FINANCIAL OPTIMIZATION

Transcription

1 FINANCIAL OPTIMIZATION A Fundamentally Better Way to Invest Ka Hm Thomas Chu Teddy Yp A thess submtted n partal fulfllment of the requrements for the degree of BACHELOR OF APPLIED SCIENCE Supervsor: R. Kwon Department of Mechancal and Industral Engneerng Unversty of Toronto March 26 th, 2009

2 Abstract The objectve of ths thess s to evaluate the performances and volatltes of the U.S. fnancal market compare to portfolos bult usng the Fundamental Indexng method. An ndex s a statstcal measure of change n an economy or a securtes market. The S&P 500, a collecton of 500 large cap common stocks traded n the Unted States, wll be used to represent the U.S. fnancal market. Portfolos wll be constructed based on the Fundamental Index approach developed by Robert D. Arnott. The Fundamental Index s the concept of weghtng companes from an economcs-centrc pont of vew, rather than a tradtonal market-centrc pont of vew. The weghtng s based on 4 metrcs book value, proft, dvdends and cash flows. Four portfolos wll be constructed base on ths concept, a pure fundamental portfolo (FI-Pure), a equally weghted portfolo (FI- Equally Weghted), a portfolo based solely on maxmzng the expected return (FI-Max Return) and a portfolo ncorporatng the Modern Portfolo Theory developed by Harry Markowtz (FI-Mnmum-Varance). Ths mathematcal model mnmzes the effects of rsk (varance) on expected nvestment portfolo returns.

3 Acknowledgements We would lke to express our grattude s to our supervsor Dr. Roy Kwon for hs expertse n the feld of Fnancal Optmzaton and hs gudance for the completon of ths thess. We would also lke to thank the Lbrarans and Lab Assstants at the Rotman Fnance Lbrary for provdng us wth assstance n gatherng exquste fnancal nformaton for the constructon of our optmzaton model.

4 Table of Contents Acknowledgements... Lst of Symbols... v Lst of Fgures... v Lst of Tables... v Lst of Equatons... v 1. Motvaton (Teddy Yp) Lterature Revew (Teddy Yp) Captalzaton Weghted Indces Fundamental Indces The Four Fundamental Factors Modern Portfolo Theory and Dversfcaton Methodology (Thomas Chu) The S&P 500 (the U.S. Equtes Market) The Fundamental Index Methodology Fundamental Index Formulaton Modern Portfolo Theory Performance Evaluaton Procedures (Thomas Chu) Gatherng Data Buldng the Fundamental Index Calculate F Buldng the Mean-Varance Model Evaluatng portfolo(s) performance... 31

5 5. Results (Teddy Yp) Effcent Fronter of the Mean-Varance Model Economc Sector Dstrbuton Sharpe Rato Turnover Analyss for Fundamental Strateges Delta Rate of Return Actual Cumulatve Return Standard Devaton Concluson (Thomas Chu) References Appendx A Appendx B... 53

6 v Lst of Symbols n x = 1 : Mathematcal notaton representng Summaton of many smlar terms E (x) : Mathematcal notaton representng the Expected Value of X, a random varable x : Mathematcal notaton representng the Expected Value of X, a random varable 2 σ : Varance of a random varable σ : Standard Devaton of a random varable σ x,y : Covarance between two random varables, x and y n = 1 x : Mathematcal notaton representng Multplcaton of many smlar terms S&P 500: Standard & Poor s portfolo of 500 U.S. large cap stocks CSCO: Stock Exchange Symbol for Csco Systems MSFT: Stock Exchange Symbol for Mcrosoft Corporaton IBM: Stock Exchange Symbol for Internatonal Busness Machne (IBM) Corporaton MS: Stock Exchange Symbol for Morgan Stanley DELL: Stock Exchange Symbol for Dell Inc

7 v Lst of Fgures Fgure 1: Portfolos Breakdown Fgure 2: Mnmum-Varance Set Fgure 3: Effcent Fronter Fgure 4: 2001 Effcent Fronter Fgure 5: Economc Sector Breakdown Fgure 6: Sharpe Rato s of the dfferent portfolos Fgure 7: Fundamental Index Turnover Analyss Fgure 8: Actual Return VS Expected Return Fgure 9: Rate of Return Fgure 10: Fundamental Index vs S&P Fgure 11: Cumulatve Rate of Return Fgure 12: Standard Devaton of all strateges... 44

8 v Lst of Tables Table 1: Mnmum-Varance Ponts Table 2: Dstrbutons of other ndces for

9 v Lst of Equatons Eq 1: Fundamental Factors Equaton Eq 2: Fundamental Weghtng Equaton Eq 3 Fundamental Weghtng Equaton (Companes wthout Dvdends) Eq 4: Composte Factor Eq 5 Total Return Eq 6: Rate of Return Eq 7: Amount Receved Eq 8: Investment Fracton Eq 9: Total Weghtng Eq 10: Total Portfolo Return Eq 11: Portfolo Rate of Return Eq 12: Expected Return for asset j Eq 13: Varance for asset j Eq 14: Covarance for asset j = 1, Eq 15: Portfolo Mean Return Eq 16: Portfolo Expected Return Eq 17: Varance of Portfolo Return Eq 18: Mean-Varance Optmzaton Problem Eq 19: Actual Portfolo Rate of Return Eq 20: Smplfed Actual Portfolo Rate of Return Eq 21: Amount Receved on Portfolo Eq 22: Total Cumulatve Return Factor... 24

10 v Eq 23: Amount Receved on Portfolo m year from Eq 24: Sharpe Rato... 25

11 1 1. Motvaton (Teddy Yp) A stock market, or equty market, s a prvate or publc market for the tradng of company stock and dervatves at an agreed prce. (Brealey et al., 182) The movements of the prces n a market are captured n prce ndces called stock market ndces. Tradtonally, ndces are weghted based on market captalzaton. Market captalzaton s a measurement of corporate or economc sze equal to the share prce tmes the number of shares outstandng of a publc company. (Brealey et al., 186) An example of a well known market captalzaton weghted stock s the S&P 500, whch was frst publshed n 1957 to represent the 500 large cap common stocks actvely traded n the Unted States. (Brealey et al., 186) However, t has been proven that the tradtonal captalzaton weghted way of ndexng has ts flaws. Captalzaton weghted ndces wll tend to subject ts funds to market turbulences. An example of such turbulences s a stock bubble, where the prces of stocks rse and become overvalued compared to ther true value. A recent event of a stock bubble was the "tech bubble", whch covered roughly from 1995 to 2002 when the stock market saw ther value ncrease rapdly from growth n the new Internet sector and related felds. ( Tech Bubble ) Ths perod was marked by the foundng of a group of new Internet based companes commonly referred to as dot-coms. (Brealey et al., 198) When the bubble crashed, nvestors that have nvested n the dot-coms lost a lot of money, ncludng those who nvested n Csco Systems. Csco Systems was the largest stock n the world based on market captalzaton at the peak of the tech bubble, t was valued at nearly $600 bllon. From 1997 to 2000, Csco s P/E (prce over earnng rato) rose from 30 to nearly 200 as nvestor s

12 2 expectatons rose even faster than Csco s fast-growng operatng results. However, Csco Systems only reported sales valued at $20 bllon, 12 month earnngs that were less than $3 bllon, cumulatve profts that were under $8 bllon, and never pad dvdends to ts shareholder. When the market crashed, of the $600 bllon, $500 bllon was gone n the subsequent 30 months. (Arnott, Hsu, and West, 9) The bubble had the same effects north of the boarder as well. Nortel saw ts prce skyrocket despte the company s postng net losses n 1998 and 1999, by whch tme ths sngle company consttuted 28% of the total value of the Canadan stock market. (Arnott, Hsu, and West, 9) Index funds are supposed to be the ultmate dversfcaton choce. An ndex mutual funds or exchange-traded funds (ETFs) whose portfolos mrror the components of the ndex should theoretcally be the optmal method to elmnate unque stock rsks. However, market captalzaton weghted ndces put more of the nvestor s money n growth (or hgh-p/e stocks) and less money n value (low-p/e stocks) (Arnott, Hsu, and West, 80). In the frst two years after the tech bubble burst, the market captalzaton weghted ndex funds (S&P 500) were down, whle the average stock was up. Ths was because the ndexes had loaded up on hgh prce growth companes and when the market crashed, the funds went wth t. (Arnott, Hsu, and West, 81) Ths thess wll show that there s a better way to construct these ndex funds such that nvestors can mnmze exposures to rsks assocated wth over-valued stocks.

13 3 2. Lterature Revew (Teddy Yp) 2.1 Captalzaton Weghted Indces Most companes lsted n the S&P 500 durng the rse of the tech bubble never had earnng n ther entre hstory. These companes were at exceptonal valuaton multples. Those multples factored nto the market captalzaton whch determned the weghts n the ndexes. (Brealey et al., 186) Therefore, f a select few stocks rapdly soar n prces, they wll compose an ncreasng porton of the ndex, much lke Csco Systems. By nvestng n market captalzaton funds, the resultng portfolo may have less dversfcaton than the broad economy, especally durng a bubble where prces move together. Investors make a common error by referrng to market captalzaton ndces for nvestment decson makng and the tendency to chase wnners. (Arnott, Hsu, and West, 35) Captalzaton-weghted ndex funds chase performance by allocatng more of an nvestor s money to recent wnners and less to recent losers. (Arnott, Hsu, and West, 35) A stock that doubles n prce gets double the weght solely because t has doubled n prce. In the recent events of the tech bubble n early 2000, some captalzaton weghtng ndex share prces were far above ther eventual ntrnsc value, whch lead to an naccurate hgh captalzaton weghtng. (Arnott, Hsu, and West, 36) Therefore, the tradtonal methods of captalzaton-weghtng ndces overweght overvalued stocks and underweght undervalued stocks. Investors hope to beat the market by buyng stocks when they are on ther way up and by sellng them on ther way down. When nvestors try to acheve superor return by

14 4 spottng and explotng patterns n stock prces, they are referred to as techncal analysts. (Brealey et al., 183) Unfortunately, a large prce rse n one perod may be followed by a further rse n the next perod, but t s just as lkely to be followed by a fall. (Brealey et al., 184) Statstcans who have studed stock prce movements know that you won t get rch by lookng for consstent patterns n prce changes. Prces wander randomly, vrtually equally lkely to offer a hgh or low return on any partcular day, regardless of what has occurred on prevous days. (Brealey et al., 184) 2.2 Fundamental Indces There are other approaches that nvestors can take to make market judgments. Ths s through lookng at other types of nformaton of companes. They can do ths through studyng a frm s busness prospects by lookng at the fnancal and trade press, the company s fnancal accounts, the presdent s annual statements, and other tems of news. Investors who analyze company nformaton are called fundamental analysts. (Brealey et al., 186) Ths approach was adopted by Robert Arnott who founded Research Afflates and made an effort to dscover ways to mnmze rsks for nvestors whle outperformng the market. Robert D. Arnott who s also the author of the book The Fundamental Index, the book n whch ths thess s based on. Research Afflates started ther research after the crash of the tech bubble. In order to avod stock bubbles, methodologes developed by Research Afflates show a dfferent way of weghng ndces. They developed a Fundamental Index whch ncorporates other metrcs to come up wth the weghtng of the ndex. Usng a multple metrcs was beleved to be able to avod bubbles. Over the past 50 years, only three fveyear perods n whch the S&P 500 has produced a negatve return have occurred. There

15 5 has been no 10-year perod for the past 50 years that has resulted n losses n the S&P 500. The worst 10-year return was a 12.3 percent gan from Furthermore, n the last 20 years, there have been only two 10-year perods wth gans of fewer than 200% ( and ). (Arnott, Hsu, and West, 11) Therefore makng the rght choces n nvestng would result n some postve gans n the long run. Research Afflates had a strong belef that constructng an ndex by company sze would prove to be a better way than of weghtng companes by ther market captalzaton n market where prcng errors exst (an neffcent market). Companes are already judged n sze based on measurements such as annual sales, profts, net assets or the number of employees whch should also be factors n weghtng. Research Afflates frst looked at the Fortune 500 for a lst of the largest companes. The Fortune 500 s determned by annual sales, not by market captalzaton. As a result of ths, they dscovered that the sales weghted approach outperformed the S&P 500 by more than 250 bass ponts annually over a tme frame of 30 years. (Arnott, Hsu, and West, 17) They then decded to look nto other factors wth depth. They concluded that these 4 metrcs should come nto play when ndexng; sales, dvdends, cash flow and book value. Incorporatng these metrcs for ndexng resulted n the Fundamental Index. 2.3 The Four Fundamental Factors Durng Research Afflates journey to fnd a better way to ndex, they encountered several key fndngs. Frstly, they notced that all of the Fundamental Index portfolos performed smlarly, whle captalzaton weghtng dd not. Each one broke the lnk between portfolo weght and prcng error. Secondly, reweghtng an exstng capweghted ndex wll only result a fracton of the potental beneft than by startng from

16 6 scratch and creatng new Fundamental Index portfolo. Lastly multple metrcs of company sze wll result n a broader and more representatve nvestment portfolo than usng a sngle metrc. (Arnott, Hsu, and West, 87) The 4 metrcs approach resulted n returns that are hgher than the average of the sngle-metrc approaches. Also, rsks of a Fundamental Index are less than the average of the ndvdual metrc approaches. Fundamental Index portfolos are constructed by equally weghtng the 4 ndvdual Fundamental Index weghts to create the RAFI (Research Afflates Fundamental Index) weght for each company. The 4 metrcs have to be factored n together n order to construct an effcent ndex. An ndex based only on dvdend payment alone wll overweght sectors of the economy that pay large dvdends (.e., utlty companes & fnancal servces) and exclude segments that do not pay cash dvdends (.e., IT and growth companes) (Arnott, Hsu, and West, 80). Ths wll tend to perform well n bear markets, and underperform n most bull markets. Also, usng just the metrc of dvdends wll have excessve exposure to banks and utlty companes. A sales based metrc overweghs stocks that have hgh sales and low margns (.e., tradng companes). Also, a sale based ndex concentrates on companes that have thnner margns and hgher volatlty. (Arnott, Hsu, and West, 85) They tend to struggle n bear market whle performng better than the other FI Strateges n bull market. A proft based ndex may lead to over- or under-exposure to companes wth hgh cyclcal ncomes because profts are lnked to the economc cycle experenced by the company. Usng only cash flow as a measure of profts may gve too much weght to

17 7 mature and slow growng companes or cash cow whle underweghtng young, fast growng companes. (Arnott, Hsu, and West, 82) A book value based ndex may lead to overexposure to companes wth aggressve accountng practces or underexpose companes wth conservatves accountng practces. Companes that are captal ntensve wth large book assets may be over-weghted n relaton to companes that are more relant on human captal or ntellectual captal (.e., bank vs. IT) (Arnott, Hsu, and West, 82) It s mportant to weght ndces based on these 4 metrcs n order to portray the effcent market. Based on these postve results, ths thess wll approach Research Afflates methodology smlarly by usng representatve measurements for each fundamental factor. The 4 metrcs used are Cash & Equvalents, Net Income, Total Revenue and Dvdend per Share. By constructng a portfolo through assessng these fundamental factors, t should show less volatlty n prces, especally n an event lke the tech bubble. However, rankng companes based on ther fundamental factors may not be suffcent for nvestors. Although a Fundamental Index may select stocks based on true company sze, t does not take nto consderaton the volatlty or rskness of the stocks. Investment rsk depends on the dsperson or spread of possble outcomes. The standard measures are varance and standard devaton. More varable returns mply greater nvestment rsks. (Brealey et al., 321) Ths also depends on the nvestor s rsk averseness. Rsk averson s the reluctance of a person to accept a bargan wth an uncertan payoff rather than another bargan wth more certanty, but possbly lower returns. (Brealey, Rchard A, 321) For example, a rsk-averse nvestor mght choose to put hs or her money nto bonds but wll have a low and guaranteed nterest rate, rather

18 8 than nto a stock that s lkely to have hgh returns and has a chance of becomng worthless. Ths thess assumes that all nvestors are rsk neutral, he or she wll prefer the asset wth less rsk for the same expected return. There are mathematcal models that can be solved to select specfc stocks n order to mnmze rsks. 2.4 Modern Portfolo Theory and Dversfcaton The Modern Portfolo Theory (MPT) proposes how to dversfy and optmze portfolos. The basc concepts of the theory are Markowtz dversfcaton, the effcent fronter, captal asset prcng model, the Alpha and Beta coeffcents, the Captal Market Lne and the Securtes Market Lne. (Brealey et al.) Harry Markowtz won the Nobel Memoral Prze n Economc Scences n 1990 for hs theory on appled mathematcs to model and analyzes the stock market. Dversfcaton s the dea that an nvestor can reduce portfolo rsk by nvestng n companes n whch are not postvely correlated. Investors can reduce ther exposure to ndvdual asset rsk by holdng a dversfed portfolo of assets. In other words, t s avodng puttng all our eggs n one basket. Ths thess wll model the Markowtz mean and varance model n dversfyng the portfolo. An effcent portfolo s expected to yeld the hghest return for a gven level of rsk or lowest rsk for a gven level of return. (Brealey et al. 320) Markowtz emphaszed that qualty of a portfolo wll be dfferent from the qualty of ndvdual assets wthn t. Thus, the combned rsk of two assets taken separately s not the same rsk of two assets together. The qualfcaton of rsk and the need for optmzaton of return wth lowest rsk are the contrbutons of Markowtz. The Modern Portfolo Theory emphaszes the trade off between rsk and return. (Brealey et al. 326) Investors know there s a rsk-return

19 9 tradeoff, n order to obtan a hgher return on nvestment; the nvestor must be wllng to take on more rsk. There s however one knd of nvestment that s consdered to be rskfree, the U.S. Treasury blls, notes and bonds; these nvestments yeld a rsk-free rate of return. There may be multple portfolos that have the same rsk. Smlarly, there may be multple portfolos that have the same expected return. Modern portfolo theory assumes that for a specfed amount of rsk, a ratonal nvestor would choose the portfolo wth the greatest return, and for a specfed level of return, the ratonal nvestor would choose the portfolo havng the same lowest rsk. A portfolo s sad to be effcent f there s no portfolo havng the same standard devaton wth a greater expected return or there s no portfolo havng the same expected return wth a lesser standard devaton. (Brealey et al. 325) Portfolo dversfcaton s an effectve tool because prces of dfferent stocks do not move exactly together. Statstcans make the same pont when they say that stock prce changes are less then perfectly correlated. Dversfcaton works best when the returns are negatvely correlated. When one busness does well, the other does badly. Unfortunately n practce, stocks that are negatvely correlated are extremely rare. (Brealey et al., 321) The purpose of ths thess s to analyze the tradtonal market captalzaton model (S&P 500), a fundamentally bult ndex, and three portfolos usng a mxed strategy of a mean varance model appled to the generated Fundamental Index. The resultng set of portfolos wll be compared based on ther return and volatlty (standard devaton),

20 10 whch wll lead to the concluson of the most effcent way of nvestng. Fgure 1 llustrates the dfferent portfolo strateges mplemented n ths thess. T-Blls (Rsk-Free Return) S&P 500 Fundamental Indexng Pure Fundamental Mnmze Varance Maxmze Return Equally Weghted Fgure 1: Portfolos Breakdown

21 11 3. Methodology (Thomas Chu) 3.1 The S&P 500 (the U.S. Equtes Market) The S&P 500 s desgned to reflect the U.S. equty markets, whch s the most wdely accepted leadng ndcator of the market. It ncludes 500 large cap stocks, whch together represent about 75% of the total U.S. equtes market. Almost all of the stocks ncluded n the ndex are among the 500 Amercan stocks wth the largest market captalzatons. It s calculated usng a market-cap weghted methodology, the weghtng of each stocks s the quotent of the total market captalzaton of the company dvded by the total market captalzaton of all 500 companes. ( S&P Indces ) The ndex s owned and mantaned by Standard & Poor's, a dvson of McGraw- Hll. The component of the S&P 500 s selected by commttee, rather than rule based, such as the Russell Even though ths could potentally lead to some bas to nclude some stock whle excludng others, the S&P 500 s generally accepted as a representaton of the U.S. equty markets, and t wll be used for the same purpose n ths thess. 3.2 The Fundamental Index Methodology When constructng any ndexes, there are three attrbutes to take nto consderatons: 1. An ndex should be a representatve of the nvestment opportunty set. Studes and mathematcal models have shown that mantanng a welldversfed portfolo of 25 to 30 stocks wll yeld the most cost-effectve level of rsk reducton. Investng n more securtes wll stll yeld further dversfcaton

22 12 benefts, albet at a drastcally smaller rate. ( Dversfcaton ) A portfolo of 35 stocks s beleved to be a good ft for the scope of ths thess. 2. An ndex should be replcable because the portfolo manager needs to be able to purchase the stocks n the prescrbed weghts. An ndex of llqud or prvately held securtes would prove dffcult to manage on a day to day bass (Arnott, Hsu, and West 52-53). The components of the portfolo wll be compose of the most common and lqud stock n the two largest stock market n the U.S. the New York Stock Exchange (NYSE) and the NASDAQ. 3. An ndex should have Low Turnover to mantan contnuty. (Arnott, Hsu, and West 53) Followng the Fundamental Index methodology would satsfy the 3 key attrbutes of constructng an ndex. The frst step n creatng a Fundamental Index s to pck a set of smply and wdely avalable non-prced based measure of company sze. (Arnott, Hsu, and West 75) Ths would create a broadly representatve, hgh lqudty, hgh capacty and low turnover ndex. Fundamental Index s to use a reasonable combnaton of fnancal varables, each of whch captures dfferent aspect of company sze. There are many great measurements of sze, the fnancal measures of sze whch resulted n Research Afflates researches were sales/revenues, profts, net assets and dstrbutons to shareholders. (Arnott, Hsu, and West 80) Sale s the amount of money that s brought nto a company by ts busness actvtes. Proft s the money a busness makes after accountng for all the expenses. Net assets are everythng of value that s owned by a person or company; t could also be used to represent the lqudty of a company. Dvdend (dstrbutons to shareholders) s the

23 13 dstrbuton of a porton of a company's earnngs, decded by the board of drectors, to a class of ts shareholders. (Brealey et al. 307) In ths thess, sale s represented by Total Revenue and proft s measured by Net Income. Whle Gross Profts s a company's revenue mnus ts cost of goods sold, Net Income s a better representaton of proft because t also takes nto consderaton other non-producton expenses (.e., operatng expenses, R&D, etc), whch s crucal to many IT and product development companes. A dstrbuton to shareholders s represented by Dvdends per share, and net assets s measure by Cash & Equvalents, the most lqud assets found wthn the asset porton of a company's balance sheet. Ths composte approach would represent a better and robust constructon than any sngle metrc constructed ndex, also they can be easly found through the company s ncome statement and/or balance sheet. The company s sze s then determned by averagng the weghts of the 4 sze metrcs. (Arnott, Hsu, and West 84) However, there are exceptons n the case of zero dvdends, snce not all companes pay dvdends. Half of Corporate Amerca s profts were pad to shareholders n the form of Dvdends, but rapdly expandng enterprses prefer to renvest profts for future growth, than payng dvdends. These companes are often concentrated n certan sectors and/or growth phases. If those companes are excluded from the ndces, a meanngful segment of the economy wll be left out. Therefore the fundamental ndex makes a specal provson for zero-yeld companes. The composte weghtng for these companes are determned by averagng the weghts of the three remanng metrcs. (Arnott, Hsu, and West 80)

24 14 The Fundamental Index wll comprse of neutral selectons of the publcly traded parts of the broad economy. The cap-weghted ndexes are neutral relatve to the market, but are not neutralty relatve to the economy. Also, Fundamental Index portfolos are able to absorb vast sums of money, just lke portfolo s based on a cap weghted ndex. In addton to ths approach, The Fundamental Index approach nvolves less dsruptve turnover than the cap weghtng approach. (Arnott, Hsu, and West 85-86) The Fundamental Index wll nherently nvolve the dea of sellng hgh and buyng low, n contrast to tradng n cap weghted ndexes where t nvolves addng new stocks to the ndex after bg run ups or droppng exstng stocks after bleak results. Ths s because a fundamentally bult ndex wll have weghtngs that are less susceptble to large, often cyclcal, surges n operatng results or sudden reversals n dvdend polcy. In addton, changes n the sze of a company on one fundamental measure may be offset by the changes n another sze metrc. (Arnott, Hsu, and West 95-96) The orgnal Research Afflates method uses a 5 years smoothng approach by takng averages of the metrcs for the last 5 years. In ths thess however, only the most recent annual value was taken nto consderaton. It s beleved that takng averages 5 years prevous wll underestmate rapdly growng companes and overestmatng underperformng companes. 3.3 Fundamental Index Formulaton For company, there are 4 metrcs that can be found on each company s balance sheet or ncome statement each year: Cash and Equvalents C, Dvdends per Share D, Net Income I, and Total Revenue R. The weghtng for C (denoted by WC ) s calculated by:

25 15 WC n C = 1 C Eq 1: Fundamental Factors Equaton The same s done for D, I, R. For a gven company, ts fundamental weghtng s calculated through the followng formula: W WC = For companes that does not pay dvdends: + WD + WI 4 + WR Eq 2: Fundamental Weghtng Equaton W WC = + WI 3 + WR Eq 3 Fundamental Weghtng Equaton (Companes wthout Dvdends) The total number of companes, n represents the top 100 cap-weghted companes for a partcular year. Once thew s calculated, the lst of company was ranked n descendng order based onw. The top 35 companes were chosen for that year. The fnal composte factor F s calculated through ths formula: F = n W = 1 W Eq 4: Composte Factor 35 = 1 The F s a renormalze weghtng based on only the top 35 companes, where F = 1. The renormalze F s the weghtng for company n the portfolo for that year.

26 Modern Portfolo Theory Accordng to the theory, t's possble to construct an "effcent fronter" of optmal portfolos offerng the maxmum possble expected return for a gven level of rsk. (Brealey et al ) Ths theory addresses the trade-off between the expected rate of return and the varance of the rate of return n the portfolo. Unlke the Fundamental Index whch s farly straght forward, there are several key concepts before proceedng to the optmzaton model Asset Return When makng an nvestment, the ntal outlay of captal s known (amount nvested), but the amount to be returned s uncertan. Ths thess focuses on strctly a sngle nvestment perod; money s nvested at the ntal tme, and payoff s attaned at the end of the perod, each perod s one year, wth a total of ten years as the study horzon. Company stocks that can be bought and sold frequently s called an asset, base on ths property, ths study wll focus on calculatng the asset return when t s bought (amount nvested) at the frst day of the year when the market opens, and sold (amount receved) at the frst day the market opens the year after. (Luenberger ) The Total Return R for year can be calculated by dvdng the amount receved X + 1 by amount nvested X : R X = X +1 Eq 5 Total Return The Rate of Return r, for the year s:

27 17 r = X +1 X X Eq 6: Rate of Return Then the amount receved X + 1 can also be calculated by: + 1 = (1+ X r ) X Eq 7: Amount Receved Short Sales It s possble to sell an asset whch the nvestor does not own through the process of Short Sellng. To do ths, the nvestor wll borrow asset from someone who owns t, than sell the borrowed asset to some else recevng X. At a later date, the nvestor wll repay the loan by purchasng the asset at X + 1 and return t to the lender. The nvestor wll make a proft of X X + 1 (rather than X +1 X ). In the case when the prces go up, the nvestor wll ncur a lost of X +1 X. (Luenberger 139) Short Sellng s extremely rsky because f an nvestor antcpated the wrong drecton; prces can rse up wth no lmt. The nvestor would ncur devastatng loss especally when tryng to Short Sell securtes wth low prces (.e., n the event of an acquston when prces goes up). On the contrary, f an nvestor s bettng on prces gong up, t can only go down to a value of zero, losng only the ntal nvestment. All models n ths thess prohbt short sellng, the theory of the Fundamental Index does not allow negatve weghtng (or else t wouldn t have been chosen). Thus each chosen stock n the portfolo must have a zero or postve weghtng.

28 Portfolo Return Suppose n year there are n assets, the total portfolo return can be calculated by summng the ndvdual n assets that makes up X. The selected amount are now denoted by n,, j= 1,2..., n such that =1 X, j = X. (Luenberger 140) Snce ths study X j j prohbts short sellng, X s always equals to or greater than zero. The amount nvested can be expressed as fractons of the total nvestments: X, j = w, j X Eq 8: Investment Fracton Where w j s the weght of asset j n the portfolo for year, and n j= 1 w, j = 1 Eq 9: Total Weghtng The overall Total Portfolo Return and the Rate of Return of the portfolo for year, R = n j= 1 w, j R, j Eq 10: Total Portfolo Return r = n j= 1 w r, j, j Eq 11: Portfolo Rate of Return Random Varable Suppose that r s a random quantty that can take on any one of an nfnte number of values wthn a certan range wth a probablty of p. Ths quantty r characterzed ths

29 19 way before ts value s known s called a random varable. In ths thess, the future unknown return of any gven asset s a random varable. (Luenberger 141) Expected Return The expected value of a random varable x s just the average value obtaned by regardng the probabltes as frequences E( x) = = x p 1. (Walpole et al. 88) However, calculatng the expected return for an asset n the stock market a year nto the future wll guarantee one thng, that s, the actual return wll most defntely not equal to the expected return. Although there are several methods of calculatng future expected return (.e., tme-seres study, forecastng, etc), these approaches are large enough to be consdered a separate thess topc by tself. In ths thess however, a much more smplfy approach s taken. The Expected Return for stock j n year 0 s calculated by takng the averages of the rate of return for the stock startng from = 5, to year = 1, and k s the number of prevous years: n E ( r 1 1 0, j ) = r, j k = 5, for an asset j Eq 12: Expected Return for asset j It s mportant to note that startng from -5 was arbtrary pcked, the expected return could be calculated wth only 2 prevous years of data. Although t s true that the greater sample sze wll provde a more accurate estmaton, t s not a good dea to calculate the average return for a company that has been consstently growng for 40 years. It would not consder the recent success and under-estmate the expected return for that gven asset.

30 Varance & Standard Devaton The expected value of a random varable descrbes where the probablty dstrbuton s centered. But by tself, the expected value does not ndcate the varablty of the expected rate of return. (Walpole et al. 95) Thus, there are other parameters whch descrbe the varablty of the expected return; that s the varance, where 2 2 σ = var( x) = E[( x x) ] or the standard devatonσ. In fnance, the volatlty (or rsk) of an asset s often refer to the standard devaton of the expected return of that asset. The standard devatonσ of company stock j n year 0 startng from = 5, to year = 1, and k s the number of prevous years s gven by:, j σ 0, j = 1 2 k r, j ( r = 5 1 = 5 k( k 1), j ) 2, for an asset j Eq 13: Varance for asset j Covarance When there are two or more random varables (.e., x and y) n year, ther mutual dependence can be summarzed by ther covarance (Walpole et al. 95), denoted by σ (, x),(, y) ; note that by symmetry, σ (, x),(, y) = σ (, y),(, x). If two random varables x and y s uncorrelated, σ 0. Ths occurs when the two random varables are ndependent (, x),(, y) = from one another. Whenσ 0, the two varables are sad to be postvely (, x),(, y) > correlated. Whenσ 0, the two varables are sad to be negatvely correlated. The (, x),(, y) <

31 21 covarance between two assets (j=1,2) n year 0 startng from = 5, to year = 1, and k s the number of prevous years s gven by: 1 1 ( 0,1),(0,2) = σ ( r,1 E( r0,1 ))( r,2 E( r0,2 )), for two asset j = 1, 2 k = 5 Eq 14: Covarance for asset j = 1, Mean Return of a Portfolo Suppose n year 0, there are n assets, wth random rate of return 0,,...,, r,1 r0,2 r0, n each wth an expected return of E ( r ) = r, 0,1 0, 1 E ( r0,2 ) = r0, 2,..., E ( r0, n ) = r0, n. As stated above, the rate of return of the portfolo s gven by the weghted sum of the ndvdual return of each asset s: r = w, 1r,1+ w,2r, w, nr, n = w, jr, j, for any gven year n j= 1 Eq 15: Portfolo Mean Return Takng the expected rate of return of both sdes gve: r n 1 ( r,1) + w2e( r,2) wn E( r, n ) = w, je( r, j ) j= 1 = E( r ) = w E, for any gven year Eq 16: Portfolo Expected Return Therefore, the Expected Rate of Return of the Portfolo s equal to the weghted sum of the ndvdual expected rate of return of each asset. (Luenberger 150) Varance of a Portfolo Return The varance of asset j n year s denoted byσ, the varance of the return of 2 the portfolo n year by σ and the covarance of the return of asset j wth asset k n year by σ (, j),(, k ). The varance of the portfolo n year s gven by: 2, j

32 22 σ = σ = E[( r σ = E[( n n j= 1 k= 1 w, j n j= 1 w r w, j, j r ) 2, j, j, kσ j, k ] n j= 1 w, j r, j, for any gven year Eq 17: Varance of Portfolo Return ) 2 ] The Markowtz Model Suppose there are n assets n year, the expected rates of return are gven by,, and the covarance are σ (, j),(, k ), for j, k = 1, 2 n. the portfolo s r, 1 r,2,..., r, n defned by a set of n weghts, w,, where j = 1,2 n, and w 1. To fnd the j n j= 1, j = mnmum-varance portfolo, the mean s fxed at some arbtrary value r *. The model wll solve for the feasble portfolo weghts wth mnmum varance that has expected mean r * : Mnmze 1 2 Subject to n j= 1 n n j= 1 k= 1 n j= 1 w w, j, k σ w r = r w, j = *, j, j 1 w, j 0 For j = 1, 2 n j, k Eq 18: Mean-Varance Optmzaton Problem The last constrant n the optmzaton problem prohbts of short sellng, by statng all of the weghtng must be postve. (Luenberger 158)

33 The Feasble Set Suppose there are n assets, these n assets can form a numerous number of portfolos, usng every possble weghtng scheme. Hence, there are portfolos wth dfferent number of assets, dfferent composton of assets, and dfferent weghtng for each asset. These portfolos are made by lettng the weghtng coeffcents w, j, for any n gven year, range over all possble combnatons such that w =1, j = 1. Ths set of ponts correspondng to the portfolos s called the feasble set. (Luenberger 157) j r r Mnmum-varance pont Mnmum-varance pont σ σ Fgure 2: Mnmum-Varance Set Fgure 3: Effcent Fronter The area rght of the bullet shape lne n Fgure 2 and Fgure 3 are the feasble set. Ths lne represents the Mnmum-Varance Set, snce for any value of the rate of return; the feasble pont wth the smallest standard devaton s the correspondng left boundary pont. The pont on the tp of the curve s the Mnmum-Varance Pont, ths pont represent the mnmum volatlty of a gven set of assets. The top half of the bullet shape lne, Fgure 3 s the effcent fronter. Ths lne represents the maxmum possble expected return for a gven level of rsk.

34 Performance Evaluaton Actual Return Whether t s F n the Fundamental Index or w, j n the Modern Portfolo Theory, the goal s to obtan the weghts for each asset n order to calculate the actual portfolo rate of return. Suppose there are n assets, let W, j represent the weghtng for an asset j n year, X, j represent the amount nvested n year and X + 1, j s the amount receved at the begnnng of next year. The Actual Rate of Return on Portfolo, for year s: r = n + 1, j w j j= 1 X, j X X, j, For a gven year Eq 19: Actual Portfolo Rate of Return or smply r = n j= 1 w r, j, j ; where r, j = X + 1, j X X, j, j Eq 20: Smplfed Actual Portfolo Rate of Return Then the amount receved X + 1 can also be calculated by: X r ) X + 1 = (1+ Eq 21: Amount Receved on Portfolo The Total Cumulatve Return Factor for a gven portfolo over m year horzon s: The amount receves R= m = 1 ( 1+ Eq 22: Total Cumulatve Return Factor X m n year m after the ntal nvestment n year s: r )

35 25 X = ( R) m X Eq 23: Amount Receved on Portfolo m year from Sharpe Rato The Sharpe rato s a measure of the excess return (or Rsk Premum) per unt of rsk n an nvestment asset or a tradng strategy. It s used to characterze how well the return of an asset compensates the nvestor for the rsk taken. ( Sharpe Rato ) The formula of the Sharpe Rato for a gven portfolo n year s gven by: r r σ, f Eq 24: Sharpe Rato Where r,f s the rsk-free rate of return (the U.S. Treasury Bll) When comparng two assets each wth the expected return r aganst the same benchmark wth return r f, the asset wth the hgher Sharpe rato gves more return for the same rsk. Investors are often advsed to pck nvestments wth hgh Sharpe ratos Delta Delta s the dfference between two numbers. In ths thess, Delta s equal to the Actual Return mnus the Expected Return on portfolo. When delta s postve, actual return exceeds expectaton. If t s negatve, the expected return overestmated the portfolo s performance.

36 26 4. Procedures (Thomas Chu) 4.1 Gatherng Data Determnng the lst of companes lsted on the S&P 500 Ths s a backward study from , so there are sgnfcant changes to the ndex lstng ten years ago from today. The S&P 500 re-ndexes ther companes several tme a year, the lst of addton/deleton of companes are publshed n ther quarterly reports. Ths thess however only focuses on annual study at the begnnng of each year, thus only the lstng at the begnnng of each year was gathered. Standard & Poor s publsh annual reports on ther product n the begnnng of each year; ths report can be found n the Robarts Lbrary at the Unversty of Toronto for every year dated back to the 1980 s. These reports were only n hard copes, so the tcker symbols was coped and looked up manually afterward Determne the top 100 by market captalzaton The goal was to use the tcker symbols to construct a company lstng for each year, sorted by total market captalzaton. However, determnng the lst was much harder than antcpated, some tcker symbols no longer exst due changes n company status (.e. acqustons, bankruptcy, merger etc). An mportant assumpton was made for these companes, f ther nformaton can no longer be found, t won t be suffcent to make the fundamental lst anyway. The Fundamental Index concept s based on evaluatng companes on ther fundamental; top 100 companes n the U.S. based on fundamentals are rarely seen nexstent n one decade.

37 27 The lst of tckers was valdated for ts exstence and ts correspondng reference to the correct company. The tcker was used to gather company market captalzaton for each year. The Rotman Fnancal Lab has equpped computers that contaned real tme and hstorcal fnancal data connected to Reuters. Reuters supply the fnancal markets wth nformaton and tradng products. Through Reuters, the market captalzaton can be downloaded through a functon called DataStream and looks up the desgnated tcker symbol and generates all the market captalzaton for each company for all 10 years. For each year, the companes are ranked based on ther total market captalzaton and the top 100 s selected for each year Gatherng the 4 metrcs for the top 100 companes through Captal IQ Once the top 100 companes were selected for each year, the 4 fundamental factors had to be obtaned for each company; these data were downloaded from Captal IQ. Captal IQ s a provder of nformaton and analytcal tools for nvestment bankers, money managers, and other fnancal professonals. It allows users to download all publc companes fnancals onto a sngle Excel fle. The user can also specfy the tme study horzon. The 4 fundamental factors requred were cash & equvalents, total revenue, dvdends per share and net ncome. Total revenue, dvdend per share and net ncome can be found n the n the company s Income Statement, cash & equvalent was found n the Balance Sheet. After gatherng all the data, t was organzed nto one spreadsheet, whch made t easer to analyze and reference when rankng the companes for each year based on the 4 factors.

38 Gatherng Adjusted Prces from Yahoo! Fnance The adjusted prces was downloaded n order to analyze hstorcal performances of any company; t s a modfcaton made to a securty's prce that takes nto consderaton the effect of a splt on the total number of shares or unts outstandng. (Brealey et al., 188) Yahoo! Fnance contans all the adjusted hstorcal prces and can be downloaded onto an Excel fle by specfyng the tcker symbol. Ths was done for the top 100 companes on the fundamental lst to analyze the Fundamental Index performance and for the S&P 500 to represent the performance of the market. After downloadng the separate Excel fles, t was organzed nto one Excel spreadsheet whch made t easer to perform calculatons (.e., expected and actual returns). 4.2 Buldng the Fundamental Index Calculate F For the top 100 companes of each year, the total fundamental weghtng W, was calculated for each company usng Excel. The W was ranked n descendng order where the top 35 companes were chosen. The Fnal Composte Factor, F, was calculated based on renormalzng the W, for the tope 35 companes. It s mportant to note that the Balance Sheet and Income Statement for each company were publshed at the end of each year. Thus, the W and F calculated for 1996 was based on the reported number at the year end of 1995.

39 Buldng the Mean-Varance Model Calculatng Expected Return, Varance and Covarance usng Adjusted Prce In ths thess, the Markowtz perod s one year, thus the expected return s calculated based on a movng average approach. The expected return of the year s therefore averagng annual return from prevous years. The expected return for the frst year, 1996, used a sample sze of 7 data ponts (annual returns from ). Annual returns were calculated usng the adjusted prce collected on Yahoo! Fnance. Snce t s a movng average approach, the expected return for the second year, 1997, used annual returns from The expected return was calculated n the same manner for the remanng years. The Varance of a company and the Covarance between companes were calculated usng the same set of data ponts that calculate the expected return of each year. There are several nstances when a company appears on the top 35 that does not have 7 years of hstorcal data (.e., CSCO, MS). In these stuatons, the movng averages of the avalable years were taken nstead. The company expected return and varance was calculated n ths manner. However, the covarance between companes can not be adjusted ths way snce both set of data ponts must be the same sze. In these stuatons, the sze of the data ponts s determned by the company wth the shorter hstory.

40 Buldng and Solvng the Mean-Varance Optmzaton Model n Excel The optmzaton model was bult and solved usng Mcrosoft Excel and Solver. Refer to Appendx A for detal nstructon on how the optmzaton was bult. The set of weghtng, w, j, for stock j n year was determned by solvng the mnmum-varance model for a gven r*. The Portfolo Return and Portfolo Varance assocated wth the set of w, j were also calculated Fndng the Mnmum-Varance Pont and Graphng the Effcent Fronter After havng the Mean-Varance model bult, the effcent fronter was constructed n order to determne the mean-varance pont (the maxmum expected mean, wth the least varance) and the correspondng company weghtng. It started at 0.01 and ncremented by 0.01 up to 1.0. The model was solved multple tmes to fnd the optmal pont wth the mnmum standard devaton for each year. To do ths, an Excel Macro was wrtten to run Excel Solver many tmes over the specfed ranges of returns. Please refer to the Appendx A for a sample macro. Wthn that 0.01 to 1.0 mean range, there may be nfeasble solutons. These solutons were taken out manually, leavng only the feasble solutons. Wth the solutons for each year, the effcent fronter can be graphed and the mnmum-varance pont was dentfed.

41 Maxmum Return on Portfolo and Equally Weghted Portfolo Addtonally, there are two strateges that are used for result comparson. The Maxmum Return portfolo only takes nto consderaton the maxmum expected return of a portfolo, thus the company wth the hghest expected return has the weghtng of 1, whle the rest has a weghtng of zero. The portfolo return and varance equals to the expected return and varance of that one company. The equally weghted strateges set the weghtng for each company to 1/35, or Evaluatng portfolo(s) performance Calculate Annual Actual Return, SD, Delta, Sharpe Rato and Cumulatve Return for each portfolo All the formulas for the calculatons are specfed n the methodology. Excel was used to calculate all the parameter of the dfferent portfolo. These results were then calculated for analyss purposes to depct trends and performances. Please refer to the Results secton and Appendx B for the graphs and tables.

42 32 5. Results (Teddy Yp) The followng secton summarzes the crtera used to evaluate the models bult based on the thess methodology: 1. Effcent Fronter of the Mean-Varance Model 2. Economc Sector Analyss of the Fundamental Index Strateges a. Dstrbuton of Economc Sectors b. Comparson wth RAFI US Large Weght and Cap 1000 Weght 3. Sharpe Rato 4. Turnover Analyss for Fundamental Strateges 5. Delta Comparson 6. Rate of Return 7. Actual Cumulatve Return 8. Volatlty (Standard Devaton)

43 Effcent Fronter of the Mean-Varance Model The mnmum-varance set was constructed by solvng the optmzaton model * through settng the expected portfolo return equal to dfferent r. An Excel Macro was used to assst the process for solvng the dfferent models for all 10 years; t generated a lst of possble feasble solutons for the entre study horzon. Ths lst was used to construct the mnmum-varance set for that year. The followng graph llustrates the outcome for 2001: Effcent Fronter Mean Standard Devaton Source: Captal IQ and Yahoo! Fnance Fgure 4: 2001 Effcent Fronter The mnmum-varance pont ths year has a mean of 0.27 and standard devaton of Fgure 4.2 concludes the mnmum-varance pont for all years. Please refer to Appendx B for the entre collecton of effcent fronters.

44 34 Year Portfolo Standard Devaton Expected Portfolo Mean Source: Yahoo! Fnance Table 1: Mnmum-Varance Ponts It was observed that the hghest volatlty came after the peak of the tech bubble n the Mean-Varance model. The peak n mean occur n 2000, when the expected return s 0.4. Ths s due to the rapd prce ncrease of the tech companes. After the burst of the tech bubble, expected return start stablzng agan, however the standard devaton sts at around 0.1 because of the fluctuaton of the prces leadng up to the burst. Later n ths secton, a comparson wll be drawn between the mnmum-varance pont of the effcent fronter for the mean-varance model and the other strateges used n the thess. 5.2 Economc Sector Dstrbuton The resulted set of companes based on the selected 4 fundamental factors was categorzed based on ther desgnated economc sectors. They were segregated nto seven man economc sectors. They are Basc Materals, Consumer Goods, Energy, Fnancal, Healthcare, Servces and Technology. Fgure 4.3 llustrates the sector dstrbuton based on the weghtng.

45 35 Dstrbuton per Sector 35.00% 30.00% 25.00% 20.00% 15.00% 10.00% Basc Materals Consumer Goods Energy Fnancal Healthcare Servces Technology 5.00% 0.00% Year Source: Captal IQ and Yahoo! Fnance Fgure 5: Economc Sector Breakdown The results ndcate that companes n the Fnancal sector domnate the ndex. Companes such as JP Morgan Chase, Ctgroup and Morgan Stanley were on the top 35 lst every year. The second economc sector that domnated the Fundamental Index was the Technology sector. Here we have companes such as IBM, Csco, Mcrosoft who dd very well leadng up to the tech bubble. The remanng sectors dd not have as much nfluence as the domnatng two, however they were well represented at a very consstent bass throughout the 10 years. Fgure 4.4 llustrates the dstrbuton comparson between two other ndexes, the RAFI US Large and the Cap 1000 Weght.

46 Comparson Economc Sector RAFI US Large [1] Cap 1000 Weght [1] Fundamental Basc Materals 4% 4% 14% Consumer Goods 25% 20% 11% Energy 10% 11% 6% Fnancal 21% 21% 23% Healthcare 8% 12% 11% Servces 19% 14% 6% Technology 14% 18% 29% Source: Captal IQ and Arnott, Robert Table 2: Dstrbutons of other ndces for 2005 Lookng at ths table, t was observed that the weghtng algns approxmately wth the other well dversfed ndexes. The fundamental strategy adopted n ths thess weghted stocks relatvely close to other ndexes. One man dfference s that the weghtng generated from the thess puts more on basc materals than servces. In Basc Materals, we fnd companes such as Chevron and Dow Chemcals and n servces we fnd companes such as Home Depot and Wal-Mart. The dfference are due to the sze of the portfolos, the RAFI US Large and Cap 1000 are comprse of 1000 stocks, where as the Fundamental Portfolos are make up of only 35. It can be concluded that for ts sze, the pure fundamental strategy generates a well dversfed portfolo that dstrbute across the economc sector just as well as any other ndces. 5.3 Sharpe Rato The Sharpe rato s used to characterze how well the return of an asset compensates the nvestor for the rsk taken. It s used to compare the dfferent strateges used to the U.S. T-Blls (rsk free rate) as a benchmark to see whch nvestment would gve more return for the same level of rsk. Investors are often advsed to pck

47 37 nvestments wth hgh Sharpe ratos because of t. Fgure 4.5 s a graph that shows the Sharpe rato between all 5 portfolos. Sharpe Rato Rato S&P 500 FI - Pure FI - Max Return FI - Mn Var FI - Equally Weghted Year Source: Captal IQ and Yahoo! Fnance Fgure 6: Sharpe Rato s of the dfferent portfolos The portfolo wth the hghest Sharpe rato over the entre study horzon s the Mnmum-Varance strategy. Ths makes sense because the goal of the optmzaton model was mnmzng portfolo rsk (varance). The year that generated the hghest Sharpe rato was 2000 for the Mean-Varance model wth a Sharpe rato of Ths s because as the market was reachng the peak of the bubble. The prces of stocks all ncreased at a very large rate wth lower varances and all the prces were movng up together. It can be observed later n ths secton that the actual rate of return before the burst of the bubble were large postve fgures. These large fgures lead to a large expected rate of return for the year 2000.

48 38 The Equally Weghted, Max Return and Pure-FI methodology had smlar outcomes wth consstent Sharpe ratos throughout the 10 years. As expected, the S&P 500 had the lowest Sharpe rato whch ndcates the lowest expected return for the same level of rsk. All the strateges that use a fundamental approach generated a hgher expected return wth the same level of rsk than the S&P Turnover Analyss for Fundamental Strateges As stated n the Methodology secton of the thess, an ndex should have Low Turnover to mantan contnuty. An ETF fund that mrrors the component of an ndex should follow the same prncple. The fgure below ndcates that 57.14% of the companes lsted n the Fundamental Index appear 8 to 10 years. Ths ndcates a portfolo bult based on the fundamental strateges has a small number of turnover. It can also conclude that the fundamental strateges have a tendency to pck companes that wll be around for the long run. Source: Captal IQ Fgure 7: Fundamental Index Turnover Analyss

49 Delta To test the accuracy of the calculated expected return on portfolo, t was compare wth actual return of the portfolo for all the strateges. Fgure 4.6 depcts the dfference between the actual performances versus the expected performance. Actual Return mnus Expected Return Delta S&P 500 FI - Pure FI - Max Return FI - Mn Var FI - Equally Weghted Year Source: Captal IQ and Yahoo! Fnance Fgure 8: Actual Return VS Expected Return Durng the rse of the tech bubble, the actual return was out performng the expected returns. However, by 1999 the market started slowng down. From 1999 to 2002, the expected returns became a lot hgher than the actual return. Ths s because the calculaton of the expected return can not recognze these changes n the market. The calculaton s based on the averages of the past and when the bubble burst, there are huge negatve dfferences between the actual return and expected return. Therefore, past

50 40 hstorcal data used to calculate those expected returns were not adequate when there s a sudden drop n prces Durng ths tme perod, a lot of nvestor lost a great amount of money. The Max Return strategy outperforms the rest of the portfolo pror to the burst. However after the peak, t can be observed that the Max Return strategy suffered the bggest negatve dfferences from the actual return. The FI-Max portfolo s the hardest to forecast due to hgh volatlty from expected return calculaton. 5.6 Rate of Return The actual rate of return of each portfolo was compared to determne how each one performs on an annual bass. Fgure 4.7 shows the rate of return results of the dfferent strateges. Rate of Return % % % Rate of Return 80.00% 60.00% 40.00% 20.00% 0.00% % % % % Year T Blls S&P 500 FI - Pure FI - Max Return FI - Mn Var FI - Equally Weghted Source: Captal IQ and Yahoo! Fnance Fgure 9: Rate of Return

51 41 The Max Return strategy gave the greatest return for the years before the tech bubble burst. Pror to the bubble burst, pckng any IT stocks from the lst of fundamental portfolo wll almost guarantee great return. However, ths model does not take nto consderaton of the volatlty of the securty. When the market crashed n 2000, the actual return of ths portfolo dropped below 0, whch generated loss to nvestors. The strategy that performed poorly after the Max Return strategy was the S&P 500 where the weghtng of the stocks are based on market captalzaton. The S&P 500 moves wth market trends, so the weghtng was placed heavly on technology companes that crashed when the bubble burst. The only strategy that had only 1 year of negatve return was the FI Mnmum-Varance strategy. Ths model made an effort to mnmze rsks and volatlty. Nonetheless, none of these strateges survved n 2002 other than the T blls whch have a rsk free return. Ths thess wanted to show that the fundamental way of ndexng outperforms market captalzaton ndexng method. Fgure 4.8 shows the dfference n actual return between the FI Pure strategy and the FI Mnmum Varance strategy compare to the S&P 500.

52 42 Fundamental Index mnus S&P 500 Return Dfference 25.00% 20.00% 15.00% Return Dfference 10.00% 5.00% 0.00% (FI - Pure) - S&P (FI - Mn Var) - S&P -5.00% % Year Source: Captal IQ and Yahoo! Fnance Fgure 10: Fundamental Index vs S&P 500 The results ndcate that both fundamental strateges outperform the S&P 500 n the ten year span. The mnmum varance strategy outperformed the S&P 500 for all except for one year n 2003 and the Pure Fundamental Index strategy was only outperformed 3 out of 10 years. 5.7 Actual Cumulatve Return The cumulatve rate of return analyss was to see how much an nvestor would gan throughout the years. Fgure 4.9 llustrates the cumulatve rate of return for all the strateges.

53 43 Actual Cumulatve Return Return T Blls S&P 500 FI - Pure FI - Max Return FI - Mn Var FI - Equally Weghted Year Source: Captal IQ and Yahoo! Fnance Fgure 11: Cumulatve Rate of Return The strategy that generated the greatest cumulatve rate of return was the Maxmum Return strategy. Ths s because ther set of stocks focuses only on the bg wnner such as Mcrosoft pror to 2000, and Dell afterward. Ths analyss s suggestng that f someone nvested n ths portfolo at the begnnng of 1996, the stock values would have went up 6 tmes n ten years. Another strategy that would have generated smlar returns would be the Mnmum Varance strategy. It acheved smlar returns, but ts return was not subjected to large fluctuaton. Also, t can be noted that all fundamental strateges outperform the S&P 500 despte the tech bubble burst; nvestors would gan on the market through long term nvestments.

54 Standard Devaton A way to analyze the volatlty of returns s though analyzng the portfolo s standard devaton. Hgher standard devatons ndcate the portfolo s more volatle and ts return s subject to prce fluctuaton. Fgure 4.10 llustrates that volatlty of all the strateges. Standard Devaton Standard Devaton S&P 500 FI - Pure FI - Max Return FI - Mn Var FI - Equally Weghted Year Source: Captal IQ and Yahoo! Fnance Fgure 12: Standard Devaton of all strateges The results ndcated that the Max Return strategy has the most rsk. For all of the 10 years, the prce s of ts set has had a hgh standard devaton than any other strategy. It also remaned hgh after the peak of the tech bubble. The Fundamental Pure strategy, Equally Weghted Strategy and the S&P 500 all had very consstent varances throughout the 10 years. As expected, the Mn Varance model generated the lowest standard devaton, snce the objectve of the mean-varance model s to mnmze varance (rsk),

55 45 these results algns wth the thess s hypothess. The T-Bll was not plotted because t s rsk free, standard devaton equals to zero.

56 46 6. Concluson (Thomas Chu) Investors have always tred to fgure out better ways to nvest. Ths thess has tested out dfferent ways of buldng a portfolo. A cap-weghted approach mrrorng the S&P 500, selectng stocks based on fundamental factors and mnmzng varance usng the Modern Portfolo Theory optmzaton model. The Pure Fundamental Index strategy outperformed the S&P 500 almost every year whch lead to a hgher cumulatve return over 10 years. In addton, the S&P 500 had the lowest Sharpe rato ndcatng that t generated the lowest return based on the same level of rsk. The fndngs of ths thess algns wth the Research Afflates fndngs as weghtng securtes through fundamental factors generate greater returns and less rsks than through the tradtonal market captalzaton strategy. The equally weghted Fundamental strategy behaved very smlarly to the Pure Fundamental strategy as t generated smlar return and standard devatons. The Maxmum Return strategy provded an upper bound on maxmum return, however ths strategy should not be used snce ts weghtng was on only one company (whch hardly defne t as a portfolo). The Mnmum-Varance strategy outperform the Pure Fundamental Strategy and the S&P 500, t has the hghest Sharpe rato and the least varance. Lookng at the cumulatve returns, the Mnmum-Varance strategy has smlar gan as the Maxmum- Return strategy through the 10 years span before, durng and after the tech bubble, but wth lower rsk than the S&P 500. The cumulatve returns on the Mnmum-Varance model generatng nearly 6 tmes the ntal nvestment. Overall, ths thess found that the weghtng through fundamental analyss s a better way of constructng portfolo snce t takes nto consderaton more factors of the

57 47 selected companes other than just market captalzaton. Ths would subject nvestors to less rsk exposed n an event of a bubble as ndcated n the results. The portfolo wth the best performance over the study horzon was buldng the Markowtz Mean-Varance model for the pool of assets determned by Fundamental Strategy; t generated hgh returns wth the lowest standard devaton. The results also ndcate that no strategy can fully protect nvestors from a bubble burst. There wll always be rsks nvolved n nvestng n the stock market; however there are strateges as presented n ths thess that can mnmze the nvestor s rsks and losses.

58 48 References Arnott, Robert D., Jason C. Hsu, and John M. West. The Fundamental Index - A Better Way to Invest. Hoboken, NJ, USA: John Wley & Sons, Inc., Brealey, Rchard A., Stewart C. Myers, Alan J. Marcus, Alzabeth M. Maynes, and Devashs Mtra. Fundamentals of Corporate Fnance. 3rd Canadan ed. Halfax, Nova Scota, Canada: McGraw-Hll Ryerson Lmted, Investopeda.com, "Dversfcaton." Investopeda.com. Forbes Dgtal Company. 11 Mar 2009 < Investopeda.com, "Sharpe Rato." Investopeda.com. Forbes Dgtal Company. 11 Mar 2009 < Investopeda.com, "Tech Bubble." Investopeda.com. Forbes Dgtal Company. 11 Mar 2009 < Luenberger, Davd G.. Investment Scence. New York, NY, USA: Oxford Unversty Press, Inc., Standard and Poor Corporaton, "S&P Indces FAQ." S&P Indces. January Standard & Poor's. 19 Mar 2009 < Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (1996): VI- X. Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (1997): V- IX. Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (1998): V- IX. Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (1999): V- IX. Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (2000): V- IX.

59 49 Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (2001): V- IX. Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (2002): V- IX. Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (2003): V- IX. Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (2004): Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (2005): Standard and Poor's Corporaton, "Standard & poor's Composte Stock Prce Index." Standard & Poor's regster of corporatons, drectors and executves (2005): Walpole, Ronald E., Raymond H. Myers, Sharon L. Myers, and Keyng Ye.Probablty & Statstcs for Engneers & Scentsts. 7th ed. Upper Saddle Rver, NJ, USA: Prentce Hall, 2002.

60 50 Appendx A Buldng an optmzaton model usng Mcrosoft Excel and Solver Ths example wll use 3 assets to demonstrate how to buld an optmzaton model n Excel. Refer to Fgure A.1 for the spreadsheet layout. All tables wth a prefx of X.2 refer to excel formulas used to calculate values of table X.1 1. Organze company s prces by column, lst t by years n descendng order (Table 1) 2. Calculate the annual rate of returns of each year for all company. (Table 2.1, refer to secton for equaton) 3. Calculate Average Return and Standard Devaton for each company (Table 3.1) 4. Calculate Covarance for each company (Table 4.1) 5. Table 5.1 represent the double summaton n the objectve functon n n 1 ( w, jw, kσ j, k ) of the mean-varance model 2 j= 1 k= 1 6. Calculate portfolo varance, standard devaton and portfolo mean. 7. Use Solver to fnd optmal pont (Refer to Fgure A.2): a. Go to tools, solver b. In Set Target Cell, select portfolo varance c. Choose opton mn for mnmzaton d. For Changng Cell, select the darken cells under Weght e. Add the 3 constrants (refer to Fgure A.2) f. Clck solve 8. If there s an optmal pont, solver wll dsplay the soluton 9. Graphng the Effcent Fronter a. The effcent fronter requres solver to run multple tmes for dfferent value of expected portfolo mean. b. Organze a table smlar to the one n Fgure A.3 c. Each tme solver s run, change the last constrant to a dfferent value d. Record the portfolo expected mean and standard devaton e. When there are enough ponts, graph t on a chart 10. to assst wth the gruelng process of runnng solver numerous tme, a macro was mplemented, refer to Fgure A.4 for the macro code, please change the cell value to the one correspondng to your spreadsheet

61 51 Fgure A.1 Sample Spread Sheet Fgure A.2 Sample Solver Input

62 52 Fgure A.3 Sample Table for Constructng Effcent Fronter Sub Macro5() ' ' Macro5 Macro Dm a As Integer a = 65 For x = -1 To 5 Range("A65").Select SolverOk SetCell:="$B$57", MaxMnVal:=2, ValueOf:="0", ByChange:="$A$49:$A$55" SolverAdd CellRef:="$A$49:$A$55", Relaton:=3, FormulaText:="0" SolverAdd CellRef:="$A$56", Relaton:=2, FormulaText:="1" SolverAdd CellRef:="$B$59", Relaton:=2, FormulaText:="$A$" & (a + x + 1) SolverOk SetCell:="$B$57", MaxMnVal:=2, ValueOf:="0", ByChange:="$A$49:$A$55" SolverSolve True Range("B58").Select Selecton.Copy Range("B" & (a + x + 1)).Select Selecton.PasteSpecal Paste:=xlPasteValues, Operaton:=xlNone, SkpBlanks _ :=False, Transpose:=False Range("A49:A55").Select Applcaton.CutCopyMode = False Selecton.Copy Range("C" & (a + x + 1)).Select Selecton.PasteSpecal Paste:=xlPasteValues, Operaton:=xlNone, SkpBlanks _ :=False, Transpose:=True SolverReset Next End Sub Fgure A.4 Sample Macro code