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the pull of the market, they induce changes in the rates of return on may“ .
tiesTlius, in Figure 6.1, if the return to the market portfoiio in a given permd w
to —5 percent, we would expect the return to the stock to be 2 percent. 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 righthand 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 SINGLEFACTOR 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
51,!
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 singietactor 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 welldiversified 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 singlefactor 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 threesecurity 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 singlefactor 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 lefthand 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 twosecurity 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
industrytype 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$__iM
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 anglefact
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 gistfactor 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 singlefactor 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 twostock 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 ﬁghtm 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 singlefactor 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: Bitire
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] EGgetImr 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 singlefactor model the beta 1’ Intoi 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 twofactor 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 capitaiizationweighted 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 longterm
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 emaii 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 ' ‘
treasuryhill 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 permgit 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 numbereight 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 highquality 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 shortterm 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},
.phascjntrodp‘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
97
.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 compet1 1 1 . . t .l I ' I
I l ale] "la t these 1 lVﬂSlUIS iieiieved “key tend to he disappomtﬁﬁi
In est01.5111 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 shortterm 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 shortterm negative serial correlation may be induced by a phenomenon
called the “bidasked 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 shortterm 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 shortterm revarsals remain suc
cessqu even when returns {or the previous month do not reflect the last day of trading. Second, there are intermediateterm (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 longterm (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 timevarying 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 PROiECTING 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 ' '
:ebggkgrice 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_p1ice 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
Bookpm“? , . 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 capitalizationweighted 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 bookprice 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 crosssectionai standard deviations below an average stock, in terms or itss 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 shortterm 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 shortterm 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 20year treasury bond. You will see an index number
(January 1982 = 100) for the yieidtomaturity 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
iii1:135:11: 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~percentagcpoint 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 twofactor
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 twofactor 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 singiefactor 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 ughOK
—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 singlefactor 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 industryWide event, what is true of the SFM estimate of the portfoiio’s ' ut rrta
b. WEILG 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 {actors linpr 1“ degBI 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
Company59 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‘ EeCted,$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) 'shu‘ 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
singlefactor 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‘riottake 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 singlefactor 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 capitalizationweighted 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 singiefactor 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 threaterrata) Plugging in the known va sz {saw 62 (r) 8202 {FM}
2 tablet; on the righthand side of the equation, we find
o (8.4) = 62%) —— 53,029,”) = .07 w (.50)2(.06) = .055
(jag) = 0203) — [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 smallgrowth 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 offdiagonal 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 :mTMNewFiitanceoom 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 1388 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—
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ual Risk and Diversification,"Journal of Financial
and Quantitative Analysis (March). Cohen, K., and Pogue, J. 196’:1 . “An Empirical Evaluation
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Business (April). Cornell, B., and Dietrich, i. K. 1978. “MeanrAhsoEute
Deviation versus LeastSquares Regression Estima—
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Quantitative Analysis (March). Eubank, A.A., and anwait, I. 1979. “How to Deter
mine the Stability of Beta Values,” Journal of Portfolio
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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
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