to uncertan prospects maes assumptons about choce behavour and argues that f these assumptons are satsfed, a decson-maer must act as f she s maxmzng expected utlty wth respect to some subjectvely held probabltes. A pont worth notng s that Savage s theory s about sngle person decson problems. If we magne a stuaton where two Savage guys, A and B, are faced wth uncertanty, there s no partcular reason why when they form ther subjectve probabltes these probabltes should be the same (as would naturally be the case n a von Neumann-Morgenstern world where the probabltes are objectvely specfed). Nevertheless, n economc modellng, t s standard to assume that, f they have access to the same nformaton, agents wll form common subjectve probabltes. Ths s called the common pror assumpton, and s often dentfed wth Harsany s (1968) development of Bayesan games. Essentally t holds that dfferences n opnon are due to dfferences n nformaton. An old dea n the economcs lterature, datng bac at least to Fran Knght (1921), s that a dstncton should be drawn between rs (.e. stuatons where t mght be possble to assgn probabltes) and uncertanty (.e. stuatons where one s just clueless). Accordng to ths defnton, the von Neumann-Morgenstern theory deals clearly wth rs. Also, whle Savage s theory does not assume nown probabltes, t s nevertheless a model of rs n ths sense people at least behave as f they are assgnng probabltes. Arrow-Debreu model was developed as a model of general equlbrum that has been fundamental to economcs and fnance. Compared to earler models, the Arrow-Debreu model bascally generalzed the noton of a commodty, dfferentatng commodtes by tme and place of delvery. For example, apples n Malaysa n July and apples n Sngapore n June are consdered as dfferent commodtes. General equlbrum theory s concerned wth the allocaton of commodtes (between natons, or ndvduals, across tme, or under uncertanty, etc.). The Arrow-Debreu model studes those allocatons whch can be acheved through the exchange of commodtes at one moment n tme. When the descrptons are so precse that further refnements cannot yeld magnable 5

11 allocatons whch ncrease the satsfacton of the agents n the economy, then the commodtes are called Arrow-Debreu commodtes. Kenneth J. Arrow (1951) and Gerard Debreu (1951) wor together to produce the frst rgorous proof of the exstence of a maret clearng equlbrum, gven certan restrctve assumptons. One of ther ey contrbutons s to ntroduce tme and uncertanty nto general equlbrum models. Frst of all, t solves the long-standng problem of provng the exstence of equlbrum n a Walrasan (compettve) system. Ths model analyses the exact stuatons of those marets that are very compettve. In economcs, Arrow-Debreu model suggests that a set of prces such as aggregate supples wll equal to aggregate demands for every commodty under certan assumptons made about the economc condtons (.e. perfect competton and demand ndependence). Wth a general equlbrum structure, the model s applcable n evaluatng the overall mpact on resource allocaton of polcy changes n areas such as taxaton, tarff, and prce control. The functons of Arrow-Debreu model can be dvded nto sx categores, asset prcng model, equty rs premum, corporate fnance, Modglan and Mller Theorem, Arrow-Debreu securty and others. An mportant feature of Arrow and Debreu (1954) s ts conscous use of explctly gametheoretc deas, prevously found n the related paper by Debreu (1952) on hs own. However, a generalzaton of the usual noton of a game s nvolved. For there s an auctoneer whose strategy choce determnes the prce vector. Gven ths choce, agents are then constraned to choose net trades wthn ther budget sets. Thus the strategc choce of the auctoneer lmts the strateges that the other players are allowed to choose. Marowtz (1952) n an artcle on Portfolo Selecton postulates that an nvestor should maxmze portfolo expected return whle mnmzng portfolo varance. Marowtz paper s the frst mathematcal formalzaton of the dea of dversfcaton of nvestments. It comes to a concluson that through dversfcaton, rs can be reduced (but not generally elmnated) wthout changng expected portfolo return. One of the most mportant aspects of Marowtz s wor was to show that t s not a securty s own rs that s mportant to an nvestor, but rather the contrbuton the securty maes to the varance of hs entre portfolo - and that ths was prmarly a queston of ts covarance wth all the other securtes n hs portfolo. Ths school of thought formed the bass of the famous Marowtz problem whch explctly addresses the trade-off between expected rate of return and varance of the rate of return n a portfolo. A major set-bac of the Marowtz model was the large data and computatonal requrements n analyss of a portfolo wth many assets. Tobn, on the other hand, realzed that nvestors have a full range of lqudty preferences, and expanded potental nvestor choces to nclude low rs assets. He stated that nvestors should frst determne ther appette for rs. Ths appette should be satsfed from the one domnant equty portfolo determned from a Marowtz optmzaton (the sngle portfolo on the effcent fronter wth the hghest return per unt of rs). Then the lqudty and safety needs are satsfed wth the local zero rs portfolo. In essence the nvestor has two bucets and need only choose how to dvde hs assets between them. So, each nvestor owns the same equty portfolo, but 6

12 tempers the lqudty needs and rs-reward profle wth dfferent proportons of zero rs assets. Hs proposton was nown as the Separaton Theorem. Treynor s paper of 1961 had the ntenton to lay the ground wor for a theory of maret value whch ncorporates rs. In essence, the ams of the paper were to: Demonstrate that the overall behavour of the agents leads to Proposton 1 of Modglan and Mller (1958) Investgate the relaton between rs and nvestment value Dstngush between nsurable rs and unnsurable rs. Treynor approached CAPM from the perspectve of corporate cost-of-captal decson mang hence strvng to understand the relaton between rs and the dscount rate. Proposton 1 of Modglan and Mller (1958) says that In equlbrum, the maret value of any frm s ndependent of ts captal structure and s gven by captalzng ts expected return at the rate that s approprate for ts class. The assumptons that Treynor gave ncluded: No taxes No maret frctons Tradng does not affect prces Agents (nvestors) maxmze utlty n the essence of Marowtz Agents are rs averse A perfect lendng maret exsts Agents have dentcal maret nowledge and agree n ther forecasts of future values Sharpe, 1964, also set out to evaluate the relatonshp between the prces of assets and ther rs attrbutes. Sharpe n 1964, noted that through dversfcaton, some of the rs nherent n an asset can be avoded so that ts total rs s obvously not the relevant nfluence n ts prce; unfortunately lttle has been sad concernng the partcular rs component whch s relevant. Sharpe (1964) and Lntner (1965) added some two ey assumptons to the Marowtz model of the mean-varance portfolo. Gven maret clearng asset prces at t-1, nvestors agree on the jont dstrbuton of asset returns from t-1 to t. And ths s the dstrbuton from whch the returns we use to test the model are drawn from. Borrowng and lendng s at a rs free rate, whch s smlar for all nvestors and does not depend on the amount borrowed or lent. These assumptons meant that the maret portfolo must be on the mnmum varance fronter f the asset s to clear. 7

13 Mchael Brennan (1970) derved the after tax CAPM where the before-tax return of stocs was postvely related to the tax burden of equty securtes. In hs model, stocs payng hgher dvdend yelds exhbt hgher rs-adjusted returns than stocs payng lower or no dvdends. Fscher Blac (1972) developed a verson of the CAPM wthout rs free borrowng and lendng; commonly nown as the zero-beta CAPM. He showed that the CAPM result of maret portfolo beng mean-varance-effcent could also be acheved by allowng unrestrcted short sales of rsy assets. Ths mples that f there s no rs free asset, nvestors select portfolos from along the mean-varance-effcent fronter. The maret portfolo thus becomes a portfolo of the effcent portfolos chosen by nvestors. Mayers (1972) shows that when the maret portfolo ncludes non-traded assets, the model also remans dentcal n structure to the CAPM. Soln (1974) and Blac (1974) extended the model to encompass nternatonal nvestment. Treynor and Blac (1973) showed how best to construct a combnaton of actve and passve portfolos. They do ths by lnng the CAPM wth Sharpe (1963) ndex model. They explan when a portfolo should choose to run an almost perfectly dversfed ndex (passvely), and how the portfolo s dversfcaton should vary wth the prospects for the stocs n whch the portfolo s nvested; they also provde the frst analyss to underpn maret-neutral hedge funds. Modern portfolo optmsaton and rs management systems are often extensons of the Treynor-Blac model. 2.2 EMPIRICAL TESTS OF THE CAPM Tests of the CAPM were based on three mplcatons of the relaton between expected return and maret beta mpled by the model: Expected returns on all assets are lnearly related to ther betas and no other varable has margnal explanatory power. The beta premum s postve; the expected return on the maret portfolo exceeds the expected return on assets whose return s uncorrelated wth the maret return. In the Sharpe-Lntner verson of the model, assets uncorrelated wth the maret have expected returns equal to the rs free nterest rate, and the beta premum s the expected maret return less the rs free rate. In the case of Treynor-Sharpe-Lntner-Mossn CAPM, the slope of ths lne should be equal to the maret rs premum, and the ntercept should be equal to the rs free rate. For the zero-beta CAPM, the slope should be less than the maret rs premum, whle the ntercept should be greater than the rs free rate. There should also be no systematc reward for non-maret rs. Blac et al (1972) performed the earlest tests of the CAPM. The tests focused on the Treynor- Sharpe-Lntner model s predctons on the ntercept and slope n the relaton between expected return and maret beta. The relatonshp between the mean excess return and the beta was lnear and ths was consstent wth some type of CAPM. It however found that: Estmates of beta for ndvdual assets were mprecse thus creatng a measurement error problem when used to explan average returns The regresson resduals had common sources of varatons 8

14 To mprove the precson of estmated betas, researchers such as Blume (1970), Frend and Blume (1970) and Blac, Jensen and Scholes (1972) wored wth portfolos rather than ndvdual assets. Usng portfolos n cross-secton regressons of average returns on betas reduces the crtcal errors n varaton problems. It however shrns the range of betas and reduces statstcal powers. To mtgate ths problem, researchers sort securtes on beta when formng portfolos; the frst portfolo contans securtes wth the hghest betas, up tll the last one whch contans securtes wth the lowest betas. Roll (1977) created a major turnng pont n the emprcal testng of the CAPM. He argued that prevous tests of the CAPM had examned the relatonshp between equty returns and beta measured relatve to a broad equty maret ndex e.g. the S&P 500. However, the maret defned n the CAPM was not a sngle equtes maret but an ndex of all wealth e.g. bonds, property, foregn assets, human captal etc. Thus the portfolo used by Blac, Jensen and Scholes was not the true portfolo. Roll also shows that unless the maret portfolo s nown wth certanty, CAPM could not be tested. Fnally, he argues that the tests of CAPM are at best the tests of mean-varance effcency of the portfolo taen as the maret proxy. More recent tests have been conducted on the CAPM. Gbbons (1982) proposed a methodology that drectly tests the restrcton on returns mposed by the CAPM. Hs method was based on Maxmum Lelhood Estmaton whch avods the need of separate steps. By estmatng the beta and the rs premum smultaneously, errors-n-varable are avoded and there s an ncrease n precson of parameter estmates for the rs premum. Hs approach stll rejected CAPM. Gbson, Ross and Shanen (1989) conducted statstcal tests to confrm whether the maret proxy s the tangency portfolo n the set of portfolos that can be constructed by combnng the maret portfolo wth specfc assets used as dependent varables n the tme-seres regresson. Other authors who tred to handle the Roll crtque nclude: Shanen (1987) and Kandel and Stambaugh (1987). They both argue that although the stoc maret s not the true maret portfolo, t must nevertheless be hghly correlated wth the true maret. Even wth ths nsght, they stll found evdence that CAPM dd not hold. Stambaugh (1982) found that even when bonds and real estate are ncluded nto the maret proxy, the CAPM s stll rejected. Other rs factors were also found to nfluence stoc prces. These nclude: Prce to earnngs rato (Basu, 1977) Company Sze (Banz, 1981) Boo-to-Maret equty (Fama and French, 1992) Other systematc nfluences (Dmson and Mussavan, 1998) A ey assumpton of Marowtz portfolo optmzaton and the orgnal CAPM was that nvestors only care about the mean and varance of one-perod portfolo returns, whch s an unrealstc assumpton as nvestors can rebalance ther portfolos frequently. Daly movements n prces of many assets cannot be explaned by CAPM. Samuelson (1969), Haansson (1970) and Fama (1970) wored on the ntertemporal portfolo choce and asset prcng models, assumng that agents mae portfolo and consumpton decsons at dscrete tme ntervals. Merton s (1973) ntertemporal captal asset prcng model (ICAPM) s a natural extenson of the CAPM whch assumes that tme flows contnuously. ICAPM has dfferent assumptons about the nvestor objectves. Merton summarzed hs result by sayng An ntertemporal nvestor who currently faces a fve per cent nterest rate and a possble rate of ether two or ten per cent next perod wll have 9

15 portfolo demands dfferent from a sngle-perod maxmze n the same envronment or an ntertemporal maxmzer facng a constant nterest rate of fve per cent over tme. The upshot s that a CAPM wll hold at each pont n tme, but there wll be multple betas; the number of betas wll be equal to the one plus number of state varables that drve the nvestment opportunty set through tme. Merton s analyss ran contrary to the basc assumpton of CAPM, that an asset has greater value f ts margnal contrbuton to wealth s greater. Breeden (1989) came up wth the Consumpton Captal Asset Prcng Model. Ths allowed assets to be prced wth a sngle beta le the CAPM; however, the beta was not measured wth respect to aggregate maret wealth, but wth respect to aggregate consumpton flow. He nssted that nvestor preferences must be defned over consumpton and not wealth. One troublng feature though was the fact that the supply sde of assets was beng assumed away, yet the demand sde of nvestors was adequately dealt wth. One mportant nsght of the ICAPM s that multple rs factors are needed to explan asset prces. Ross (1976) developed the arbtrage prcng theory (APT) as an alternatve model that could potentally overcome the CAPM s problems; and retan the underlyng message of the CAPM. The major dea of APT was that only a small number of systematc nfluences affect the long term average return of securtes. Ross s APT was based on factor models whch are multple factors that represent the fundamental rss n the economy. Multfactor models allow for many measures of systematc rs. Each measure captures the senstvty of the asset to the correspondng pervasve factor. Ross sad that the APT was more of an arbtrage relaton than an equlbrum relaton. If the factor model holds exactly and assets do not have specfc rs, then the law of one prce mples that the expected return of any asset s just a lnear functon of other assets expected return. (If ths was not the case, arbtrage would be tang place). When assets have no specfc rs, all asset prces move n locstep wth one another and are therefore just leveraged copes of each other. Ths result s much harder when assets have specfc rs. It s possble to form portfolos where the specfc rs s dversfed away. To acheve full dversfcaton, an nfnte number of securtes s requred. Wth a fnte set of securtes, each of whch has specfc rs, the APT prcng restrcton wll only hold only approxmately. APT requres factor choce, number of factors, nterpretaton of factors etc whch was a hotly contested debate. One of the earlest emprcal studes was by Roll and Ross (1980), uses factor analyss. Ths was a statstcal technque that allows the researcher to nfer the factors from the data on securty returns. The advantage of the factor analyss technques s that the factors determned from the data explan a large proporton of the rss n that partcular dataset over the perod under consderaton. The dsadvantage s that factors usually have no economc nterpretaton. Roll and Ross concludes by argung an effort should be drected at dentfyng a more meanngful set of suffcent statstcs for the underlyng factors. An alternatve to factor analyss s by usng observed macroeconomc varables as the rs factors. Chen et al (1986) argued that at the most basc level, we can magne some fundamental valuaton model determnes the prce of assets.e. the prce of a stoc wll be the correctly dscounted expected future dvdends. Thus the choce of factors should nclude any systematc nfluences that mpact future dvdends, the way traders and nvestors form expectatons and the rate at whch nvestors dscount future cash flows. 10

16 An example s the Unted States stoc prces are sgnfcantly related to: Changes n ndustral producton The spread between the yeld on short-term and long-term government bonds: Ths s nterpreted as a proxy for busness cycle. The spread between low-and-hgh grade bonds: Interpreted as a proxy for overall busness rs n the economy. Changes n expected nflaton Changes n unexpected nflaton Arguments aganst the APT nclude: Shanen (1982, 1985), he asserted that for ndvdual securtes the approxmaton mpled by Ross was so mprecse, that t maes t mpossble ever to test whether the APT s true or false. Shanen argues further that snce the expected return for any securty or portfolo s related only approxmately to ts factor senstvtes, to get an exact prcng relatonshp, addtonal assumptons are needed. He mantans that researchers who test the APT by assumng that the restrcton holds, even for securtes, are actually testng an equlbrum form of APT. In concluson, both CAPM and APT have fundamental lmtatons to any emprcal verfcaton. Fama et. al (1993) explan the dfferences between the returns on the New Yor Stoc Exchange (NYSE) and Natonal Assocaton of Securty Dealers (NASD). Stocs on the NYSE have hgher average returns than the stocs of smlar sze on the NASD durng the test perod. They use Fama and French three-factor model to explan the dfference. Ther analyss demonstrates that reason for ths varaton s the dfference between the rs of the stocs, whch s captured by Fama and French three-rs factor model. Fama et al. (1993, p.37) argue that stocs wth hgh senstvty tend to be frms wth persstently poor earnngs, whch lead to low stoc prce and hgh boo-to-maret equty ratos. Stocs wth low senstvty to the boo-to-maret rs factor tend to have persstently hgh earnngs whch lead to low BE/ME. They conclude that boo-tomaret rato s the most mportant rs factor that explans the dfference n returns between NYSE stocs and NASD stocs. Frazzn (2013) and Pedersen (2013) have presented a model named Bettng aganst Beta that fnds evdence that long leveraged low-beta assets and short hgh-beta assets produce sgnfcantly postve rs-adjusted returns. 11

17 CHAPTER 3: METHODOLOGY 3.1 MARKOWITZ MEAN-VARIANCE PORTFOLIO THEORY Harry Marowtz (1952, 1959) developed hs portfolo-selecton technque, whch came to be called modern portfolo theory (MPT). The mean-varance portfolo theory was desgned to construct the optmal portfolo based on the dea that between rs and return there s a postve relaton. Marowtz proved that nvestors should create ther portfolo n order to offer them a maxmum expected level of return for a gven level of rs or, a mnmum level of rs for a gven expected level of return. The Marowtz model s based on several assumptons concernng the behavour of nvestors and fnancal marets namely: A probablty dstrbuton of possble returns over some holdng perod can be estmated by nvestors. Investors have sngle-perod utlty functons n whch they maxmze utlty wthn the framewor of dmnshng margnal utlty of wealth. Varablty about the possble values of return s used by nvestors to measure rs. Investors care only about the means and varance of the returns of ther portfolos over a partcular perod. Expected return and rs as used by nvestors are measured by the frst two moments of the probablty dstrbuton of returns-expected value and varance. Return s desrable; rs s to be avoded 1. Fnancal marets are frctonless. Marowtz showed n hs theory that stocs are related to each other and that the rs can be decreased through dversfcaton. He was the frst to clearly and rgorously show how the varance of a portfolo can be reduced through the mpact of dversfcaton. The proof s very smple: f one taes 2 stocs and he wll calculate the correlaton coeffcent, the value of ths coeffcent would be less than one and f the respectve stocs are ncluded n a portfolo, the overall rs of ths portfolo would decrease. Every possble asset combnaton can be plotted n rs-return space, and the collecton of all such possble portfolos defnes a regon n ths space. The lne along the upper edge of ths regon s nown as the effcent fronter. Combnatons along ths lne represent portfolos (explctly excludng the rs-free alternatve) for whch there s lowest rs for a gven level of return. Conversely, for a gven amount of rs, the portfolo lyng on the effcent fronter represents the combnaton offerng the best possble return. Mathematcally the effcent fronter s the ntersecton of the set of portfolos wth mnmum varance and the set of portfolos wth maxmum return. 1 Marowtz model assumes that nvestors are rs averse. Ths means that gven two assets that offer the same expected return, nvestors wll prefer the less rsy one. Thus, an nvestor wll tae on ncreased rs only f compensated by hgher expected returns. Conversely, an nvestor who wants hgher returns must accept more rs. The exact trade-off wll dffer by nvestor based on ndvdual rs averson characterstcs. Ths mples a ratonal nvestor wll not nvest n a portfolo f a second portfolo exsts wth a more favorable rs-return profle -.e., f for that level of rs an alternatve portfolo exsts whch has better expected returns. Usng rs tolerance, we can classfy nvestors nto three types: rs-neutral, rs-averse, and rs-lover. Rs-neutral nvestor s do not requre the rs premum for rs nvestments; they judge rsy prospects solely by ther expected rates of return. Rs-averse nvestors are wllng to consder only rs-free or speculatve prospects wth postve premum; they mae nvestment accordng to the rs-return trade-off. A rs-lover s wllng to engage n far games and gambles; and adjusts the expected return upward to account for the fun of added rs. 12

18 Expected Return A Marowtz portfolo model s one where no added dversfcaton can lower the portfolo's rs for a gven return expectaton (alternately, no addtonal expected return can be ganed wthout ncreasng the rs of the portfolo). The Marowtz Effcent Fronter s the set of all portfolos of whch expected returns reach the maxmum gven a certan level of rs. C D A Mnmum Varance Portfolo (MVP) B A Investment Opportunty 0 Standard Devaton Fgure 1: Investment opportunty set Every possble asset combnaton can be plotted n rs-return space, and the collecton of all such possble portfolos defnes a regon n ths space. The lne along the upper edge of ths regon s nown as the effcent fronter. Combnatons along ths lne represent portfolos (explctly excludng the rs-free alternatve) for whch there s lowest rs for a gven level of return. Conversely, for a gven amount of rs, the portfolo lyng on the effcent fronter represents the combnaton offerng the best possble return. Mathematcally the effcent fronter s the ntersecton of the set of portfolos wth mnmum varance and the set of portfolos wth maxmum return. The area wthn curve ABCD s the feasble opportunty set representng all possble portfolo combnatons. Portfolos that le below the mnmum-varance portfolo (pont B) on the fgure can therefore be rejected out of hand as neffcent. The portfolos that le on the fronter BA n the Fgure 1 would not be lely canddates for nvestors to hold. Because they do not meet the crtera of maxmzng expected return for a gven level of rs or mnmzng rs for a gven level of return. Ths s easly seen by comparng the portfolo represented by ponts A and A. Snce nvestors always prefer more expected return than less for a gven level of rs, A s always better than A. Usng smlar reasonng, nvestors would always prefer B to A because t has both a hgher return and a lower level of rs. In fact, the portfolo at pont B s dentfed as the mnmum-varance portfolo; snce no other portfolo exsts that has a lower standard devaton. The curve BC represents all possble effcent portfolos and s the effcent fronter 2, whch represents the set of portfolos that offers the hghest possble expected rate of return for each level of portfolo standard devaton. 2 The effcent fronter wll be convex ths s because the rs-return characterstcs of a portfolo change n a non-lnear fashon as ts component weghtngs are changed. (As descrbed above, portfolo rs s a functon of the correlaton of the component assets, and thus changes n a non-lnear fashon as the weghtng of component assets changes.) The effcent fronter s a parabola (hyperbola) when expected return s plotted aganst varance (standard devaton). 13

19 Expected Return The best choce among the portfolos on the upward slopng porton BC of the fronter curve s not as obvous, because n ths regon hgher expected return s accompaned by hgher rs. The best choce wll depend on the nvestor s wllngness to trade off rs aganst expected return. Relaxng the assumpton of no short sellng n ths development of the effcent fronter nvolves a modfcaton of the analyss of the effcent fronter of constrant (not allowed short sales). If the number of short sales s unrestrcted, then by a contnuous short sellng of the lowest-return asset A and renvestng n hghest-return asset C the nvestor could generate an nfnte expected return. The effcent fronter of unconstrant portfolo s shown n Fgure 2. C D D B wthout short sales wth short sales R A 0 Standard Devaton Fgure 2: The effcent fronter of unrestrcted/restrcted portfolo The upper bound of the hghest-return portfolo would no longer be C but nfnty (shown by the arrow on the top of the effcent fronter). Lewse the nvestor could short sell the hghestreturn securty C and renvest the proceeds nto the lowest-yeld securty A 3, thereby generatng a return less than the return on the lowest-return assets. Gven no restrcton on the amount of short sellng, an nfntely negatve return can be acheved, thereby removng the lower bound of B on the effcent fronter. Hence, short sellng generally wll ncrease the range of alternatve nvestments from the mnmum-varance portfolo to plus or mnus nfnty 4. RETURN Gven any set of rsy assets and a set of weghts that descrbe how the portfolo nvestment s splt, the general formulas of expected return for n assets s: n E( r ) w E r (F.1) P 1 3 Ratonal nvestor wll not short sell a hgh-return asset and buy a low-return asset. Ths case s just for extreme assumpton. 4 Whether an nvestor engages n any of ths short-sellng actvty depends on the nvestor s own unque set of ndfference curves. 14

20 where: n w = 1.0; 1 n = the number of securtes; w = the proporton of the funds nvested n securty ; r, r = the return on th securty and portfolo p; and P E = the expectaton of the varable n the parentheses. The return computaton s nothng more than fndng the weghted average return of the securtes ncluded n the portfolo. RISK The varance of a sngle securty s the expected value of the sum of the squared devatons from the mean, and the standard devaton s the square root of the varance. The varance of a portfolo combnaton of securtes s equal to the weghted average covarance 5 of the returns on ts ndvdual securtes: n n 2 rp p w wj r rj (F.2) Var Cov, 1 j1 Covarance can also be expressed n terms of the correlaton coeffcent as follows: Cov r, r (F.3) j j j j where = correlaton coeffcent between the rates of return on securty, r, and the rates of j return on securty j, r j, and, and j represent standard devatons of r and r j respectvely. Therefore: Var r n n w w (F.4) p j j j 1 j1 Overall, the estmate of the mean return for each securty s ts average value n the sample perod; the estmate of varance s the average value of the squared devatons around the sample average; the estmate of the covarance s the average value of the cross-product of devatons. CALCULATING THE MINIMUM VARIANCE PORTFOLIO In Marowtz portfolo model, we assume nvestors choose portfolos based on both expected return, Er ( p), and the standard devaton of return as a measure of ts rs, p. So, the portfolo selecton problem can be expressed as maxmzng the return wth respect to the rs of 5 Hgh covarance ndcates that an ncrease n one stoc's return s lely to correspond to an ncrease n the other. A low covarance means the return rates are relatvely ndependent and a negatve covarance means that an ncrease n one stoc's return s lely to correspond to a decrease n the other. 15

21 the nvestment (or, alternatvely, mnmzng the rs wth respect to a gven return, hold the return constant and solve for the weghtng factors that mnmze the varance). Mathematcally, the portfolo selecton problem can be formulated as quadratc program. For two rsy assets A and B, the portfolo conssts of w, w, the return of the portfolo s then, The weghts should be chosen so that (for example) the rs s mnmzed, that s Mn P wa A wb B 2wAwB AB A B w A A B for each chosen return and subject to w w 1, w 0, w 0. The last two constrants A B A B smply mply that the assets cannot be n short postons. Above, we smply use two-rsy-assets portfolo to calculate the mnmum varance portfolo weghts. If we generalzaton to portfolos contanng N assets, the mnmum portfolo weghts can then be obtaned by mnmzng the Lagrange functon C for portfolo varance. Mn n n 2 p ww jj j 1 j1 Subject to w1 w2... w N 1 C w w rr W 1 j1 1 n n n jcov j 1 1 (X.8) n whch 1 are the Lagrange multplers, respectvely, s the correlaton coeffcent between r and r j, and other varables are as prevously defned. The effcent set that s generated by the aforementoned approach (equaton X.8) s sometmes called the mnmum-varance set because of the mnmzng nature of the Lagrangan soluton. If we add a condton nto the equaton X.8, whch s be subject to the portfolo s attanng some target expected rate of return, we can get the optmal rsy portfolo. Mn n n 2 p WW jj j 1 j1 j n Subject to * WE R E, where * 1 E s the target expected return and n 1 W 1.0 The frst constrant smply says that the expected return on the portfolo should equal the target return determned by the portfolo manager. The second constrant says that the weghts of the securtes nvested n the portfolo must sum to one. The Lagrangan objectve functon can be wrtten: 16

22 n n n n * C w wjcovrr j 1E w E r 21 w 1 j1 1 1 Tang the partal dervatves of ths equaton wth respect to each of the varables, w1, w2,..., wn, 1, 2and settng the resultng equatons equal to zero yelds the mnmzaton of rs subject to the Lagrangan constrants. Then, we can solve the weghts and these weghts are represented optmal rsy portfolo by usng of matrx algebra. If there no short sellng constrant n the portfolo analyss, second constrant, w 1.0, n should substtute to w 1.0, where the absolute value of the weghts w allows for a 1 gven w to be negatve (sold short) but mantans the requrement that all funds are nvested or ther sum equals one. The Lagrangan functon s C w w rr E w E r w 1 j1 1 1 n n n n * jcov j 1 21 (X.10) If the restrcton of no short sellng s n mnmzaton varance problem, t needs to add a thrd constrant: w 0, 1,, N The addton of ths non-negatvty constrant precludes negatve values for the weghts (that s, no short sellng). The problem now s a quadratc programmng problem smlar to the ones solved so far, except that the optmal portfolo may fall n an unfeasble regon. In ths crcumstance the next best optmal portfolo s elected that meets all of the constrants. DEMERITS OF MARKOWITZ THEORY Even though the theory of Marowtz was spectacular and useful n ths feld, t had some nconvenences. The calculaton of portfolo return together wth the weghted portfolo asset may perfectly accurate f t s just for two to three stocs. But when t comes to the addton of stocs D, E and so on up to a thousand or more, t becomes too long, tme consumng, and almost mpossble to calculate as the number of asset correlatons becomes more and the calculaton wdens up nto several algebrac numeratons of every return varance beng extended to three or more securtes. Further, t was done tang nto account a very abstract concept n economcs of expected utlty model of Von Nuemann and Morgenstern (1953). The economcal practce has shown that models constructed based on the dea of utlty are very dffcult or even mpossble to apply. Also, the mathematcs beyond of mean-varance s very sophstcated, whch maes the applcaton dffcult when the portfolo conssts of a great number of shares. Specfcally, to estmate the benefts of dversfcaton would requre that practtoners calculate the covarance of returns between every par of assets. Fnally, crtcs of the model argued that t s a statc one whch made the results to be based. n 1 17

23 3.2 MATHEMATICAL DERIVATION OF CAPITAL ASSET PRICING MODEL The bass of the CAPM s the portfolo theory wth a rsless asset and unlmted short sales. We do not consder only the decson of a sngle nvestor, but aggregate them to determne a maret equlbrum. Addtonally to the assumptons for portfolo theory, we have to add that all nvestors have the same belefs on the probablty dstrbuton of all assets,.e. agree on the expected returns, varances and covarances. If all nvestors agree on the characterstcs of an asset the optmal rsy portfolo wll be equal for all nvestors, even f they dffer n ther preferences (rs averson). Because all assets have to be held by the nvestors the share each asset has n the optmal rsy portfolo has to be equal to ts share of the maret value of all assets. The optmal rsy portfolo has to be the maret portfolo. Moreover all assets have to be maretable,.e. all assets must be traded and there are no other nvestment opportuntes not ncluded nto the model. Table below summarzes the assumptons. No transacton costs and taxes Assets are ndefntely dvsble Each nvestor can nvest nto every asset wthout restrctons Investors maxmze expected utlty by usng the mean-varance crteron Prces are gven and cannot be nfluenced by the nvestors (compettve prces) The model s statc,.e. only a sngle tme perod s consdered Unlmted short sales Homogenety of belefs All assets are maretable Table 3.2.1: Assumptons of the CAPM 18

24 Every nvestor j (j = 1,....., M ) maxmzes hs expected utlty by choosng an optmal portfolo,.e. choosng optmal weghts for each asset. Wth the results of Arrow Pratt measure of rs averson we get j j 1 2 max E[ U ( R )] max N p U N p z j p { x } 1 { x } max N p z j p { x} 1 2 max 1 x z x x N N N N j { x} N for all j 1,..., M wth the restrcton x 1. The Lagrange functon for solvng ths problem can easly be obtaned as 1 N N N N 1 Lj x z jx x 1 x The frst order condtons for a maxmum are gven by L N j z x 0, 1,..., N, x j 1 L N j 1 x 0 1 for all j 1,..., M. Solvng the above equatons for µ gves N N a jx z jcov R, xr K1 1 z jcov R, R p z j p Wth p 0 we fnd that hence we can nterpret that λ as the expected return of an asset whch s uncorrelated wth the maret portfolo. As the rsless asset s uncorrelated wth any portfolo, we can nterpret λ as the rs free rate of return r: r z j p From (3.6) we see that the expected return depends lnearly on the covarance of the asset wth the maret portfolo. The covarance can be nterpreted as a measure of rs for an ndvdual asset (covarance rs). Intally we used the varance as a measure of rs, but as has been shown n the last secton the rs of an ndvdual asset can be reduced by holdng a portfolo. The rs that cannot be reduced further by dversfcaton s called systematc rs, whereas the dversfable rs s called unsystematc rs. The total rs of an asset conssts of the varaton of 19

25 the maret as a whole (systematc rs) and an asset specfc rs (unsystematc rs). As the unsystematc rs can be avoded by dversfcaton t s not compensated by the maret, effcent portfolos therefore only have systematc and no unsystematc rs. The covarance of an asset can be also nterpreted as the part of the systematc rs that arses from an ndvdual asset: x x Cov R, R Cov x R, R N N N p p p Cov R, R Var R 2 p p p p Equaton (3.6) s vald for all assets and hence for any portfolo, as the equaton for a portfolo can be obtaned by multplyng wth the approprate weghts and then summng them up, so that we can apply ths equaton also to the maret portfolo, whch s also the optmal rsy portfolo: r z 2 p j p Solvng for z j and nsertng nto (3.6) gves us the usual formulaton of the CAPM: Defnng we can rewrte (3.8) as r r p r r p β represents the relatve rs of the asset (σp) to the maret rs (σ 2 ). The beta for the maret portfolo s easly shown to be 1. We fnd a lnear relaton between the expected return and the relatve rs of an asset. Ths relaton s ndependent of the preferences of the nvestors (zj), provded that the mean-varance crteron s appled and that the utlty functon s quadratc. Ths equlbrum lne s called the Captal Maret Lne (CML). Fgure 3.1 llustrates ths relaton. For the rs an nvestor taes he s compensated by the amount of µp r per unt of rs, the total amount (µp r)β s called the rs premum or the maret prce of rs. The rs free rate of return r may be nterpreted as the prce for tme. It s the compensaton for not consumng the amount n the current perod, but wat untl the next perod. Equaton (4.9) presents a formula for the expected return gven the nterest rate, r, the beta and the expected return on the maret portfolo, µp. However, the expected return of the maret portfolo s not exogenous as t s a weghted average of the expected returns of the ndvdual assets. In ths formulaton only relatve expected returns can be determned, the level of expected returns,.e. the maret rs premum, s not determned by the CAPM. Although we can reasonably assume µp to be gven when nvestgatng a small captalzed asset, we have to determne µp endogenously. p 20

26 Thus far we only consdered the maret portfolo, but an nvestor wll n general not hold the maret portfolo (optmal rsy portfolo). As we saw n secton 3 the optmal portfolo wll be a combnaton of the maret portfolo and the rsless asset, where the shares wll vary among nvestors. We can now add another restrcton to our model. The return on the optmal rsy portfolo has to be such that the maret for the rsless assets has to be n equlbrum. The amounts of rsless assets lent and borrowed have to be equal: M j1 x jr 0 where xjr denotes the demand of the jth nvestor for the rsless asset. Wth ths addtonal maret to be n equlbrum t s possble to determne µp endogenously. We herewth have found an equlbrum n the expected returns. These expected returns can now be used to determne equlbrum prces accordng to secton 1. Nevertheless ths result remans to determne only relatve prces. The rs free rate r s not determned endogenously. Although t can reasonably be assumed that r can substantally be nfluenced by monetary polcy, especally for longer tme horzons t s not gven. CRITIQUE OF CAPM The assumptons underlyng the CAPM are very restrctve. The problems assocated wth the use of the mean-varance crteron and the quadratc utlty functon have already been mentoned n secton 4.2. Some restrctons such as the absence of transacton costs and taxes, unlmted borrowng and lendng at the rs free rate and short sales have been lfted by more recent contrbutons wthout changng the results sgnfcantly. One restrcton however that mostly s not mentoned n the lterature s that the assets have to be lnearly dependent. Ths lnearty, also used n portfolo theory s mpled by the use of the covarance, whch s only able to capture lnear dependences approprately. Ths lnearty of returns rules out the ncluson of dervatves that mostly have strong non-lneartes n ther pay-offs and have become an mportant tool for nvestment n recent years. By excludng such assets the need to nclude all assets s volated. Besde these theoretcal crtques emprcal nvestgatons show a mxed support for the CAPM. There exst a large number of emprcal nvestgatons of the CAPM usng dfferent econometrc specfcatons. Early nvestgatons manly supported the CAPM, but more recent results show that the CAPM s not able to explan the observed returns. If other varables, such as boo-maret ratos, maret value of a company or prceearnngs ratos are ncluded the beta has no sgnfcant nfluence on the observed returns. Le the portfolo theory the CAPM s a statc model,.e. the nvestment horzon s assumed to be only a sngle perod. As has been ponted out, after every perod the portfolo has to be rebalanced even f the belefs do not change. Ths rebalancng wll affect the maret equlbrum prces and hence the expected future returns 21

27 and rs prema. Therefore a dynamc model wll capture the nature of decson mang more approprate, such a modfc aton s gven wth the Intertemporal CAPM to be presented later. Another crtcal pont n the CAPM s that uncondtonal belefs (means, varances and covarances) are used, the nvestors are not able to condton ther belefs on nformaton they receve. A drect mplcaton of ths s the assumpton that belefs are constant over tme. Many emprcal nvestgatons gve strong support that belefs are varyng over tme, of specal mportance s the beta. But also the rs averson and the rs premum have been found to vary sgnfcantly over tme. These aspects are taen nto account n the Condtonal CAPM that wll be presented n the next secton. The CAPM further taes the amount an nvestor wants to nvest n the assets as exogenously gven. However the amount to nvest s a decson whether to consume today or to consume n the future. The nfluence of consumpton s ncorporated nto the Consumpton-based CAPM. Fnally the CAPM explans the expected returns only by a sngle varable, the rs of an asset relatve to the maret. It s reasonable to assume that other factors may as well nfluence the expected returns. We wll therefore dscuss the Arbtrage Prcng Theory as an alternatve to the CAPM framewor. The Intertemporal CAPM also allows for more rs factors. Another assumpton remans crtcal for the CAPM: the assumpton that all assets are maretable. Some nvestment restrctons due to legslaton n foregn countres are taen nto account by the Internatonal CAPM, but assets such as human captal are not maretable. Therefore the maret portfolo cannot be determned correctly. Roll (1977) showed that the determnaton of a correct maret portfolo s mportant to acheve correct results. Only small devatons from the true maret portfolo can bas the results sgnfcantly. Other nvestgatons showed that small devatons do not have a large mpact on the results. It remans an open queston whether the assumpton of all assets to be maretable s restrctve or not. As a result of these crtques the CAPM has been modfed and today a wde varety of extensons exst. Alternatves that use a dfferent approach to asset prcng are not frequently found and had, wth the excepton of the Arbtrage Prcng Theory, no great mpact as well n applcatons as the academc lterature. The remanng sectons wll gve an overvew of the extensons of the CAPM as well as alternatve approaches. 22

28 CAPM EXTENSIONS 3.3 THE ZERO-BETA CAPITAL ASSET PRICING MODEL: Ths model was developed by Blac (1972) and s applcable to marets where rs-free asset s partally or completely restrcted. Zero Beta CAPM was generated for relaxng the assumpton of a rs-free captal asset and the assumpton that nvestors can borrow and lend on the bass of a rs-free nterest rate. Ths s because even though most nvestors can mae nvestments from an unlmted amount of rs-free assets; they however cannot borrow at the same amount lmtlessly. Zero beta model ndcates that a rs free nterest rate s not necessary n order for CAPM to be vald. Investors eep dfferent rsy portfolos; however all such portfolos tae place on the effcent fronter. Unon of the portfolos at the fronter at the same tme taes place at the fronter. Lnearty of the model s stll vald and the beta coeffcent contnues to be the measure of systematc rs. However, one lmtaton of ths model s the requrement that there s no restrcton on short sellng. Snce the correlaton coeffcents of many of the assets n the maret are postve, t s almost not possble to establsh a Zero Beta portfolo wthout short sellng. For any fronter portfolo p, except the mnmum varance portfolo, there exsts a unque fronter portfolo wth whch p has zero covarance. Usng the two-fund theorem, any portfolo can be wrtten as a combnaton of 2 fronter portfolos, any portfolo the nvestors choose can be a combnaton of the maret portfolo and ts zero-beta counterpart. By the spannng property of the fronter portfolos, any portfolo on the fronter can be obtaned as a lnear combnaton of only two portfolos on the fronter. Therefore we have two-fund separaton even wthout the rs-free asset. An mportant assumpton behnd the Zero-Beta CAPM s that short-sales are possble. To obtan zero-beta portfolos we typcally would have to short sell some assets. If there are short-sales constrants the Zero-Beta CAPM fals to hold. The stages n Zero Beta CAPM model analyss are the same; the only dfference s the fact that a portfolo return, whose correlaton wth the maret s zero taes place nstead of a rs-free asset return. Snce zero beta portfolo return s uncertan n ths case, the second-pass regresson equaton s compared wth CAPM, arranged as: In ths case, t s evdent that 0 Er estmate of m value. On the other hand, ( ) E r E r E r r z m z 1 E r E r E r z m r coeffcent s an estmate of 1 R and 23 z value and s an values are estmates of uncertan real parameters of captal asset. Therefore, dfferng from Standard CAPM, t has to 0 r 1 be z 0. Then, t has to be equal to /1 rz R. Snce z value s uncertan, the 0 hypothess, whch has to be tested, s equalty of all /1. Chou (2002) used Wald test statstc n testng of ths hypothess and found that the Internatonal zero-beta CAPM was vald n at least 16 OECD over the perod

29 E(r) Q R M E(r ) Z Z Snce any combnaton of two effcent fronter portfolos s also effcent, the average (maret) portfolo wll also be effcent here, as depcted by pont M. Moreover, the Zero Beta model must now apply, because the maret portfolo s effcent and all nvestors choose rsy portfolos that le on the effcent fronter. As a result, the ray from the expected return on the effcent portfolo wth zero correlaton wth M (and hence zero beta) to the effcent fronter, wll be tangent at M. Ths can happen only f: r f (1 t) < E(r Z ) < r f More generally, consder the case of any number of classes of nvestors wth ndvdual rs-free borrowng and lendng rates. As long as the same effcent fronter of rsy assets apples to all of them, the Zero-Beta model wll apply, and the equlbrum zero-beta rate wll be a weghted average of each ndvdual's rs-free borrowng and lendng rates. In the zero-beta CAPM the zero-beta portfolo replaces the rs-free rate. Snce the zero-beta portfolo s uncorrelated wth the maret portfolo, then we must have: Cov( Rz, Rm) z Var( R ) m Cov( R, R ) 0 hence z 0 z m 24

30 3.4 THE FAMA-FRENCH THREE-FACTOR MODEL The Fama and French three-factor asset prcng model was developed as a response to poor performance of the CAPM n explanng realzed returns. Eugene Fama and Ken French (1992) presented emprcal arguments aganst the CAPM model showng that the cross secton of average equty returns n the US maret shows lttle statstcal relaton to the βs of the orgnal CAPM model. They dscovered that anomales relatng to the CAPM are captured by the three-factor model. They base ther model on the fact that average excess portfolo returns are sensble to three factors namely: excess maret portfolo return; the dfference between the excess return on a portfolo of small stocs and the excess return on a portfolo of bg stocs (SMB, small mnus bg); and the dfference between the excess return on a portfolo of hgh-boo-to-maret stocs and the excess return on a portfolo of low -boo -to -maret stocs (HML, hgh mnus low). Ths s formulated as below: E R R E r r E SMB E HML f m m f SMB HML The coeffcents n ths model have smlar nterpretatons to beta n the standard CAPM. β M s a measure of the exposure an asset has to maret rs (although ths beta wll have a dfferent value from the beta n a CAPM model as a result of the added factors), β SMB measures the level of exposure to sze rs and β HML measures the level of exposure to value rs. In ths equaton, SMB (small mnus bg) s the dfference between the returns on dversfed portfolos of small and bg maret captalzaton stocs, referred to as the sze premum, and HML (hgh mnus low) s the dfference between the returns on dversfed portfolos of hgh and low Boo-to-Maret stocs. Ths s the value placed by accountants on a company as a rato relatve to the boo equty (BE) dvded by ts maret equty (ME). CAPM s assumpton of a sngle rs factor explanng expected returns has been crtczed. Fama and French (1992, 1993, 1995, 1996, 1998) proposed an alternatve prcng model whch ncorporates these three factors as proxes for rs: the maret, the sze, and the value factors. Although the Fama-French model has been adopted both by most practtoners and academcs n fnancal ssues pertanng to portfolo management, captal budgetng, and performance evaluaton, the three-factor model suffers from many drawbacs. In fact, whle fnancal data exhbt mult scalng,.e. are a combnaton of dfferent mult-horzon dynamcs, the Fama-French model s a sngle scale model that studes the relaton between rs factors and expected returns on a global scale nvestment horzon. Furthermore, most studes neglect the assets long-term holdng perod focusng manly on short-term analyss though t s crucal to tae nto account the tendency of some nvestors to hold stocs over the long-run. The model fts two addtonal rs factors to the CAPM n order to explan the return varatons better and cure the anomales of the CAPM. Fama and French (1996) pont out that the model captures many of the varatons n the cross -secton of average stoc returns, and t absorbs 25

31 most of the anomales that have plagued the CAPM. In the same study they argue that the emprcal success of ther model suggests that t s an equlbrum prcng model, a three -factor verson of Merton s (1973) ntertemporal CAPM or Ross s arbtrage prcng theory. Ths model s better than the CAPM to estmate expected returns, and captures n a better way the varaton n average returns for portfolos formed on sze, boo to maret, and the others factors, for whch CAPM s not effcent. From a theoretcal perspectve, the man shortcomng of the three factor model s ts emprcal motvaton, as the SMB and HML explanatory returns are not motvated by predctons about state varables of concern to nvestors. 26

32 3.5 THE CONDITIONAL CAPITAL ASSET PRICING MODEL As we saw n the last secton the performance of the tradtonal CAPM s only poor. It has been proposed that the reason may be the statc nature of the model and the therewth followng assumptons of a fxed beta and fxed rs prema. Allowng the beta to vary over tme can be justfed by the reasonable assumpton that the relatve rs and the expected excess returns of an asset may vary wth the busness cycle. It therefore s reasonable to use all avalable nformaton on the busness cycle and other relevant varables to form expectatons,.e. to use condtonal moments. Ths gave rse to the Condtonal Captal Asset Prcng Model. DERIVATION OF THE MODEL We assume that the CAPM as derved n the last secton remans vald f we use condtonal moments nstead of uncondtonal moments: where t 1 t t t t t E R t 1 E r t 1 E RM t 1 E r t 1 t1 R denotes the return of asset at tme t, t t Cov R, RM t1 t Var R Mt 1 t r the rs free rate of return at tme t, the return of the maret portfolo at tme t and Ωt 1 the nformaton avalable at tme t 1. For notatonal smplcty we defne t 1 t 0 E rt 1 R E r E R t1 t t t t1 1 M t1 t1 M t1 0 as the expected rs free rate and the expected rs premum, respectvely. We can rewrte (5.1) as ER t t1 t1 t1 t1 0 1 We want to explan uncondtonal returns because expectatons cannot be ob- served. Tang expectatons of (5.5) we get wth the law of terated expectatons: t t1 t1 t1 t1 t1 E R E 0 E 1 E Cov 1, The rs premum, γ t 1, and the rs, β t 1, wll n general be correlated. To 1 see ths magne a recesson n whch future prospects of a company are very uncertan, then the beta wll be relatvely hgh as other companes, le consumer goods or utltes that form a part of the maret portfolo, are affected much less. The 27 t R M

33 rs premum wll also be hgh to compensate the owners of the asset for the hgher rs n a recesson as the asset forms part of the maret portfolo. In a boom relatons are lely to change. Hence rs premum and rs wll be correlated. Let the senstvty of the condtonal beta to a change n the maret rs premum be denoted by ϑ: t1 t1 Cov, 1 t1 Var 1 We further defne a resdual beta as: t 1 t 1 t 1 t 1 t 1 E 1 E 1 The frst two terms represent the dfference of the condtonal and the uncond- tonal beta. The frst term adjusts ths dfference for the devaton of the rs premum from ts uncondtonal value. We fnd that t 1 t 1 t 1 t 1 t 1 E E E E 1 E 1 0 t1 t1 t1 t1 t1 t1 t1 t1 t1 E 1 E 1 E E 1 E 1 1 E t 1 t 1 t 1 t 1 t 1 t 1 t 1 t E 1 E 1 Cov, 1 E E 1 E 1 E 1 t 1 t 1 t 1 Cov, 1 Var 1 t 1 t 1 t 1 t 1 Cov, 1 Cov, 1 0. We can now solve (3.8) for the condtonal beta: E E t t t t t 1 1 We further can solve (3.7) for t1 t1 Cov, 1 and nsert the expresson nto (3.6) to obtan: t t1 t1 t1 t1 E R E 0 E 1 E Var 1 The excess returns turn out to be a lnear functon of the expected beta and the senstvty of the beta to a change n the rs premum. We want to express ths 28

34 relaton wth usng the uncondtonal beta. Equaton (5.5) also remans vald for the maret portfolo, so that we get wth t 1 1: M ER t t1 t1 M t1 0 1 and from the defnton of t 1 0 : E R M t t1 t t We defne ε t as the resdual from relaton (5.1): wth t t t 1 t t 1 t E 1 t 1 E R t 1 E 0 t 1 E RM 0 t1 E R t 1 t 1 t 1 t 1 t t 1 t M 0 0 t1 t1 t1 t1 1 1 t t t t t 1 t t 1 t 1 t E R M t 1 E R R M t 1 E 0 RM 0 R M t1 E R R E E r R t t t t M t1 t1 M t1 0 By the law of terated expectatons we have: E E E t t t 1 0 E R E E R t t t t M M t 1 0 t t1 t t t1 E 1 E E RM 0 t1 t t t 1 E E RM 0 t1 t1 t t t 1 E t 1 E RM 0 t1 29

35 0 By nsertng (5.11) nto (5.15) and solvng for R t we get R R E R E R t t 1 t t 1 t 1 t t 1 t 1 t 1 t t 1 t 1 t 0 M 0 M M 0 From the defnton and the propertes of the covarance we get wth (3.21): Cov R, R Cov, R Cov R, R E Cov R E, R t t t 1 t t t 1 t t 1 t t 1 t 1 t 1 t M 0 M M 0 M M M Cov R R t t1 t1 t M 0, M Cov R, Cov, Cov R, E Cov R E, t t1 t1 t1 t t1 t1 t1 t t1 t1 t1 t1 0 M 0 M Cov R t t1 t1 t1 M 0, It has been shown by Jagannathan/Wang (1996, pp.38 ff.) that the last term n (5.22) and (5.23) becomes zero f we assume the resdual betas, η t 1, to be uncorrelated wth maret condtons. We defne t t Cov R, R M t Var R M as the tradtonal maret beta whch measures the uncondtonal rs and t t1 Cov R, 1 t1 Var 1 As the premum beta whch measures the rs from a varyng beta. We can substtute (5.24) and (5.25) nto (5.22) and (5.23), respectvely, and obtan after solvng for β and β γ : M t1 t t t1 t t t t t t Cov, 0, R M Cov RM 0, R Cov R E R M M t1 E t t t Var RM Var R M Var R M 30

36 M t1 t1 t t1 t1 t t1 t1 t1 t1 Cov 0, 1 Cov R 0, Cov R E 1, M 1 t1 E t 1 t 1 t1 Var 1 Var 1 Var 1 If we defne b b 10 t 1 t Cov 0, R M, b t Var R M Cov, t1 t t1 Var 1 11, b Cov R Var R Cov R t t 1 t M 0, RM t M t t1 t1 M t1 Var 1,, b 12, b M Cov R E, R t Var R M t t t t t M M Cov R E, t t 1 t 1 t 1 t t1 Var 1 b b b 10 20, bb B b b we can wrte (5.26) and (5.27) n vector form as t1 E b B t If E b b ( d d ) b 1 and ϑ are lnear dependent,.e t1 E d0d1we get from (5.28): b b d b d b ( ) ( ) By nsertng (5.29) nto (5.12) we get t t 1 t 1 t E R 1 E 0 E 1 d0 d1 Var 1 t1 t1 t1 t1 E 0 E 1 d0 E 1 d1 Var 1 t1 t1 t1 t1 E d1 Var t t 1 b10 b11d 0 E 1 d1 Var 1 E 0 E 1 d 0 b b d b b d a a 0 1 wth t1 t1 E d Var b b d t1 t1 a0 E 0 E 1 d0 b b d and a 1 t1 t1 E 1 d1 Var 1. b b d

37 In ths case the condtonal CAPM has the same form as the uncondtonal form t 1 presented n the last secton. But f E s not a lnear functon of ϑ we are able to nvert equaton (5.28): t1 E 1 1 B b B By rewrtng (5.12) we get n vector form: t1 E t t1 t1 t1 E R E 0 E 1 Var 1 Insertng (5.31) becomes t t1 t1 t1 1 t1 t1 1 E R E 0 E 1 Var 1 B b E 1 Var 1 B c11c 1 12 Wth B, c21c22 e E c Var c t1 t t 1 t 1 t 1 1 e0 E 0 E 1, Var 1 B b, t1 t1 and e E c Var c we can rewrte (5.33) as E R e e e t The expected return depends lnearly on the maret rs and the rs of a change n the maret rs,.e. t depends on two dfferent beta. Although we have the dependence on two beta, ths model s not a specal case of other models that wll be dscussed later havng mult-beta structures. The orgnal form had a sngle (condtonal) beta, only to derve the uncondtonal form ths second beta turned out, no other source of rs than the maret rs has been added. Ths second beta s due to the unobservablty of expectatons. Any rs factor that we can determne can change over tme,.e. we could derve the condtonal verson for varous sources of rs, when determnng the uncondtonal form for every rs factor such a second beta would turn out, hence the Condtonal CAPM s a generalzaton of the uncondtonal form and not a generalzaton to nclude other rs factors. EMPIRICAL RESULTS Models wth tme varyng betas and rs prema have attracted ncreased attenton n recent years. The reason on one hand s the emprcal evdence that covarances, varances and rs prema are not constant over tme. On the other hand gves the poor performance of the tradtonal CAPM rse to modfcatons of ths model. By ntroducng tme varyng betas and rs prema the condtonal CAPM remans to be a statc model from ts nature. Although a dynamc model would capture realty more approprate, the condtonal CAPM performs much better than ts uncondtonal form. The model by Fama/French (1992), whch adds other factors le the boo-to-maret 32

38 rato or frm sze, does not perform approxmately well and ncludng other factors nto the condtonal CAPM does not mprove the results much, suggestng that ther factors are of no real mportance. Jagannathan/Wang (1996) also showed that by ncludng other rs factors, dfferent from those of Fama/French (1992), the performance can be ncreased sgnfc antly by usng monthly nstead of yearly data. Ths gves rse to other models allowng for more general rs factors and dynamc models. Summarzng can be sad that the Condtonal CAPM fts the data much better than the tradtonal CAPM, but especally for explanng asset prces n the short run, t fals. 33

39 3.6 THE ARBITRAGE PRICING THEORY Besde the problem of dentfyng the maret portfolo and the crtques concernng the mean-varance crteron, a crtcal pont n the concept of the CAPM s the aggregaton of all rss nto a sngle rs factor, the maret rs. Ths aggregaton s useful for optmal or at least well dversfed portfolos, but for the explanaton of returns of ndvdual assets ths aggregaton may be problematc. It s well observable that assets are not only drven by general factors le the maret movement, but that ndustry or country specfc nfluences also have a large mpact on returns. Ths secton presents an alternatve to the CAPM, the Arbtrage Prcng Theory (APT ) as frst ntroduced by Ross (1976). DERIVATION OF THE APT Assume that the returns are generated by the followng lnear structure: R where R denotes the realzed return of asset, µ the uncondtonal expected return, δ a vector of dfferent rs factors, β a vector representng the nfluence each rs factor has on the asset return and ε an error term summarzng the effects not covered by the model. We mae the followng assumptons on ths structure: E 0 E 0 These assumptons state that the nfluence of effects not covered have on average no nfluence on the returns,.e. there s no systematc bas. The factors havng an nfluence on the returns are assumed to be normalzed,.e. only devatons from ther average values are consdered, the effect of the level of these factors are summarzed n µ. We do not have to assume that the ε are ndependent of each other, t s suffcent f they are not too dependent on each other, such that the law of large numbers can be, j j that appled. We need for all E j Cov j 0 By the law of large numbers we fnd a vector xn x xn Lm x where n n,..., ' 1 such that,..., 1 n e. n a well-dversfed portfolo the error terms have no nfluence on the return of a portfolo. We therefore nterpret x as the weght asset has n such a portfolo. 34

40 In general for any portfolo x we have xr x x x Where R R,..., 1 Rn x x,, 1,..., n 1,..., n. The term x denotes the rs from the common factors that cannot be elmnated by dversfcaton,.e. the systematc rs, whle x ε can be elmnated by dversfcaton..e. t s the unsystematc rs. An arbtrage portfolo s defned as a portfolo wth no rs, no net nvestment, but a postve certan return. Hence n an arbtrage portfolo nether systematc nor unsystematc rs must be present and the followng restrctons apply: where 1,...,1 portfolo: x x x. Insertng these condtons nto 7.4 gves us for an arbtrage xr x In equlbrum we must assure that no arbtrage s possble,.e. x If we contnue to apply the mean-varance crteron, whch s not mpled by the APT but mostly done for conventonal reasons, the condton of no arbtrage possbltes assures that wth the ntroducton of more assets the effcent fronter does not converge to a vertcal lne n the, plane..e.. The assumptons underlyng the APT as presented by Schneller (1990, pp. 2 ff.) and Ross (1976, pp. 347 and 351) are summarzed n table

41 The returns are assumed by nvestors to follow equaton (3.1) Investors are rs averse wth a fnte Arrow-Pratt measure of rs averson No transacton costs or taxes exst No restrctons on short sales for any asset In equlbrum no arbtrage possbltes exst For at least one asset the possble loss from holdng the asset s lmted to t < Every asset wants to be held by nvestors,.e. the total demand for every asset s postve Homogenety of belefs,.e. all nvestors expect the same µ < and agree on β Table 3.6.1: Assumptons of the APT Let us consder an arbtrage portfolo fulfllng restrctons (3.5) and the expected return of ths portfolo s x c where c m t wth m = x0µ denotng the expected return of the optmal portfolo of the nvestor. From the lmtaton that µ < for all assets we now that m <. We wll therefore consder an arbtrage portfolo whose certan return s larger than the expected return of the optmal portfolo plus the maxmum loss assocated to the asset wth lmted lablty,.e. the arbtrage portfolo should be preferred to all other portfolos. We do not assume ths portfolo to have no unsystematc rs as t s not necessarly well dversfed, we only assume t has no systematc rs. Before dervng the optmal arbtrage portfolo we have to state some prelmnares from probablty theory. Ross (1976, p. 359) states that f ỹn a n quadratc mean, then E [U (ỹn)] U (a) n quadratc mean. Suppose that the unsystematc rs x V x, V denotng the covarance matrx, of the arbtrage portfolo converges to zero n quadratc mean.e. xvx 0. We then fnd that wth the above result 36

42 E U x R t U x t U c t U m By the ε-crteron of convergence there exsts a number of assets n such that E U x nr t U m n hence the optmal portfolo x0n wth ts return mn would no longer be optmal. Hence x V x does not converge to zero and there exsts a number a > 0 such that xvx a We now can determne the optmal arbtrage portfolo. From the theory of portfolo selecton we now that the only effcent portfolo for a gven expected return s the portfolo whch has the lowest varance (rs). The rs of the arbtrage portfolo s only the unsystematc rs x V x. We have to apply the restrctons x x 0 0 x 0 By defnng W = (µ, β, ι) and g = (c, 0, 0) we can rewrte these restrctons as xw g The Lagrange functon shows up to be L xvx 2 xw g The frst order condtons for a mnmum are L 2Vx 2W 0 x L 2 xw g 0 whch solves to Vx W xw g Solvng (7.14) for x and nsertng nto (7.15) gves We further fnd 1 W V W Ross (1976, pp. 357 f.) has proofed that there exsts a vector a and a number A > 0 such that wth a g = 1 37 g 1 1 xvx xw g gw V W g a 0

43 Wa Wa A If we defne ca = (1, γ, ρ) we have a g = 1 and (7.18) becomes n ca As (7.19) has to hold for any n, also for n = ths mples that on average 2 0 A fnte number of assets may not fulfll (7.20) but all of the remanng assets have to fulfl t, hence nearly all assets have expected returns accordng to what mples If we assume asset 1 to be a zero-beta portfolo,.e. x1β1 = 0 then the return of ths asset becomes n the absence of systematc rs: x R x x x x11 x x x1 Hence we have found that R1 r ρ s the return of the zero-beta portfolo. If there s a rsless asset t s the return of ths asset. Assume now that there exst portfolos whch have only a rs on one factor, l, and that ths rs s a unt,.e. x β l = 1 and x β s = 0 for all s 6= l. In ths case (7.22) becomes wth nsertng (7.24) and x ι = 1 as normalzaton: x xr x r or 38

44 x r r where µ l x µ s the expected return of the portfolo wth only a unt rs n factor l and no rs n the other factors. Therefore we can nterpret γ l as the rs premum for havng one unt of rs n factor l. β l then represents the amount of rs asset has from factor l. By nsertng (7.24) and (7.26) nto (7.22) we get r r Where 1,..., The exact relaton only holds for an nfnte number of assets, but as we fnd a very large number of assets the devaton wll be very small. At a frst glance we could nterpret the APT as a generalzaton of the CAPM to a multbeta model. The same structure of the result n equaton (7.27) as n the CAPM seems to support ths vew. But t has clearly to be ponted out that the models dffer substantally n ther assumptons. The CAPM s concerned to fnd an equlbrum of the maret by holdng optmal portfolos as mpled by portfolo theory, whereas the APT fnds ths equlbrum by rulng out arbtrage possbltes. We wll address ths dstncton between the CAPM and the APT n the next secton that provdes a very smlar extenson of the CAPM. The APT allows to nclude other sources of rs than only the maret rs, e.g. ndustry specfc factors. Furthermore a maret portfolo has not to be determned, consequently we also do not face the problem of excludng assets from the consderatons, wth a suffcent number of asset ths relaton should hold. Also other measures of rs than varances and covarances could be used, how to determne the betas s not predetermned by ths theory. As noted n secton 4.2, Fama/French (1992) and Fama/French (1993) fnds evdence that other varables are able to explan the observed returns better than the maret rs. The APT could be a framewor to fnd a justfc aton of ther results on a sound theoretcal bass. The next secton wll therefore gve a short overvew of the emprcal fndngs concernng the APT. EMPIRICAL EVIDENCE The frst problem to solve n applyng the APT s to dentfy the rs factors δ. Rs factors can ether be constructed by fndng a portfolo of assets that has a hgh correlaton wth a certan rs, ths portfolo s called a factor portfolo, or by usng other varables, such as macroeconomc data, e.g. GDP. The advantage of the former approach s that expected returns for the rs factor, µ l can easly be determned from the maret and the rs β can also be estmated from maret data. The dentfcaton of these parameters for macroeconomc data mposes much more dffcultes. Ths s the reason why n most cases factor portfolos are used. 39

45 To dentfy the systematc rs and hence the characterstcs of the factor portfolos we can ether use theoretcal consderatons or statstcal methods to dentfy these rss. Wdely used statstcal methods are factor analyss and prncpal components method. There exsts a large number of surveys nvestgatng the explanaton of asset returns usng APT. The factors mostly dentfed n these studes are related to dvdends or earnngs, boo-maret relatons, the sze of a company and the varance of asset returns. Most nvestgatons show that three to fve factors are suffcent to explan the observed returns, addng more factors does not mprove the result substantally. The nvestgatons cted gve evdence that the APT can explan the observed returns qute good for long and medum tme horzons. For tme horzons below one year they are not able to explan the data adequately. Compared to the present value model the tme horzon can be reduced sgnfcantly from four to about one year, but as n the CAPM there reman many effects that cannot be explaned suffcently. The assumpton of a lnear relaton between the assets n the CAPM by the covarances s replaced by the assumpton of a lnear relatonshp wth rs factors. Le n the CAPM ths assumpton lmts the theory as nonlnear assets, le dervatves, cannot be modelled adequately. The advantage of the APT n ths case s that t s not necessary to form a maret portfolo and to nclude these assets, t enables to exclude human captal or real estate. It enables also to restrct the analyss to a certan group of assets, provded that the number of assets s suffcently large that the approxmaton n (7.27) holds. The more assets are ncluded the more precse the fndngs should be, wth restrctng to only a few assets the prcng relaton does not brea down as n the CAPM, t only becomes less precse,.e. we should fnd more nose. 40

46 3.7 THE INTERTEMPORAL CAPITAL ASSET PRICING MODEL The models of asset prcng consdered so far had one feature n common: they were all statc. The amount nvested nto assets was fxed for a gven perod of tme and the amounts nvested nto each asset could not be changed. At the end of a tme perod t was assumed that the nvestors consume ther wealth. A more realstc settng would be to allow nvestors to change the amounts nvested nto each asset and also to wthdraw a part of ther nvestment for mmedate consumpton. Not only the amount nvested nto each asset but also the fracton of the wealth nvested nto assets wll become an endogenous varable, whle n the prevous models ths fracton was fxed. The Intertemporal Captal Asset Prcng Model (ICAPM ) as frst developed by Merton (1973) taes these consderatons n account. THE MODEL In the ICAPM we assume a perfect maret,.e. all assets have lmted lablty, we face no transacton costs or taxes, assets are nfntely dvsble, each nvestor beleves that hs decson does not affect the maret prce, the maret s always n equlbrum, hence we have no trades outsde the equlbrum prces, nvestors can borrow and lend wthout any restrctons all assets at the same rate. Further assumptons are that tradng taes place contnuously,.e. all nvestors can trade at every pont of tme. All varables that can explan the prces and prce changes of the assets (the state varables) follow a jont Marov process. The state varables are further assumed to change contnuously over tme,.e. no jumps are allowed. If we let P denote the prce of asset at tme t, Ωt the nformaton avalable at tme t t t and h the number of tme unts, we have for the expected return of asset, µ, and the varance of the returns, σ 2, per unt of tme: 1 P E h th P P t h t t Pt h _ P t E h t h Pt If we assume that µ and σ 2 2 exst and are fnte and that lm h 0 0.e. by tradng at every pont of tme (wth a very short tme horzon) the uncertanty cannot be elmnated. t 41

47 All assets have lmted lablty No transacton costs and taxes No dvdends are pad All assets are nfntely dvsble All nvestors beleve that ther decsons do not nfluence the maret prce All trades tae place n equlbrum Unrestrcted borrowng and lendng of all assets at the same condtons Tradng taes place contnuously Uncertanty cannot be elmnated by a contnuous revson of the portfolo The state varables follow a jont Marov process The state varables change contnuously Table summarzes the man assumptons of the ICAPM. Defne y (t) to be d N (0,1) dstrbuted, we then can wrte the return dynamcs mpled by (8.2) wth solvng for P th Pt P t P Pt h h y t t h Pt Tang the lmt of (8.3) wth respect to h we get the dfferental equaton of the return process: dp P pt h pt lm dt dt y t h0 p t Defne dz to be a Wener process: dz y t dt Insertng (3.5) nto (3.4) gves us the result that returns follow an Ito process: dp P dt dz 42

48 Expected returns and varances of returns are also assumed to follow an Ito process: d adt b dq d fdt gdx where dq and dx are two, not necessarly ndependent, Wener processes. Equatons (3.6)-(3.8) are assumed to form a jont Marov process. For the further analyss we assume that we have n rsy assets and one rsless asset, numbered n + 1. The rsless rate of return can change over tme,.e. we have n 1 0, n1 r, bn 1 0. Unle n the CAPM the nvestor does not maxmze hs termnal utlty at the end of a gven tme perod (hs tme horzon), but the utlty over the whole tme perod wth length T > 0. The nvestor wll maxmze the functon T p t p T E[ u ( C ( t)) e dt U ( W ( T )) e ], 0 where U denotes the utlty of ndvdual, C ts consumpton at tme t and ρ the dscount factor for future utlty. The frst term denotes the present value of consumpton from tme 0 to tme T and the second term denotes the present value of the utlty from termnal wealth W (T ), or termnal consumpton. Merton (1990, pp. 124 ff.) derves the dynamcs for the wealth W (t) at any pont of tme. It conssts of the number of assets that have been hold at a prevous pont of tme, N (t h) multpled by ts present prce, P : t n1 t 1 W ( t) N ( t h) P. In the same moment of tme he chooses a new portfolo,.e. a new number of shares, N t and the optmal consumpton per unt of tme, C (t)h. Hs wealth becomes Solvng (8.13) and (8.14) for C (t)h gves n1 t 1 W ( t) N ( t) P C ( t) h. n1 ( ) ( ( ) ( )) t 1 C t h N t h N t P 0, or wth shftng the tme perod by h: 43

49 ( ) n1 ( ( ) ( )) th 1 C t h h N t N t h P n1 ( N ( t) N ( t h))( P P ) 1 n1 1 By lettng h 0 (7.16) and (8.13) become th t ( N ( t) N ( t h)) P t n1 t 1 W ( t) N ( t) P, n1 n1 t t 1 1 C ( t) dt dn ( t) dp dn ( t) P. Dfferentatng (8.18) by usng Ito s lemma gves us wth (8.19) n1 n1 n1 ( ) ( ) t ( ) t ( ) t dw t N t dp dn t dp dn t P n1 1 N ( t) dp C ( t) dt. t The frst term represents the captal gans made from nvestng nto assets and the last term are the losses n wealth due to consumpton. N () t Pt Defne w () t W () t as the fracton of wealth nvested nto asset after consumpton. Insertng ths defnton nto (3.20) we get: dp dw ( t) w ( t) W ( t) C ( t) dt. n1 t 1 Pt Insertng (8.6) gves us the Ito process governng wealth: n1 1 dw ( t) w ( t) W ( t) dt n1 1 w ( t) W ( t) dz C ( t) dt n1 W ( t) w ( t)( r) r] dt 1 n1 1 C ( t) dt W ( t) w ( t) dz. 44

50 Defne further X = (X1,..., Xm) to be the vector of state varables, whch we assume to follow an Ito process: dx Fdt GdQ Where dq ( dq,..., dq ) s a Wener process, wth ν j denotng the correlaton q m between dq and dq j, η j between dq and dz j. It s further F ( f1,..., f m ) and G = (g1,..., gm). After these prelmnares we now can solve the maxmzaton problem of equaton (8.12). For determnng the maxmum we have to fnd the optmal amount of consumpton, C (t) n every moment of tme and the optmal share of the remanng n1 wealth to nvest nto each asset, { w ( t)}. We defne a performance functon J ( W, t, X ) as 1 max T p (,, ) n [ ( ( )) C ( t),{ w ( t)} t 1 J W t X E U C e d as we can rewrte (8.24) as p T U ( W ( T )) e ]. p T J ( W, T, X ) U( W ( T)) e t max T p (,, ) n [ ( ( )) C ( t),( w ( t)) t 1 J W t X E U C e d J ( W, T, X ) ]. wth v T t applyng the mean-value theorem states that there exsts a * t [ t, T] such that t * T p * p t t U ( C ( )) e d U ( C ( t )) e u. We can expand J (W, T, X) n a second order Taylor seres around (W, t, X): J ( W, T, X) J ( W, t, X) J W J v J X W t X 1 1 J W J W X J v WW WX tt 1 JWtWv J XtXv J XX X 2 where the sub-ndces denote the dervatve wth respect to ths varable evaluated at ( W ( t), t, X ( t)), W W ( T) W ( t) X X T X t. and 2, 45

51 For smplcty we assume that the dervatves J Wt JW and J X do not vary wth tme.e. J 0whch s reasonable f we assume v to be not too large. Insertng (8.27) and Xt (8.28) nto (8.26), elmnatng J (W, t, X) and dvdng by v gves us: W v max p t* n E[ U ( C ( t*)) e J C ( t),{ w ( t)} W Jt 1 2 X 1 W W X J X JWW JWX v 2 v v X Jttv J XX t ] v Tang the lmt as v 0, hence t t, we get as W and X follow a contnuous Ito process: max n C ( t),{ w ( t)} 1 2 p t dw dx 1 dw W t X WW [ U ( C ( t)) e J E J J E J E dt dt 2 dt 2 dwdx 1 dx JWX E J XX E ] 0. dt 2 dt Wth the contnuty of the Ito process we get E[dW ] = E[dX] = 0 and therewth E[dW 2 ] = Var[dW ] and E[dX 2 ] = V ar[dx]. wth the notatons of (8.22) and (8.23) we fnd that dx J E m X J f dt 1 2 m m dx XX j j j dt 1 j1 J E J v dwdx m n JWX E J W v Ww j j j j dt 1 j1 2 m n dw JWW E JWW w wj jw dt 1 j1 where the subndces and j are for the dervaton wth respect to the s and j s state varable. We further have Insertng these results nto (8.30) gves n1 dw W w r r C dt 1,, 2,., [ U( C ) e J C ( t),{ w ( t)} t max n 1 p t 46

52 J W w r r C n1 W 1 1 j f J w w W m m n WW j j j1 m n 1 j1 1 j1 J W v Ww j j j j m n 1 Jj jvj ] 0. 2 Defnng the functon n bracets for notatonal smplcty as and denotng the dervatve wth respect to C as C and wth respect to condtons for a maxmum: w as U J 0, C C W n m 2 Jw rw JWW wj jw J jww j j j1 j we get the (n+1) frst order Condton (8.32) states the usual result that margnal utlty of mmedate consumpton (UC ) has to equal margnal utlty from deferred consumpton (JW ). n d w W as the If we defne V j as the covarance matrx of the assets, j 1 j j 1 demand vector for the assets of the nvestor j j 1 nm, j j A, j1 n, as the vector of expected returns, as the covarance matrx of the asset returns and the state varables J J W j w, H j J J and n H H we can rewrte (8.34) after dvdng by W n j 1 WW vector form. WW 0. A r Vd H n Solvng for the demand we have: 1 d A V r H V 1 Wth v j as the (, j)th element of V 1 * and g we can rewrte (8.35) for an ndvdual asset as 47 j j j n m n * d A r H j j j j. j1 j1 1 The demand for an asset conssts of two parts. The frst term s the demand smlar to that of the statc CAPM, t s due to an effc ent nvestment by mean-varance

53 maxmzng. A ncludes a term for rs averson of the nvestor and µj r s the rs premum. The second term s an adjustment made for hedgng aganst an unfavourable shft n the state varables. Ths adjustment can be ether postve or negatve, dependng on the sgn of C x whch forms a part of H. If the state varables do not vary over tme, t turns out that H = 0 and the results are dentcal to the standard CAPM. In general, however, the state varables wll change over tme. For smplcty we now assume that only one state varable vares over tme, e.g. the rs free nterest rate r. In ths case (8.36) becomes n m * d A r H. j j j jr j1 j1 Assume there exsts an asset, the nth asset, that s perfectly negatve correlated wth the state varable,.e. ρnr = 1. Such an asset could be a bond, whch s rsless n terms of default and whose value only depends on the nterest rate. Because of the varyng nterest rate the bond would no longer be rsless n ts value. We fnd such that p p p p, j jn n jn p p * jn j n jn j j j jn j n n. By usng (3.39) the second term n (3.37) becomes element of the nverse of σ we fnd that f = n g H. Because v rj s an n r r vj jn n j 1 n j1 j jn 1 and otherwse n j1 j jn 0. Therewth we can smplfy (3.37) to n j j j1 d A for 1,..., n 1, d A H. n n nj j j1 n 48

54 We defne now three portfolos: Portfolo 1: Conssts of all n rsy assets wth weghts n j1 j1 j n n j 1 are equal to the weghts of the optmal rsy portfolo n portfolo theory. j j, whch Portfolo 2: Conssts only of the nth asset. Portfolo 3: Conssts only of the rsless asset. By constructon the compostons of the portfolos do not depend on the preferences of the nvestors. Let λ ( = 1, 2, 3) denote the fracton of hs total wealth nvestor nvests nto each of these portfolos. The total demand for the frst n 1 assets s gven by λ 1 δ W and for the nth asset by (λ 1 δ n +λ 2 )W. These demands have to satsfy the condtons (8.42) f an nvestor has to be ndfferent between nvestng nto all n assets drectly or nvestng only n the three portfolos: n n 1 j j j 1 W 1 W d A n j n j 1 j 1 j 1 j j Solvng 1 and 2 gves W d A H. n 1 n 2 n nj j j1 1 j1 n n A 1 j j W H 2. W n Instead of drectly nvestng nto all n rsy assets, an nvestor wll be ndfferent to choosng a combnaton of the optmal rsy portfolo and the nth asset that has the hghest correlaton wth the changng state varable. The remanng wealth s nvested nto the thrd portfolo consstng only of the rsless asset. Ths result s nown as the Three-Fund Theorem. It s the dynamc equvalent of Tobn s Separaton Theorem. Portfolos one and three ensure the nvestor to have a mean-varance effcent holdng of assets,.e. hs holdngs are on the effcent fronter of the CAPM. But as the state varable changes over tme stochastcally, the effcent fronter shfts over tme, portfolo two hedges aganst an unfavourable shft. In the general case where m 1 state varables can change over tme, t has been shown n Merton (1990, pp. 499 ff.) that (m + 2) portfolos are combned by the nvestors, hence t s nown as the (m+2)-fund Theorem. The propertes n the general case do not change, one portfolo remans to be the optmal rsy portfolo, one,, n 49

55 conssts only of the rsless assets and the other m portfolos consst only of the asset havng the hghest correlaton wth one of the state varables. We can now derve the equlbrum expected returns n the case of m = 1 state varables. Solvng (8.34) for µ rι gves 1 H Vd. A A As ths relaton holds for every nvestor ndvdually, t also has to reman vald when aggregatng the demands of all K nvestors. Let n and D D 1. K K K,, K 1 K 1 K 1 A A H H D d We can rewrte (8.45) as 1 H VD. A A As we have assumed that trade only occurs n equlbrum, the total demand always has to equal the value of all assets, n1 M : D M. D We defne w as the fracton of the th asset n the total maret value, whch M because of the equlbrum assumpton has to equal relatve demand. In scalar form (8.46) becomes for = 1,..., n: 1 n M H w j j A j1 A M A n j1 H n r wj j A n If we denote the maret portfolo by the subndex M we have n n j, j j and M w j M w j1 j1 becomes: M H M. A A Multplyng by wj and summng over all = 1,..., n gves M 2 H Mn M 2 H M M M M. A A n A A For = n (8.47) becomes M H n nm n. A A 2 n M w j 1 j j M and (8.47) 50

56 By solvng (8.49) and (8.50) for we get M A M M n, 2 2 n n M M nm n M M nm H 2 nm M 2 M 2 n. A n M M nm n M M nm Insertng ther results nto (8.47) gves the equlbrum expected returns after rearrangng terms: n M nm 2 n M M nm 2 M M M 2 n M M nm n M 1 2 M n, wth 2 1 n M nm 2 M M M and. 2 2 n M M nm n M M nm The excess returns compensate nvestors for the systematc rs le n the CAPM and addtonally for the rs of an unfavourable shft n the state varable Even assets whch have a beta of zero,.e. are uncorrelated wth the maret portfolo, may have returns hgher than the rsless rate of return because of the exposure to an unfavourable shft n the state varable. Ths relaton can be observed n realty and could not be explaned by the CAPM. By allowng n general m 1 state varable to change over tme, Merton (1990, p. 510) has shown that (8.53) n general becomes m j 1 M j j1.e. every rs of a state varable s compensated ndvdually. The ICAPM extends the CAPM to a dynamc envronment. The results are smlar to those of the APT, but t has the advantage that the rss can be determned from characterstcs of the assets. Cox et al. (1985) derve a smlar result n a much more general framewor than provded here.. 51

57 EMPIRICAL EVIDENCE Emprcal nvestgatons nto the performance of the ICAPM are not frequently found. The reason s that at a frst glance the ICAPM and the APT loo ale from ther results. The ICAPM only has a fxed rs factor, the maret portfolo. In the APT the maret portfolo s not a rs factors as t has been assumed that an nvestor s perfectly well dversfed and hence the only source of rs are common factors. In the ICAPM portfolos have not to be perfectly dversfed, nor the maret portfolo nether the portfolos havng the hghest correlaton wth the state varables, hence we could nterpret the rs from the maret portfolo as arsng from mperfect dversfcaton. If all portfolos are perfectly dversfed and the state varables equal the common factors, the ICAPM collapses to the APT. We could therefore vew the APT as a specal case of the ICAPM. Therefore the APT and ICAPM are often treated ale, despte ther dfferent theoretcal foundatons. The nvestgaton by Doo/Edelsten (1991) explctly uses the ICAPM as ther reference model. They use the uncertanty about future nflaton and real producton as state varables. By usng monthly data for the perod they show that excess returns are sgnfcantly affected by changes n these state varables. They further fnd changes nflaton uncertanty to be hghly persstent, whereas changes n real producton uncertanty are only temporary. Therefore they fnd that changes n nflaton uncertanty nfluence asset prces sgnfcantly, whereas real producton uncertanty has no mportant nfluence. Wth these results Doo/Edelsten (1991) can explan the behavour of the stoc maret n ther observed tme perod qute well. But as most models t fals to explan short-term movements of prces. The ICAPM faces the same problem of dentfyng state varables as does one n the APT wth dentfyng common factors. Fama (1998) addresses ths problem of determnng state varables. 52

58 3.8 THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL When we want to apply the ICAPM to explan the behavour of asset prces we face the problem of dentfyng the relevant state varables, the theory does gve no hnt how to choose the relevant varables. The Consumpton-Based Captal Asset Prcng Model (CCAPM) as frst developed by Breeden (1979) develops the ICAPM further wthn the same theoretcal framewor to aggregate the rss from shftng state varables nto a sngle varable, consumpton. DERIVATION OF THE MODEL The assumptons underlyng the CCAPM are dentcal to those of the ICAPM developed n the last secton. The nvestors are also assumed to maxmze the functon already stated n equaton (8.12) 0 T E U C t e dt U W T e 0 p t p T wth respect to consumpton C (t) and the portfolo composton w. 1 The only addtonal assumpton we have to mae concerns consumpton. We assume that there exsts only a sngle consumpton good and that the state varables nfluence also the consumpton,.e. consumpton becomes a stochastc varable. Ths nfluence could e.g. be through nfluencng the prce and therewth the number of goods that can be bought wth a gven wealth. The frst order condtons for maxmzng (9.1) have already been derved n secton 4.7. We restate them here for convenence slghtly modfed: n J W U C, J J d w W V V. W 1 XW 1 JWW JWW If we derve (9.2) agan wth respect to the state varables X and wealth W we get J U U C WX CX CC X J U U C WW CW CC W Insertng these results nto (9.3) we get,. Uc U C w W V V 1 CC X 1 UCCCW UCCCW 1 C z V V, C W 1 X 1 CW 53

59 Wth z CV W gves U U C W as the Arrow Platt measure of rs averson. Premultplyng (9.6) by CC. C Vw W z C W X The term V w W represents the vector of covarances of the asset returns wth a change n the wealth. We wll denote V W Vw W. Rearrangng (9.7) gves. z V C C W W X The covarances of the asset returns wth the nvestors consumpton can be derved usng Ito s lemma: Insertng (9.9) nto (9.8) gves: K K 1 C 1 V V C C K W X. C W. z V C As (9.10) holds for every nvestor we can aggregate ther decsons. Defnng z z andv V 4 gves us C or for a sngle asset, z V C. z C Equaton (3.12) has also to be fulflled for any portfolo. Let such a portfolo be denoted by the subndex M, t has not necessarly to be the maret portfolo. Wth the weghts w M of ths portfolo we can defne the portfolo s expected return and n n M 1 MC 1 M M covarance wth consumpton as w and w C. Multplyng (3.12) wth and addng over all assets gves us. z M MC Solvng ths equaton for z and nsertng nto (3.12) gves after rearrangng C M. MC By defnng betas wth respect to aggregate consumpton n the conventonal way as C and we can rewrte (3.14) as C MC 2 MC 2 C C 54

60 C MC. M Suppose there exsts a portfolo M that s perfectly correlated wth changes n aggregate consumpton, then (9.15) reduces to. C M The rss of an unfavourable shft n the state varables that determned the excess returns n the ICAPM have now been aggregated nto a general rs factor, aggregate consumpton. The dfferent rs factors all nfluence current and future consumpton, hence they can be aggregated n ths way. The term s also called the maret prce of consumpton rs In order to nterpret ths result we have to remember that the present utlty, s, besde termnal wealth, affected by the amount of consumpton and ts tmng. As we further can reasonably argue that.e. the hgher the wealth the hgher consumpton, and wealth s affected by these returns, consumpton s also affected by the returns of the assets held. Assume now that a state occurs that reduces current consumpton and the expected return of an asset,.e. βc 0. The reduced current consumpton reduces the nvestors present utlty. Because expected future returns of the asset are also reduced, future expected wealth s reduced and hence future expected consumpton, what reduces utlty further. When holdng the asset an nvestor faces ths rs of an unfavorable shft n the state varables that nfluence hs future consumpton, hence he has to be compensated for ths rs. The same argumentaton can be used for a negatve beta. EMPIRICAL INVESTIGATIONS An emprcal nvestgaton of the CCAPM faces several measurement problems concernng aggregate consumpton: The statstcal data on aggregate consumpton capture not consumpton drectly, but the expendtures for consumpton goods and servces. As the goods purchased are not necessarly consumed mmedately (e.g. they can be stored for later consumpton or can be consumed over tme le consumer durables), the data wll be based, although as a result of aggregaton the bas wll be reduced, but t may lead or lag the busness cycle. The data avalable are not for an nstant of tme, but denote consumpton for a certan perod, at least a month or a quarter, whereas asset prces are avalable on a daly or even ntraday bass. The data are generated from samples and we face the problem of samplng errors. Compared to the use of a maret portfolo as n the CAPM the use of consumpton data also has a vrtue. Whle the maret portfolo that s determned typcally does not nclude mportant assets le real estate or human captal, consumpton data cover a much larger fracton of the effectve consumpton. 55

61 Besde the data problem an emprcal nvestgaton faces many econometrc problems, e.g. the determnaton of the portfolo wth the hghest correlaton wth a change n aggregate consumpton (also called the Maxmum Correlaton Portfolo), whch s the equvalent to the maret portfolo n the CAPM. For an overvew of these econometrc consderatons see Breeden et al. (1989). Breeden et al. (1989) fnds that excess returns of a zero beta portfolo are small, as predcted by the theory, whereas they are relatvely large n the CAPM model. They report further that the maret prce of consumpton rs s postve by observng data for the perod usng quarterly and monthly data. The lnearty of excess returns and consumpton rs (βc ) s only rejected for more recent sub perods ( ) where the data qualty s mproved. Ths suggests that data qualty s a crucal factor for nterpretng the results. Summarzng can be sad that the results show a wea support for the CCAPM, a stuaton smlar to the results concernng the CAPM. As other models presented thus far the CCAPM s especally not able to explan the behavour of asset prces n the short run, but ths may also be the consequence of mssng consumpton data for perods below a month. Ahn/Cho (1991) show that wth a tme varyng rs averson, where the rs averson depends on past returns, the CCAPM becomes much more consstent wth the data. 56

62 3.9 THE INTERNATIONAL CAPITAL ASSET PRICING MODEL The prevous models of asset prcng mplctly assumed that all nvestors are located n the same country and consder only assets n ther home country. In realty, however, we fnd nvestors n dfferent countres and they also nvest a part of ther wealth abroad. In such a framewor we therefore should consder the nfluence of exchange rates, dfferent tastes for consumpton across countres and barrers to foregn nvestment. These ponts were ncorporated nto the Internatonal Captal Asset Prcng Model (Internatonal CAPM ) as developed by Stulz (1981b), Stulz (1981a) and Stulz (1995). NO DIFFERENCES IN CONSUMPTION AND NO BARRIERS TO FOREIGN INVESTMENT Stulz (1995) provdes a model where nether the nvestors have dfferent tastes across countres nor dfferent costs of nvestng at home and abroad are faced. The assumptons underlyng ths model are very smlar to those of the ICAPM, they are lsted n table We assume that the prce of asset, denomnated n the currency of country j, Pj follows an Ito process: dp P j j dt dz j j j, where µj denotes the expected nomnal return of asset n currency j, σj the j standard devaton of ths return and dzj a standard Wener process. The prce of the consumpton good n country j, P C, also follows an Ito process: dp P C j C j jdt jdz j, where πj denotes the expected nflaton rate n country j, σπj the nflaton rate s varance and dzπj a standard Wener process. The real prce of an asset n terms of the consumpton good s gven by: P r j Pj P C j and can be shown by Ito s lemma to follow Pj d C dpj Pj P P j P j C j 2 j j j, j j dt jdzj jdz j, where σj,πj denotes the covarance between the nomnal asset return and the nflaton n ths country. The frst term denotes the expected real return of asset 57

63 2 n country j,. As the sum of two Wener processes s agan a Wener j j j j, j j process we defne a Wener process jdzj jdzj jdz j and get an Ito process for the real prce of the asset: dpj dt dz P j j j j. We have now formulated exactly the CAPM derved n secton 4.3, only substtutng nomnal returns by real returns. Wth the law of one prce we now that real prces and returns are equal n all countres and we can wrte (10.5) as dp dt dz P From the CAPM we now that n ths case W,. where the superscrpt r denotes real varables and s the expected real return on the world maret portfolo. In real terms there are no dfferences between countres and for the decsons t s of no mportance where an nvestor and an asset s located. It s more convenent to wrte the returns n nomnal than n real returns. Assume therefore that there exsts an asset that s rsless n nomnal terms and that has a zero beta wth the real world portfolo return. Ths asset also has to fulfl (10.7). By replacng µ r by ts orgnal expresson we get 2 j j j, j j 0. Rearrangng and nsertng nto (10.7) gves us j j, j j, j j W. The last expresson on the on the rght sde denotes the real excess return of the world maret portfolo. As we now from the dervaton of the real excess return ths s gven by Wj j W 2 2 W j, j j j j rj, j j Wj W j j, j j, j. Stulz (1995, p. 205) maes now the assumpton that for all assets σj,πj = 0,.e. the nomnal asset returns are uncorrelated wth the nflaton rate. Wth ths assumpton we see from (10.9) and (10.10) that the real and nomnal access returns concde. The beta of the real returns also equals the beta of the nomnal returns, Hence (10.9) becomes 58

64 j j j W j j. We therewth found that wth the assumpton of no dfferences n tastes and no nvestment barrers the CAPM as derved n secton 4.3 remans vald n an nternatonal settng. The maret portfolo becomes the world maret portfolo. We can vew the world as an ntegrated maret, dfferent currences and borders have no nfluence wth the assumpton of Purchasng Power Party. We wll further on consder the nfluence dfferent tastes and nvestment barrers have on the expected returns of assets. Dfferences n consumpton Stulz (1981a) extends the CCAPM to the case that nvestors and assets are located n dfferent countres and that the tastes of the nvestors dffer between these countres but are dentcal wthn a country. For smplcty we assume that there exst only two countres, the home country and the foregn country, denoted by an asters at the varables. There exst no barrers to nternatonal nvestment. To model the dfferent tastes n the countres we assume that there exst dfferent consumpton goods n the home country and n the foregn country and at least one consumpton good s consumed n both countres. For the goods consumed n both countres the law of one prce apples as before,.e. wth an exchange rate of e between the two currences we have for these consumpton goods: P C C* j epj. The prce for the jth domestc good n domestc currency s assumed to follow an Ito process: dp P C j C j jdt jdz j. where µ C denotes the expected prce change, σ C the varance of the prce change and dz C a j standard Wener process. The exchange rate also follows an Ito process: de e edt, edze wth µe as the expected change n the exchange rate, σe ts varance and dze a standard Wener process. We assume that there are n rsy assets that are traded among countres, hence the law of one prce apples to all assets. Addtonally each country has a rsless nomnal asset n hs currency wth an nterest rate of r and r, respectvely. These rsless assets are also traded between the countres as the (n + 1)th asset, but as the exchange rate may vary t s not rsless n the other country. The rsy foregn assets are assumed to follow an Ito process n foregn currency: 59

65 dp P * * * * dt dz * * * * *, Where s the expected return n foregn currency, the standard devaton of these returns and a standard wener process. Usng the law of one prce for these assets the prce n domestc currency s gven by and by applyng Ito s lemma the dynamcs of the foregn assets n domestc currency are gven by dp dp de de dp P P e e P * * * * * * * * * * Analogous equatons can be derved for the domestc assets. For the rsless assets n domestc currency these dynamcs are gven by. dp P n1 n1 rdt, dp P n1* n1* * dt de. e Defne the excess returns of a foregn asset n domestc currency, H, as the return of ths asset fnanced by borrowng abroad at the rate of r. We get by usng (10.16) and (10.18): dh dp dp H P P * * n1* * * n1* dp de dp * dt. P e P * * * * * * * * The dynamcs of the exchange rate does not affect the return as the nvestor has fully hedged ths rs by fnancng hs nvestments abroad. Hence the exchange rate rs wll not be compensated as t can be dversfed, the only rs s the development of the return n foregn currency. Equaton (10.19) for a foregn nvestor nvestng nto a domestc asset becomes: dh dp de dp H P e P * dt. * For a domestc nvestor nvestng at home we have: dh H dp dt. P The excess returns of foregn and domestc nvestors dffer only n the term..e the covarance of the change n the exchange rate and the return of the asset n domestc currency. We further get the excess returns for a domestc nvestor for the foregn rsless asset by (10.17) and (10.18) as: 60

66 dh H * n1 * n1 de e * dt dt * n1 * n1 * de dt e 2 * dt dh de de H e e After these prelmnares we are now able to derve the model of nternatonal asset prcng. Unle n the CCAPM presented above we now have more than one consumpton good. Let q denote the quanttes consumed by domestc nvestor of consumpton good j, then we have n C C j j j1 C P P,. where C denotes the consumpton of the th domestc nvestor, P C the vector of domestc consumpton good prces and q the vector of consumpton for each consumpton good. Besdes the determnaton of the optmal amount spent for consumpton, the nvestor now addtonally has to fnd the optmal allocaton the dfferent consumpton goods. We therefore defne the utlty functon as, max q U C P P u q C C max C u q ( C P q. q The frst order condtons for ths maxmzaton are uq P C, UP q where UC s the shadow prce of ncreased consumpton. Hence (10.25) and (10.26) become. u q U P C C, U P U q C. The optmal portfolo for a domestc nvestor has been derved n equatons (9.2) and (9.3) to fulfl the followng condtons: J W U C, J J d w W V V. W 1 XW 1 JWW JWW 61

67 We now assume that the frst K state varables are the log-prces of consumpton goods. Usng (10.24) as the utlty functon and applyng the theorem of mplct functons we get from (10.29) by dfferentatng: J U C U W, np CC np C, np. Let Y denote the vector of the state varables that are not prces of consumpton goods,.e. X = (lnp, Y ), therewth we can wrte as J npw U CC C np U C np U C n J P XW J UCCC U Y CCC X 0 YW by usng the results and notatons of the CCAPM. Insertng (10.32) nto (10.30) and contnung as n 4.8 gves us J U C w W V V W 1 CC X 1 JWW JWW 1 UCnP 0 WW V J m C z 0 CWw W V 0 CX 62 1 z C 1 X U V V 1 CnP 1. UCCCW V CW CW 0 We defne α as the share of total consumpton nvestor spends for good j Pq j C C j j and m as the margnal share from an ncrease n total consumpton: q m P P q C C j j j j j C By usng these defntons we get wth (10.28): P np. C C C UC, np UCP U C PCP P Uccq Ucqc CC U P q U P qc CC C C C U C U m C where α and m are the vectors of the budget and margnal shares. Insertng nto (10.33), multplyng by V CW and rearrangng gves,

68 V C C 0. Defne as the prce of a baset of consumpton goods that contan exactly one unt of asset g that s consumed n both countres and the expendture for each asset s respectvely. We therewth have n the frst Case unts of asset j and n the second case unts n these basets. Let be the vector of covarances of the asset returns wth changes n m 0, V And the vector of covarances of the returns wth changes n V 0. Thus we can rewrte (10.37) as C. m z V V C V We can now aggregate (10.40) over all M domestc nvestors and get wth M M M C M z z z, C C, P P andp : m P 1 m C z m C. z V V CV The term 1/Pm denotes the real value of a margnal ncrease n domestc consumpton as the result of a change of the value of the portfolo domestc nvestors hold,.e. Pm depends on the preferences of the domestc nvestors. For a foregn nvestor we get the equvalent equaton n foregn currency: z V V C V * * * *. * m* * C* * * In order to aggregate domestc and foregn nvestors both equatons have to be denoted n the * same currency. Defne z F ez*, C F ec*, P F F * ep and P ep as the correspondng values n domestc currency. Wth Ito s lemma we get: m V V V V * * F F *1/ * * e, e V V V V * m* F F */ * * e, em m V V 1 V V C * C* F F *1/ ec * C * e e m F. 63

69 From (10.22) and (10.23) we see that for the excess returns of the rsless assets of the other country we have dh dh de H H e * n1* n1 * n1* n1 and from (10.20) and (10.21) for the rsy assets: dh dh de dp H H e P *. * 2 By defnng L we get from (10.46) and (10.47): I n L * * V. * e It can further be shown that for any stochastc varable y that LV y V. * y Multplyng (10.42) wth L and nsertng (10.43), (10.45), (10.48) and (10.49) we get F F F F. m C z F V V C V Ths relaton s now expressed n domestc currency and can therefore be aggregated wth equaton (10.41). By defnng z W z z F, P W P P F W F and C C C we get m m m W W W W. m C z W V V C V By denotng the real world consumpton we get wth Ito s lemma: P V P V V C V W W W c W W W. C / P C Insertng nto (10.51) and rearrangng gves: V W m P z W W V c. Ths relaton has to hold for any asset and any portfolo M (not necessarly the maret portfolo): W P M V W V Mm W z Solvng for the preference parameter and nsertng nto (10.53) gves the fnal relatonshp W m c M Vc W. Mm V M V. V c M 64

70 The portfolo M can freely be chosen, a useful approach n lne wth the tradtonal CAPM s to choose the world maret portfolo of all jontly rsy assets and the rsless asset of the other country. The left hand sde of equaton (10.55) denotes the real excess return of the asset, on the rght sde the frst term denotes the real excess return of the reference portfolo and the last term denotes the covarance of the nomnal asset returns wth world real consumpton relatve to ths relaton of the reference portfolo. The real expected excess returns are hgher the hgher ths covarance s. A company that produces a product whose demand depends heavly on aggregate consumpton would have a hgher covarance. The excess returns depend on the product the company produces and the country n whch ther product s demanded. Ths model showed that dfferent tastes across countres do not affect the allocaton of assets and ther real expected returns, they only depend on the relaton to world aggregate consumpton,.e. aggregated tastes. These fndngs are very smlar to those of the CCAPM. By usng a portfolo M as reference whose re- turns are perfectly correlated wth changes n aggregate consumpton, (10.55) equals the CCAPM n real terms. To get dfferences n the allocaton of assets, e.g. the observed bas towards domestc assets we have ether to devate from the law of one prce for consumpton goods and assets, e.g. by ntroducng transportaton costs or to ntroduce barrers to nternatonal nvestment, what wll be done n the next secton. 65

71 BARRIERS FOR INTERNATIONAL INVESTMENT Stulz (1981b) provdes a model of nternatonal asset prcng n the presence of barrers to nternatonal nvestment. Le n the last secton he assumes two countres, n each country there are n and n assets located, such that n+n = N. We assume that foregn nvestors are free to nvest n ther home country or abroad wthout facng any restrctons, whereas domestc nvestors can nvest nto domestc assets wthout any restrctons, but f they want to nvest n foregn asset have to pay a tax of θ per tme perod and nvested unt of wealth. Ths tax can ether be a transacton tax, but t can also be nterpreted as costs of obtanng addtonal nformaton. The expected return from holdng a foregn asset s reduced to µ θ and from beng short n an asset to µ θ. The tax of θ has not to be pad on the net poston of a foregn asset, but on every poston ndvdually,.e. holdng one unt of a foregn asset long and one unt short results n a tax of 2θ. Wth these condtons we assume every nvestor to optmze hs portfolo holdngs accordng to portfolo theory presented n secton 4.2. If we denote the covarance matrx of all domestc and foregn assets by V and w the long and v the vector of short postons of the th domestc nvestor the varance of a portfolo s gven by p. V w V w The return of ths portfolo conssts of the expected returns from holdng the assets, (w v ) µ, subtractng the taxes for holdng foregn assets, (w + v ) ιnθ, where ιn denotes a vector wth zeros n the frst n and ones n the last n rows, and the return from holdng the rs free asset w w n w 1. p From portfolo theory we now that to f the effcent fronter we have to mnmze (10.56) subject to the constrants Wth λ denotng the Lagrangean multpler for the frst constrant we get the followng frst order condtons wth L denotng the Lagrange functon: L w L n 0, V w n 0, V w L w w 0, L 0. 66

72 By combnng (10.58) and (10.59) we get Vw, n Or wth V denotng the th column of V,.e. V(w v ) denotng the covarance of the th asset wth the portfolo w v, and 1 F () = 1 f the asset s foregn and the nvestor s domestc 0 f the asset s domestc or the nvestor s foregn We can rewrte (10.62) as F V w F 1 1. For domestc assets and foregn nvestors (10.64) reduces to j Vjw. If the second nequalty n (10.64) holds strct we now from (10.58) and (10.60) that N j1 w V w 1 0 j j j F holds only f Otherwse ths sum must be postve by the restrcton that and the second term beng postve. Equvalently wth (10.61) the strctness of the frst nequalty n (10.64) mples If both nequaltes hold strctly the asset s not held by the nvestor. Hence one of the two nequaltes has to be an equalty. Defnng for a domestc nvestor 1 for a foregn asset held long () = 0 for a domestc asset -1 for a foregn asset held short n we can wrte V w. Dvdng (10.67) by (10.65) gves us j j V w V w for j. The N net demands w v are the only unnown varables n these N 1 equatons and the relatve demands of the assets can be determned. As no preferences enter equaton (10.68) the relatve demand for every domestc nvestor only depends on the propertes of the assets and 67

73 hence all nvestors have the same relatve demand for rsy assets,.e. have the same optmal rsy portfolo. But as we wll see t has not to be the world maret portfolo. The optmal rsy portfolo for foregn nvestors wll be dfferent of that for domestc nvestors unless the term θπ () equals zero,.e. no barrers for nvestment exst. Nevertheless all foregn nvestors wll demand the same optmal rsy portfolo. Suppose that there exsts a foregn asset whose return s uncorrelated wth any other asset, domestc or foregn,.e. σj = 0 for 6= j. Suppose further that the excess return s zero, as t s n the tradtonal CAPM,.e. µ r = 0. Then (10.64) becomes w. 2 For holdng v the asset long we need As λ can be shown to be postve, ths mples w < 0. In the same manner holdng the asset short mples < 0. Both results volate the assumpton of nonnegatvty of the postons. Therefore an asset wth such a property s not traded by domestc nvestors. Ths result shows that not all assets are held by domestc nvestors. Defne now such that (10.64) becomes 1F Q V w 1 q. F As for domestc assets 1F () = 0 we fnd that unless and for foregn assets q + Q = 2θ. Defnew s as the vector of the share the assets have on total world wealth W W and let wth W as the wealth of nvestor defne D D F T D T D T W, T T, T T,,, q q1,..., q D F D D F N q q D T T T and, Where D and F denote summng over all domestc and foregn nvestors, respectvely. Multplyng (10.62) by W and summarzng over all nvestors gves D F n. V w W T T T q DF DF Insertng the above defntons and usng that θιn = q = 0 for foregn nvestors we get D T 1 Vw W T T n T q D F D F T T T T DF s w D F T T T T q D D F D n D F D T T D T 68

74 q D T T q D F D D n D T D T F D D D n q. Multplyng (10.71) by w s and defnng as the varance of the world maret portfolo as the tax to be pad by a domestc nvestor for holdng the world maret portfolo and we get wth as the expected return on the world maret portfolo: W T T q m Vw Defne 2 2 W D F D D m m m m m s From (10.74) and (10.75) we see that wth barrers to nternatonal nvestment an asset wth a beta of zero wth the world maret portfolo may not have the same expected return as the rs free asset f t s a foregn asset. And a beta of one does not ensure to receve the maret return. 69. as the vector of betas of the assets wth the world maret portfolo. Solvng (10.72) for and nsertng nto (10.71) we get D 1 D q D m D q D. n m m m Wth θ = 0 we get the usual relatonshp of the CAPM as derved earler. For domestc assets ths reduces to m D D m m qm. We fnd a lnear relatonshp between the beta of the asset wth the world maret portfolo and the excess return of the asset. The slope can be ether smaller or larger than n the CAPM, where t s For a foregn asset we get from (10.73): q q. m D D D D D m m m We ft a smlar Securty Maret Lne as for domestc assets, t has the same slope, but t s shfted. If the asset s hold long, the second nequalty n (10.64) becomes an equalty and by nspecton of (10.70) we see that q = 0 and hence = P π q = 0. The SML s shfted by γ D θ upwards. If the asset s held short t follows n the same way that Q D = 0 and hence from q D + Q D = 2θ that = 2θ and therefore the SML s shfted downwards by γ θ. We ft to have three SML, one for domestc assets, one for foregn assets held long and one for foregn assets held short. An asset not held by domestc nvestors fulfl both nequaltes n (10.64) strctly, hence we fnd that for those assets 0 < q < 2θ and the assets plot between the two SML for foregn assets. Fgure 10.1 llustrates these fndngs.

75 At last we wll show whch propertes an asset must have that t s held by domestc nvestors. Aggregatng (10.64) over all domestc nvestors after havng multpled by W gves n n. T V w W T D D D Defne G D as the fracton of total wealth nvested nto rsy assets by domestc nvestors and G F by foregn nvestors N 1 G D w W W D 1 F 1 N F W 1 G w W and w D v D the vectors of fractons nvested nto each rsy asset by domestc nvestors. The nequaltes n (10.76) have to hold strctly for assets not hold by domestc nvestors. For foregn assets not held we get. T G W V w W T D D D D D D From (10.65) we smlarly get V F F w, D D D F F F T F F F GW V V w andv V w denote the covarance of asset wth the portfolo held by domestc and foregn nvestors, respectvely. Solvng (10.78) for µ r and nsertng nto (10.77) gves us or after rearrangng G W G W T V T G W V T V T T T F F F F D F D D D D D F D F F,, G W T G W F V. F T D D F F D V D Defnng D D D F F G W F G W and enables us to rewrte ths as D F T T D D F F V V. Foregn assets that have a small covarance wth the optmal rsy portfolo of the domestc and foregn nvestors wll not be held by domestc nvestors. These assets then wll also have a small covarance and hence a small beta wth the world maret portfolo, whch s a weghted average of these portfolos. The smaller the tax s the more assets wll be traded, f θ = 0 all assets are traded as predcted by the CAPM 70

76 The reason that assets wth a small beta wth the world maret portfolo are not held s that ther expected benefts are too small to overcome the costs. The dversfcaton effect can also be acheved by assgnng a hgher fracton to the rsless asset n the optmal portfolo. Also foregn assets that have covarances close to those of domestc assets wll not be traded, although ther beta may dffer sgnfcantly from zero. These foregn assets can easly be substtuted by domestc assets wthout mposng the costs. These nterpretatons show that t s very lely to f domestc nvestors to hold more domestc assets. EMPIRICAL EVIDENCE Most emprcal nvestgatons test the segmentaton or ntegraton of asset marets,.e they want to f or reject barrers to nternatonal nvestments. In general nether evdence for segmentaton nor ntegraton can be found at sgnfcant levels. The reason may be that these barrers n most cases affect nvestors from dfferent countres or dfferent types of nvestors of the same country not equally. Dfferent tax structures, nvestment quotas and other factors dffer wdely between countres and types of nvestors. Ths wll nfluence the behavour of asset returns n a much more complcated way than modelled here. Addtonally globalzaton and lberalzatons change the rules permanently such that t s dffcult to nvestgate asset returns over tme. These dffcultes mae t very problematc to fnd support for or aganst the model. The only substantal evdence can be derved from assets that dffer only n ther avalablty to foregn nvestors, e.g. the A- and B-shares of the Shenzen Stoc Exchange n Chna. It can only be sad that the more wdely an asset s avalable to foregn nvestors the hgher ts prce s. Asset prces and exchange rates are jontly log-normal dstrbuted There exsts a sngle consumpton good or equvalently a common baset of consumpton goods The consumpton good s contnuously and costless traded between the countres There are n rsy assets that are traded contnuously and costless There s an asset that s rsless n real terms,.e. n terms of the consumpton good There are J countres There exst no transacton costs, taxes, transportaton costs or tarffs No restrctons on short sales No barrers to nternatonal nvestment Investors are prce taers and have the same nformaton Investors are rs averse For the consumpton good the law of one prce apples,.e. the prce s equal n all countres adjusted only by the exchange rate (Purchasng Power Party, the exchange rate changes accordng to the dfference of nflaton n the countres. Table 3.9.1: Assumptons of the Internatonal CAPM 71

77 3.10 THE PRODUCTION-BASED CAPITAL ASSET PRICING MODEL The CCAPM and the ICAPM presented n the last sectons too the productons sde of the economy as gven and only modelled the demand. Emprcally strong evdence has been found that stoc returns forecast GDP growth very well. By nvertng ths relaton hgh expected GDP growth n the future should result n hgh asset returns now. By tang the demand sde (consumpton) as gven and modellng the supply sde (producton) of the economy Cochrane (1991) developed the Producton-based Asset Prcng Model (PAPM ). THE MODEL We assume a dscrete tme settng where a sngle good s produced n a fnte number of states. We further have a sngle asset whch pays a dvdend n every perod dependng on the state. The dscount factor also depends on the state. From secton 4.1 we now that the fundamental value of the asset wth the current state as the only source of nformaton s gven by t Pt E ps Ds, Tt t where Pt denotes the prce at tme t, ρ the dscount factor of nvestor n state s t, Dτ the dvdend and s t the current state. Defnng the return as usual we get R P D P t1 t1 t t1. Pt Usng (11.2) t can easly be verfed that s t Pt E pt 1 1 Rt 1 Pt s. By nsertng (11.2) nto (11.3) and teratng by nsertng (11.3) agan, we get s t Pt E pt 1 1 Rt 1 Pt S s t E pt 1 pt 1 Dt 1 s s s t E pt 1 pt 2 1 Rt 2 Pt 1 Dt 1 S s s t E pt 1 pt 2 pt 2 Dt 2 Dt 1 S... t E ps DTS, t t 72

78 hence (11.3) and (11.1) are equvalent. By dvdng (11.3) by Pt we get s t E pt 1 1 Rt 1S 1. For later convenence we rewrte n terms of the consumpton good. Lets denote the prce for the consumpton good at tme 0 for delvery at tme t + 1 f a certan state occurs (contngent contract). We assume that such a prce exsts for every state and date,.e. we assume a complete maret. The real prce of ths clam n terms of the good at tme t n state s t s gven by p C s t1 P P C 0 C 0 s t s 1 We also can express f we are n a certan state s t by t 1 t Where s s usng (11.6) gves t 1 1 t P s P s p s s C t C t s t t 0 0 1,. denotes the probablty that state s t+1 occurs gven state s t. Rearrangng and p p C t1 s t1 t1 t s ( s s. We now turn our attenton to the producton of the good. The total producton, yt, conssts of the producton of a sngle consumpton good, ct, and nvestments I t yt ct It The total producton s assumed to follow a certan producton functon y. t t t t f, l, s, where t denotes the captal stoc and lt the labour nput, whch we assume to be constant over tme. The ncrease n the captal stoc follows t1 t, It, where n general g(t, It) 6= t + It as we also account for adjustment costs of new nvestments,.e. costs le educaton of employees. 73

79 Dfferentatng (11.10) and (11.11) totally wth subscrpts denotng the dervatves we have: d t1 I t dit, d t 1 d I t 1 di, t2 t1 t1 dy f t 1 d. t1 t1 We can vew the captal stoc as fxed for a gven perod and labour nput s fxed over the entre tme by assumpton, hence the margnal product of captal depends only on the producton of the consumpton good (sales) as the costs are fxed. It further depends on the adjustment costs of new nvestments n ths perod. Ths margnal product of captal has to equal the return on nvestment by standard neoclasscal theory: c t1 1 Rt 1 I t. t 1 We assume now that the captal stoc for the perods t + 2 and followng s fxed and that the frm has to decde whch amount to nvest n perod t and whch amount to nvest n perod t + 1. Ths mples the restrcton dt 2 0. By nsertng (11.13) and rearrangng we get di q t1 d t 1 t q 1 1 I t By usng (11.9),(11.12),(11.14) and (11.17) we get from (11.15) c I t t1 1 Rt 1 t 1. dy di I t di t1 t1 t I t t1 f t 1 I tdi I tdi I t1 t t di t If we assume the company to maxmze the expected profts by choosng the approprate nvestment strategy, standard neoclasscal theory suggests that margnal costs and benefts have to equal. By (11.10) the total producton n a perod s gven by the captal stoc as the only varable. Total producton can only be dvded between the producton of the consumpton good 74 t1 f t 1 I t. I t1

80 and new nvestments. An ncrease n nvestment by dit has to be accompaned by a decrease n sales of consumpton goods wth the same amount, hence margnal costs of ncreasng nvestments are P C (s t )dit. The margnal benefts are the ncreased producton of the consumpton good n the next perod, due to a rse n total producton and a reducton n future nvestment (the captal at t + 2 s fxed to a certan amount),.e. the margnal benefts are C t1 P s E dc t 1 t t1 s s s t. So the frst order condton for a proft maxmum s P s P s di E dc s s s C t1 C t t t t1 t t1 Insertng (11.9), (11.14) and (11.17) gves C t1 P s C t t1 t P s dit E f 1. t 1 t t I t dits s s It1 Dvdng by P C (s t )dit and usng (11.18), (11.6) and (11.8) we get t E pt 1 1 Rt 1 s 1.. By comparng ths frst order condton for a proft maxmum on the return on nvestment wth the asset prcng condton (11.5) we see that they both loo ale. By wrtng these condtons n another form we get t1 t t1 t s s pt 1 Rt 1 s s pt 1 Rt t1 t1 s s It s obvous that the return on nvestment s a lnear combnaton of asset returns and vce versa. In order to prevent arbtrage,.e. the company may sell assets short and nvest the proceedngs to obtan a rsless proft or vce versa, the two expected returns must equal: E R. t1 E Rt 1 Havng shown the equalty of expected asset returns and return on nvestment, we now can relate the asset returns easly to the growth rate of GDP,.e. the busness cycle. Hgh expected growth of GDP wll ncrease the return on nvestment and hence expected asset returns. The hgh expected returns for the near future wll lead asset prces to ncrease as shown n secton 4.1. Ths result wll gve the result of the stoc maret leadng the busness cycle, but ths lead should not too far as the precson of expectaton decrease wth the forecast horzon. 75

81 We further have the ratonale for asset prcng that the expected asset return should equal the return on nvestment. EMPIRICAL EVIDENCE An emprcal nvestgaton nto the performance of the PAPM can use equaton (11.23). Crtcal to the nvestgaton s the determnaton of the return on nvestment, the return on assets can easly be derved from the maret. Ths return can only be determned on the bass of the balance sheet, whch s only publshed quarterly, n Europe often only sem-annual or annual, whle stoc returns are avalable daly. Further the data n the balance sheet may be based due to poor accountng standards and actons taen for fscal measures,.e. the data qualty may only be poor. We therefore face smlar problems as n the CCAPM. The nvestgaton undertaen by Cochrane (1991, pp. 223 ff.) usng quarterly data for the perod show nevertheless a hgh and sgnfcant correlaton between asset returns and returns on nvestment. The returns on nvestment are more volatle than asset returns, suggestng that the arbtrage s not complete due to transacton costs or maret ncompleteness. Cochrane (1991, pp. 232 ff.) further fnds that the asset returns forecast GDP growth by about 9 months, a not too long perod of tme as predcted by theory. These results suggest that n the longer run (for perods longer than a quarter) the stoc maret s drven by the busness cycle, where for nvestgatng sngle assets or ndustres the busness cycle has to be decomposed nto the parts relevant for ths nvestgated assets. As other varables of nterest to the expected returns, e.g. nterest rate or consumpton n the CCAPM, are also closely lned to the busness cycle, ts mportance for explanng asset returns s further ncreased. What remans an unsolved problem s the explanaton of asset returns n the short run. Here only the Condtonal CAPM, especally the ARCH specfcatons, gve promsng hnts. 76

82 SUMMARY Ths chapter gave an overvew of the man models used n asset prcng, ncludng some more recent developments. Ther am s to determne the fundamental value and/or an approprate expected return. Most models relate expected returns to rss nvestors have to bear and have to be compensated for. They dffer manly n the rs factors they allow to enter nto the model. Although a large number of rs factors have been proposed n the lterature and many models are used, none of the presented models or any other models used by practtoners are able to explan the observed asset prces or returns suffcently. Whle most models wor qute well for tme horzons of more than a year, they fal n explanng short term movements. Only the Condtonal CAPM, especally f covarances are modelled wth a GARCH process, shows a satsfactory ft wth the data, but the GARCH specfcaton msses any economc reasonng. The exstence of many anomales observed n asset marets, e.g. seasonal patterns, cannot be explaned by any of these theores. They are subject to a large feld of theores tryng to explan a certan effect or a group of effects. To explan anomales often even the assumpton of ratonal actng nvestors, a central element n economc theores, s questoned n these efforts. As there exsts no generally accepted model of asset prcng the opnons of nvestors concernng the value of an asset dverge substantally. Access to relevant nformaton and a superor model are central elements to mae extraordnary profts n the maret. Therefore not only academc research but also securtes companes mae huge efforts to mprove the models. 77

83 Captal Asset Prcng Model: Sharpe-Lntner Verson In 1964 (Sharpe) and n 1965 (Lntner) contnued the wor of Marowtz and constructed the famous Captal Asset Prcng Model (CAPM). Bascally, the model was developed to explan the dfferences n rs premum across assets. The CAPM shows clearly that these dfferences are generated by the dfferences n the rsness of assets, that s, the hgher the rs of an asset the hgher the rs premum demanded by nvestors. Wth ths extenson, Sharpe-Lntner asset prcng model bult the relatonshp between rs and expected return of fnancal assets based on the nvestors rs profle for the frst tme n the fnance hstory. Rs-free Rate Accordng to Damodaran (2006), the rs-free rate refers to the zero coupon government bond rate whch matches the analysed cash flow tme perods. For a long term analyss, a long term government rate s used as the rs-free rate of all cash flows, whle n short term analyss, t s recommended to use entrely the short term government securty rate as the rs-free rate. An nvestment should have no default rs n order to be rs-free. Maret Rs Premum Damodaran (2006) stated that n an average rs nvestment, the maret (rs) premum s the premum that nvestors demanded for nvestng wth relatvty to the rs-free rate. In other words, ths premum demanded s the benchmar or the mnmum amount of money needed to be exceedng by the expected return on nvestment. Moreover, there are three condtons for a maret premum, whch s; the premum should be greater than zero, there s an ncrease wth the rs averson of the maret nvestors, and an ncrease n the rsness of the average rs nvestment. The CAPM Equaton The general equaton of the model s: where: f r r r m r r - expected return of stoc β ι relatve rs of share rm - expected return of the maret portfolo r f rs-free nterest rate f Ths formula wll be the man tool for testng the hypothess H1: The maret rs premum s postvely related wth the expected return. What CAPM says s that the equlbrum n the captal marets s characterzed by only 2 numbers:- () the return for watng.e. r f ; () the extra return.e. rm - r f. 78

84 A very mportant consequence of ths model s the separaton theorem, whch says that n the captal marets the rs has two components: dversfable (non-systematcal) rs and non-dversfable (systematcal) rs. When prcng, the only sgnfcant rs s the systematc one, snce nvestors can just get rd of the non-systematc rs through dversfcaton. Sharpe & Lntner show that the true measure of rs s the well-nown coeffcent, beta computed as follows: Beta (β) Where: Cov (R, Rm) represents the Covarance of the ndvdual stoc return wth the maret return, And Var (Rm) represents the Varance of the maret return. Emprcal evdence was n favour of CAPM and the model became extremely famous n the modern portfolo theory. Thngs were clear: stocs wth beta lower than 1 were consdered passve stocs and stocs wth beta hgher than 1 were consdered aggressve and rsy. Dependng on ther appette towards rs, nvestors would choose the stocs n ther portfolo accordng to the value of beta. Though, some crtcsms of CAPM emerged. One very nown crtc belonged to Fama and French. In 1992, they dscovered a negatve relatonshp between rs and return. Snce then, a very mportant queston was rased, s beta dead? The concluson of these ssues s that whle academc debate stll rages on, the CAPM may stll be useful for those nterested n the long run. Wth ths development, the accompanyng assumptons of CAPM were stated as: (a) all nvestors are rs-averse ndvduals, who maxmse the expected utlty of ther end of perod wealth, (b) the nvestors are prce taers and have homogenous expectatons about asset returns that have jont normal dstrbuton, (c) there exst a rs-free asset such that nvestor may borrow or lend unlmted amounts at the rs-free rate, (d) the quanttes of asset are fxed, also all assets are maretable and perfectly dvsble, (e) asset marets are frctonless and nformaton s costless and smultaneously avalable to all nvestors, and (f) there are no maret mperfectons such as taxes, regulatons, or restrctons on other sellng (Attya Y. Javed, Alternatve Captal Asset Prcng Models: A Revew of Theory and Evdence). Concludng the Sharpe-Lntner model, the CAPM model can be wrtten as: E (R ) = R f + β ((E (R m ) R f ) 79

85 The Equaton of rs premum can be re-wrtten as: E (R ) R f = β ((E (R m ) R f ) The verson of CAPM by Sharpe and Lnter has dscovered a new feld of fnance study, whch s the reason why the model has been wdely used and tested based on both theoretcal cases and emprcal cases (Attya Y. Javed, Alternatve Captal Asset Prcng Models: A Revew of Theory and Evdence) 80

86 CHAPTER 4: APPLICATION OF CAPM IN THE INSURANCE INDUSTRY One of the man objectves of fnancal nsttutons s maxmzng shareholder value and hence the man focus s on how the frm s captal s allocated and utlzed. Captal allocaton n ths context refers to the determnaton of the amount of a frm's equty captal that s assgned to each project or lne of busness undertaen by the frm. However, frms are usually concerned about captal allocaton n the context of prcng and project selecton, e.g., to determne the proporton of the frm s overall cost of captal that must be contrbuted by each lne of busness n order to maxmze frm value. Ths dscusson of captal allocaton s conducted n the context of the nsurance ndustry. Varous technques have been suggested for allocatng equty captal. However, most of the technques dscussed are perfectly general and can be appled n other ndustres as well. Captal allocaton s perhaps of specal nterest to fnancal frms such as nsurers. For such frms, the prncpal provders of debt captal (nsurance reserves) are also the frm s prncpal customers. Unle the holders of bonds and other (non-nsurance) debt captal, nsurance polcyholders cannot protect themselves aganst the nsolvency of specfc debt ssuers by holdng a dversfed portfolo. Unle the dversfed bond nvestor, the typcal polcyholder reles upon one nsurer (or at most a few, n the case of lfe nsurance) for each type of protecton purchased (e.g., auto nsurance, homeowners nsurance, health nsurance, etc.). Most nsurance polces are purchased not as an nvestment but to protect aganst adverse fnancal contngences. Thus, nsolvency rs plays a specal role n the nsurance ndustry, and captal s held to assure polcyholders that clams wll be pad even f larger than expected. In dscussng captal allocaton, t s mportant to eep n mnd that the nsurer s entre captal s avalable to pay the clams arsng from any specfc polcy or lne of busness. If the nsurer becomes nsolvent t s the entre company that enters banruptcy the frm does not go banrupt lne by 81

87 lne. Nevertheless, t s often useful to thn of captal as beng allocated by lne of busness for prcng, underwrtng, and other types of decson mang. Tang a close loo at captal allocaton n the context of the nsurance ndustry s also useful to elucdate the nteracton between fnancal decson mang and the rs-based captal rules appled to the nsurance ndustry by regulators. Captal allocaton s also related to recently emergng concepts such as rs adjusted return on captal (RAROC) and economc value added (EVA), whch have become mportant management decson mang technques for both fnancal and non-fnancal frms. USING CAPITAL ALLOCATION TO MAXIMIZE VALUE Why allocate captal? The motvaton for anythng a frm does should be to maxmze shareholder value, that s, to ncrease the maret value of equty captal. Unfortunately, ths straghtforward and powerful objectve s often overlooed n practce. Through extensve dscussons wth people n the nsurance ndustry, t appears that many frms are managng to ther GAAP (generally accepted accountng prncples) balance sheets and ncome statements, wth the objectve of showng healthy GAAP earnngs and/or maxmzng the value of GAAP equty. Of course, the frm needs to be cognzant of ts GAAP performance because of the mportance of accountng results to fnancal analysts and traders. However, t s a mstae to lose sght of the frm s true msson the maxmzaton of maret value. Marng to maret should play a crtcal role n the frm's nternal decsons. Captal allocaton can be used to facltate and mprove the measurement of the economc proftablty of busnesses wth dfferent sources of rs and dfferent captal requrements. In the nsurance ndustry, t s customary to defne busnesses n terms of lnes of nsurance, for example, the commercal lablty lne or the auto lablty lne. Although ths s the tradtonal approach, nsurers need to step bac and thn carefully more about the ssue of «what s a busness?» n desgnng captal allocaton and performance measurement systems. 82

88 Sometmes the banng lterature tals about depost accumulaton or ganng demand deposts as one busness, and mang loans as another busness. In ths conceptualzaton, the economc concept of transfer prcng s used, whereby the ban s loan orgnaton busness wll borrow money from the depost accumulaton busness and pay an mplct rate of nterest n return for havng funds to nvest n loans. Ths approach could also be used n nsurance. The underwrtng operaton could be consdered as a funds-generatng busness n whch money s beng borrowed from polcyholders. The underwrtng operaton then would lend the funds to the nvestment busness n return for a transfer prce. In ether of the above lnes of busness concepts, the maturty and duraton characterstcs of the debt captal and the nvestments resultng from the nsurer s dfferent busnesses must be recognzed. Thus, funds generated by ssung long-tal lablty polces are lely to lead to dfferent nvestment objectves than funds rased by ssung short-tal property nsurance polces n order to manage the rss of duraton and convexty. One cannot assume that the long-term lablty lne has an asset portfolo that loos le the company; t has to be managed to meet the frm s overall objectves n terms of nterest rate rs. Duraton and convexty management s extremely techncal. Therefore, specal care must be exercsed n gvng a partcular busness credt for the money t generates, whle, at the same tme, chargng t for the use of captal. Ths s the case for both the debt and equty captal needed to operate the busness. The prmary ln between captal allocaton and value maxmzaton s to enable the frm to measure performance by lne of busness to determne whether each busness s contrbutng suffcently to profts to cover ts cost of captal and add value to the frm. To measure the cost of captal, t s necessary to determne the captal requred to offer each type of nsurance relatvely rsy lnes typcally requre more captal than less rsy lnes. For example, one mght wonder whether the commercal lablty nsurance busness s mang an adequate proft, that s, whether the nsurer 83

89 should be chargng hgher or lower prces than at present, whether t should ext ths busness or perhaps devote more captal to t. Insurers can maxmze value by sheddng unproftable busnesses as well as by dentfyng proftable new projects. By wthdrawng from unproftable lnes, the nsurer may be able to ncrease the maret value of equty, even as revenues declne. The ultmate objectve should not be revenue growth, but maxmzng net worth. To provde a framewor for the dscusson of captal allocaton methodologes, t s helpful to provde a smple mathematcal statement of the captal allocaton problem. Defne x as the proporton of the frm s equty captal allocated to busness, where x s between 0 and 1. Thus, x ndcates the proporton of captal whch s allocated to busness and, therefore, the amount of captal allocated to busness s C, whch s the total captal, C, multpled by x. If the frm has N busnesses, then N N x 1 and C C 1 1 That s, the sum of the captal allocated to all of the frm s busness wll be less than or equal to the frm s total captal. Whle t may seem surprsng that a frm may not assgn all of ts captal to ts busnesses, n fact some leadng-edge researchers argue aganst allocatng all of the captal and favor, nstead, an allocaton whch results n less that 100 percent beng assgned (Merton and Perold 1993). We return to ths ssue n the dscusson below of the use of opton models to allocate captal. Once captal has been allocated by lne, how can the resultng allocatons be used to maxmze frm value? One approach that has receved consderable attenton s to calculate the rs-adjusted return on captal (RAROC). RAROC s defned as the net ncome from a lne, dvded by the captal allocated to the lne. That s, RAROC Net Income C where C s the captal allocated to lne of busness.. The numerator of the RAROC formula, net ncome, also needs to be defned carefully. It may seem obvous, but t really s not once t s 84

90 consdered n economc terms. Bascally, net ncome n the RAROC formula should be after taxes and nterest expense. Even though nterest expense s a banng term, t also apples to nsurers n the form of underwrtng loss. That s, the nsurance maret mplctly dscounts the loss cash flows for the tme value of money, meanng that the underwrtng proft s negatve n most cases. The negatve underwrtng return, whch s analogous to nterest expense, needs to be taen out when calculatng the return from a lne of busness. Once the RAROC for a lne of busness has been calculated, how does the frm now whether the lne s current rs-adjusted return s adequate? The rs-adjusted return should be compared wth the cost of captal for busness, where the cost of captal s obtaned usng an approprate asset prcng model. If the rs-adjusted return equals or exceeds the cost of captal, then contnung to devote resources to ths lne of busness s consstent wth the goal value-maxmzaton. However, f the rs-adjusted return s below the cost of captal, the lne of busness s reducng the frm s maret value. In ths crcumstance, the frm should tae some acton to mprove the stuaton such as reprcng the nsurance, tghtenng underwrtng standards, or wthdrawng from the lne of busness. A slghtly more formal way of determnng whether a partcular lne of busness s addng to frm value goes under the name of economc value-added (EVA). 1 Economc value-added measures the return on an nvestment n excess of ts expected or requred return. EVA sees to dentfy lnes that create value for the frm. EVA s net ncome mnus the cost of captal, or hurdle rate, for a certan busness, multpled by the captal allocated to the busness: EVA = Net Income - r C where r = cost of captal (hurdle rate) for busness. Thus, f EVA 0, wrtng the lne of busness s consstent wth value maxmzaton, whle f EVA < 0, the lne s destroyng frm value. The EVA formula can be changed slghtly to put the results n rate of return format, creatng a measure called the economc value added on captal (EVAOC). EVAOC s defned as EVA dvded by the captal 85

91 allocated to a lne,.e., EVAOC = (Net Income / C ) - r Ths s smlar to RAROC, except that the lne s cost of captal s subtracted. Agan, f EVAOC s postve, the lne s creatng value for the frm. An mportant detal s how to determne the cost of captal for busness. Ths too can present a problem n the nsurance ndustry due to data lmtatons. One approach proposed by fnance researchers to estmatng the cost of captal for a lne of busness s the «pure play» technque. The pure play approach estmates the cost of captal by fndng other frms that offer only one lne of busness. The cost of captal for a busness n the multple-lne frm can then be based on the cost of captal of mono lne frms wrtng only that specfc type of busness. Ths approach s problematcal n the nsurance ndustry because t s dffcult to fnd frms that wrte only one lne of busness. Even f such a frm or frms could be found, the underwrtng rs characterstcs of the pure play frms could dffer sgnfcantly from those of a gven lne of busness wrtten by the multple lne frm. An alternatve to the pure play technque s the use of so-called full-nformaton betas to determne the cost of captal (Kaplan and Peterson 1998). Ths estmaton technque uses data on conglomerates (frms that wrte several lnes of busness) to conduct regressons that permt the estmaton of the cost of captal by lne. Another problem n the nsurance ndustry s the lac of qualty data. An nsurer thnng about mplementng VaR, EVA, and the other economc methodologes dscussed here needs to thn about revsng ts data system to capture the data requred to mplement the methodologes. Insurers should be desgnng data systems that allow them to report underwrtng results frequently, for example, on a monthly bass. Data qualty s crucal. Wth nadequate data, even a perfect model wll fal. Most nsurers do not have the necessary data to mplement VaR, RAROC, and EVA at the present tme. 86

92 CAPITAL ALLOCATION TECHNIQUES Varous methodologes have been developed that could provde the foundaton for a system of captal allocaton. The followng s by no means an exhaustve lst. Many other proposals can be found throughout the related lterature. I frst provde an overvew of the methods and then go nto more detal n dscussng each method separately. OVERVIEW OF CAPITAL ALLOCATION TECHNIQUES Regulatory Rs-Based Captal. In the Unted States, regulators have developed a formula to calculate the rs-based captal of nsurers. A company s rs-based captal s used to defne the mnmum captal t must hold n order to avod regulatory nterventon. The regulatory thresholds or acton levels are determned by the rs-based captal rato, whch s the rato of the company s actual captal to ts rs-based captal (see Cummns, Harrngton, and Nehaus 1993). If the nsurer s actual captal s greater than 200 percent of ts rs-based captal, no regulatory acton occurs. However, f actual captal falls below 200 percent of rs-based captal, varous regulatory actons are taen, dependng upon how far actual captal falls below 200 percent. An nsurer s rs-based captal s computed by a formula desgned to requre more captal for rser companes. Some nsurers actually use the regulatory rs-based captal formula n allocatng captal for purposes of managng the frm. In my opnon, ths s unwse, because the regulatory charges are of questonable accuracy and are based on boo rather than maret values. Furthermore, the regulatory charges gnore mportant sources of rs such as nterest rate (duraton and convexty) rs, as well an the nsurer's transactons n the dervatves maret. Even f the charges were accurate, they would be accurate only for the average frm n the ndustry. Consequently, n the case of frms wth boos of busness havng above or below-average rs, the regulatory charges would produce napproprate allocatons of captal. The result s that busnesses may be charged for too much or too lttle captal, leadng to sub-optmal decsons. 87

93 The Captal Asset Prcng Model. The second approach nvolves usng one of the oldest of modern fnancal theory technologes, the captal asset prcng model (CAPM). Ths s not the best soluton to the problem, but t may provde a helpful benchmar based on a famlar methodology. At the very least, the use of the CAPM allows managers to compare the preferred method to the results generated by a classc technque. Value At Rs (VaR). Ths value at rs concept has become very popular n the banng and nvestment banng communtes where there s a need to examne the rs exposure of the frm s tradng boo for foregn exchange, bonds, etc. Smply stated, the value at rs (VAR) s the amount the frm could lose wth a specfed small probablty, such as 1 percent, n a specfed perod of tme. The measurement of value at rs from currency and securtes tradng has advanced rapdly, thans n part to daly and even more frequent data on exchange rates and asset prces whch allow for very accurate and sophstcated calculatons of VaR. 3 VaR s also lely to be very useful to nsurers and, n fact, s closely related to tme honoured nsurance and actuaral concepts such as the probablty of run and the maxmum probable loss. Unfortunately, the applcaton of the most sophstcated VaR technques requres very frequent data (monthly data s an absolute mnmum), but nsurance prces and losses are not observed wth suffcent frequency nor n a maret context. Most nsurers do generate such data nternally. Usng sophstcated tools such as Value at Rs requres an ntegraton of the captal allocaton methodology wth data processng and nformaton systems to ensure that pertnent and useful data are generated to provde nputs for the Value at Rs models. Margnal Captal Allocaton. Margnal captal allocaton s a term that can be appled to the captal allocaton technque proposed by Robert Merton and Andre Perold (1993) and to a related technque developed by Stewart Myers and James Read (1999). Both technques are based on the opton prcng model of the frm. In the optons vew of the frm, the value of the polcyholders clam on the frm s equal to the present value of losses mnus the value of the nsolvency put opton. The 88

94 nsolvency put opton s the expected loss to polcyholders due to the possblty that the frm wll default. The smplest opton nterpretaton of the frm nvolves a one-perod, two-date model where the frm ssues polces at tme 0 and clam payments occur at tme 1. If assets exceed labltes at tme 1, the frm pays the losses and the equty owners receve the resdual value (the dfference between assets and labltes). However, f assets are less than labltes, the nsurer defaults and the polcyholders receve the assets. The payoff to polcyholders at tme 1 s thus equal to: L - Max[L- A,0], where L = losses, A = assets, and Max[L-A,0] s the payoff on the nsolvency put. The Merton-Perold (M-P) and Myers-Read (M-R) captal allocaton technques tae a dfferent vew of what s meant by a «margnal» approach to captal.allocaton. Stated smply, the M-P approach vews margnal captal allocaton n terms of what happens to the nsolvency put opton f entre lnes of busness are added to or removed from the frm. The M-R approach s margnal n the nstantaneous ntepretaton famlar from calculus,.e., they allocate captal based on what happens to the frm s nsolvency put opton n response to very small changes n the labltes of the lnes of busness wrtten by the frm. The M-P method may leave some captal unallocated, whle the M-R method maes a unque allocaton of 100 percent of captal. We now turn to a more detaled examnaton of how The Captal Asset Prcng Model s utlzed n nsurance. THE CAPITAL ASSET PRICING MODEL The CAPM states that the return on equty or cost of captal for a frm s determned by the followng formula: re = rf + ße [rm - rf] where re = cost of equty captal, rf = default rs-free rate of nterest, 89

95 rm = expected return on the "maret", and ße = the frm s beta coeffcent = Cov(re,rm)/Var(rm). where Cov( ) = the covarance operator and Var( ) = the varance operator. How can the CAPM be used by an nsurance company to mae prcng and nvestment decsons? Project decsons can be made by decomposng the beta coeffcent to determne the betas by lne of busness. For example, let s consder an nsurer wth two lnes of busness. Its net ncome would be: I = raa + r1p1 +r2p2 where I = net ncome, ra = return on assets, r1, r2 = rates of return on underwrtng from lnes 1 and 2, A = assets, and P1, P2 = premums from lnes 1 and 2. Next ntroduce the balance sheet dentty, whch says that assets are equal to equty, plus the labltes generated by the two lnes. Then dvde by equty to express the result as a rate of return: re = ra(e+l1+l2)/e+r1p1/e+r2p2/e Then the Beta coeffcent can be decomposed as follows: ßE = ßA(1+1+2) + ß1s1 + ß2s2 where ßE, ßA, ß1, ß2 = betas for frm, assets, and nsurance rs of lnes 1 and 2, 1, 2 = lablty leverage ratos for lnes 1 and 2, = L /E, = 1, 2, and s1, s2 = premum leverage ratos for lnes 1 and 2, = P /E, = 1, 2. The calculaton uses the property that the covarance s a lnear operator. The formula for the decomposton of ßE shows that the beta of the frm, whch drves the requred 90

96 return on equty, s the beta of assets tmes 1, representng the nvestment of equty captal, plus the lablty leverage ratos for lnes one and two. Then the formula adds on the beta of each ndvdual lne's underwrtng returns multpled by the lne-specfc premum to surplus rato. So we fnd a theoretcal justfcaton for the tradtonal rule of thumb leverage rato that has been used for years n the nsurance ndustry -- the premum to surplus rato. The model can be solved for the requred rate of underwrtng return on each lne of busness: r = - r f + β (r m r f ) for lnes of busness = 1 and 2. Thus, each lne mplctly pays nterest for the use of polcyholder funds (the term - r f ) and receves a rate of return based on the systematc rs of the lne (the term β (r m r f )). The CAPM result has the followng mplcaton: It s not necessary to allocate captal by lne usng the CAPM, but rather to charge each lne for at least the CAPM cost of captal, reflectng the lne's beta coeffcent and leverage rato. Costs of captal based on other asset prcng models, such as the arbtrage prcng theory (APT), have smlar mplcatons. Although the CAPM provdes a useful way of conceptualzng the contrbutons of the frm s lnes of busness to the return on equty, there are at least three mportant problems wth ths model. (1) The CAPM only rewards the frm for bearng systematc underwrtng rs, that s, underwrtng rs that s correlated wth the maret portfolo. However, nsurers also need to be concerned about extreme events,.e., tal rs, that s smply not prced n the CAPM model. Ths s mportant n vew of the role of nsurers as fnancal ntermedares, where a frm s prncpal credtors are also ts customers. (2) Lne of busness underwrtng beta estmates are dffcult to estmate gven the data currently avalable (Cummns and Harrngton 1988), although progress has been made n estmatng costs of captal n the more recent lterature (e.g., Lee and Cummns 1998). (3) Research has shown that rates of return are drven by other economc factors besdes beta coeffcents (e.g., Fama and French 1996). Thus, sole relance on the CAPM would gnore mportant determnants of the cost of captal. The prmary role for the CAPM s to serve as a benchmar to compare wth the results of 91

97 other estmaton methodologes. If the two methods yeld drastcally dfferent results, t would be approprate to chec the methodology and data carefully before proceedng. ADDITIONAL ISSUES Another mportant ssue has to do wth the economc cost of the frm s overall captal as well as the captal allocated to ndvdual lnes of busness. The captal of fnancal nsttutons such as nsurers s nvested n maretable securtes. If captal marets are effcent and frctonless, the nvested funds wll earn the equlbrum maret rate of return and thus wll be cost-less to the frm. However, the exstence of varous maret and nsttutonal mperfectons lead to frcton costs whch mply that the captal nvested wll not earn the full far maret return requred to avod a loss to the nsurer. Varous types of frcton costs are present that create costs for nsurers that reduce the returns from the nvestment of ther captal. The three most mportant sources of costly captal to nsurers are: Agency and nformatonal costs. It s well-nown that managers of frms can behave opportunstcally, and thus fal to realze the owners' objectve of value maxmzaton. In addton, adverse selecton and moral hazard are endemc to nsurance marets and wll create costs to the extent they cannot be controlled through an nsurer s prcng and underwrtng decsons. The Federal ncome tax system leads to double taxaton of nvestment ncome; and, as a result, nvestng n securtes through an nsurance company produces lower nvestment returns than nvestors could realze by nvestng drectly n the maret. And regulaton, and especally the rs-based captal system, mposes costs on nsurers n the form of a regulatory opton on the nsurer s assets. The opton s created because the RBC system gves regulators the legal rght to seze control of the nsurer when ts assets stll exceed ts labltes. Other regulatons such as nvestment restrctons also may lead nsurers to hold neffcent portfolos, further reducng returns for any gven level of rs. 92

98 The exstence of maret frctons means that a spread develops between the returns that could be earned by nvestng drectly n captal marets and the returns actually earned on the captal held by nsurers. It s ths spread cost that must be taen nto account n determnng whether lnes of busness are earnng the approprate rates of return. Of course, the rs of ndvdual lnes also s mportant n determnng ther cost of captal. Usually, the type of rs recognzed n the cost of captal context s systematc maret rs, determned by an asset prcng model. It s also nterestng to consder the role of regulatory rs-based captal n the context of a margnal allocaton system. Wth a well-desgned margnal captal allocaton system, s regulatory rs-based captal relevant? It can be, because there s a potental cost mposed on the frm by the regulatory rs-based captal system. For most nsurers, and reasonably small EPD targets, no cost wll be realzed, snce the nsurer s total captal generally wll be greater than the regulatory captal requrement. We also consder the followng two cases: Regulatory captal for one or more lnes exceeds the margnal captal allocated to these lnes but the nsurer s overall captal s greater than rs-based captal. In ths case as well regulatory costs are not lely to be ncurred because the rs-based captal test s appled to the entre frm and not by lne of busness. Ths concluson would have to be modfed f rsbased captal acton level tests were to be appled by lne. Regulatory rs-based captal exceeds the frm s overall captal, ncludng both allocated and unallocated captal. In ths stuaton, regulatory penaltes wll apply. Thus, the concluson s that regulatory captal usually wll not be a problem, even f one or more lnes of busness have allocated captal that s less than the by-lne rs-based captal, as long as the frm s overall captal exceeds ts overall rs-based captal. 93

99 The fnal pont to be made s a caveat. Insurers should use cauton n desgnng and mplementng a captal allocaton system. The use of an napproprate system s lely to lead to rejecton of some projects that should be accepted and acceptance of some projects that should be rejected. Systems that use stand-alone captal are lely to be partcularly harmful. However, the adopton of an approprate margnal captal allocaton system can have sgnfcantly benefcal effects on the maret value of the frm, by enablng the frm to dentfy projects and busnesses that are creatng value for shareholders as well as those that are destroyng frm value PAST RESEARCH OF CAPM Numerous research on CAPM have been conducted n the developed world wth a majorty of studes revealng that the model fals the emprcal test n most marets. Fama and French wrote a paper on CAPM s wanted dead or alve where they crtczed the model s relance on one factor model to explan rs. They added new varables to beta whch revealed better results but stll used beta as one of the varables. Therefore they showed that CAPM was not entrely wrong but needed to be strengthened. CAPM has been tested n the Kenyan maret by several research students n partcular an MSc Fnance student at JKUAT tested for the perod The test drew conclusons that CAPM was ncompatble to the Kenyan stoc maret and recommended a CAPM test over a longer duraton and usng a larger sample. A recent test was also undertaen from January 2008 December 2013 revealng nconsstences n the model. The student recommended tang a larger sample of frms. In ths study, we analyse 47 equtes spannng over a decade and further conduct a beta-predcton usng CAPM for the year endng 31 st December

100 CHAPTER 5: DATA ANALYSIS & RESULTS The study was conducted on the publcly lsted stocs at the NSE over the ten years rangng from January 2004 December The daly stoc prces and stoc maret ndces were obtaned from the NSE whereas the weely rs-free rates of return were acqured n the form of 90-day Treasury Blls rates from the Central Ban of Kenya. The number of shares lsted over the perod was as follows:- The development of the rs-free rate of return over the decade: 95

101 Whereas the NSE 20-Share Index performed as follows: Snce the 91-day Treasury Blls were released by the government on a weely bass, the NSE stocs have also been derved weely for consstency. 5.1 NSE THROUGH THE YEARS The Narob Stoc Exchange has grown tremendously over the last decade from a maret captalzaton of Kshs bllon as at 1 st January 2004 to Kshs 1.92 trllon as at 31 st December 2013 representng a % ncrease through transactons such as ntal publc offers, rghts ssues, offers for sale, cross-lstngs, ntroductons, mergers, acqustons and delstngs. The equty turnover ncreased to Kshs 155 bllon n The NSE 20-Share Index has mproved from to approxmately 180% ncrease. The hghest closng year ndex was 5200 n year On 10 th November 2004, the central depostory system was commssoned whch automated for the frst tme n Kenya s hstory the process of clearng and settlement of shares traded n the captal maret. The NSE mplemented lve tradng on ts own automated tradng systems (ATS) n September The NSE also mplemented ts Wde Area Networ (WAN) platform n December 2007 facltatng remote tradng whch meant broers and nvestment bans no longer requred to trade from the floor of the house but through termnals n ther offces lned to the NSE tradng 96

102 engne. In an effort to provde nvestors wth a comprehensve measure of the performance of the stoc maret the NSE ntroduced the NSE All-Share Index (NASI) n February 2008 to complement the NSE 20-Share Index. The NASI s calculaton was based on maret captalzaton, mplyng that the ndex level reflected the total maret value of the consttuent stocs. The ntroducton of Safarcom through an Intal Publc Offer (IPO) pushed the number of shares lsted on the bourse to over 55 bllon shares, from the prevous 15 bllon and maret captalzaton to reach Kshs 1.28 trllon. The IPO had a 532% subscrpton level and was the frst IPO n whch ctzens of the East Afrcan Communty were accorded the same treatment as domestc nvestors and t was the largest IPO n Sub-Saharan Afrca at the tme. In 2011, the equty settlement cycle moved from the prevous T+4 cycle to the T+3 where nvestors were able to get ther money three (3) days after the sale of ther shares. The Narob Stoc Exchange Ltd also changed ts name to the Narob Securtes Exchange and reclassfed equtes under ten ndustry sectors n lne wth nternatonal best practse. The NSE later launched the FTSE NSE Kenya 15 and FTSE NSE Kenya 25 Equty Indces. Growth Enterprse Maret Segment (GEMS) was launched n 2013 and Home Afra, a real estate company was the frst company lsted on the GEMS. As a result of ts ntatves to ncrease company lstngs and dversfy asset classes, the NSE was raned the Most Innovatve Afrcan Stoc Exchange for Further, the NSE was raned the thrd most performng captal marets n Afrca for year 2013 (at 48.6 percent rate of return) behnd Ghana Stoc Exchange (49.3 percent) and Malaw Stoc Exchange (71 percent). 5.2 TESTING CAPM ASSUMPTIONS The CAPM test was conducted usng the SPSS statstcal pacage whch has capacty to handle large quanttes of data and hence best suted for the NSE data of over 25,000 prces. To carry out the study, 47 shares were dentfed as beng consstently traded on the NSE from years The CAPM assumpton for normalty was tested on SPSS and the followng results obtaned: Normalty assumpton All weely returns for the stocs, maret and rs-free rates were evaluated usng the Shapro-Wl test but found to be sgnfcant at a p-value of The sgnfcance values were all at 0.00, hence we rejected the null hypothess and assumed that the data dd not follow a normal dstrbuton. 97

103 5.3 TESTING CAPM The dea behnd CAPM s that nvestors need to be compensated n two ways: tme value of money and rs. The tme value of money s represented by the rs-free (R f ) rate and compensates the nvestors for placng money n any nvestment over a perod of tme. The other half of the formula represents rs and calculates the amount of compensaton the nvestor requres for tang on addtonal rs. Ths s calculated by tang a rs measure (beta) that compares the returns of the asset to the maret over a perod of tme and to the maret premum (R m -R f ). If a company has a beta of 3, then t s sad to be three tmes more rsy than the overall maret. Therefore, the CAPM s an equaton that ndcates the requred rate of return one should demand for holdng a rsy asset as part of a dversfed portfolo, based on the asset s beta. The hgher the beta the hgher the expected rate of return as the nvestor s rewarded for tang on rs that cannot be dversfed away. If CAPM ndcates a rate of return that s dfferent from that predcted usng other crtera (such as P/E ratos or stoc charts), then one should, n theory, buy or sell the asset dependng on the relatonshp of the dfferent estmates. The model argues that there s only one sngle source of systematc rs ts beta. However, the nvestor has the ablty to dversfy rs by collectng uncorrelated assets (mplyng a beta of zero) whch lowers the expected rate of return due to the reduced rs. The captal asset prcng model (CAPM) s expressed as follows: E(R ) = R f + β [E(R m ) R f ] where: E(R ) s the expected return on the captal asset R f s the rs-free rate of nterest such as nterest arsng from government bonds β (the beta) s the senstvty of the expected excess asset returns to the expected excess maret returns, E(R m ) s the expected return of the maret [E(R m ) R f ] s sometmes nown as the maret premum (the dfference between the expected maret rate of return and the rs-free rate of return). Expressed n terms of ther rs premum, we fnd that:- E(R ) R f = β [E(R m ) R f ] Whch states that the ndvdual rs premum equals the maret premum multpled by β. Combnng the varables we get: 98

104 E(Z ) = B E(Z m ) Where: E(Z ) = E(R ) R f E(Z m ) = E(R m ) R f Snce E(R f ) = R f Removng expectatons by the effcent maret hypothess: Z = B Z m The regresson equaton wll be: Z t = α + β Z mt + ε t The sample data of ths study are gathered through NSE Stocs, whch selected 47 stocs to be the samples of the analyss n the NSE. The tme perod s the weely data set from January 4, 2004 to December 31, In the examnaton part of ths study and to nvestgate CAPM and ts factors, a two-step regresson wll be appled to determne the varables. In the frst step, a calculaton wll be conducted to obtan the beta of the 47 stocs of the sample. Ths step s a tme seres analyss, whch conducted a regresson between the return of the stocs and the return of the maret ndex n excess of rsless rate of return order to obtan the beta. The second step s a cross-secton regresson, whch targets every stoc n the sample. In ths step, the regresson wll be conducted between the beta obtaned n the frst step and the average return of the maret, n order to obtan the Securty Maret Lne of the studed maret. Ths study attempts to dentfy whether the result of the second step s the same as the predcted result of CAPM. A total of 47 ordnary least squares regressons of the CAPM equaton were performed usng the 10- year data for the NSE securtes, the rs-free rate and the maret return (NSE 20-Share) to determne the respectve betas of each of the stocs. 99

105 The excess asset returns over the rs-free rate were computed for each stoc over the testng perod on one hand, and the maret premum values for each wee derved on the other. The rs and maret premums were regressed and the followng were the results: Stocs R squared Constant (α ) Sgnf. Constant Estmated Beta (β ) Sgnfcance Level Eaagads Ltd Ord 1.25 AIMS Kauz Ltd Ord Rea Vpngo Plantatons Ltd Ord Sasn Ltd Ord Kapchorua Tea Co. Ltd Ord Ord 5.00 AIMS Wllamson Tea Kenya Ltd Ord 5.00 AIMS The Lmuru Tea Co. Ltd Ord AIMS Car & General (K) Ltd Ord CMC Holdngs Ltd Ord Marshalls (E.A.) Ltd Ord Sameer Afrca Ltd Ord Barclays Ban of Kenya Ltd Ord CFC Stanbc of Kenya Holdngs Ltd ord Damond Trust Ban Kenya Ltd Ord Housng Fnance Co.Kenya Ltd Ord Kenya Commercal Ban Ltd Ord Natonal Ban of Kenya Ltd Ord NIC Ban Ltd Ord Standard Chartered Ban Kenya Ltd Ord Express Kenya Ltd Ord 5.00 AIMS Hutchngs Bemer Ltd Ord Kenya Arways Ltd Ord Naton Meda Group Ltd Ord Standard Group Ltd Ord TPS Eastern Afrca Ltd Ord Uchum Supermaret Ltd Ord Ath Rver Mnng Ord Bambur Cement Ltd Ord Crown Pants Kenya Ltd Ord E.A.Cables Ltd Ord E.A.Portland Cement Co. Ltd Ord KenolKobl Ltd Ord Kenya Power & Lghtng Co Ltd Ord Total Kenya Ltd Ord Jublee Holdngs Ltd Ord Pan Afrca Insurance Holdngs Ltd Ord Centum Investment Co Ltd Ord

106 Cty Trust Ltd Ord 5.00 AIMS Olympa Captal Holdngs Ltd Ord A.Baumann & Co Ltd Ord 5.00 AIMS B.O.C Kenya Ltd Ord Brtsh Amercan Tobacco Kenya Ltd Ord Carbacd Investments Ltd Ord East Afrcan Breweres Ltd Ord Kenya Orchards Ltd Ord 5.00 AIMS Mumas Sugar Co. Ltd Ord Unga Group Ltd Ord The Betas ranged from to The estmated betas were all sgnfcant wth p-values < than The alphas (constant) ranged from to wth 64% of the values beng sgnfcant: H o : α = 0 H 1 : α 0 Hence we conclude that 64% of the alphas were sgnfcantly dfferent from zero. CAPM requres that the ntercepts of the estmated regresson lnes should be 0. Further, the R-squared ranged between 1.90% %, ndcatng a wea relatonshp between the rs premum (response varable) and the regresson (ftted) lne. 101

107 5.4 THE SECURITY MARKET LINE The next step wll be to estmate the Securty Maret Lne (SML) by regressng the average excess returns over the rs-free rate n to ther betas estmated above. The Securty Maret Lne s therefore: E(R ) = R f + β [E(R m ) R f ] Re-arrangng to calculate the rs premum: E(R ) - R f = β [E(R m ) R f ] E(Z ) = B E(Z m ) Usng estmated betas ths tme as the dependent varables we have: Where: E(Z ) = ɣ 0 + ɣ 1 β + ε ɣ 1 = maret premum (slope) If CAPM holds, then the y-ntercept (constant) should be ɣ 0 = 0. A graph of the average rs premum and estmated beta revealed a postve correlaton: 102

108 The regresson coeffcents were as follows: Coeffcents a Model Unstandardzed Coeffcents Standardzed Coeffcents B Std. Error Beta t Sg. 1 (Constant) Estmated Beta a. Dependent Varable: Average Excess returns The y-ntercept ɣ 0 was wth a maret premum (slope) ɣ 1 of Despte the constant beng negatve, the value was sgnfcant at p-value < 0.05 hence we reject the null hypothess that: H o : ɣ 0 = 0 H 1 : ɣ 0 0 Subsequently, we shall perform a further non-lnearty test. As predcted by CAPM, assets returns are lnearly related to ther betas. We shall test ths hypothess by addng an term addtonal term to the prevous equaton:- E(Z ) = ɣ 0 + ɣ 1 β + ɣ 2 β 2 + ε Term β 2 s smply the second power of our estmated beta. If the model holds and the lnear relatonshp between beta and returns s strong, then addng the beta square should not nfluence the prevous results. Hence ɣ 2 = 0. Addtonally, a further chec can be performed for the non-systematc rs. The CAPM argues that the only rs that matters to nvestors s measured by beta. Any other factors do not matter. To ths end, we shall add on another parameter to the regresson equaton: E(Z ) = ɣ 0 + ɣ 1 β + ɣ 2 β 2 + ɣ 3 RV + ε The RV term stands for the varance of resduals of an asset. We obtan the value from the frst regresson used to estmate betas. Resdual varance wll therefore be the measure of rs not accounted for by beta. If CAPM holds then: ɣ 3 = 0 103

109 After performng the multple regresson above, the followng results emerged: Coeffcents a Model Unstandardzed Coeffcents Standardzed B Std. Error Beta Coeffcents t Sg. (Constant) Estmated Beta Beta Squared Varance of resduals a. Dependent Varable: Average Excess returns From the SPSS results, t was determned that the value of ɣ 2 = 0 wth a p-value of Hence we have no reason to reject the null hypothess and conclude that the value s zero. Further, ɣ 3 = 0 wth a sgnfcant p-value of less than Therefore, we reject the null hypothess and conclude that the value s sgnfcantly dfferent from zero. 104

110 5.5 PREDICTING EX ANTE BETA BASED ON CAPM We also set to fnd out f CAPM s a relable predctor of ex ante beta. Usng the formula:- E(R ) = R f + β [E(R m ) R f ] E(R ) - R f = β [E(R m ) R f ] And mang beta the subject: β = [E(R m ) R f ] / E(R ) - R f We utlzed a statc/equlbrum forecast of stoc and maret returns and assumed the stocs stay constant over tme, based on ther long-run hstorcal average for the last 10 years. The rs-free rate of return s assumed to be the 1 year Treasury bond rate at the begnnng of the year e. 1 st January The ex ante beta results for year 2014 were as follows: Stocs Ex-Post Returns ( ) Annualsed Hstorcal Returns Rs- Free 1 yr Treasury Bond R-Rf Ex Ante Beta, B() Eaagads Ltd Ord 1.25 AIM Kauz Ord Kapchorua Tea Co. Ltd Ord Ord 5.00 AIM Lmuru Tea Co. Ltd Ord AIM Car & General (K) Ltd Ord CMC Holdngs Ltd Ord Marshalls (E.A.) Ltd Ord Sameer Afrca Ltd Ord Barclays Ban Ltd Ord CFC Stanbc Holdngs Ltd ord Damond Trust Ban Kenya Ltd Ord Housng Fnance Co Ltd Ord Kenya Commercal Ban Ltd Ord Natonal Ban of Kenya Ltd Ord NIC Ban Ltd Ord Standard Chartered Ban Ltd Ord Express Ltd Ord 5.00 AIM Hutchngs Bemer Ltd Ord Kenya Arways Ltd Ord Naton Meda Group Ord Standard Group Ltd Ord TPS Eastern Afrca (Serena) Ltd Ord Uchum Supermaret Ltd Ord

111 ARM Cement Ltd Ord Bambur Cement Ltd Ord Crown Berger Ltd Ord E.A.Cables Ltd Ord E.A.Portland Cement Ltd Ord KenolKobl Ltd Ord Kenya Power & Lghtng Co Ltd Ord Total Kenya Ltd Ord Jublee Holdngs Ltd Ord Pan Afrca Insurance Holdngs Ltd Ord Centum Investment Co Ltd Ord Olympa Captal Holdngs ltd Ord A.Baumann & Co Ltd Ord 5.00 AIM B.O.C Kenya Ltd Ord Brtsh Amercan Tobacco Kenya Ltd Ord Carbacd Investments Ltd Ord East Afrcan Breweres Ltd Ord Kenya Orchards Ltd Ord 5.00 AIM Mumas Sugar Co. Ltd Ord Unga Group Ltd Ord NSE 20-SHARE INDEX - (1966 = 100 )

112 After computng the ex-ante betas, we collected the NSE data for the respectve stocs over the year as well as the correspondng NSE 20-share ndexes. The objectve was to calculate ex post betas and determne f the beta forecasts matched wth the actual stoc betas n year Tests of normalty revealed the followng: Tests of Normalty Kolmogorov-Smrnov a Shapro-Wl Statstc df Sg. Statstc df Sg. KapchoruaTeaCoLtdOrdOrd500AIM LmuruTeaCoLtdOrd2000AIM CarampGeneralKLtdOrd CMCHoldngsLtdOrd MarshallsEALtdOrd SameerAfrcaLtdOrd * BarclaysBanLtdOrd * CFCStanbcHoldngsLtdord DamondTrustBanKenyaLtdOrd * HousngFnanceCoLtdOrd KenyaCommercalBanLtdOrd * NatonalBanofKenyaLtdOrd NICBanLtdOrd StandardCharteredBanLtdOrd * ExpressLtdOrd500AIM HutchngsBemerLtdOrd KenyaArwaysLtdOrd * NatonMedaGroupOrd StandardGroupLtdOrd * TPSEasternAfrcaSerenaLtdOrd * UchumSupermaretLtdOrd ARMCementLtdOrd * BamburCementLtdOrd CrownBergerLtdOrd EACablesLtdOrd * EAPortlandCementLtdOrd KenolKoblLtdOrd * KenyaPowerampLghtngCoLtdOrd TotalKenyaLtdOrd JubleeHoldngsLtdOrd PanAfrcaInsuranceHoldngsLtdOrd CentumInvestmentCoLtdOrd * OlympaCaptalHoldngsltdOrd ABaumannampCoLtdOrd500AIM BOCKenyaLtdOrd BrtshAmercanTobaccoKenyaLtdOrd CarbacdInvestmentsLtdOrd * EastAfrcanBreweresLtdOrd KenyaOrchardsLtdOrd500AIM MumasSugarCoLtdOrd UngaGroupLtdOrd NSEALLSHAREINDEXNASI01stJan * NSE20SHAREINDEX * Rsfreerateofnterest91dayTblls EaagadsLtdOrd125AIM KauzOrd * *. Ths s a lower bound of the true sgnfcance. a. Lllefors Sgnfcance Correcton Usng the Shapro-Wl test results, about 16 of the 43 stocs le above 5% level of sgnfcance whch mples that only 37.2% of the data s normal. Ths can be explaned by the low amount of data used n the study as only one year perod of 52 wees was consdered. 107

113 The weely excess stoc and maret returns were regressed and the followng betas were computed: Partculars Ex Ante Beta, B() Ex-Post Beta (2014) Absolute Error Mean Absolute Percent Error Eaagads Ltd Ord 1.25 AIM % Kauz Ord % Kapchorua Tea Co. Ltd Ord Ord 5.00 AIM % Lmuru Tea Co. Ltd Ord AIM % Car & General (K) Ltd Ord % CMC Holdngs Ltd Ord 0.50s % Marshalls (E.A.) Ltd Ord % Sameer Afrca Ltd Ord % Barclays Ban Ltd Ord % CFC Stanbc Holdngs Ltd ord % Damond Trust Ban Kenya Ltd Ord % Housng Fnance Co Ltd Ord % Kenya Commercal Ban Ltd Ord % Natonal Ban of Kenya Ltd Ord % NIC Ban Ltd Ord % Standard Chartered Ban Ltd Ord % Express Ltd Ord 5.00 AIM % Hutchngs Bemer Ltd Ord % Kenya Arways Ltd Ord % Naton Meda Group Ord % Standard Group Ltd Ord % TPS Eastern Afrca (Serena) Ltd Ord % Uchum Supermaret Ltd Ord % ARM Cement Ltd Ord % Bambur Cement Ltd Ord % Crown Berger Ltd Ord % E.A.Cables Ltd Ord % E.A.Portland Cement Ltd Ord % KenolKobl Ltd Ord % Kenya Power & Lghtng Co Ltd Ord % Total Kenya Ltd Ord % Jublee Holdngs Ltd Ord % Pan Afrca Insurance Holdngs Ltd Ord % Centum Investment Co Ltd Ord % Olympa Captal Holdngs ltd Ord % A.Baumann & Co Ltd Ord 5.00 AIM % B.O.C Kenya Ltd Ord % 108

114 Brtsh Amercan Tobacco Kenya Ltd Ord % Carbacd Investments Ltd Ord % East Afrcan Breweres Ltd Ord % Kenya Orchards Ltd Ord 5.00 AIM % Mumas Sugar Co. Ltd Ord % Unga Group Ltd Ord % AVERAGE FORECAST ERROR % Based on these results we computed error measurement statstcs to measure the forecast error and determne whether a sgnfcant dfference exsts between the ex ante and ex post betas. The Mean Absolute Devaton (MAD) revealed an error of whereas the Mean Absolute Percent Error (MAPE) shows a massve 1392%. 109

115 5.6 TESTING THE ZERO-BETA CAPM A frst-pass regresson was conducted on the 47 stocs usng raw returns nstead of the excess returns tested n the standard CAPM. Beta estmates were obtaned wthout deductng rs-free nterest rate. R R t mt t Stocs ( ) R squared Constant (α ) Sgnfcance Level Estmated Beta (B) Sgnfcance Level Eaagads Ltd Ord 1.25 AIMS Kauz Ltd Ord Rea Vpngo Plantatons Ltd Ord Sasn Ltd Ord Kapchorua Tea Co. Ltd Ord Ord 5.00 AIMS Wllamson Tea Kenya Ltd Ord 5.00 AIMS The Lmuru Tea Co. Ltd Ord AIMS Car & General (K) Ltd Ord CMC Holdngs Ltd Ord Marshalls (E.A.) Ltd Ord Sameer Afrca Ltd Ord Barclays Ban of Kenya Ltd Ord CFC Stanbc of Kenya Holdngs Ltd ord Damond Trust Ban Kenya Ltd Ord Housng Fnance Co.Kenya Ltd Ord Kenya Commercal Ban Ltd Ord Natonal Ban of Kenya Ltd Ord NIC Ban Ltd Ord Standard Chartered Ban Kenya Ltd Ord Express Kenya Ltd Ord 5.00 AIMS Hutchngs Bemer Ltd Ord Kenya Arways Ltd Ord Naton Meda Group Ltd Ord Standard Group Ltd Ord TPS Eastern Afrca Ltd Ord Uchum Supermaret Ltd Ord Ath Rver Mnng Ord Bambur Cement Ltd Ord Crown Pants Kenya Ltd Ord E.A.Cables Ltd Ord E.A.Portland Cement Co. Ltd Ord KenolKobl Ltd Ord Kenya Power & Lghtng Co Ltd Ord Total Kenya Ltd Ord

116 Jublee Holdngs Ltd Ord Pan Afrca Insurance Holdngs Ltd Ord Centum Investment Co Ltd Ord Cty Trust Ltd Ord 5.00 AIMS Olympa Captal Holdngs Ltd Ord A.Baumann & Co Ltd Ord 5.00 AIMS B.O.C Kenya Ltd Ord Brtsh Amercan Tobacco Kenya Ltd Ord Carbacd Investments Ltd Ord East Afrcan Breweres Ltd Ord Kenya Orchards Ltd Ord 5.00 AIMS Mumas Sugar Co. Ltd Ord Unga Group Ltd Ord The stocs were then arranged accordng to beta sze and portfolos of 8 stocs each formed from the bggest to the smallest to mnmze measurement errors. The second-pass regresson establshed wth the results of the model s: z p 0 1 p p 111