Introduction. Chapter 7 - An Introduction to Portfolio Management

Transcription

1 Introducton In the next three chapters, we wll examne dfferent aspects of captal market theory, ncludng: Brngng rsk and return nto the pcture of nvestment management Markowtz optmzaton Modelng rsk and return CAPM, APT, and varatons Estmatng rsk and return the Sngle-Index Model (SIM) and rsk and expected return factor models These provde the framework for both modern fnance, whch we have brefly dscussed already, as well as for quanttatve nvestment management, whch wll be the subject of the next secton of the course Lecture Presentaton Software to accompany Investment Analyss and Portfolo Management Seventh Edton by Frank K. Relly & Keth C. Brown Chapter 7 Chapter 7 - An Introducton to Portfolo Management Questons to be answered: What do we mean by rsk averson and what evdence ndcates that nvestors are generally rsk averse? What are the basc assumptons behnd the Markowtz portfolo theory? What s meant by rsk and what are some of the alternatve measures of rsk used n nvestments? 1

2 Chapter 7 - An Introducton to Portfolo Management How do you compute the expected rate of return for an ndvdual rsky asset or a portfolo of assets? How do you compute the standard devaton of rates of return for an ndvdual rsky asset? What s meant by the covarance between rates of return and how do you compute covarance? Chapter 7 - An Introducton to Portfolo Management What s the relatonshp between covarance and correlaton? What s the formula for the standard devaton for a portfolo of rsky assets and how does t dffer from the standard devaton of an ndvdual rsky asset? Gven the formula for the standard devaton of a portfolo, how and why do you dversfy a portfolo? Chapter 7 - An Introducton to Portfolo Management What happens to the standard devaton of a portfolo when you change the correlaton between the assets n the portfolo? What s the rsk-return effcent fronter? Is t reasonable for alternatve nvestors to select dfferent portfolos from the portfolos on the effcent fronter? What determnes whch portfolo on the effcent fronter s selected by an ndvdual nvestor?

3 Background Ideas As an nvestor you want to maxmze the returns for a gven level of rsk. The relatonshp between the returns for the dfferent assets n the portfolo s mportant. A good portfolo s not smply a collecton of ndvdually good nvestments. Rsk Averson Gven a choce between two assets wth equal rates of return, most nvestors wll select the asset wth the lower level of rsk. Rsk averson s a consequence of decreasng margnal utlty of consumpton. Evdence That Investors are Rsk-Averse Many nvestors purchase nsurance for: Lfe, Automoble, Health, and Dsablty Income. The purchaser trades known costs for unknown rsk of loss. Yelds on bonds ncrease wth rsk classfcatons, from AAA to AA to A Lottery tckets seemngly contradct rsk averson, but provde potental for purchasers to move nto a new class of consumpton. 3

4 Defnton of Rsk But, how do you actually defne rsk? 1. Uncertanty of future outcomes or. Probablty of an adverse outcome Markowtz Portfolo Theory Old adage s: Don t put all your eggs n one basket. But, how many baskets should you use? And how what proporton of your eggs should you put n each basket? Harry Markowtz wrestled wth these questons fgured out a way to answer both of them Earned a Nobel Prze n Economcs (1990) for hs efforts Markowtz Portfolo Theory Quantfes rsk Derves the expected rate of return for a portfolo of assets and an expected rsk measure Shows that the varance of the rate of return s a meanngful measure of portfolo rsk Derves the formula for computng the varance of a portfolo, showng how to effectvely dversfy a portfolo Provdes both: the foundaton for Modern Fnance a key tool for Haugen s New Fnance 4

5 Assumptons of Markowtz Portfolo Theory 1. Investors consder each nvestment alternatve as beng presented by a probablty dstrbuton of expected returns over some holdng perod. Assumptons of Markowtz Portfolo Theory. Investors mnmze one-perod expected utlty, and ther utlty curves demonstrate dmnshng margnal utlty of wealth. I.e., nvestors lke hgher returns, but they are rsk-averse n seekng those returns And, agan, ths s a one-perod model (.e., the portfolo wll need to be rebalanced at some pont n the future n order to reman optmal) Assumptons of Markowtz Portfolo Theory 3. Investors estmate the rsk of the portfolo on the bass of the varablty of expected returns. I.e., out of all the possble measures, varance s the key measure of rsk 5

6 Assumptons of Markowtz Portfolo Theory 4. Investors base decsons solely on expected return and rsk, so ther utlty curves are a functon of only expected portfolo returns and the expected varance (or standard devaton) of portfolo returns. Investors utlty curves are functons of only expected return and the varance (or standard devaton) of returns. Stocks returns are normally dstrbuted or follow some other dstrbuton that s fully descrbed by mean and varance. Assumptons of Markowtz Portfolo Theory 5. For a gven rsk level, nvestors prefer hgher returns to lower returns. Smlarly, for a gven level of expected returns, nvestors prefer less rsk to more rsk. Markowtz Portfolo Theory Usng these fve assumptons, a sngle asset or portfolo of assets s consdered to be effcent f no other asset or portfolo of assets offers hgher expected return wth the same (or lower) rsk, or lower rsk wth the same (or hgher) expected return. 6

7 Expected Rates of Return For an ndvdual asset: sum of the potental returns multpled wth the correspondng probablty of the returns For a portfolo of assets: weghted average of the expected rates of return for the ndvdual nvestments n the portfolo Computaton of Expected Return for an Indvdual Rsky Investment Probablty Possble Rate of Return (Percent) Exhbt 7.1 Expected Return (Percent) E(R) = E(R ) = RsP s where: Ps = theprobablty of states occurrng R = thereturn on stock n states s n s S Computaton of the Expected Return for a Portfolo of Rsky Assets Weght (W) (Percent of Portfolo) Expected Securty Return (R) Expected Portfolo Return (W X R) E(Rpor ) = E(R port where : W = the E(R ) n ) = WE(R ) = 1 percent of the portfolo n asset the e pected rate of ret rn for asset Exhbt 7. 7

8 Varance (Standard Devaton) of Returns for an Indvdual Investment Varance s a measure of the varaton of possble rates of return R, away from the expected rate of return [E(R )]. Standard devaton s the square root of the varance. Varance and Standard Devaton of Returns for an Indvdual Investment Formulas: Varance (Standard Devaton) of Returns for an Indvdual Investment Possble Rate Expected Exhbt 7.3 of Return (R ) Return E(R ) R - E(R ) [R - E(R )] P [R - E(R )] P Varance ( σ ) =.0050 Standard Devaton ( σ ) =.036 8

9 Standard Devaton of Returns for a Portfolo Formula: Two-Stock: More than two stocks: Portfolo Standard Devaton Calculaton Any asset of a portfolo may be descrbed by two characterstcs: The expected rate of return The expected standard devatons of returns A thrd characterstc, the covarance between a par of stocks, also drves the portfolo standard devaton Unlke portfolo expected return, portfolo standard devaton s not smply a weghted average of the standard devatons for the ndvdual stocks For a well-dversfed portfolo, the man source of portfolo rsk s covarance rsk; the lower the covarance rsk, the lower the total portfolo rsk Covarance of Returns Covarance s a measure of: the degree of co-movement between two stocks returns, or the extent to whch the two varables move together relatve to ther ndvdual mean values over tme 9

10 Covarance of Returns For two assets, and j, the covarance of rates of return s defned as: σ or σ j j = = ( ) ( R E ( R )) R j E ( R j ) R ( ) ( R s E ( R )) R js E ( R j ) s S dp, j P s Covarance and Correlaton The correlaton coeffcent s obtaned by standardzng (dvdng) the covarance by the product of the ndvdual standard devatons Covarance and Correlaton Correlaton coeffcent vares from -1 to +1 Covj rj = σ σ where : r = the correlaton coeffcent of returns j σ = the standard devaton of R j j σ = the standard devaton of R t jt 10

11 Correlaton Coeffcent It can vary only n the range +1 to -1. A value of +1 would ndcate perfect postve correlaton. Ths means that returns for the two assets move together n a completely lnear manner. A value of 1 would ndcate perfect correlaton. Ths means that the returns for two assets have the same percentage movement, but n opposte drectons Correlaton Coeffcent Parameters vs. Estmates Unfortunately, no one knows the true values for the expected return and varance and covarance of returns These must be estmated from the avalable data The most basc way to estmate these s the naïve or uncondtonal estmate uses the sample mean, sample varance, and sample covarance from a tme seres sample of stock returns Typcal tme seres used s last 60 months (5 years ) worth of monthly returns More sophstcated methods for estmatng these wll be dscussed n subsequent chapters 11

12 Estmaton of Average Monthly Returns for Coca-Cola and Home Depot: 001 Computaton of Monthly Rates of Return Closng Closng Date Prce Dvdend Return (%) Prce Dvdend Return (%) Dec Jan % % Feb % % Mar % % Apr % % May % % Jun % % Jul % % Aug % % Sep % % Oct % % Nov % % Dec % % E(RCoca-Cola)= -1.81% E(Rhome Depot)== 1.47% Exhbt 7.4 Estmaton of Standard Devaton of Returns for Coca-Cola and Home Depot: 001 Coca-Cola Home Depot Date R - E(R) [R - E(R)] Rj - E(Rj) [Rj - E(Rj)] Jan % % Feb % % Mar % % Apr % % May % % Jun % % Jul % % Aug % % Sep % % Oct % % Nov % % Dec.01.% % Varance= / 1 = Varancej= / 1 = Standard Devaton = / = Standard Devatonj = / = Tmes Seres Returns for Coca-Cola and Home Depot: % 0.00% 15.00% 10.00% 5.00% 0.00% -5.00% % % -0.00% Jan.01 Feb.01 Mar.01 Apr.01 May.01 Jun.01 Jul.01 Aug.01 Sep.01 Oct.01 Nov.01 Dec.01 Coca - Cola Home Depot 1

13 Scatter Plot of Monthly Returns for Coca-Cola and Home Depot: 001 Home Depot 5% 0% 15% 10% 5% 0% -0% -15% -10% -5% -5% 0% 5% 10% 15% -10% -15% -0% Coca-Cola Estmaton of Covarance of Returns for Coca-Cola and Home Depot: 001 Date Return (%) Return (%) R - E(R) Rj - E(Rj) [R - E(R)] X [Rj- E(Rj)] Jan % 5.50% -3.01% 4.03% Feb % % -6.76% % Mar % 1.51% -1.69% 0.04% Apr.01.8% 9.8% 4.09% 7.81% May.01.6% 4.65% 4.43% 3.18% Jun % -4.08% -.87% -5.54% Jul % 6.63% 0.9% 5.16% Aug % -8.70% 10.94% % Sep % % -1.56% % Oct.01.0% -0.36% 4.01% -1.83% Nov %.16% 0.7% 0.69% Dec % 9.35%.% 7.88% E(RCoca-Cola)= -1.81% 1.47% Sum = E(RHomeDepot)= Covj = / 1 = Corr(j) = Covj / (stdev() * stdev(j)) = Combnng Stocks wth Dfferent Returns and Rsk Assets may dffer n expected rates of return and ndvdual standard devatons Negatve correlaton reduces portfolo rsk Combnng two assets wth -1.0 correlaton reduces the portfolo standard devaton to zero only when ndvdual standard devatons are equal 13

14 Combnng Stocks wth Dfferent Returns and Rsk Asset E(R ) Case Correlaton Coeffcent Covarance a b c d e W σ σ Constant Correlaton wth Changng Weghts Asset E(R ) 1.10 r j = 0.00 Case.0 W 1 W E(R ) f g h j k l Constant Correlaton wth Changng Weghts Case W1 W E(R) E(Φport) f g h j k l

15 - Portfolo Rsk-Return Plots for Dfferent Weghts E(R) 0.0 Wth two perfectly correlated assets, t 0.15 s only possble to create a two asset 0.10 portfolo wth rskreturn along a lne 0.05 between ether sngle asset R j = Standard Devaton of Return 1 Portfolo Rsk-Return Plots for Dfferent Weghts E(R) 0.0 Wth uncorrelated 0.15 assets t s possble to create a two j asset portfolo wth 0.10 lower rsk than k ether sngle asset 0.05 h g 1 R j = 0.00 f R j = Standard Devaton of Return Portfolo Rsk-Return Plots for Dfferent Weghts E(R) 0.0 Wth correlated 0.15 assets t s possble to create a two j asset portfolo 0.10 between the frst k two curves 0.05 h g f R j = R j = R j = Standard Devaton of Return 15

16 Portfolo Rsk-Return Plots for Dfferent Weghts E(R) Wth R j = f 0.0 negatvely g correlated h assets t s 0.15 possble to j R j = create a two k R 0.10 j = asset portfolo 1 wth much R j = lower rsk than ether sngle asset Standard Devaton of Return - Portfolo Rsk-Return Plots for Dfferent Weghts E(R) 0.0 R j = R j = f R j = k R j = R j = 0.00 Wth perfectly negatvely correlated assets t s possble to create a two asset portfolo wth almost no rsk j Standard Devaton of Return h g Exhbt 7.13 The Effcent Fronter The effcent fronter represents that set of portfolos wth the maxmum rate of return for every gven level of rsk, or the mnmum rsk for every level of return Fronter wll be portfolos of nvestments rather than ndvdual securtes Exceptons beng the asset wth the hghest return and the asset wth the lowest rsk 16

17 Effcent Fronter for Alternatve Portfolos E(R) Effcent Fronter B Exhbt 7.15 A C Standard Devaton of Return The Effcent Fronter and Investor Utlty An ndvdual nvestor s utlty curve specfes the trade-offs he s wllng to make between expected return and rsk The slope of the effcent fronter curve decreases steadly as you move upward These two nteractons wll determne the partcular portfolo selected by an ndvdual nvestor The Effcent Fronter and Investor Utlty The optmal portfolo has the hghest utlty for a gven nvestor It les at the pont of tangency between the effcent fronter and the utlty curve wth the hghest possble utlty 17

18 Selectng an Optmal Rsky Portfolo ) E(R port U 3 U U1 Exhbt 7.16 Y U 3 U U 1 E( σ port ) X Estmaton Issues Results of portfolo allocaton depend on accurate statstcal nputs Estmates of Expected returns ( n estmates) Standard devaton ( n estmates) Correlaton coeffcent ( [n(n-1)/] estmates) Among entre set of assets Wth 100 assets, 4,950 correlaton estmates Wth 500 assets, 14,750 correlaton estmates Estmaton rsk refers to potental errors Typcally only have between 60n and 60n observaton data ponts from whch to obtan estmates Estmaton Issues Wth assumpton that stock returns can be descrbed by a sngle market model, the number of correlaton nputs requred reduces to the number of assets, plus one Sngle ndex market model: R = a + b R + ε t mt b = the slope coeffcent that relates the returns for securty to the returns for the aggregate stock market R mt = the return for the aggregate stock market durng tme perod t t 18

19 Estmaton Issues If all the securtes are smlarly related to the market and a b derved for each one, t can be shown that the correlaton coeffcent between two securtes and j s gven as: σ m rj = bb j σ σ where σ m = the varance of returns for the aggregate stock market j Implementaton Issues If portfolos on the effcent fronter are optmal, why don t all nvestors use Markowtz portfolo optmzaton? Three key problems complcate mplementaton: 1. Too many nputs requred. Use of estmates can lead to error maxmzaton 3. Relance on hstorcal data to obtan estmates Implementaton Issues 1. Too many nputs requred Lmt use to asset allocaton or small-scale problems Use factor models to obtan / develop correlaton estmates Allows for use wth much larger scale problems E.g., all 1700 stocks that Value Lne follows Smplest factor model s the sngle-ndex market model 19

20 Implementaton Issues. Use of estmates can lead to error maxmzaton Introduce addtonal hard constrants n optmzaton process E.g., Haugen, p.?? Use stratfed samplng Precludes optmzed portfolo from beng too dfferent from the benchmark portfolo Use portfolo resamplng to fnd average optmal portfolo gven range of possble estmates Implementaton Issues 3. Relance on hstorcal data to obtan estmates True parameters not only never known, but also not constant over tme Factor models and stratfed samplng help here, too Factor model relatonshps tend to be more stable than relatonshps between ndvdual stocks Stratfed samplng constrants prevent portfolo from fallng too far behnd benchmark even when estmated relatonshps change Mnmum Varance Portfolo (MVP) 1. Two-stock case: σ = w σ + w σ + w w Cov port Multple stock case: σ port n n n = wσ + ww jcov = 1 = 1 j= + 1 j = w Σw 0

21 The Internet Investments Onlne Future topcs Chapter 8 Captal Market Theory Captal Asset Prcng Model Beta Expected Return and Rsk Arbtrage Prcng Theory 1