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If: return were 15 percent instead, we would expect the stock 3 return to be 10p difference in the stock’s expected return can be attributed to the difterence of the market from one period to the other. . I _ . U , ‘ rElie second type of event that produces variability a security s return m- Eactor model is micro in nature. Micro events have an impact on indiwdual ti fire or the resignation or death of a key I vidual firm alone. They are assumed to have no effect on other firms, and the. effect On the value of the market portfolio or its rate of return. Micro events the rate of return on the individual security, however. They cause the stock: duced by the market portfolio in the period. Micro events, therefore, are, p cause the appearance of residuals or devrations from the characteristic lin Other types of events have been assumed away by the modei. One ragga ferred to as an industry event, an event that has a generalized impact on in firms in a given industry but is not broad or important enough to ]have a fizgmfi impact on the general economy or the vaiue of the market portfo io. Ev. ‘ nature also may, conceivably, cause the appearance of a resrdual, but the smgl model assumes residuals are always caused by more events. I. The foregoing scenario is consistent With the assumption that the resid shock terms for different companies are uncorrelated With one another, as d fig Figure 6.2. The residuals will tie uncorrelated if they are caused by micro c it affect the individuai firm alone but not other firms. _ I As noted, the singie—factor mode! assumes all the numbers in the :3 matrix for the returns on securities can be accounted for by the fact the: .e_ stocks responds, to some degree, to the puii of a Single commonfacilor, (to; fact, given the assumption of the singiewfactor modei, we can write e _ tween any two securities] and K, as GENO}, rig) = BJBKOEG’M) The right-hand side of this equation is the product of'three terms. "the thirst: variance of the rate of return to the market, specifies the magnitude 0 if, movement or the strength of its pull; the first two, which are'thc beta Ergo: securities, specify the extent to which each of the two securities respon s THE SINGLE-FACTOR MODEL’S SIMPLIFIED FORMULA FOR PORTFOLIO VARIANCE ‘fil . . . .. -a On the basis of the foregoing assumptions and conditions, we can derive a: ancill— . formula for the variance of a portfolio. We begin by noting that after pas;If best fit through points representing pairs of returns betwsen security or po ‘ ‘ ‘ ' - 11 W0 1 Alternativciy, you may want to account for the macro factors mdiVidualiy. Endtillfilfsa:§a};l;mdu multiiactor model with factors such as changes in interest rates, inflation, an in . CHAPTER 6 Faeroe MODELS ‘i 3'? Residual for stock J in time t 5-1,! 8 0 9 . r 9 . .b . . 8 fiestdual for Stock K .I . a in time! I GK; 0 ,i o o , - a 0 o and market returns, as in Figure 6.1, we can always split the variance of the return on a security or portfoiio into two parts: ‘52 (1’ } = 3262 UM) 4“ Ci2 (6) .1 Total variance = Systematic risk + Residual variance (6 ) The first term on the right—hand side of Equation (6.1) is called the systematic risk of the investment. Under the assumptions of the singie-tactor model, it accounts for that part of the security‘s variance that cannot be diversified away. This part of the security’s variance is contributed to the variance of even a well-diversified portfolio of many different stocks. The second term is called residual variance or unsystematic risk. It represents the part of a Security’s totai variance that disappears as we diversify. Be cause of residual variance, the variance of a portfolio is iess than the weighted average of the variances of the securities in the portfolio, We can see from the equation that variability in return to a stock may account for two components. The systematic risk accounts for one part of the total variability This part results when market movement puils the security along its characteristic line. Note that systematic risk itseif is the product of two termsThe first term involves the secu- rity’s beta, which tells us the extent to which the security responds to the up and down pull of the market. The second term is the market’s variance, which tells us the extent to which the market is pulling up and down. The remaining component of a security’s variance is the residual variance. This accounts for the part of the variability that is due to deviations from the characteristic line. Thus, when we think of the total variability in a security’s returns under the single-factor model, part of it is due to movement by the security along its characteristic line and part of it is due to deviations from the charac- teristic line. 138 PART ll PGRTFDLEO MANAGEMEN? Equation (6.1) hoids for both an individual security and a portfolio. Rem, equation for the case of a portfolio, We get 020‘?) m sithM) «i 020;?) At this point, we need equations for the beta factor and residual Variance? folio as functions of the characteristics of the securities we put In the poms“; we have these equations, we can substitute them for portfolio beta and resrdu ance and obtain a more simple, alternative expressron for portfoho variance :- findin the minimum variance set. I o ' _ . I 'I'lige beta factor for a portfolio of M securities rs a sunple weighted average of the stocks in the portfoiio, where the weights are the relative amounts inv each security. M B?” Z x131 lei Portfolio beta = Weighted average of security betas Thus if we have two stocks, one with a beta of 1.00 and the other with a betad and we invest 75 percent of our money in the stock. With the larger beta and 25 ' k the ortioiio would have a beta of .75. . . m thlflgwhgrigigde; the Fforrnula for the residual variance of a portfoho. To date what the residual variance is, we can use the same procedure we used to gate. variance of the portfolio’s returns (as opposedto its reszdutrls) in the mar owl That is, we could employ the covariance matrix for the reStduals or; :1 various For the case of a three-security portfolio, the matrix would look i: e t is. $4 In xc Security A B C xA A 62(8)») va(sa,sA) gfleosfij} xs 3 cm (3.4, 5n) '3 (Ba) 2 Visas 96 xc C COV (54: 5c) 00" (55': 3c) 5 (8c) Now the assumption of the single-factor model comes into play. The (EV between the residuals on any two securities is assumed to be equal to zero}5 :1 assumption, all the covariances in the matrix that are off the dragforlta ‘2; m equai to zero. This means, to compute the resrdual variancp of a por; o to, ac go down the diagonal of the matrix, taking each security 5 resgiua vagrath : tiplying it by the portfoiio weight at the top of the column an again hits are: weight at the left-hand side of the row. Because both these two wing 5m ‘ the portfolio weight for the security itself, we have the follong re 3 i011 p. M o2 (a?) m 2: x1262 (of) M1 Portfolio residual “Weighted average” of security residual d variance = variances where portfolio weights are square Thus the residual variance of a portfolio is also a weighted average ($150215), residhal variances of the securities in the portfolio. However, this tune, 1n average, we square the portfolio weights. CHAPTER 6 moron MODELS 1 39 Given the assumption of uncorrelated residuals among securities, the residual vari— ance of a portfolio begins to disappear as the number of securities in the portfolio is increased. Consider the residual variance formula, and suppose we have a large num- ber of securities, each with a residual variance equal to 10 percent. If we invest half our money equally in two of the securities, the residual variance of the two-security port— folio is 5 percent according to the foregoing formula. 5203,19) ""z (-502 X .10) + (.502 X .10) m 35 in the same sense, if we invest a third of our money in each of three of the securities, the residual variance oi the portfolio would be 3.33 percent and so on, as shown by the solid curve in Figure 6.3. 4 1 ’ As we diversify, the residual variance of the equally weighted portiolio apm preaches, but never quite reaches,izer'o.'1l1is is because the residuals in the portfolio are presumed to be uncorrelated, and the good individual things that are happening to some of the securities are being offset by the bad things happening to others. Some are above their characteristic lines, but others are below; the rcsiduai of the portfolio, being the average of the residuals of the individual securities, can be expected to be quite small if the number of securities is iarge. In fact, when we are dealing with a portfolio that is weighted equally among the various securities, the residual variance of the port- folio is equal to the average residual variance of the stocks, divided by the number of securitiesi‘in the portfotio. Of course, as the residual variance of the portfolio gets smalier and smatler, the correlation of the portiolio’s returns with the market gets larger and larger, as shown in Figure 6.4. These relationships depend crucially on our assumption that the residuals for difw ferent Securities are uncorrelated. SuppoSe this is an invalid assumption. Suppose that industry-type events frequently occur, and the covariance between the residuals fur Portfolio residual variance % (SI “Noah Carleech ""‘--—~.—.......... ‘~ Gov (ex. ex} = o I'l- k‘h‘m CGVlEJ,€xl<0 _u .1. .2. .J_. rm, 11:1:th 41w$__i-M 3 4 5 5 7 3 9 10 11 12 13 Number of stocks in portfolio 140 Portfolio coefficient of determination PART H PORTFOLLO MANAGEMENT 1.00 L..J L4 1.. I _..L L___l. 1 wt. 5 .J. I i Lml L_l 8 10 12 14 16 28 d 2 4 6 - . a Number of stocks in portfolio ‘ ' ' -dia different securities is typically posztzve‘and notzero. In tcll'ns ttlhepiifitiveg ments of the matrix for residual covariances Will be orn.dual ‘yaflance an ' -' del formula for port 0 1o res: ‘ If we follow the single factor mo I A d “mate the mm restéual ‘ l of the matrix, we Will un eres _ go down the diagonal ‘ l ' use will be larger than the angle-fact e ortfolio.The actual residue vana ' _ ' I ' I e toili: usp it is based on its assumption, becaus: 1t gs éhmoficggagcfid , i . ' g at are off the diagonal. The relations tp ' bmk :21 timber of securities in the portfolio may really look like the upper Figu§:§piise on the other hand, the covariances between Elie fifsidusaésff‘: tt 1 ' ' ' . This might e e ca ‘ ’ ’ o ulation are typrcally negative ' US- Fflsidg bigcgmfaanies that are very competitive. In this case, anby evgrfift t1:11: d. :31- one of the companies is negative for the edger. If thrilnulrlrgggisve the Sin ’ ' re ormnau y , riance matrix for the reszduals are p ' _ ‘0- :igcfeivgaives an overestimate of the true resrctual variance. of 51:6 p:;:;o:eme residual variance, obtained by summing the products obtaine dim the déa . matrix will be smaller than the sum Detained bydsgnply of securities if” ’ ' ' A ’dual variance an e nu - ctual relationship between res: I I folio might then look like the lower Frokentflirlle osf Siweighmd averag ' ' or o 101 To summarize, the beta factor 0. a p i the ff betas of the securities in the portfolio, where the weights are quSLItothe Sing ur money we invest in each security. The restdual varianpr ums time in _ :IlOdBl is assumed to be glyen by a similar weighted average, at 1 . ' ' ‘ hts. _ we 5 care the portfoer weig . the averflg’rficnowqthat in the context of the Singlefactor model we can riSk a an investment, including a portfolio, into two components, sys e have defi naivarlance as in Equation (6.2). Substituting the expressrons w CHAPTER 6 Faeroe MODELS 141 portfolio’s beta and residual variance, we obtain the sin gist-factor model’s simplified formula for portfolio variance: u M _ M 020'?) 2 2 x1 [if 02(rM) + Z xfozflsj) J=1 1:1 Total portfolio W Portfolio systematic + Portfolio residual variance m risk variance However, the reduction in the complexity of the model comes at a price. As we said before, the variance number'obtained from the Markowitz formula is perfectly accurate, given the accuracy of the covariance estimates. The model makes no assump- tions regarding the process generating security returns. The single-factor model, on the other hand, assumes the residuals, or deviations from the characteristic line are un— correlated across different companiesfl‘he variance number obtained from the single— factor model, therefore, is only an approximation of the true variance. Even if the estimates of beta and residual variance that we feed into the model are perfectly accu— rate, the estimate of portfolio variance we obtain from the model is only as accurate as our assumption regarding the residuals. It is obvious that the assumption isn’t strictly accurate. After all _ , suppose some- thing good happens to General Motors. This has an immediate im pact not only on General Motors itself but also on the company’s suppliers and competitors. Many com- panies would be affected simultaneously, some positively and others negatively. The residuals that appear for these firms would not be independent but rather would be generated by a common event. We know, therefore, that the residuals are correlated to some degree. We hope, however, that the degree of correlation is small enough that the inaccuracy of the single»factor model’s portfolio variance equation doesn’t transcend its relative efficiency. AN EXAMPLE WHERE THE SENGLE«FACTOR MODEL WORKS Consider two hypothetical stooks, Blue Steel and Black Rubber. In Table 6.1 are the rates portfolio, and for an equally weighted port- e. The two-stock portfolio is assumed to be g of each period. Given this, the return for tocks in each period. 0 are plotted against the returns 2 the beta factor for Blue Steel is of return for these companies, for the market folio of the two stocks for five periods of tirn rebalanced to equal weights at the beginnin the portfolio is a simple average of the returns to the s The returns for each stock and for the portfoli for the market in Figures 6.5, 6.6, and 6.7. Note the Marlee: Porg‘bh'o Blue Steel TwowSmck Portfolio Period 17,, rs I]: 1 30% 30% 42.5% 2 40 60 50 3 20 50 40 4 35 45 27 .5 36.25 5 25 15 22.5 18.75 142 PART 11 PORTFOLIO MANAGEMENT Return to 8§ue Steel 0 F's ° / 50% / /.9 40 / / / a 30 / / 20 // / o / 3 // L... 11...... L__.. to... L._ ,M 4%” fit)- 0 10 20 30 40 50% . 110;. y“ I I // Market return fight-m 5% F Return to Black Rubber f'n B ,u’ 50% Ii / / / f .a' 40'- x / / 30 ’13/ o r f , 2 .. 29/? /.x / I 10 l— / / / l .....l. J— l Lam” l _.....l NJ... 1 m ,- “20 ""10 0 10 20 30 40 SD 60% Market return ~ I Freya q I; 00... ’ :01 e 01. M ' l to .50 and the betaf 1.00, the beta for Black Rubber 18 aqua , I ;§::It;1: average of the two, or {15.1310 Intercept of the portfolio (15 the weighted average of the intercepts on Blue Steel (10 percent) an ercent . I ' (20 iilecall groin Chapter 3 the general statrsttcal proceélure forf 0:111:35, ’ ' ‘ ' betWeen the actua rates 0 re variance: First compute the dtfierences _ d e W - t the investment to pro 00 ,gl . ment and the rates of return you expat: I ‘ an teristie line and the market return for the period. The difference for 3" woulcf be equal to if; " (44+ BrMJ) .\‘ CHAPTER 6 FAgToR MODELS Return to portfolio 0' / 9/ O 00% e / / o / / 30" x // 20!— ,u/ o (b ". .24 1 I: I .iV ' x A . 14 1. l...“ L. J. J [m .1 -—20 W10 0 10 20 39 4G 50 66% Market return ’54 143 The differe’iices for each period are then squareé and the squareci eiffer'ences summed. The sum is divided by the number of periods observed, less 2. Therefore, the residual variance of Blee Steel can be competed as [.30 w (.10 + 1.00 x 30)]2 + [.00 - (.10 + 1.00 x 40)]? + [.50 — (.10 + 1.00 x 20)]2 + [.45 m (.10 -1~ 1.00 x .35);2 + [.15 m (.10 + 1.00 x .25);2 .1000 .1000/ (5 — 2) a: .0333 and the residual variance for Black Rubber as [.55 w (.20 + 0.50 x 30)}2 + [.40 ~ (.20 + 0.50 x 40)]? + [.30 m (.20 + 0.50 >< 20)]2 4- {.275 — (.20 + 0.50 x 55)}2 + [.225 — (.20 + 0.50 x 25)]2 .0600 0600/ (5 — 2) = .0200 The residual variances for the portfolio are given by (.425 — (.15 + 0.75 >< 30)]2 + {.500 — (.15 + 0.75 x 40)}2 + [.400 ~ (.15 +0.75 >< 20)}2 + [.3625 — (.15 ~a~ 0.75 x 35)]2 + [.1875 — (.15 + 0.75 x 25)}2 W .0399 0399/ (5 m 2) = .0133 CHAWER 6 FACTOR MODELS ’l 45 9GR1’FOL10 MANAGEMENT t Mt PART 11 forms to the value predicted by the sing} The portfolio’s residual variance con modei, a weighted average of the residual variances of each steak, where we portfolio weights. .0133 = (.50)2 >< .0333 4- (.50)2 >< .0200 o the correlation 95 Under the assum ‘ ' Ption 0f the sh 1 mated by oin d . ‘8 3"factor model, t; ‘ - g g own the diagonal of the covariance Eaiiiszicjfzaltgarlance can be esti» 1' e resid 1 Z W 2 LEE! 5. G (a?) "" xu X 020%) + x; X 02(83) 032 = .25 x .0732 + .25 x .0548 ecause the example was constructed 3 duals was equal to zero. AN EXAMPLE OF A POTENTEAL PROBLEM WITH THE SINGLE—FACTOR MODEL with the single-factor model, consider sh To illustrate the potential problem example. Suppose we have two stocks, Unitech (U) and Birite (B).The 5th following characteristics: Beta Residual Variance Unitech 0.50 .0732 1.50 .0548 Bitite Market index variance .0000 the variance of the two stocks can be written a. sks and residual variances: <52 (r1) ” 0362010+ 6% (er) Unitech .0882. e .502 x .060 + .0732 .1898 = 1.50?— >< .060 + .0548 That happens 0 between the resl for the residuals flap) : .032qu x .50 x .59 x101“ = 0392 Under the assum .- .. . H . Won of tee Sin' ’ the equally weighted purtfofia asgie {actor model, we would estimate the variaa 1? cc 0' 2 I? i o (rp) fill: X 520M) + 62(sp) .092 m 1.00 X .060 "l- .032 This is really an underestimate f . we use the Markowit - . 0 the {We Portfolio ‘ - 2 tech . , Var:aace.To ‘ . reterns by the portfolio Weilgggegmffitplying each element in thélgcdvih? true variance, r - 3 {WC swam riance matrix of X .50 X + 50 X -50 X .0594 + .50 x .50 x .0594 W .0992 The difference I) ctwce i . factor model (SFM) ésiggfiéual Portfolio variance and our estim t 0"" underestimate a t: using the s' of the residual . Ingle- variance. Given this information, their respective systematic ri Birite auce matrix for the rates of return to the Unitech Birite a. E 0 3‘9 0 n 7:: V3 p. U) a: U} V) E fl m G.- .n The covari Markow' - Itz vargan __ , Ge SFM Varlance Stacks Unitech .0882 .0594 Bizite .0594 .1898 0992 ‘ ~ .092 Actual residuai v ' m .0072 arzance - SFM resfl ,_ .0392 u .032 1 “31 Vanancem 0072 the covariance between gle~factor model, of the duct of their betas and the variance Interestiri Ely most r ' ’ P Ofessmnal managers focus on optimizing t k mo i313 ermr in rotation [G a gt . ~ 00k mark ' TIElelng error can be tailgfldex, suchas the S&P 500 rather tha - - Tatum to your target market .asdthe differences between your 0 ritvcilanmy Of remm‘ . m ex‘ In 0p timiZiflg. managers attifriptotho return and the - 0 minimize track- Under the assumption of the sin the two stocks is equal to the pro C0V0'Ln r3) = BUX 03X 020M) .0450 = .50 X 1.50 X .060 fer than thi . The actual covariance between the rates of return is grea I or the two stocks are positively correlatedThe any means the residuals f for the residuals is in fact assented to be given by Stocks United: Bit-ire gliittZCh In the Singlecf M _‘ ‘ Single factorasgglelrigrorizhlzve aunbute the covariances betwee tad Portfouo 0f the two :10 two 01" more factors, 3: atmdex 1“ 5‘ multifactor model weir:1 ebremms on “mks to . 0W1] EGgetI-mr because thefigzeifor example, we assume stoaks It'l ate the covariances Siinultaneflufly m50000ng to two :30: gigs: :13 and ‘ actors Now suppose we form an equally weigh factor of the portfolio is given by 5;»:qu 50+xsx BB 1.00 = .50 X .50 + .50 X 1.50 PART ii PORTFOLIO MANAGEMfiNT M. where 13“ is the stock’s inflation beta. It measu ‘ - pected .changes in the rate of inflation. The terrfzfli: trifespuorigf 2f the St industriai production in any given period, and [3 11 measures it: :iecg ’ unexpected changes in the growth rate in indusirial production “inks: changes because the price of the stock is likely to be affected oni b 6 8a -. trial production not aiready anticipated by investors and discouiitdhc'han the stocic. Just as, in the context of the single-factor model the beta 1’ Into-i by relailng the returns of the stock to the returns to the market ind Elmer Is of prevrous periods, so in a multifactor model the betas can be estiex over a I the stock’s returns both to the unexpected change in inflation and Said ‘- growth rate in industrial production. One way of obtaining numbt e um ter series is to take the difference between the actual! rates of infiatioiiZing)F industrial production and the average rates forecasted by some group sional economists. 6 Similar to the procedure used in Cha ' I _ pter 4, With our two-factor d risk of a portfolio of securities can be computed using the fo ‘ m0 6} With our two—factor model, you would compute four products. Add them up aid you have an estimate of the amount of portfolio variance that‘s attributabie t0 the portfolio 5 responses to movements in the two factors. With more factors s'ni ll" 3' I have a larger matrix and more products. ’yoa E I” C By examining the matrix, We can see the advantage in using factor models to mate the risk of a portfolio. Suppose, instead of a factor model you simply use a pasi series of returns to compute the covariances of returns between siocics. On the one hand. 7. the more months into the past you use, the iower is the sampling error for your estimate- E- _ 0n the other -‘ it is that she have a problem. ‘ _ ‘ A chance that the return observations are irrelevant to the current Situation.2 CHAPTER 6 FACTOR MODELS 14'] hand, the further into the past you go in sampiing returns, the more likely nature of the firms behind the stocics has significantly changedThUS, you Going further back in time reduces sampiing error but increases the However, if you empioy a risk factor modei, you can cffectiveiy address this problem. The nature of the underlying macroeconoiny is probably iess subject to dramatic change than is the character of an individual firm. if you believe this, you can estimate the infla- tion and industrial production betas {which are firm dependent) for the individual stocks in your portfolio over a relatively short period into the past and the variances and covari— ances between inflation and industrial production over a considerably ionger period.3 To obtain an estimate of total portfolio variance, add, to your estimate of system- atic risk, your estimate of residual vafiance. if you assume that you have accounted for all sources of the correlations of return with the various factors in your model, the corw relations between the remaining residuals can be assumed to be zero, and portfolio residual variance can be calculated in accord with Equation (6.3). To simplify matters, assume the residuais are uncorrelated and, in addition, that the rate of return to the market and the unexpected growth rate in industrial production are also uncorrelated with each other. Given this, we can Write the variance of a portfolio of stocks as carpi = a: P sis) + [52' p (52 (g) + 6%,.) Total = Systematic risk Systematic risk Residual (65) variance (inflation) (industrial production) variance The inflation beta for the portfolio is again a weighted average of the inflation betas of the stocks in the portfoliofl'he portfolio’s beta with respect to unexpected changes in industrial production is also a weighted average. If we now assume the residuais on any two stocks are also uncorrelated with each other, as with the singic—factor modei, we can write the residual variance of a portfolio as M 02¢?) a Z )cfo2 (SJ) J==1 The final equation for residuai variance is based on the presumption that we have now fully considered ali the factors that account for the interrelationships among the returns on stocksThis being the case, the residuals for different companies will now be uncorrelated. If we should find, to our dismay, that the covariances between the resid— uals are stiii significantly different from zero, we haven’t taken into account ali the rei- evant factors. We need to move to a trifactor model or beyond. The search ior such factors is now a matter of intense interest among practitioners in investments. The best evidence to date seems to indicate that the covariances among stock returns can be explained by as many as four or five factors. I What about increasing the number of observations by shortening {he return period to, say, a day rather than a month? The problem will: this is that the returns may become uonsynchrcnous with each other. Most stocks don’t trade continuously during the day. If your returns are measured from close to close, you may have a probiern. The closing price on one st0ck may be from a trade at midday; the close on another from the East minute of the trading day. There may have been a significant change in the general levci of stock prices between the final trade on the one stock and the other. 'ihus, the returns aren‘t really comparable, leading to an error in your estimate of the covariance between the two stocks. 3 it should he noted that if you estimate the betas and varianccslcovariances over the same period, you wili find that you compute the same answer for portfolio variance as you would if you simply computed the variance directly from the returns to the proposed portfoiio during {he period. CHAPTJER 5 FACYOR MODELS 149 148 PART 11 PORrFOLlO Manaesmsnr * IN THE WEE $MALL HOURS Ableary~eyed David Olson drains the last of his Coke tually increases in price by 1 percent for the mom turns his attention to his com— the unexpected change is taken to be 3 percent. Cm is a dummy variabie, which takes on a value of one {or 1987 and zero otherwise. _ from its can and then re 1%., and David has been writ- the month of October pater terminal. "it’s 4:00 A. mg code for 17 hours without a significant break. As and his mind slowly turns to curly factor model is estimated by regressmg the factors an the monthly returns to the the hours pass, try. For some countries, a particular factor may 1101 an important determinant of the countries' rel Therefore, the {actor is not included in the coum factor model. Canada, Malaysia, spaghetti, David fights to “straighten out the strands” ocess logical thought. so that they might continue to pr David is director of systems development at Haur gen Custom Financial Systems. His firm designs and builds customize ftware for financial banks, and insur— Mexico, Singapore. and d financial so the North American '. consulting firms, brokerage houses, ance companies. The software is used by these instituu United States are ail in tail to their clients nomic region. Apparently, Malaysia and Singapor included because of their tr :1 analysis or to re for asset alloca— d develOps systems measurement, and the risk man— agemcnt of equity portioiios. His bachelor’s degree in computing science and his MBA. with emphasis on finance give him the combination of skills required to bring the compiex toois and theories of modern in« vestrnent theory into the hands of the professionai in— vestor. in addition knowledge of the to a thorough principles of finance, res the skills to David’s job requi put these principles to work writing user—friendiy. in- uter codes. tions for their ow as a service. Davi countries. North American each country are t each factor and the flex. It the relations sign is in boldface. A ship. Thus, the returns month are positively re change rate relative to a weighted average of the countries of the region in the same month and vious month. R2 indicates the hip is statistically signitican n L indicates a lagged. rel to Canadian stocks in 3 fraction of a count: lained by movements teractive comp David is working at 4:00 A.M. because he’s under a tight deadline. He is on the finai stages of a particulariy returns that can be exp compiex international asset allocation modei. gion’s factors. Thus, Canada shows the strong, The model is based on a factor model that divides tionship to the factors, and Mexico the weakest} the World into three economic regions: North Amer— Interestingly, in the context of the mo ica, Europe, and the Far East. Each region has a set of Kong and Australia are considered to be merit mp— the European economic region. and Japan Sta i the Far East economic uch as aggregate consu For each month. the d by taking a weighted countries of the as the only member 0 ah the other major co The stock returns of are more closeiy reia this geographic region factors of other economic regions. Once the factor models are identificd.t need in the following ways: factors for the fortitw macroeconomic factors, 5 tion and industrial production. value of any given factor is foun ayerage of the factor across the region. The weights are based on the level of total consumption in each country. A given country is as monthly returns to signed to a particular region it the ck index are l. The values for the its capitaiization-weighted common sto found to be best expiained by changes in the region’s macroeconomic factors. year can be estimated using time son The table shows the factor models for the North Of course, time series models are not mic region. The factors are listed in in forecasting uneXpeeted changes in factors or the returns to securities. S dictably, but thei American econo the left column. Most of the Unexpected changes are based on realized deviations from the predictions of a statistical model. For exam- ple, if the statisticai modei predicts that the price of oil will Eaii by 2 percent in a particular month, and oil ac~ m are self»exp1anatory. tend to move unpre merits can be used to reliably prod“?C like the price of oil and When time series foreca for ' magi: 2:23;: are imported to the factor , c returns to tit ‘ in m I e various {1 " the szirglgiimlng year can be estimatesgtiuies e sensitivities t ’ { I 0 th estimated in the factor model a varlables 4 2, ibfilogerlage houses, etc.) ‘ o e I I ‘ mists who specialize in foiSeecfausltiiiav: acme— ; M ggaphlc regions of the worldfheigi it: geo‘ ‘ (I: the hfuture vaiues for the factors sagas“ ' r I roug t to the model and transiat d ' 6 forecasts of future expected rein 8 mm by country. m 3, Expected returns are estimated b omimcgze: 2: Zghe model 's meanwvariarice of cwmfies n , e used to buiid portfolios _ given velatiuwith marrimumfexpected return is a maior progggfilii’: vom'fimy 0f mmm heca use the volatilities (jiggling are ing 1 of th ‘ «a . in North An: hi h_ I. . griargfifayslgigity Of MBXico’ for exampic is . rcent. c0 ’ 20 percent in the United 3:221:23“ to 1355 than . In ot ‘ her parts of the internatiOnai asset allocation m odel the user mom _ , I can run ex ‘— seenasrfiostulatlng different world cod)de ' mm“ 1 s (time paths for the factors) Thmm Simmfécantfihnrque called Monte Carlo en, , e user can se ' mama I ~ :3 the likcl e v across $u§:::el:§il£t3kporlfoii03 (conssirliicrticd: . - . I eiy ortf l' ' IS caicu 'p (“o erf SCEISEENliatetd on the basrs of estimaied 3:322:11? ea h y omovements in the fact ‘ 010 c proposed scenario om'under David i s about three hours away from corn pletin g 31 hat i135 {flied the last font mfln h 0 i1 ' a Illod :3 t S t 13 WOZk ife. W" 1th a neon deadline that gives him the o p. portunit ' y for five hours of sleep. he wiii take adva t :1 age e couch in the next office. err‘can Economic Region Can Mal Mex Sgp US + _. _ __ _ +Lfl, +1.1 + “L3 HLI + + + + +L€l +L1 + + , +L1 ~Ll “L1 7 + +L1 + —L1 "L0, ~L1 + + + — +L1 + + .53 m w — W .30 25 H 2.03 ‘ .30 15.3 3.16 2.03 2.30 .2 5.2 7.3 12.} 150 E$T§MATING PORTFOL! . A MULTiFACTOR MODE]... Recall our earlier disco ple, we es PART ll ?OR?FOLIO MANAGEMENT O VARIANCE USING AN EXAMPLE ssion of Unitech (U) and Birite (l3) corporations. In the a}; . . . cks Unfilth the Vatia C y W lite E] “if 1 of the I) I. i. :1 e of an Bquall 61g 6 U 010 tWU 51 I ‘f ‘1 model The Single'faCi—Or mode} underestllnated the actuai Vang“ ac 01' . CHAPTER 8 Faeroe MODELS ll 5? If you recall, this is the answer we got when We computed the variance using the Mar— kowitz technique. We get the correct answer this time becauso the example has been constructed assuming a doubEe—factor framework. the single b t ortfolio ecause . tglii: factor was apparentiy inadequate for ex stOcks The actual covariance betWeen tiredtei variance predicted by the Single—factor mo . 2 OOV{rU,r3)> {flux 08 X0 rM .0594>.50><1.50X .06 ' ' he presence of (W ' esuluals is caused by t - o _ suppose the covariance between :hfaie of inflation and industriai production ' ' atcd Changes in m . - - - ' eduction an tors, 5?: {ifizliiigitocks with respect to the inflation and industrial pr betas or true residual variances are assumed to be 'tively correlated. A ‘ for the two stocks were post ha remduals piaining the covariance be ,_,. a m :3 § (‘3’ given by ‘ Residual Industrial Production Bela Inflation Beta 03 .20 . Unitech .50 1.40 '05 Birite 1.50 ed changes in industrial production is 6 person ' t tvarrance of unexpec , . . be 3 ercent. Ass'nrriizZtgfa the index of unanticipaie‘t inflat‘on '5 assumed to P varra The variance of each stock can be expressed as Systematic risk . . 1.~ = (Industrial production) + (Inflation) + Resrdua v + slain) + site) + 1.44 X .03 + .03 .16 X .03 + .05 if $8 Total variance 6%) a agate...) .25 x .05 2.25 x .06 + I ain form an equally weighted portfolio of these two stoc, gilt respect to the two indices is given by = .50 X .50 + .50 X 1.50 Unitech: .0882 = Birite: .1898 w Suppose we a portfolio’s betas w ' ' ‘ 1.00 Industrial production beta. Inflation beta: .80 = .50 X 1.20 + .50 X .40 at the residuals are now truly uncorrelatedf,:1;xe tr,“ be computed as the weighted average 0 _ h , C] P 1 h - Vfiflallces Of t 6 two StOCkS Where We 5 “are; the Oltfoho WE g ts ‘ 2 .05 m .02 Portfolio residual variance = .50? X .03 + .50 X Le ated as the sum of the two sys Given the assumption th ance of the portfolio can The total variance of the portfolio is estim terms and the residual variance. Systematic risk . . + 3 (industrial production) -i- (Inflation) 2 $62 + o {e Residll Total variance 2 + 2 r = etc s) (09(6): m 1.002 xeoe + .8on .03 + .02 jams arouses sea ESTiMA‘l‘iNG exercise RETgRN Securities have different expected rates of return. Across classes of securities, such as bonds and stocks, the evidence is consistent with the notion that the primary deter- minants of these differentials are“ differentials in risk. Short—term bonds are far less risky than stocks, and because of‘this, they carry lower expected rates of return. For classes of securities, sample means of past rates of return are a good starting point for estimating future expected rates of return. Analysts usually start there, and then they make subjective adjustments to the estimates based on contemporary economic conditions that are now different from the past. For example, in making an estimate of the future rate of return on stocks, you might want to consider thefact that the stock market is less volatile now than it was in the 3930s. Lower voiatility might induce investors to invest in stocks at lower expected rates of return. If this is the case, thegyrices of StOCiCS may have risen such that expected equity returns are lower now than they were in the 19303. Thus, you might want to adjust the long-term sample mean return on stocks downward to account for this. Similarly, since 1980 the volatilities of interest rates and the prices of long—term bonds have increased. Thus, the expected returns on long—term bonds may now be greater than their long- term averages. Whiie sample mean returns serve as a good starting point in estimating the ex- pected returns of classes of securities, they are poor indicators of differentials in expected return between securities within c class, such as common stocks. The past returns to individual stocks have been affected by myriad idiosyncratic events that are unlikely to repeat themseives in the future. In addition, the contemporary character of an indi— vidual firm may differ considerably from its character in the distant past. Because of this, we must set aside sample means and look for a better approach. Factor models can also be used to estimate expected returns. Here you try to esti- mate the tendencies of the market to produce differential returns (payoffs) to stocks with differential characteristics (exposures). You then project the future payoffs to the various characteristics and then relate them to a stock’s individual exposures to the characteristics to produce an estimate of overall expected future return. To illustrate the influence of one factor on expected return, over the long run the stocks of smaller firms have tended to produce greater rates of return.This may be due to the fact the investors want higher returns on small stocks because they consider them to be less liquid or more risky. In any case, the payoff to size has been negative over the long term. (The larger the company the lowor its return.) Thus, if you are esti» mating the expected return to a stock with low exposure to size (it is relatively small), its expected rate of return may be increased if you are projecting a continuation of the negative payoff to size. FIRM CHARACTERISTiCS (FACTORS) THAT iNDUCE DIFFERENTIALS lN EXPECTED RETURNS One can profile a stock, and the company behind the stock, by many different charac— teristics—risk, stock liquidity, and so on. In a given period of time, say a month, the 152 PART i1 ARBITRAGE The sound of the wind and the rain smashing against the large pane of glass 890 feet above downtown Los Angeles ramped up to the point of distraction. Dennis Bein looked up from his terminal to See several over- lapping sheets of water cascading down the window across from his desk. It was another Sunday in January. More work than usual this time because it was tirne to update the weights for ANALYTIC Investor's US. Long/Short Eq- uity portiolio. At least he wouldn’t he preoccupied by thoughts oi dropping another iii—footer into the cup on his favorite seventh green. ’Ihat cup was, undoubtedly, as flooded as the one he had left on the table this mom lug, outside the Oak Tree Cafe, just below the office. He was in the prooess of constructing two com- plementary stock portfolios. Complementary in the some that their monthly returns couid he expected to be highly correlated with each other. Dennis was em- ploying the Barra risk factor model to ensure that this would be the case. He adjusted the weights in each stock so that the “bets” on industries and sectors were very similar for each portfolio. At the same time, he was also making sure that the sensitivities (betas) with respect to the various Barra risk factors were not out of line between the one portfolio and the other. Dennis paid special attention to two of the factors in particuiarmthc {actor related to the monthly payoff to the relative market capitalization of the stocks in the portfolio and the factor rotated to the monthly payoff to the relative cheapness of the stocks Dennis knew that, no matter how carefully he constructed the two portfo- lios, it was practically impossible to reduce the tracking error {volatility of the differences in their monthly re- turns) to less than 3 percent on an annualized basisThis was true because that much of the tracking error could be attributed to market forces that were unstable and unmanageable. population of US. stooks will individually produce a wide array of returns.th the month, stocks of particular types will smaller rates of return. For example, especially in a month with strong markei. stocks that are riskier might ten In using an expected return factor model, we will measure every month and develop a history of the monthly payoffs to the V merits of a stock’s profile. From this history we will project the expected Va various payoffs into the next period.'Ihen, by interfacing the elements of ‘11 to find that, ii] pofih PORTFOLIO MANAGEMENT d to have larger returns. All of the stocks in both portfoiios were amen those included in the S85? 50!} Stock Index. They wage all fairly large companies Nevertheless, the process De ' ' his was using to construct the portfolios took into C03. sideration the likely market impr ct that his trades Eng ‘ have on the prices of the stocks he was buying and 5e ing. Dennis would e-maii his trade list to Bear Stems this evening. The trades would be executed early in the day on Monday. Execution was very important. You didnzf want to temporarily bump the prices up or down because the “rebound” would work against your performance I Although the two portfolios were matched in te, opposed in terms of Dennis’s optimizer contained estimates of the ex. pected return (for the next month) for each stock m the 3&3? population. These expected returns had be estimated Saturday morning using a separate echoted return factor model’I‘he model employed more than factors that profiled the characteristics of each sto Each stock’s profile was interfaced with the expcc monthly payoff to each factor to obtain the ove‘ monthty expected return for the stock.The two poi lios were constructed to achieve the highest puss difference in their expected returns while simult ously maximizing correlation. _ ANALYTIC would shortmsell the stocks in the expected return portfolio while buying the stocics in high.'Ihe prooeeds from short selling would be inVes in cash. The relative positions in the two portf would be adjusted until their net heta factors at; the market were zero. The combined portfolio wool market neutral. Movements in the returns to th 5' shouldn’t afEect the longishort portfolio’s perfo " The monthly return would always be equal to th‘ ference between the return to the long and thee plus the return earned on the cash investment. tend to ptOCil-“Ie 1- these tendI3 CHAPTER 6 The simulated co -' ‘ treasury-hill returns) of 1:112:21?ng :5 shfjvmye 'm ure old. The strategy has produced an im refz'm Eg- nuahzed return of more than 33 perm-git 551'“: an" annualized volatiiity of return of slighti mow”?l an 3 percent. Nearly all the volatility is atf'ributhlillahél: the tracking error between the long and the short. 7 portfolios. Fncron MODELS 1 53 Thi has an Sihpulddbe an easy product to market. First it second ‘tphc e return greater than that of stochs immune}; as a risk level similar to that oi short— t I “email prgaépfffrp; govermncnt bonds. Finally it i: ‘ . ‘ estock market [all , «have: . t ‘ . s,althou h rt stols thl suffer a deterloration in the value:g 0:13:26: holdings the gain ' , s m the short osilio ' should offset the losses in the [gage us for this fund w1% “2% N o a s a 0 N “I e v w ‘_ 43 0 cu sassas§§§3$5§§8ggflngm0N m m a) m m (a m 0 C C) r- I'" N V w sawmmsssaacggggeE D) O) a) O) at g a T 1 Year Am“! Standard 3 Years Since Inception Hem“ Damion annual Standard Annual S as. Lenwsmn Equ' etum Deviation Return Dlaagdflard VI Tb”: 159.39% 3.40% 14.10% 3569’ a an 223/ v c 13.57” vafue Added :4 37; 0.08 5.24% 0.06% 5 eat/o 314% v o 3.42% 9.27% 355% I [a 015% . 8.27% 314% ’ stock 5 overall expected return for current profile with the various exp ect ’ ed payoffs, we can obtain an estimate of the the coming period“ PART li PORTFOLIO MANAGEMENT 3513‘ PHASES AND FACTORS Bob Marchcsi’s number-eight iron arched perfectly into the ball, sending it Skyward with just the right amount of backspin. It hit just above the eighteenth hole, popped back, and rolled three feet in front of the cap. BINGO! What a day this had been. A warm sunny day—truly rare for Scotlandw—a light breeze, and with this sore bird, he'd be 10 over on St. Andrews, one of the toughest courses he'd ever played. In the days of his life, this will surely place in Bob’s top 10. Bob Marchesi is president of DeMarche Associ— ates Based in Kansas City, DeMarche is one of the major pension fund consulting firms. DeMarche ad~ vises hundreds of pension funds on the selection of in- vestment managers and the measurement and analysis of their performance. He also advises on the optimal allocation of investments over broad classes of assets, such as domestic common stock, real estate, and gov- ernment securities. Much of the asset allocation analysis is based on the optimization techniques of Markowitz. Estimates of expected return and standard deviations of the asset classes are entered into the optimizer. So are es— timates of the correlations between the monthly re- turns to the asset classes. An efficient set is produced, and an allocation is recommended on the basis oi the plan’s risk tolerance. More recently, the analysis had been extended to cover more detailed asset group- ings. For example, DeMarche classifies the US. equity population into various indiccs based on the quality or risk oi the stocks, size, and growth histories of earn- ings per share as well as other factors. In some cases, they rise their optimizers to determine how much of a pension fund’s equities should be invested in each type of stock. The various elements of a stock’ families. Risk Differences in the risk ofsto forthcoming chapters we will learn a the nature of risk and predict the nature 0 ile it is important for you to of their widespread use in the investment community, in building techno the best forecasts of future return, it may be best to rely on a risk measures. These might include the sensitivity to mar sensitivities to other macroeconomic variab volatility (standard deviation) of retarp, future return. Howover, wh DeMarchc anaiyzes the asset allocation decision‘ terms of both shortutun and long—run expectations f0 return. For the iongwruu picture, it bases expected mm ' to the asset classes first on the leng‘tcrm track records 0 the asset classes in producing returns for their invesgé 7 It makes subgectivc modifications in these long—rem alized returns when it is clear t over time in the overall investment climate make the past an inaccurate guide to the future, 1) Marche uses index or factor models to estimate sh run expected returns to the asset classes. The factors at usually macroeconomic variables, such as the rate 9 hy DcMarche includ curred Elation. One of the models osed the following set of factors: 1. The rate of return on a treasury bill {I bill} 2. The difference between the rate of return 3. Unexpected changes in the rate of inflatio consumer prices {inflation 4. Unexpected percentage changes in industn duetion (ind. prod.) 5. The ratio of dividend to market price fort 500 in the month preceding the return (yie 6. The difference between the rate of return low- and high-quality bond (confidence) ' tell DeMarche the implications on expected :3; forecast oi the economic climate (the facto bout widely accepted the s profile might be grouped i f the relationship between risk; understand these in hot returns (if! les (industrial productio the relative amount of debt in on a short-term and iong‘terrn government ). r val more com??? I m scgtlsrind. He’d been on the road throughout the Unite tates promoting his firm’s newest approach t asset allocation for pension funds. in this approaclo BeMarche overlays the factor model with a market}, .phascj-ntrodp‘l. DiMarche researchers have identified _ our 18 the We ases oft ‘ " " n the direction; moment 1:: Easyfgil‘; which are based ., estimates the factor sensitivities se arat l mgs per share: prices and cam- p e y over eaCh hat changes have 6 L1, The initial phase of a hull market .2. The intermediate phase of a bull market CHAPTER 6 FACTOR MODELS l 55 Bob was enjoying a well~descrv ed rest playing golf 3. 'Ihe final phase of a b all market 4. The bear market I ' ‘ . sensitixfézstmgiy’ f” a given Who of stoek, the factor ’ mews fr 5 can Change dramatically as the market om one phase to the next. Thus, DeMarche of t “51:11:01: market phases. The accompanying table I actor sensmvities for large versus small stocks in goiflg from abe . . . of a bull market, at market ‘0 “‘3 lnltlal phase WM Phase I Small Stocks M 6 45 Large Slacks 5mg” Stacks L '34 is, _1 .21 5 16 arge Stocks W582 as ‘86 5.81 d. prod, ‘ —2.4s ' .92 ‘ ‘54 05 "3‘23 —2 20 1. ' . ' nfidcnce 51 —.16 0‘} an . ".63 m 43 ~18 00 “~ ‘ 2.4 ' 21 Wm 22 1.45 ‘ .20 vestmcnts that maxim” mes expected retur ' ' posare for the forthcoming year. R given “8k ex‘ S y . 0 far, the market reactlon to the new approach star the forthcomin ' has be ' - g erred ‘ ell quite favora P (usually a year).'Ihen one of many quantitatitieiflificmrfiham mafia: is 'ionst at have estab— ate expected retnr . ‘ as for the asset [ and Sin . C asses (55631 as ' . . all stocks) on the hasrs of the factor sensi~ ‘1 DeMarche as a leader in the pensiomconsulting 1“ 1 the Phase The - -. - se expected {cf Rd to 3“ o131i¥ni2cr to d - “ms can‘thcn be Bot» lines u h' etermme the mix of in. a 10 indeed! i3 IS putt.Aha! Ioday turned out so be £5, a at :3" (9 f) C H "l (b :1 ... El 9-7 .1 3"? N m '2} 5‘ S3 m G I . , calculate the typ1cally experienced in such a phase. and 2:1: fi . . cations in these factor averages to reflect expcc St! UCtlH'e, the Variablhty Of 3. 111111 Can}!!! 5 [361 Shale and SO UH. COHeCtl ely you would 5 g ’ ? expect £113 [)3 to these “SIC Vanables {(5 hf} posltllde, Wlth StGCkS halal]; gle sures to the ' ' ' L. I ' variety of rislc attributes having greater expected rates f lquldlty inferencerin the liquidr' 0 ramm- ater expo- ancing their ortf ' ty ofskaS are 3130 0m“. i ' at the dealer?’ bidoglrfigrgerzfgustkbuy at the dealers’ :sked pariichthgictsm In Habak- . - e 1 -as ed 3 d W must sell exElected impact of the tr prea serves as part f h ‘ we on the , _ _ 0 t e cost of tradm Th want to htly a neat d pace of the SIOCK Is also im - g. e g 631 Of the Stock” you may have to raise thggrtiigtalggat 1:th you ve e cor— . . f h I d‘ d t C‘s lent aSked PRICE 1“ Gide: t0 attlact the 1131111381 0 S 3133 you Wain I] I I . 1V1 Lia S 0 act I Zed 5‘ g l d ale Cha] 8 l by WE CI difffilln ilquldlty- IO k663i) the expat:th Iates 0f lat“: H net mmensurat ' e across different stocks of differing liquidity stocks must 156 PART ll PORTFOLIO MANAGEM 93 9. 0 M V! E a: .... C!» E in trading. F ' . have gross expected returns that reflect the cost of the mmme of dady trading: a “Ines [he iotal nurnber of sha . at} h 1 . . . . w I ‘ tive to total market capitalization (price per sha twice the amount of msmut a , I e o outstanding), the bid—asked spread as a percentag ' he like. I Dwmlrisgigfiigtéd States, mutual funds have grown to the e sub'cct to the cash bl 'nfluence on the prices of stocks. These funds at } sidera e 1 C II! 1 to at“! out (if the fund AS EIIVeStOIS “HOSE E11016 1110116 , the Blus 0 ing n stock with the proceeds. As investors withdraw fun mum funds demand liquidity (is. M . . . - meet the 03511 33mm v ' nrd stocks. To. date the“ mvesgi’lgtesytgave a ca§Jtive demand for the relatively ilq . their investmfin - lative liquidity silo ds becomes greater, re I I‘ . ence of mutual fun ‘ ‘ “Balm across dim}. 67mm that tlfidlefli: determining the relative expectedtpigzsécmrs mg represem Piayka géiigfall you should expect the payoffs to the va g the greater Liqmdity ha- iiizfli'ials in liduidity to be nega tive, with stocks havin lower expected rates of return. point where they exerts ‘ ’ 'tudc‘of ' ‘ ' to the relative magrn hea mess in price indica _ _ g t flute? riot: ll]: current cash flows available to tie {1:13:131 es: 8 '10 ness in price include the relations rp d. flung Cheap e, cash flow per share, i nin s er shar _ h I factors “$38315; érgwrh stocks tend to have high rat; ablecause these cash flows are expected to gro e willing to pay a relatively highp lues for these ratiosfl‘hrs is: elatively slowly in the four for these stocks today. I much higher Cheapness Factors rc rent market price in re vestors. Factors rcprepe rent market price 0 3;: share, book value per share, price to current cash flows. This is. veswrs at higher levels in the future. Thus, m(i to have low va these stocks today. Value stocks ten tad to grow I the cash flows of value stocks are expec 11 [V33 “IS 318 t E Wiillllg t0 pay a {Blanvely high price In Carrel“: IBSEaECh has shown {hat Vagfle SLOCkS have 33 8d return than growth stocks subject of much controvcr . therefore, are more ri . ‘ ' that this is true, ’ e that the Pr “dilequkmitggldrey? discussed earlier. Others, however, behev int 6 ns c . . , O V e OCKS Ie unexpeCt d and: sys 61:13:31 out 3 Lu? t 3111 SE a . ' tors overt , of competition in a . . believe that “was , b 1 eve that the forces L“, two markets 6 I S of overreac ponems girovggzzfii driva profits to normal 1<2v£cls.1‘~d‘“’Cate - - ten 0 . A ' 0 0 EZlSeve that by Projectlng relatwely rapid rates in. AS the forces of compet-1 1 1 . . t .l I ' I I l ale] "la t these 1 lVflSlUIS iieiieved “key tend to he disappomtfifii In est-01.5111 gI‘O into P 33' a 1 ' ‘dends and ' ' ks and the future div: fitabihty of growth stoc , en 1999). I repflitssfittdlcekfignd to be smaller thanfréxpccthilfifip: 2:3; to “valuelgr on e ' a o s to at Hmspecme Of Whethegdfiesgodld expect these payoffs to be neg‘ ' r overreaction, . . I eat “Sh fl from; mitigng the highest prices in relation to curr stoc 5 future expected rates of return. factors indicating cheapo :S' ’6‘ H ,.r 8 [M ’15 g Q 3‘ Et G 0“. 11 p0 n ' o ' ' drcatc the pr rowrh potential in ‘ a ' 1 Factors related to g j I e 1 Cl 8 r tth n ;:6:!;B future growth in a stock 5 Gish fiows Th y n 11d a ’ ' era in fnaefitwerathings to book equity, the raung 0p g CHAPTER 5 Facroa MODELS 157‘ operating income to total sales, the ratio of total sales to total assets, and past rates of growth in the various measures of cash flow. Given the magnitude of the factors that ‘ reflect cheapness in price the greater the growth potential for the cash flows, the greater the expected future rate of return. Thus, you would expect the payoffs to the growth potential factors to be collectively positive. Technical Factors Technical factors describe the history of the rates of return to a stock. Recent research indicates the existence of at least three separate relationships between the nature of the past history of return and a stock’s future expected return. First, there appears to be very shortmterm (1 month) reversal patterns in returns. If a stock went up significantly in priceiast month, there will he a tendency for the stock to come back down in price next month (Jegadeesh 1990). These short-term reversal pat- terns may be caused by price pressure induced by investors attempting to buy or sell large amounts of a particular stock'quickly. An investor attempting to sell quickly may drive the price of the stack beiow its fair valueihis being the case, the stOck can be ex— pected to recover and return to its fair value shortly thereafter. The opposite would be expected to happen to a stock driven ab0ve its fair value by a significant buyer. It is also possible that short-term negative serial correlation may be induced by a phenomenon called the “bid-asked bounce.” Stocks fluctuate bctvveen bid and asked prices. Because of this, security returns measured over adjacent intervals may exhibit negative serial correlation (Roll 1984). Jegadecsh (1990), who discovered the short-term reversal pat- tern, argues that the bias due to bid»asked bounce is likely to be small. Moreover, he finds that trading strategies that attempt to exploit short-term revarsals remain suc- cessqu even when returns {or the previous month do not reflect the last day of trading. Second, there are intermediate-term (6 to 12 months) inertia patterns in stock returns, with stocks showing a tendency to repeat their performance over the previous 6 to 12 months in the next 6 months. This may be due to the market’s tendency to underr‘eact to initial reports of unusually high or low rates of profitability by firms. An initial good (bad) quarterly earnings report tends to be followed by one or two more. Failing to recognize this, the market underreacts to the first report and then completes its reaction as the next two are reported in the 6 months that follow (see Jegadeesh and Titman 1993). Finally, there are long-term (3 to 5 years) reversal patterns in stock returns (legadeesh and Titman 1993).Tliis may be due to the fact that the market over- reacts to a chain of positive {negative} reports of good (bad) earnings numbers. Believ- ing that the chain will continue into the future for an extended period, investors drive the price up (down) to high (low). Consistent with our discussion here, as competitive forces come into play, the stocks that went up (down) in price in the past tend to reverse their performance in the future. Some contend that these technical patterns aren’t the product of market under« and overreaction. They believe, instead, that risk premiums on stocks become larger and smaller over time. Risk premiums in expected returns become larger and smaller as the risk of stocks becomes larger and smaller. Risk premiums may also change as investors‘ sensitivity to risk grows and declines. Both the levels of risk and risk aversion may change with the business cycle. As we Ineve into a recession, the risk of common stocks may increase; we also become poorer, so our aversion to taking on risk may become stronger. Given this, the expected returns to stocks may be higher in recessions and lower in booms. To the extent that changes in prosperity occur in reguiar time patterns, the tech- nical factors that we see in the history of stock returns may be induced by time—varying risk premiums. Irrespective of whether the serial patterns that have been observed in returns are caused by inefficient markets or time-varying risk premiums, you should PART l1 FORTFGLlO MANAGEMENT M", _..,v ~w~wwW W s to be {a} negative: the ayoff . m 233$: perfgrmance in the past (a) 1 mon ' tivel and (c) negatWe, respec I 12 months, and (c) 3 to 5 yea 158M *0 positive? ( ) (more ACTOR enroFFs needs to estimate the tender; such as firm size, to system no PROiE-CTING F eturn factor model, one sures to different factors, EsTIMATtNG A . . cred r In bufldmg an EXPE . ~ expO stocks With difiemg . the ma nitude of produce differing. estimate the relationship bleatrgfélmh 5ay,g;iaru1ary 19 SHPPOS" on d'fferentstooks m an ’ t. nshi is s Jrminced by 1 . , f January. The rain 10 P . reported returns? tie as of the begtnning 0 _ “rm a measure of 3 their book to p;1§:t;apoint represents a particular flf?;et:fln in January 1988 16‘, Figure 63E“? lotted horizontally and the fag fits all“? b§°13£§ {lifefpret the horizontal grief a firm a . . B ' ' :ebggk-grice ratio that IS 1'5 Staggffrfcrent firms. in this month if:th , . . . - atio acrOS ‘ _ o to 00 "P . the variability mg}: I roduce higher returns, thus, the 1:3}; off 04, in dicafing m- pr'rce ratio tendel mg month the Sropg is actually 60:53 b00k_p1-ice ratio,i 1988 is pomweh sriandard deviation increase m a 5;? an ce . month, for 6 id be cape“ad to go up by 4 per' f ences in the returns pi this month coil isn’t the 011;); detennmant 0f 0:1” 3:8 Payoffs to the variety Book-pm“? , . us; estima 3 . ' 11 sunultaflco Y ' firm charactens given month’ so wgafalrcteristiCS—S The Payoffs ’E0 the £23122; many months. uidityiazid 3:51;;me m book.pricc was in Fggurgxafifie the payoff to sit mate 35 ions factors. of . ’ . . offs to th6 V31” .. s tefldfll history 0f the Pg; in recent decades, With smaller iii; of the 19705.1:1 the over mOSt {mills United States this was true for m of return. It Total return: 3’" C 9 Line 0! best 1}! ".50 ..; 1.5 2.0 "mo mo 5 on 0.5 in — 1 .5 "1.0 ' Book to price " sis. sing multiple regression analy 511,55 can be done 1’ CHAPTER 6 Facros MoosLs 159 ever, as pension funds and other institutional investors moved funds into portfolios that were designed to replicate a capitalization-weighted portfolio of largest US. stocks (usually the S&P 500 Index), the payoff to size tended to become positive, with the larger stocks producing higher returns under the price pressure of the pension funds.6 Thus, the payoffs to the various firm characteristics have interesting histories, and one can use the information in those histories to make projections of the magnitudes of the future payoffs in future periods. The projections might be based on simple mov- ing averages of the magnitudes of the payoffs in trailing periods, or they might be based on more complex statistical time series models. The experiments reported in this chapter employ simple averages of the payoffs observed in the months prior to the month in which expected return‘is to be estimated. For example, suppose one estimates the forthcoming payoff to size based on the simple average of estimated payoffs 5over the past 12 months. We want to estimate the ex— pected return to a particular stock, and We begin by estimating the component of the total expected return that is attributable to the relative size of the company. Based on the crossnsectional variability across different firms, the company we are interested in is 1.00 standard deviation below the size of an average company in the marketAssume the estimate of payoff to size in the next month is “2.00. (For each standard deviation below the market average in size, expected return increases by 2 percent.) Given this, the compgnent of the stock's expected return attributable to its relative size is: Factor exposure >< Projected payoff m Expected return component 4.00 SD. X —2.00% u 2.00% Thus, based on its relative size alone, we expect this stock to produce a 2.00 percent greater rate of return than an average stuck in the forthcoming period.We would now obtain similar expected return components for all the other factors in our model. If you are employing an expected return factor model, with a wide variety of fac— tors profiling thc characteristics of the individual stocks, you might employ, for each stock, a spreadsheet like the table that follows for Green River Paper Company. Spreadsheerfor Expected Return (Green River Paper Company) Factor Green River exposure a Projected payoff = Return component Size —2 Standard deviation 2: —2.00% m 4.0% Volume m5 Standard deviation r »«1.00% = 5% % Debt —1 Standard deviation * +1.50% "~1- 4.5% Total expected return: 2.60% Only three factors are explicitly represented in the spreadsheclflhe others are part of the computation of total expected return, but they are not explicitly represented. In looking at the first factor, we see that Green River is a relatively small company. Look- ing back at Figure 6.8, we see the cross section of stocks with respect to book-price in January 1.988. At the beginning of the month for the spreadsheet, there is also a cross- scctional distribution for company size. Within that distribution, Green River happens H 6Chan and Lakonishok (3993) show that between 1977 and 199i being a member of the S&P 500 con- tributed an average of 2.}? percent to a stock’s rate of return. PART lE PORTFOLIO MANAGEMENT CHAPTER 6 FACTOR Mongs ' 1 fit 160 to be 2 cross-sectionai standard deviations below an average stock, in terms or its-s Based on the average of the estimated payoffs for the last 12 months, we project. ll be —2 percent per unit of standard deviation 0% size, 3' the payoff for next month wi Green River is 2 standard deviations below average, we increment up its expg return by 4 percent because (a) it is small, and (b) smali stocks have tended to paid larger returns. Green River is also less iiquid than an average stock by .5 standard deviano the cross seetion.Again, based on the last 12 months, we estimate the payoff to liq ity to be -1 percent per unit of standard deviation. So Green River’s expected rem boosted .5 percent because (a) it is relatively illiquid, and (b) Eiquid stocks have ten to produce lower rates of return. Finally, after considering the effects of other factors not explicitiy shown 1;, spreadsheet, we reduce the expected return by 1.5 percent because Green River reiatively small amount of debt in its capital structure, and more levered {truism tended to produce higher returns over the past 12 months. ‘ After adding all the components (including those not explicitiy considezea conclude that given its overall profile and our projections of the individual pay" those components Green River has an expected return that is 2.6 percent greater an average stock .7 Cumulative performance I79 I r I 1 l‘ 4- - ‘ r 85 81 62 63 3‘3 85 ‘86 ‘37 '88 '89 '90 '91 ‘9? '93 A TEST OF THE ACCURACY OF EXPECTED RETURN FACTOR MODELS Even though risk factor models of the type described eariier in this chapter t business, expected return factor models are more popular in the investmen relatively more accurate in their predictions. To see how accurate, we shat! ru using the 3,600 largest stocks in the US. population. As we did with a single factor in Figure 6.8, We wiii simultaneously estima individuai payoffs to an array of 70 factors for the 12 months of 1979.8 first month of 1980, the 12 payoffs of 1979 are averaged individ For each stock, these individual projected factor payoffs are multiplied by the e __ of the profile going into 1980, in accord with the spreadsheet discussed earlier. have an expected return for each stock for January 19891116 stocks are ranked expected returns and formed into deciles of approximately 300 stocks each. has the highest expected return and docile 1 the lowest. We then observe how the deciles actually perform in this first month.'ih the same thing for February. Dropping the payoffs for January 1979 and addin offs for January 1980, we again take an average of the trailing 12 and multiply the new eiernents of each stock’s profile for February 1980 to get expected r the stocks in the next mouthffhe stocks are reranked and again formed into How do the deciles perform? The logarithm of the reaiize'd, cumuiative returns to are 6.9. Note that the deciles correctly order themseives almost immediately first few months of the test. The annualized returns by 6.10. A line of best fit is passe of the iine is actualiy 37 percenti Astonishingly, individual years of the test. :1 m :1 *4 H. O H 0 to n Aanuaiized return, % It must be ' . rapidly (we? tiriiéutigsgleofiames of the stocks in the individual deciles are chan in turnover woutd consume y . 0.511.} trade for fre‘h {he tradittg Costs associated withgth'g However, in Haugen (199351t“¥fi°a“‘ amount of the return spreads across the scene”: effectively be emgloyed if:1): )1: is shown that In the factor model, expected returns ea: managed and Hausa t. e Context; 0f POMOIEO Optimization, where turn ' c ton costs are accounted for. In this context, portfolios COIEStIuCtEG that have ICIUHIS 3] Elli C y 0 g 1 anti highe] than the SQOCk. {fidex f F of its 7 An average stock is one that is equal to the mean oi the cross section for each element 3A multiple regression procedure is empioyedfisce Haugen (199%) for details. PART lE PORl’FOLlO MANAGEMENT 1;; “I its fi‘glgiumfllh Ml borne FACTOR MODELS ro SEMULA'TE invssrmsnr PERFORMANCE . . . . . am . Factor models can be used to simulate possibilities ford 13V?:tmfifta}::;0:§1ne e In D ' ' ' ' that i n exrs . ' ' ven for securities or industries . - . 1 . i I gerlfifs Of indie: the return to an asset class or a portfolio in period t IS again llilfiarly ssurn ' ‘ ' 6.6 . related through time, to several factors (I a 1 through it) as1n Equation ( ) .- rlz E3111: "3" B2174: n!- ' H "l— lsnInu + 8r es of the returns to the periodic values for} the return that is unrelated to the tack, ter in the book, we will asaurne that q; Here the betas represent the sensitiviti factors and a: represents the component of In the examples that follow here and la are five factors: . ' I 1 The monthly percentage change in industrial production { 1) . . I ) 2. The monthly rate of inflation { 2 3. The difference in the monthly return to long— boncls (13) l 4. The difference in the mo the same maturity (Ia) ' . r - 5 gifts monthly percentage change in the price of oil (15) ' - vol of interest rates. In periods 0 The third {aetolisrigy‘lcsaiigfiefinggdél 1lie low relative to short-term boa rates, the m; resents changes in investor confidence. More confident fourth fast?“ tigrnates of the probabilities of default on corporate bonds;i 1 is {magma the” aifidence should increase the prices of corporate bonds an trip m myssmr CO turns on government bonds, which are not sub3ect to define . “flame E9 the tm is to estimate the five beta factor values for a particular rfiv The film Stleiep investment’s monthly returns on the monthly values for t by regressglg roduces monthly values for the unexplained compolpe e The regrefim; idation {6.6) as well as an estimate of its volatility over t e p- rang: ifillnalsst‘ilme that uneilpiained components of return have a zero ex and are normally distributed. Now we estimate a possrb ' I ' t. ' riod in the more distant pas _ I “Ineligible past period1 we can observe the sequential history of th ' ' eri uiti ly the five factor observations for the first month 11; tliprsptarsfiopuh’ Ifirlictcfbetas. This calculation gives us the component of t c ciated with each factor. ' We then “puil” an observation dd. ' .By a ing ned com orients oi return . E3151:E first mogth’s return. The prooess is ropeatecltfoértilcp:1 secon ' e as . . obtain a sequence of possrblc returns for th p p the WBb sue I ' ' ' lt be done, you can go to To see how eaSfly fins mjgol the area labeled Modern Investment The? Fmame’com‘ once there, go t Then go to the area labeled Sessions. Copy or into our computer. _ I in You fillzgaigndustrie: into the directory in your computer called Option on your C: drive. and short-term government nthly return to corporate and government bonds to sequence of future returns to an investment 0v ed by 3 tot from the assumed, normal distribution of __ . . n 6 Six corn orients, we obtain a _ Eh p d month, and - CHAPTER 6 Factors MODELS “E 63 Now go to the i’rograrns section of Windows and run PManager. Go to Open un- der File, and select and open the file Industries. Bring up the window Expected Return r and note that We have assumed a 10 percent base expected return for all industries. This is the assumed expected return. if all factor values in Equation (6.6) are taken to be zero. Now open the window Select Period under Historical and Simulate. You will see a graph of the period February 1968 through June 1998 with recessions shaded in gray. Select Factor and highlight the 20-year treasury bond. You will see an index number (January 1982 = 100) for the yieid-to-maturity on the bond. The period August 1979 through October 1981 has beenPsclected on the Screen. (Other periods can be selected by pointing the arrow to the broken iines and right—clicking to remove and left—clicking to restore. But, for now, let’s workgwith the period already selected.) Highlight Factor and then Zoom In to bring up the period selected'l‘hen highlight Simulate under Historical and'Sirnulate. You should see sequences of possible returns for the period {given the realized values for the factors) appear on the screen for the industry stock indexesuwred for the banking industry and green for the health care in— dustry. When the simulations are finished, three lines will appear for each industry. Ninety percent of the sequences fell within the two outer tines. The median sequence value is the middle line. Nop bring up the window Draw Graph under Back Test and Simulate. This shows the actual cumulative return to the two industries in the period. Highlight Tile under Window and the three graphs should be placed side by side.’I‘he actual resuit under Back Test may be viewed as one of the possible simulated resuits under His— torical Simulation. Note that the actual and expected possible results for the banking industry are much worse than for health care during this period. To see why, bring up the 20~year bond yield on the recessionmshaded graph. Interest rates rose sharply during this period.This was bad for the interest—sensitive banking industry, and, at the same time, the health care stocirs were iargely unaffected by the recession. To see the results for other industries, bring up the Select Portfolios window under Simulate. You can observe the results for only two industries at a time. °3° SUMMARY Factor models can be used to predict portfolio volatility and expected return. Volatilw ity factor models are based on the presumption that the covariances between security returns are attributable to the fact that security prices respond to varying degrees to the pull of economic and financial variables like the return to the market index, infiam tion, industrial production, and so on. Volatility models have the advantage of being potentially more accurate in forecasting while at the same time being less computa- tionally demanding. Expected return factor models employ firm characteristics that can be used to pre- dict the relativc returns within stock populations. These factors can be classified into characteristics that describe the relative risk of a stock, its relative iiquidity, the magnitude of its price in relation to current cash flows, the potential for growth in those cash flows, and the performance history of its rates of return.The components of expected return are the products of a stock's exposure to a particular factor (such as the size of the firm) and the projected payoff to the factor (e.g., to what extent will small firms pro- duce greater returns than large firms in the forthcoming period). W PORTFOLIG MANAGEM ENT pgRT H M AM i? t ewiactot mode 0:. QUESTION SET 1 . . 'n '. the foundation of. the at if the single—gamer m 0 d 61 in tion serves as I sum mm 1- What ass“ f pllowing information anithleaapd 2?}; 2' Given} the O variance between stoc 5 what is the co B1 fl .85 10 {51 = 1.30 11 tfactor = .09 12‘ Variance of the marke 13. .n ,_ 3‘ Assume the follow1 8 Residual Variance __ 01 W Stock X '06 ' t at a portfolio of X and Y is cons . , _ 61. for Y: the single—factor mod I lvariance as o. i 2}" a) 1:2 to € 52 on M i: 9.. Also assurne nd a 1:3 werg _ A: 3What is the resuiua . v ' n fight residual variance of the po b. D assumption? had ‘ 3‘1 ose you 4 tuggion 0’E the W of the portfoiio it I 1 folio without the singlewfactor m ' ' ' m J’s return as; t d the following relationship for int estima e turn on a market factor: U = ldf u a return on the marhet f;?t(:;ts{§:? a. If this}: ected change in fun;1 I sraphie m iii-1:135:11: the name given ttJ) t cctgml Tatum ’s a 2- What might account for arms of the equ . ' t two t . on the ham Of theofijiIS'tg data for Quemcms 5 county 8 (1 RES“!!! I an REE S B I a. P a . Bighth pottfoilo of A and B IS fOImfld. ' v icient for the portfolio. variance of the petite ‘ ming the ortfolio assu Of phe following table. Assam resent?“ Pbeing different f anon? through 8' Suppose an equally if 5. What is the beta coe 6. Compute the residual ’1'. Compute the variance . . t the missrng columns in (D H- c”; 4 a g. ('5 cs 4' 'n 6. ' 8. I‘ll} lket factor (M) to be .031 - angysrcinanc mar . Correlatmn Beta ‘ Risk WISE?“ of,- with M Securi‘)’ i W 1 .006 '93 i: 2 .006 ‘0 if 3 .005 k? t = p ' . s stemaflc Us ' 9. What is the meaning of ‘m 3’ sh 15‘ I{ef€'r E0 the folk)ng data f0: Questlons HIIOU 14. 15. 16. 17. 18. 19. CHAPTER 6 FACTOR MODELS 165 Correlation coefficient between stocks A and B = .50 Standard deviation of the market factor (M) e .10 Correlation ofStock with M Standard Deviation Stool: A 0 .10 Stock B 0.5 .20 . What are the beta values for A and B? . What is the covariance between and B, assuming the single—factor model? What is the true covariance bétvveen A and B? Suppose a portfolio was constructed, with weights of At) forA and .60 for B. What is the beta of this portfolio?‘ Compute the variance of the portfolio in Question 13, assuming the Markowitz model. Compute the variance of the model. What is a factor model (either a single plish? What is the potential advantage son with the single~factor model? Suppose you employed a two—factor model to estimate the for the percentage return on stock K portfolio in Question 13, assuming the single~factor factor or multifactor) supposed to accoun of the multifactor approach, in compari— following relationship rK= .5 + .8rM+ .2g+ 8K where W represents the percentage return on the market the unexpected growth rate of industrial production. a. If the market index's return is 5 percent and the unexpected growth of indus- trial production is 2 percent, what return would you expect for stock K? b. What kind of change in stock K’s return would you expect if there were to be no change in g and a two~percentagc-point decrease in r M9 Write the formula for the variance of a portfolio, assuming that a two—factor model has been used to explain returns and that the covariance between the factors is zero. Also, write the general expression for the portfolio’s residual variance. If the two—factor model is really appropriate to account for the intern reiationships among returns on individual stocks, what simplification occurs in the general expression for the portfolio’s residual variance? Compute the variance of stock X using the expression derived from the two-factor model and the following information.'lhe two factors consist of the return on a market factor and a factor of unexpected growth in industrial production. index and 3 represents Stock X’s market beta = .75 Stock X’s growth beta = .40 Growth factor variance m .10 Market factor variance = .08 Stock X ’s residual variance = .03 Refer to the following data for Questions 21 through 25. A two-factor model is being employed, one a mark the growth of industri et factor (M) and the other a fac tor of unexpected changes in a] production (g). CHAPTER 6 FACTOR MODELS 'l 6‘? Pontromo MANAGEMENT . ‘ ‘ 3. How are the Markowrtz and factor models used in portfolio SEICCHOH? 4. A new law in Brazil makes the construction and operation of steel factories a real bargain; however, the steei may only he used in the construction of South American automobiles, which rapidly become popuiar in the United States. If you were managing a portfolio that had three classes of securitiesmdrugs,ser- vices, and machinery manufacturingmmand used the singiewfactor modei, wouid PART ii I Residual Variance Market Beta Growth Beta .05 2 ‘ .62 Stock 1 .1 Stock 2 . d 12 Rat factor —~ . Variance of the mar {h £80m; = .10 you expect the systematic or unsystematic variance to be affected by this event? Variance of the gmw d 2 m .02 5. Under the singie-factor model the relationship betWeen returns to the market ' sis of stOCkS 1 an and returns to a security or a portfolio is expressed by the equation arianCB between fwd“ M and 3 fl 0 GOV Covariance betwwn r 32...: A + BrMi + £1 . . - 1 and . . " .‘ . '. 9 Compute the variance of stock 1 Weighted POYithO 0f Sim“ a. What iS the name given this regresszon line? 2 . . d an equaliy had constructe f this po 21' Assume you 'dual variance 0 . r 6 two cofipiiiggfliicigiihlphfying assumption of th 3 . a. residual covariance. rtfol‘ro in two ways: is. Define each term in the e ugh-OK —factor model about C;- c. What type of event is assumed to cause period—touperiod movement along this line? What term in the equation accounts for this variabiiity‘? . . CE. I I I I - | treSlduai comman- d' What type (if event Produces devrauons from this line? Explain. What term in ifying assumption abou ' h ut makifig the gimp] th beta for an equally We‘lghted P the equation accounts for this variabiiity? b_ Wit 0 d the grow . . , . . . . me the market beta 3“ h Vans 6. What is the Single—factor models key assumption, and what does it imply about 22- Egggcks 1 and 7» ' “folio of stocks 1 and 2. compute t ‘3 the reSidual returns to securities in a. portfolio? . a “any weighted P0 ode} about 7. Suppose you are managing a portfolio consxsting entirely of aerospace stocks. is 23' Eggs? in two ways: sumption of the two-{actor m the single-factor model likely to accurately estimate the portfolio’s residual vari- PO . 'm lify'mg as ance? Explain. along “16 $1 . P . 'duai covana A I . _ I a' M ‘ a1 covariance. I . . assumption 2113013t I381 . m um 8. if,1n a portfolio of stocks, those of a given industry res 0nd in a similar manner resxdu . h 53mphfymg _ a sgngie 01' m , . kingt 9’ rigor“) variance by to an industry-Wide event, what is true of the SFM estimate of the portfoiio’s ' ut rrta b. WEI-LG ant to compute 130 on w ' del? 24' 3211133; than by the MarkowrtZamd beta v m' as how we wou‘rd arrive mg; to Specuiate on W I £11510] (:3: ‘lllifliniatioli FurthEE, - I“ g’hlstoilcai ‘fflorxn'a it)“ It) {:Siiii ate beta. ' i b ‘ {Data Oi futuEe Expel 1151 1 . . . . I {ac-tors linpr 1“ deg-BI Hurling . Wh are liqm 27 difiZrentiais? 28. Describe some 0 29. What is the impa 30. What is the cheapne Cheapness fame; nee between growth st t is the differe ‘ . evadefi 31. ngpness facmfl e‘? State the empirical I ' hnic . What do the tee 32 technical factors. resid uai variance ? 9. Suppose the stock of two highly competitive companies is held in a portfolio. Would the SFM over— or underestimate this portfolio’s residuai variance? 10. Current research has shown that value stocks have earned much higher rates of return than growth stocks in recent decades. What are some of the potential explanations of this phenomenon? 11. Payoff to the size factor was negative in much of the 19705 in the United States. However, it tended to become positive in the 1980s. Explain why. 12. When using factor models to run simulation on investment performance, we usu- ain make some assumptions on the unexplained components of expected return. State one potential assumption and describe how it is applied in the simulation. Refer to the following data for questions 13 and 14. The expected returns of stocks A and B are affected by three factors. The factor exposure and the projected payoff are described in the following table: has alue for a stock I tentiai difficult: ensures. . ? W m d on the liquidity of stocks. e some of the measures f the iiquidi ct of mutual fun _ ss iactor‘? Describ of th ocks and value stocks in to al factors dcscrib r is 1.0 standard d}? -- ” . Factor Projected Fayed“ Exposure to A (5“; day) Exposure w B (std day) : are , . Tradin volume 415‘?“ 2.0 ~1.0 stock A 5 ex?” 5k factor 13 8 0 33_ Suppose 3‘36 Projected payoff t0 Eletum? Price/book 4% -1.0 3.0 mean: an. - R factor to A s 614W" , 106 Several Size “2% 1.5 was non 0f “1‘5 “S r rowth Potential? 13656“ 34' What is thiggcmr 0 g osme) has a positive 13. Compute the total expected returns of A and B and interpret your resuits. growm may the layetage ratio (:70 dam exp 14. What are the return components of the size factor to A and B? lain W 35- Expected return 0f StOCkS‘ ex? ‘3’ ANSWERS "r0 QUESTION SET 2 2 expe 1. A covariance factor is t icali an index of securit rices silch as the S&P 500 Tron! SET . r and an ya it y p . _ do QuES . nee between a covariance €30“) d a factor ex Index, or a macroeconomic variable such as the rate of industrial production. 1' What is the dlfiifiere nee between a factor PaYOff an ' ere 2. What is the di NT PART ll l’onrrotto MANAGEME t systematic risk. They ac ‘ ks. different stoc tics of differ sens ‘ to predict the relativ 168/ Covaria tions th Ex ecte . . ' 61 as theifrelatrvc size, their r covariance returns to 2. Factor payoff duce differen stocks wi ex osurcs d . Faster p eir different; in exp betas, others 5 3. in general, of securities tics of a optimize the por t 4. The systema only the auto ste S. a. b. 6. The single-{ac nce factors are s at exist between different port th high mar l market betas. laining th the Markowita (usually co indivr ‘ mats, tfoho. ic ris e1 industry. The regression is called r = rate . . k ho) if the that = beta is the slope returns to a seed rate of retu . m the mar a1 a: rate of return him the act“ ’ ' . . t to w Mg. = Eesrduai, the extegifiers from the expBCtfi L m events, C| giccause of move for this variation. d. Vertical devi Company-59 differ from the expo accounts for this related for one to degrees, to the pa exp That is to say, solely to the com returns are the resu CUICCS 0 t0 ‘ the returns tens charac ‘ E0 the' or their relative t can be used facmrs. Expeidims in a forthcoming period. _ . f for stocks With (:1: Kampm’ in - cies the tendon S am larger returns eturns during a t I ket betas w mail. Some stock and factor the I turnon stocks 'dual securities an ations eci‘iic fac cte ecurit ‘ ressed as the s of indmdual the residual returns eiate escribe the vari al returns. 5 have l t 1 a ' used for asse . at is o optnruz is are used t mod tine expressed b the characteristic lane. of return to a seen A =5 point at which the folio) in period i resid tor model the cause then influen it of firm— perio ill usually pro dels for portfodli )- FaCtOT me fine the investment in e rity (or portfo arket. . _ m to the In hot in period L tors cause th d of time. Fo d dctcrmi k would be 3‘ EeC-ted,$1llce [he dfivelop ins 13 “lama astutels L“: a 50 L c If ‘BES O TE=A* BrM,i+ char d rate of return uai variance. assumes that th niy: Each security is y’s beta duce ous characterist For example, so . u arge market caprtahr locatio tiou ‘ _ a 0 select, Within each ach that will rity (or portfolio) . . . . e " acmmhc hm Brltlturn to the security ( cted rate of . expe in the perro ate of return to y the equation Si during a returns to B assumed to respond dot the degree of response on of this assumptl Code], maxi) 'sh-u‘ of the covariance cc of macro specific nucroevents. ics of stocks the 1 e stocks have m ations, others sum in period i. rcepts depen th among i events in the econ W count for the corral "c": 4 E: 91 .4- G d is zero. ‘ tt no the exten flaw ge to changes return to the scour n. d rate of retur e market, are assume ‘15th lineThe term Br W individuai se rrelated. ent firms, such «'9 9’3) s32; “c 3a cuts would affect m fparts and th dent (1",) an Ea" ,inv ‘ ' 'ties' Vidual securi Rd} omy. Rest CHAPTER 6 Faeroe Mopsts 169 7. 10. 11. 12. 13. 14. 2. Industry events, such as an industry—wide rise in labor costs, could affect all the stocks in the portfolio but not have an appreciable effect on the market. "Ihere- fore, the covariance between the residuals of the stocks may be nonzero. The single-factor model, however, ignores the covariance of residuals among individ- ual stocks and, consequently, will misestimate both the residual variance and total variance of the portfoiio. When returns to two stocks of a given industry change in the same direction in response to an event that affects the entire industry but not the general economy, the covariance between the residuals of firms in the industry is likely to be nonzero. Since the singlewfactor model ignores any covariance between the residuals for different stocks, it doesv‘riot-take this covariance into account and, conse- quently, it overestimates the= portfolio residual variance and total variance. When two companies are highly competitive, what is gained by one is usuaily lost to the other.Tnus, the covariance between their returns is likely to be negative. The single-factor model ignores this negative covariance between stocks and, thus, overestimates the portfolio residual variance. One explanation is that value stocics are “fallen angeis” and therefore are more risky. As a result, the premium returns to these stocks are expected and required. Another interpretation is that the premium returns to value stocks are unex~ pected and systematicaily come as a surprise to investors. investors overreact to the past records of success and failure by firms. One potential explanation to investor’s overreaction to success is that forces of competition in a line of busi— ness tend to quickly drive profits to normal levels. By projecting relativeiy rapid rates of growth for long periods into the future, investors in growth stocks may drive prices too high. In the 19803, pension funds and other institutional investors moved funds into portfolios that were designed to repiicate a capitalization-weighted portfolio of ' largest US. stocks (usually the S&P 500 Index). This drives up demand for iarge stocks. Because of the price pressure of the pension funds, the payoff to size tended to become positive. Usually, the unexplained components of expected return are assumed to have a zero expected return and foliow a certain distribution, such as a normal distribu— tion. To apply this assumption in the simulation, we first calculate the return based on factor exposure and factor payoff, then we puli an observation from the assumed, normal distribution of the unexpiained components of return. By adding this to the calculated return, we get an estimated return observation. Repeating this procedure many times, we are able to obtain a sequence of the returns. Totai expected return to A m 2(—0.5%) + (wt}(~1%) + l.5(—2%) e —3%, Similarly, total expected return to B = 4.5%. These are the extra expected returns relative to an average stock. Size factor contributes ~3% to A’s return and 1% to 3’5 return. «:0 PROBLEM SET Given the following information and the assumption of the singie-factor modei, what is the beta factor of stock 1? B2 ” 1.20 620.114) = .3162 cm (rll ’2) = .99 PORTFOLIO MANAGEMENT E 70 PART EE M CHAPTER 6 FACTOR MODELS 1171 Refer to the following table for Problems 2 through 7. CW0? r2) m fizfizggirnr) Portfolio Expected Stocks Weight Beta Return 020-) s By rearranging the te W rms, We . A .25 .50 .40 .07 can 80”" for B .25 .50 .25 .05 [3 m (30., (r1, r2) 3 W "—-——-—--«~mw C .50 2 1.00 r .21 .07 [32620]”) o to) = -66 a _a9% 2. Given the assumption of the singleufactor model, what is the resrduai variance 2. We know 1200362) each of the foregoing stocks? 3. What is the beta factor of the three—stock porttoiio? 4. What is the variance of the portfolio? 5. What is the expected return on the portfolio? 6. Given the actual (Markowitz) covariance between the stocks’ returns, actual portfolio variance? or threat-errata) Plugging in the known va sz {saw 62 (r) 8202 {FM} 2 tablet; on the right-hand side of the equation, we find o (8.4) = 62%) —— 53,029,”) = .07 w (.50)2(.06) = .055 (jag) = 0-203) — [swam a .05 — (.se)1(.06)= .035 o (cc) m 520C) w 035%,”) m .07 ~ (1.0)2(.06) m .010 3- The beta factcir f .- or the orti‘ol’ ‘ ‘ - stocks. From the text, WI; kmW10 rs Simply the weighted average beta of the three 8 :r' a) a», a Cov (r A, r B) 2 .020 Gov (rA, rc) m .035 Cov (r3, rc) = .035 7. Why might the actuai covariance differ from those found using the single—fat model formula? Refer to the foiiowiag data for questions 3 through 11. Four factors are identified to contribute to the expected return of stock A, B The following table lists the factor payoff and individual exposure to each fac M '31:: ijfij J=1 Therefore, Exposures (Std dev) BPfl {45A 4" 35,953 + x 38 C C Factor Projected Payoff A B C 4 = (Ease) + 25) (‘50) + (50) (1 00) m 75 ' —— ° P3. .0 G. . ' . ' ' m ' Size. 29A} 0 2 5 The variance of the portfolio can be split int Trading volume «3 /0 ~21} 1.0 1.0 and restduai variance. 0 two Components, systematic risk PIE “1.5% 2.0 —2.G 1.9 2 _ 2 0" (Ta) — £31902 (no) + (slop) "/0 debt 1% ii} 0.5 "0.5 8. Find the extra expected return above the average to stocks A, B, and C one has the highest expected return? 9. Why does PIE factor have a negative projected payoff? 10. Is stock A roost iikely to be a small-growth stock, a smallnvalue stock, al growth stock, or a largewaiue stock? What about B? ovariame matrix M 31. Suppose you find an additioaai factor, a growth potential factor with a p g _ _ payoff of 2%.You estimate that stock C’s exposure to this factor is 0.58: ReSidual variance = 02 (a )= Z 2 2 P Jml x10 (51) deviation above the mean. How will this change the expected return of N t h . 0 e t ' at the weights used are the square of the portfolio weights Using the residuai va ’ ‘ rtances com ' aai variance of the three—stock portfolijiloEEd m Pmblflm awe can find me Fwd- From Probiem 3, we know the beta of the variance of the market, we can if “‘5 StOak portfolio is .75. Knowing this and ‘ ad the portfolio’s systematic risk. T ' Systematic risk = swam = (.75)2(.06)m .0338 he portfolio residuai variance uader the sin sum of the eiements on the diagoflal in the cgle—factoz' model is the weighted #0 0 03 E SET 9 AN$WERS '1' PR L M O2< __ 2 g 2 8P) " w (a) "t oozes) + xtoztac) m (.25)2(.055) + (.25)2(.035 2 ' z .0081 )+ (.50) (.010) 1. Given the assumption of the single—factor model, we can write the cm between any two stocks as 172. CHAPTER 6 FACTOR MODELS ’i '73 FORTffDLlO MANAGEMENT we can now find the variance of the portfoho. . « v ation. _ with this info:m 2 20 )+ 62 (a?) z 3338 + .0081 — .0419 0 (rp) 7: l3?“ M a weighted average of the ected rate of return on the portfolio is x 5. The E Pd returns on each stock in the portfoiio. e expat EU )= 35215024) + xBE(" 13) 4' xCE(”c) P = {.25)(.40) + {.25)(.25) + (.50)(.21) = .2675 or 26.75 % I I 0110 variance includes the off-diagonal terms m he terms airing the diagonal. kowitz) portt actual (Mar. 6' Efariance matrix as wall as 1 M2 vr ) 52(77): Z xlxKCO (7.1: “($1 2 2 o (rc) 2. 2 :x:c;2(rA)+XEO‘ (rB)+xC ( r) +2xAxBCov(rA,rB)-l~2xAxCO)v rA, C + ZxBxCOov (r3, re) $ (.257107) + (.25)? (.05) + (507(07) + 2125) (.25)(.020) + 2125) (.50) (.035) 1- 2(.25)(.50)(.035) Mal k0 W in) pox [£0110 V a! la! 166 Can diffel! f! 01 [E the! {)GI {£0110 V 7’ The “Ml (the single—factor model if the singic ‘ v 1- ance am ' ’ cks. f i ' on the portfolio S sto forallthecoal g f) ( /) ( I /) Eh acted return to A = {*3 "2 0 "l‘ “2 "1 0 +2 :1 5 0t +1 1 B :2 ~1.5%, extra exp I search has shown that stocks With Therefore, the payoff o_ t 3 exp 8' ISgiiniiiariy, extra 01; ' ' a cheap 'ce/earnrngs 1s 9. Eheapness factors tend to pro gamers is negative. han the average and high PIE, A is most hire- - v ' much smailer t ' l ewame stock. _ 10' “11313123; st0ck. B is more likely to be a 31% % x 9.5} from 5111 - extra 651 GC‘C Ie—[uul 0: C W1] 6 lnCEeaSe y 1 1/0 ( The additional factor. pected return to ness factor. Re duce lower returns. 11. urea PROBLEM SET 5 and an. itie of return to four secur _ Emiowmg table represents :mTMNewFii-tanceoom wrth the s d wealth. use: 13:: 11%: annual returns {robin £26293; 0 l ' ontainingt e 05 19 Pilgtiétggfigieigrities as Weil as the tabtes for the _ d devia f the efficient 0:0 COMP 1. The I Said EfficientSe model parameters. deviation 0 Standard Draw a diagram 0 0 H’s .. set in return/stander ortfolios. P Annual Rates of Return (%) 2 3 4 Index I “9.04- Year ~5 52 1.31 43.14 42%: 27.33: 1 14.62 15.8}. 17.86 37.35 16.457 2 13-88 211777 5.18 20. 3 Year Index I 2 3 4 4 2.10 46.92 40.00 w28.47 4.75 5 2.31 -26.80 37.61 32.71 "23.34 6 16.00 37.72 33.77 33.76 44.78 7 10.01 26.26 28.25 29.20 4.56 8 11.35 ~4.81 9.80 10.53 21.26 9 15.95 7.39 16.95 15.31 25.51 10 16.13 21.82 58.47 31.91 15.38 11 20.94 32.70 30.51 37.38 12.53 12 4.18 , , H422 W510 47.01 47.93 13 11.96 ' " 20.72 4.93 21.87 5.74 14 7.66 . 23.00 28.27 22.17 22.84 15 5.17 '7 15.83 --5.45 "9.31 1.67 For probicm 2, please refer to file industses. 2. One type of mutual fund is calied a specialized sector fund, which concentrates on a particular industry. Suppose you have investments in four such mutaat funds A, B, C, and D, with the investment focus on the automobile, banking, construc» tion, and drug industries, respectively. current time is June 1998;31011 would like to progcct the expecteti return of each fund for the period of July 1998 to June 1999. You have the foilowing predictions of the economic factors for the next year: Industrial Growth 30—Day T—Bifl Yield Oil Price 6% 6% 29 The numbers are enduorwperiod prediction, assuming a linear path from the cur» rent period. For exampie, the Industriai Growth window wilt look like 10 a: Percentage change from previous 12 months in the index cl industna} production G 9706 9703 9710 9732 9802 9354 9806 9396 9815 9812 9902 9904 9996 Dale 95% Con! lnl E15345 513% Conf int ’i 74 PART 1! Ponrrouo MANAGEMENT You employ a multifactor model with respect to these macroeconomic factors f6 project the fund performance next year. a. Sup factor beta. What are the short—ran expected returns for each fund over the next year? Plot the efficient frontier and identify the position of each fund, i). Now you expand the beta estimation to cover the period from 1983.07 to 1998.06. Repeat part a. c. What can you conclude from parts a and b? *3: REFERENCES Black, F., lensen, M. C., and Scholes, M. 1972. “The Capi— tal Asset Pricing Model: Some Empirical Tests,” in Studies in Theory of Capital Markets, ed. M. C. Jensen. New York: Hacger. Binnie, M. E. 1971. “On the Assessment of Risk," Joan nal of Finance (March). Brennan, M., Chordia,T., and Subrahrnanyam, A. 1999. “Cross-Sectional Determinants of Expected Returns,” Journal of Financial Economics. Brenner, M, and Smidt, S. 1978. “Asset Characteristics and Systematic Risk,” Financial Management (Winter). Chan, L, and Lakonishok, I. 1993. "Are Reports of Beta’s Death Premature?” Journal of Porg‘olio Man- agement (Summer). Chen, S. 1981. “Beta Non—Stationarity, Portfolio Resid~ ual Risk and Diversification,"Journal of Financial and Quantitative Analysis (March). Cohen, K., and Pogue, J. 196’:1 . “An Empirical Evaluation of Alternative Portfolio Selection Models," Journal of Business (April). Cornell, B., and Dietrich, i. K. 1978. “MeanrAhsoEute- Deviation versus Least-Squares Regression Estima— tion of Beta Coefficients,” Journal of Financial and Quantitative Analysis (March). Eubank, A.A., and anwait, I. 1979. “How to Deter- mine the Stability of Beta Values,” Journal of Portfolio Management (Winter). Fania, E. F. 1973. “A Note on the Market Model and the Two Parameter Model,” Journal of Finance (December). Frabozzi, F. 1., and Francis, J. C. 1978. “Beta as a Random Coefficient,” Journal of Financial and Quantitative Analysis (March). Frankfurter, G. M. 1976. “The Effect of ‘Markct Indices‘ on the Ex—Post Performance of the Sharpe Portfolio Selection Modei,”lournal of Finance (lune). Haugen, R. A. 1999a. The New Financem'l‘he Case for an Over-reactive Stock Market, Upper Saddle River, NJ : Prentice Hall. Haugen, R. A. 19991). The Inefficient S lock Marketw What Pays Offand Why, Upper Saddle River, NJ: l’rentice Hall. ’13 O m (D '-< O I: l: m (it ,... D" (‘9 E. O )1 B n) i: O 3:! W O B H \0 \O {,3 O 5.3 H O H \0 O on O U\ PI- 0 (‘9 w s‘ m m- a El £5. Haugen, R. A., and Baker, N. L. 1996. “Commonaiity the Determinants of Expected Stock Returns,” J9 ml of Financial Economics (Joly). Hill, N. C., and Stone, B. K. 1980. “Accounting Betas, Systematic Operating Risk, and Financial Leverag Risk Composition Approach to the Determinants Systematic Risk,” Journal of Financial and Quanh rive Analysis (September). Jegadecsh, N. 1990. “Evidence of Predictable Behan of Security Returns," Journal of Finance (July). Jegadeesh, N, and Titman, S. 1993. “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency," Journal of Financé (March). . King, B. n 1966. “Market and Industry Factors in st Price Behavior,” Journal of Business (January) Klemkosky, R. C., and Martin, J. D. 1975. “The A merit of Beta Forecasts,” Journal of Finance (September). ' Lindahl-Stevens, M. 1978. “Some Popular Uses and Abuses of Beta,” Journal of Portfolio Manageme (Winter). '9 McCiay,M.1978.“Thc Penalties of incorring Urisy tematic Risk,” Journal of Portfolio Management (Spring). " Robichek,A. A., and Coho, R. A. 1974.“'111e Eco _ V nomic Determinants of Systematic Risk,”]ouma of Finance (May). Roenfeldt, R. 1..., Griepentrof, G. 1..., and Pfiaum,_ _ 1978. “Further Evidence on the Stationarity of B Coefficients,” Journal of Financial and Qannti Analysis (March). _ R011, R. 1984. “A Simple Implicit Measure of the Effective Bid-Asked Spread in an Efficient M Journal of Finance. ' Rosenberg, B. 1974. “Extra-Market Components Covariance Among Security Returns," Journa Financial and Quantitative Analysis (March): Rosenberg, B.,and Guy, J. 1976.“Beta and lave:t 2r ! ’31 t: :3 G.- c; 3 a :5 H. E. m 3.. 9-1 21 .7: Se :3 n n a a. he :5 E. 9. ca ‘6‘ E a :1 (July—August). new CHAPTER 6 FACTOR MOSfiLS " scheles, Mi, and Williams, 3. 1977. “Estimating Beta from " _ Nonsync ironous Data," Journal 0 Fl ' i M (December). If f nancia Econom— Sharpeflill. F. 1963. “A Simplified Model of i’ortfolio - ABaEYSIS,” Management Science (January). Theobald, M. 1981. “Beta Stationarity and Estimation Period: Some Analytical Results,” Journal of Financial and Quantitative Analysis (December). 1'75 Umstcad, A., and Bergstrom, G. L. 1979. “Dynamic Estimation of Portfolio Betas," Journal of Financial and Qimnlltarive Analysis (September). Weinstein, M. 1981. “The Systematic Risk of Corporate Bonds," Journal of Financiala d ‘ ' ' (sewerale n QuantitativeAnalysw ...
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