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Unformatted text preview: The Correlation Structure of
Security Returns 'fHE SiNGLE—iNDEX MODEL in the previous three chapters of this booit we oudined the basics of modern portfolio the
ory. The core of the theory, as described in these chapters, is not new; in fact, it was pre
sented as early as 1956 in Markowitz’s pioneering article and subsequent book. The reader,
noting that the theory is over 50 years old, might well ask what has happened since the thew
cry was developed. Furthermore, if you had knowledge about the actual practices of ﬁnan
cial institutions, you might well ask why the theory took so long to be used by ﬁnancial
institutions. The answers to both these questions are closely related. Most of the research
on portfolio management in the last 50 years has concentrated on methods for implement—
ing the basic theory. Many of the breakthroughs in implementation have been quite recent,
and it is only with these new contributions that portfolio theory becomes readily applica—
bic to the management of actuai portfolios. In the next three chapters we are concerned with the implementation of portfoiio theory.
Breakthroughs in implementation fall into two categories: The first concerns a simpliﬁca—
tion of the amount and type of input data needed to perform portfolio analysis. The second
involves a simpliﬁcation of the computational procedure needed to calculate optimal port
folios. As will soon become clear, these issues are interdependent. Furthermore, their res—
olution vastly simpiiﬁes portfolio analysis. This results in the ability to describe the
problem and its soiution in relatively simpie termsmterrns that have intuitive as well as
analyticai meaning, and terms to which practicing security analysts and portfolio managers
can reiate. In this chapter we begin the problem of simplifying the inputs to the portfolio problem.
We start with a discussion of the amount and type of information needed to solve a port~
folio problem. We then discuss the oidest and most widely tised simpiiftcation of the port
folio structure: thc singieiudex model. The nature of the model as weil as some estimating
techniques are examined. In Chapter 8 we discuss aiternative simpliﬁed representations of the portfolio problem.
In particular, we will be concerned with other ways to represent and predict the correlation
structure between returns. Finally, in the last chapter dealing with implementation we will
show how each of the techniques that have been developed to simplify the input to portfo
lio analysis can be used to reduce and simplify the calcuiations needed to find optimal
portfolios. 130  (HAWER 7 THE CORREMTJON STRUCTURE OF SECURETY RETURNS 1 3 1 Most of Chapters 7 and 8 wili be concerned with simpiifying and predicting the corre
lation structure of returns. Many of the single— and multi—index models discussed in these
Chapters were developed to aid in portfolio management. Lately. however, these models
have been need for other purposes that are often viewed as being as important as their use
'm portfolio analysis. Although many of these other uses will be detailed later in the book.
we briefly describe some of them at the end of this chapter and in Chapter 8. 1H5 ENPUTS TO PORTFOLIO ANALYSIS {.et us return to a consideration of the portfolio problem. From earlier chapters we know
that to define the efﬁcient frontier we must be able to determine the expected return and
standard deviation of return on a portfbiio. We can write the expected return on any port folio as  N
E5 = 2 xiii. (7.1)
.ml
While the standard deviation of return on any portfolio can be written as
2/2
N N N
0;» = XXI20‘} +22X1'Xj056jpg‘j {7.2)
[:1 (:1 1:1
jar These equations define the input data necessary to perform portfolio analysis. From
Equation (7.1) we see that we need estimates of the expected return on each security that
is a candidate for inclusion in our portfolio. From Equation (7.2} we see that We need esti
mates of the variance of each security, plus estimates of the correlation between each pos
sible pair of securities for the stocks under consideration. The need for estimates of
correlation coefﬁcients differs both in magnitude and substance from the two previous
requirements. Let’s see why. The principal yob of the security analyst traditionally has been to estimate the future per—
formance of stocks he or she follows. At a minimum this means producing estimates of
expected returns on each stock he follows,1 With the increased attention that “risk” has received in recent years, more and meta ana—
lysts are providing estimates of risk as well as return. The analyst who estimates the
expected. return of a stock should also be in a position to estimate the uncertainty of that
return. ' Correiations are an entirely different matter. Portfoiio anaiysis cails for estimates of the
pairwise correlation between ail stocks that are candidates for inclusion in a portfolio.
Most ﬁrms organize their analysts along traditional industry lines. One analyst might fol—
low steel stocks or, perhaps in a smailer firm, ail metal stocks. A second analyst might fob
low chemical stocks. But portfolio analysis calls for these analysts not oniy to estimate
how a particular steel stock will move in relationship to another steel stock, but also how
a panicoiar steel stock will move in {elationship to a particular chemical stock or drug
stock. There is no nonoverlapping organizationai structure that ailows such estimates to be
directly produced. 'Whether the analyst’s estimates contain information or whether one is better off estimating returns from an equi
librium model {such as those to be presented in Chapters 10, l3. and 14) is an open question. We have more to say
about this later. However. the reader should note that portfolio selection modeis can help to answer this question. 1 3 2 rarer 2 someone matrsss The problem is made more complex by the number of estimates required. Most ﬁnan
ciai institutions follow between 150 and 259 stocks. To employ portfolio analysis. the
institution needs estimates of between 150 and 250 expected returns and 150 and 250 vari
ances. Let us see how many correlation coefﬁcients it needs. If we ict N stand for the num
ber of stocks a firm follows, then it has to estimate pa for all pairs of securities i and j. The
ﬁrst index i can take on N values (one for each stock); the second can take on (N — 1) val—
ues (remember j as i}. This gives us MN m 1) correlation coefficients. However, since the
correlation coefficient between stocks 2‘ and j is the same as that between stocks j and 2‘, we
have to estimate oniy N(N “ l)/‘.’2 correlations. The institution that follows between 159
and 250 stocks needsbetween £1,175 and 31325 correlation coefﬁcients. The sheer num
ber of inputs is staggering. It seems uniikely that analysts will be abie to directly estimate correiation structures.
Their ability to do so is severely limited by the nature of feasible organizational structures
and the huge number of correlation coefﬁcients that must be estimated. Recognition of this
has motivated the search for the development of models to describe and predict the corre
lation structure between securities. In this chapter and in Chapter 8 we discuss some of
these modeis and examine empirical tests of their performance. The models developed for forecasting correlation structures fall into two categories:
index models and averaging techniques. The most widely used technique assumes that the
co~rnovernent between stocks is due to a single common influence or index. This model is
appropriately sailed the singie—index model. The single—index model is used not only in
estimating the correlation matrix, but also in efficient market tests (discussed later} and in
equiiibrium tests, where it is cailed a returngenerating process. The rest of this chapter is
devoted to a discussion of the properties of this model. SlNGLEENDEX MODELS: AN OVERWEW Casual observation of stock prices reveals that when the market goes up (as measured by
any of the widely available stock market indexes), most stocks tend to increase in price,
and when the market goes down, most stocks tend to decrease in price. This suggests that
one reason security returns might be correlated is because of a common response to mar
ket changes, and a useful measure of this correiation might be obtained by relating the
return on a stock to the return on a stock market index. The return on a stock can be writ
ten as2 R: Wt +¥3tRm a, is the component of security is return that is independent of the market’s perform»
ance——a random variable. R,“ is the rate oi return on the market index—a random variable.
E». is a constant that measures the expected change in Ri given a change in Rm.
This equation simply breaks the return on a stock into two components, that part due to the market and that part independent of the market. [5, in the expression measures how
sensitive a stock’s return is to the return on the market. A (3, of 2 means that a stock’s return 2The return on the index is identical, in concept, to the return on a common stock. It is the return the investor
Wonk? earn if he or she held a portfolio with a composition identical to the index. Thus, to compute this return.
the dividends that would be received from holding the index should be calculated and combined with the price
changes on the index. ~, (HAPTER 7 THE CCRRELATlON STRUCTURE OF SECURH’Y RETURNS 1 33 is expected to increase (decrease) by 2% when the market increases (decreases) by 1%.
Similarly, a B of 0.5 indicates that a stock’s return is expected to increase (decrease) by 712—
of “1% when the market increases (decreases) by 1%.3 ‘ The term n, represents that component of return insensitive to (independent of) the
return on the market. It is useful to break the term a, into two components. Let or; denote
the expected value of a, and [at e, represent the random (uncertain) e‘iement of a1. Then {ii =oti+ci where er has an expected value of zero. The equation for the return on a stock can now be
written as R: m Diff EliRm “l” 61 Once again, note that both 2, and Rm are random variables. They each have a probability
distribution and a mean and standard deviation. Let as denote their standard deviations by
(re, and cm, respectively. Up to this point we have made no simplifying assumptions. We
have written return as the sum of several components but these components, when added
together, must by deﬁnition be equai to total return. it is convenient to have 2,» uncorrelated with Rm. Formally. this means that p, cov(ei Rm ) : E[(ei " ORR)" " Era] = 0
If e, is uncorrelated with R,,,, it impiies that how well Equation (7.3) describes the return
on any security is independent of what the return on the market happens to be. Estimates
of on, Bi, and 0%, are often obtained from time seriesregression analysis.4 Regression
analysis is one technique that guarantees that e, and R», will be uncorrelated, at least over
the period to which the equation has been ﬁt. All of the characteristics of singleindex
models described to this point are deﬁnitions or can be made to hold by construction.
There is one further characteristic of singleindex models: it holds oniy by assumption.
This assumption is the characteristic of single~index models that differentiates them from
other models used to describe the covariance structure. The key assumption of the singleindex model is that e, is independent of e; for ail val~
ues of i and j or, more fortnaily, Emmi} "~= O. This impiies that the only reason stocks vary
together, systematically, is because of a common co—movemcnt with the market. There are
no effects beyond the market (e. g., industry effects) that account for comovement among
securities. We will have more to say about this in our discussion of multiindex models in
Chapter 8. However, at this time, note that, unlike the independence of an and Rm, there is
nothing in the normal regreseion method used to estimate on, 13;, and o3, that forces this to
be true. It is a simplifying assumption that represents an approximation to reaiity. How
Well this model performs wili depend, in part, on how good (or bad) this approximation is.
Let us summarize the singleindex model: BASIC EQUATION R! m a; .1. Big”: + e: for all stocks i = 1, ..., N 3We are illustrating the singleindex model with a stock market index. it is not necessary that the index used be
a stock market index. The selection of the appropriate index is an empirical rather than a theoretical question.
However, anticipating the resuits of future chapters, the results should be better when a broadvbased market”
weighted index is used. such as the 3&1” 500 index or the New York Stock Exchange index. “ibis will be discussed in more detail later in the chapter. ; 34. mar 2 someone mamas BY CONSTRUCTION 1. Mean of ei m E(e,~) = 0 for all stocks t' = i, N
BY ASSUMPTiON
1. index unrelated to unique return: for at; stocks 1' ﬂ 1, .., N E[€t(Rm W 111)} = 0
2. Securities only related through common
response to market: E(eiej) m 0 for all pairs of stocks 1' = 1, ..., N
andj=1,...,Nbutintj BY DEFINITION
1. Variance of e; "—r E{eg}2 m 0%. 2. Variance of Rm = E(Rm “ 1%,? = 0’?” for all Stocks r‘ w i, ..., N In the subsequent section we derive the expected return, standard deviation, and covariance
when the single—index model is used to represent the joint movement of securities. The
results are 1. The mean return, R: oe + 1321i".
2. The Variance of a security’s return, o? = Bio?" + egg.
3. The covariance of returns between securities 1' and j, [If]? m {SeSpit. Note that the expected return has two components: a unique part or, and a marketvreiated
part BER“. Likewise, a security’s variance has the same two parts, unique risk o3, and
market—related risk B? 032. In contrast, the covariance depends oniy on market risk. This is
what we meant earlier when we said that the singleindex modei implied that the only rea»
son securities move together is a common response to market movements. In this section of the text we delive these results.
The expected return on a security is = 'i" ERR»: + 6i] Since the expected value of the sum of random variables is the sum of the expected val
ues, we have E(R,) e E{o'.i) + autism) + 3(a) or; and B; are constants and by construction the expected value of e. is zero. Thus, E(R,~) m oti + [itEm Result 1
The variance of the return on any security is o? a E(R; _ it.)2
Substituting for R,» and R} from the expression above yields o? e E[(oc, +13,R,,1 +e;)(ot;+[3;lim)12
Rearranging and noting that the ads cancei yields
(5,? = fife(Rm  Emit €f]2 Squaring the terms in the brackets yields a} a stem. — if + mews", — FM] + r«:(e,}2 cmP'fER 3' THE CORREMTiON STRUCTURE OF SECURJTY RETLJRNS 1 35' Recall that by assumption (or in Some cases by construction) Hal(Rm v“ ﬁmﬂ m 0. Thus, ’ (31‘2 a BrgE<Rm "“ Em )2 “1‘ 522 = 3525;: + 5% . Result 2
The covariance between any two securities can be written as
e=nasnrnt
Substituting for Rf, RE, R1, and EJ yields Off = Eﬂﬁxi + ﬂight + ei) — (OLE + BEE)” ‘ [(0% t ﬁi‘Rmf‘TeJ)” (0% + Mail} Simplifying by canceling the 063 and combining the terms involving B’s yields anaeesaunnewnnnl Carrying out the multiplication
6,]: 3 v“ ‘Rjnf 'i BJE[€i(Rm " Em i neeewsnaei Since the last three terms are zero, by assumption Gij = [5:15 15:27: Result 3 , These results can be illustrated with a simple example. Consider the returns on a stock
and a market index shown in the ﬁrst two cotumns of Tabie 7.1. These returns are what an
investor might have observed over the prior ﬁve months. Now consider the values for the
single—index model shown in the remaining columns of the table. Column three just repro—
duced column one and is the return on the security. In the next Section of this chapter we
will show you how to estimate on and Bi. For now asseme that B; = 1.5. This is what it
would be equal to if we applied the ﬁrst estimation technique described in the next section.
Then from result 1, o.. m 8  6 = 2. Since the singleindex model must hold as an equal—
ity, e, (coiurnn 6) is just deﬁned in each period as the value that makes the equality hold or: 3i 2 RE “(at +l3rRm) ' For example, in the first period the sum of Otf and BER,n is 8. Since the return on the secum
rlty in the ﬁrst period is 30, e, is +2. Tobie 7.1 Decomposition of Returns for the SingleIndex Model WWW l 2 3 4 5 6 Month Return on Stock Return on Market Rt : ee t BgRmt e,»
1 i0 4 10 = 2 + 6 + 2
2 3 2 3 w 2 + 3  2
3 35 8 15 m 2 + 12 + l
4 9 6 9 = 2 + 9  2
5 3 0 3 w 2 i 0 + 1
Totai 40 20 40 = 10 + 30 + 0
Average 8 4 8 w 2 + 6 + 0
Variance 20.8 8 20.8 = {i + 18 + 2.8 MW 1 36 EAR? 2 PORT?OLIO ANALYSSS The reader should now understand where all the values of the singleindex model come
from except 3,: B. divides return into marketrelated and unique return. When ii. is set
equal to 1.5. the market return is independent of the residual return e,. A lower value of e,
ieaves some market return in e. and the covariance of el with the market is positive. A B:
greater than 1.5 removes too much market return and results in a negative covariance
between e; and the market. Thus the value of B; is unique and is the value that exactly 36p»
arates market from unique return, matting the covariance between R", and e, zero. The
reader can calculate the covariance between columns 2 and 6 in Table 7.1 and see that it is indeed zero.
Before leaving the simple example, let’s apply the formulas return on the security is presented earlier. The mean Ems/sag using the formula from the singleindex model.
it} not; dried—3," =2+i.5(4)28
The variance of security i is calculating from the formula derived for the singieindex
modet.
o? = bio?“ + 63;.
= {1.5)2(8)+2.8
m 20.8 Calculating the variance of the security directiy from column 2 of Table 7.1, we see that the answer is 20.8, identical to the answer produced by the equation above.
Having explained the simple example, we can turn to the calculation of the expected
return and variance of any portfolio if the singleindex model holds. The expected return on any portfolio is given by
.R‘P = Z Xﬁr
Substituting for R, we obtain
at: = Zine. +EXiBrRm o4)
i=1 (:1 We know that the variance of a portfolio of stocks is giVen by Substituting in the results stated above for 0% and of}, we obtain .N N N N
at, = z Xﬁpgoﬁ, + E 2 XIXJiiilijo%1+ Exﬁegf (7.5) i=1 [:1 int i=1 jil There are many alternative ways of estimating the parameters of the single—index model.
From Equations (7.4) and (7.7) it is clear that expected return and risk can be estimated for
any portfolio if we have an estimate of at; for each stock. an estimate of {3, for each stock.
an estimate of 0%; for each stock. and. ﬁnally. an estimate of both the expected return (Em) OMPT‘ER 7 THE CORRELATION STRUCTURE OF SECURITY RETURNS f 37 and variance (032") for the market. This is a total of 3N + 2 estimates. For an institution fol
lowing between 156 and 250 stocks, the single—index model'lrequires between 452 and 752—
estimates. Compare this with the 1§,175u—31,12.5 correlation estimates or 1: 475—31 625
total estimates required when no simplifying structure is assumed. Furthermore notehhat
there is no requirement for direct estimates of the joint movement of securities ,onl esti~
mates of the way each security moves with the market. A nonoverlapping organiziion 1
structure can produce all the required estimates. a The model can also be employed if analysts supply estimates of expected return for each
stook, the variance of the return on each stock, the Beta (5,} for each stock and the vari
ance of the market return.5 This is 3N "l 1 estimates. This alternative set oftestimat h
the advantage that they are in more familiar terms. as as We have discussed means and variances, before. The only new variable is Beta The Beta
is gulfply a mejsure of the sensitivity of a stock to market movements. . e one we iscuss alternative wa s ' ' ' ' Characteristics “the Sing§e_index mzdeif estimating Betas, let us examine some of the CHARACTERISTICS OF THE SINGLEJNDEX MODEL iDeﬁne the Beta on a portfoiioﬁp as a weighted average of the individual Bis on each stock
’1?th portfolio where the weights are the fraction of the portfolio invested in each stock
en I N
i3? = 2 XiBi
£=l
Similarly deﬁne the Alpha on the portfolio or}: as N
{1p = EXEOIZ
i=l Then Equation (7.4} can be written as
E.» = 0i.» + 31°17»: If the portfolio P is taken to be the market portfolio (all stocks held in the same to o 
trons as they were in constructing Rm), then the expected return on P must he R p Fforrn
the above equation the only values of B]: and or)» that guarantee R» m Em for any choice of
Rm )3 or,» equal to G and [3p equal to 1. Thus, the Beta on the market is 1 and stocks are thought (3f as being IIIOIB 01' 1355 “Sky than the InaIket accoxdlno [0 Whathel the” Beta 13
s
a S'I‘he fact that thcs ' :5 m '
c mpu a cqulvatent to those discussed earlier is ea T
 . sy to show. he expccm
used directly to estimate the expected return on a portfolio d mums can be N
Ev = 2X};
{El “i6 est'mates 0f the V U V arket and s c e on each stock can be
ariance Of return on a $930k, the ariance 0f 3’12 m
I v I 3 B ta
“5‘56 ‘0 derive estimates Of its residual risk by noting that . a? = lit25%: + (53; in Eddillon ‘ I W t s 311d Va: ances and
I , {ills structure 15 natural f0: 2]} ' ' '
ose lUSiltuliOnS that an: anal S 5‘ estimate I
‘ . y 5 of mean '
I! 0‘13] 8831113288 of correlations 0r covariances. l 1 38 PART 2 PORTFOUO ANALYSIS Let us look further into the risk of an individual security. Equation (7.5) is N N N N
at e 2 Kate. + 2x363. + Z EXszﬁeBJct
I=l i=1 5:: 1:1
jet In the double summation i at j. if i m j. then the terms would he Xinsioﬁp But these are
exactly the terms in the ﬁrst summation. Thus, the variance on the portfolio can be written as N N N
5%: = "i" i=1 j=l [:21 Or by rearranging terms N N N
0% = 2333; EXij 0%: +2.35%
i=1 1:1 i3: Thus, the risk of the investor’s portfolio could he represented as N
at: a hint + 2 X363:
i=1
Assume for a moment that an investor forms a portfolio by pincng equal amounts of
money into each of N stocks. The risk of this portfolio can be written 386 1 N1
2" 2 2 __ __ 2.
op— PomIN iE—lNoﬂ Look at the last term. This can be expressed as UN times the average residual risk in the
portfolio. As the number of stocks in the portfoiio increases. the importance of the average residual zisk. N 2
Est N i=1 diminishes drastically. In fact, as Table 7.2 shOWs. the residuai risk fails so rapidly that most of it is effectiveiy eiiminated on even moderateiy sized portfolios.7
The risk that is not eliminated as we hold larger and larger portfolios is the risk associ
ated with the term Bp. If we assume that resident risk approaches zero. the risk of the port— folio approaches
1/2 N
0P =[ 1206;}; = $90»: =0'm ZXiﬁi
l‘ﬁl Since cm is the same, regardless of which stocic we examine, the measure of the contri«
hution of a security to the risk of a large portfolio is 6;. 6Esaulining the expression for the variance of portfoiio P shows that the assumptions of the singieindex model are inconsistent with 0%: w «,2». However. the approximation is very close. See Fame (1968) for a detailed dis» Cussion of this issue.
Tie the extent that the singleindex model is not a perfect description of reality and rcsitiuais from the market model are corrciated across Securities. residual risk does not fall this rapidly. However, for most portfoiios the
amount of positive correlation present in the residuals is quite small and residual risk declines rapidly as the num ber of securities in the portfoiio increases. (“AFTER 7 THE CORRELATION STRUCTURE 0:: SECURITY RETURNS
1 3 9 Table 7.2 Residual Ri $1 and Portfolio Size Residual (Variance) Exped Number as a Percent of the Residual ’
of Securities in a OneStock Portfolio with Eatint i 100 2 50 3 33 4 25 5 20 i0 . 10 20 ' ,' . S 100 ' 1 1000 I. t‘ ' 0‘]
W The risk of an ludividual security is Ego; + 0%. Since the effect of 02 on portfolio risk
can be made to approach zero as the portfolio gets larger, it is commoziﬂto refer to 3 A
drvetstftable risk.8 However, the effect of [ﬂog on portfolio risk does not diminishoral :3
gets larger.'Since GE. is a constant with respect to all securities, [3' is the measure of a as
rity’s nondiversiﬁable risk.9 Since diversifiable risk can be eliminated by holding at; enough portfolio. [31 is often used as the measure of a security’s risk. ESTIMATING BETA Eve estimates of Beta for a security or a portfolio. On the other hand, estimates of future
eta couid he arrived at by estimating Beta from past data and using this historical Beta as
an estimate of the future Beta. There IS evidence that historical Betas provide usequ inforn Because of this, even the firm that wishes to use analysts’ subjective estimates of futu Betas should start with (supply analysts with) the best estimates of Beta avaiiable from h're
torical data. The analyst can then concentrate on the examination of inﬂuences that am
expected to change Beta in the future. In the rest of this chapter we examine some of ti:
techniques that have been proposed for estimating Beta. These techniques can be cias ' ﬁed as measuring histories? Betas, correcting historical Betas for the tendency of hist 8'1
cal Betas to be closer to the mean when estimated in a future period and co ' Sm
hlstorlcal estimates by incorporating fundamental ﬁrm data. : necnng Estimating Historical Betas
In Equation (7.3) we represented the return on a stock as
Rt 2 05f + Bar: 'i' 6; orTlgns equation is expected to hold at each moment in time. although the values of «1 B
og. might differ over time. When looking at historical data, one cannot directly observej 3 .
9An alternative nomenclature calis this nonmarket or unsystematic risk.
An alternative nomenclature calls this market risk or systematic risk. 140 PART 2 noarrouo ANALYSES on, 8;, or 03,. Rather, one observes the past returns on the secarity and the maricet. If on, [3,,
and GE, are assumed to be constant through time, then the same equation is expected to
hold at each point in time. in this case, a straightforward procedure exists for estimating
on, Bi, and 03;. Notice that Equation (7.3) is an equation of a straight line. if 0%, were equal to zero, then
we could estimate or, and B, with just two observations. However, the presence of the ran
dom variable a, means that the actual return will form a scatter around the straight line.
Figure 7.} illustrates this pattern. The vertical axis is the return on security i and the hon
zontal axis is the return on the market. Each point on the diagram is the return on stock 1
over a particular time interval, for exampie, one month (I) plotted against the return on the
market for the same time interval. The actuai observed returns lie 0n and around the true
relationship (shown as a solid tine}. The greater oi}, the greater the scatter around the line,
and since we do not actually observe the line, the more uncertain we are abont where it is.
There are a number of ways of estimating where the line might be, given the observed scat
ter of points. Usaaily, we estimate the iocation of the line using regression analysts. This procedure could be thought of as ﬁrst plotting R5, verses Rm, to obtain a scatter of
points such as that shown in Figure 7. 3. Each point represents the return on a pamcaiar
stock and the return on the market in one month. Additional points are obtained by plotm
ting the two returns in successive months. The next step is to fit that straight line to the data
that minimized the sum of the squared deviation from the line in the vertical (RR) direc—
tion. The slope of this straight line wonid be our best estimate of Beta over the period to
which the line was ﬁt, and intercept would be our best estimate of Alpha {calm I More formally, to estimate the Beta for a ﬁrm for the period from t = l to r = 60 Via
regression analysis use [a E:)(Rwﬁmll mt Fire ‘Gif R;, and R,,,, come from a bivariate normal distribution, the unbiased and most efﬁcient estimates of or, and B;
are those that come from regressing R" against Rm, the procedure described above. cHAPT‘ER '3' THE CORREMTION STRUCTURE OF SECURITY RETURNS I 4,] and to estimate Aipha use”
" {xi : Rf: ’giRmt To learn how this works on a simple example let us return to Table 7.i. We used the data
in 7.1 to show how Beta interacted with returns. But now assume that all yon observed was
columns 1 and 2 or the return on the stock and the return on the market. To estimate Beta,
we need to estimate the covariance between the stock. and the market. The average return on
the stock was 40/5 m 8, whereas on the market it was 20/5 = 4. The Beta value for the stock
is the covariance of the stock with the market divided by the variance of the market or inane we) 5
m: t=t 5 I. \
gamer/s The covariance is found as follows: Stock Return Market Return Month Minus Mean Minus Mean Value
1 (10 — 8) x (a u» 4} _ o
2 (3 w 8) X (2 —— 4} m 10
3 (15 —— 8) x (3 w 4} = 28
4 (9 m 8) X (6 — 4) m 2
5 (3 — a) x (0 w 4) = an
Total 60 The covariance is 60/5 = 12. The variance of the market return is the average of the sun;
of squared deviation from the mean a}, 404—4)? +0.41)? +(8—4)2+(6—4}2 +(0m4ﬂ/Sm8 Thus Beta = 12.18 = 1.5 This value of Beta is identical to the number used in constructing Table 7.1. Alpha can be computed by taking the difference between the average security return and
Beta times the average return on the market. a, =8~{l.5)(4)=2 “We other statistics of interest can be produced by this analysis. First, the size of oi, over the estimation period
can be found by looking at the variance of the deviations of the actuai return from that predicted by the modei: 1 60 2
<53, "(er +135Rmii] {:1 Remember that in performing regression analysis. one often computes a coefficient of determination. The coef
ﬁcient of determination is a measure of association between two variables. In this case. it would measure how
muoh of the variation in the return on the individuai stock is associated with variation in the return on the mar— ket. The coefﬁcient of determination is simply the correlation coefﬁcient squared, and the correiaticn coefficient
is equal to Cyint _ﬁi5g'n_ Gm
W_W_§i.._ pm UgGm 650”, o; 1 42 PART 2 romrouo ANALYSIS The values of ct; and 13; produced by regression anaiysis are estimates of the true on and
[3, that exist for a stock. The estimates are subject to error. As such, the estimate oi st, and
B; may not be equal to the true or; and B; that existed in the period.12 Furthermore, the
process is complicated by the fact that on and Bi are not perfectly stationary over time. We
would expect changes as the fundamental characteristics of the ﬁrst: change. For example,
[3; as a risk measure should be related to the capital structure of the ﬁrm and, thus, should
change as the capital structure changes. Despite error in measuring the true [3, and the possibility of reai shifts in 9; over time, the
most straightforward way to forecast B, for a future period is to use an estimate of B, obtained
via regression anaiysis from a past period. Let us take a look at how well this works. Accuracy of Historical Betas The ﬁrst iogicai step in looking at Betas is to see how much association there is between
the Betas in one period and the Betas in an adjacent period. Both Biume (1970) and Levy
(i971) have done extensive testing of the relationship between Betas over time. Let us look
at some representative results from Blume’s {1970) study. Blurrie computed Betas using
time series regressions on monthly data for nonoverlapping sevenyear periods. He gener
ated Betas on single stock portfolios, Zwstock portfolios. 4stock portfolios, and so forth up
to 50stock portfolios and for each size portfoiio examined how highly correlated the Betas
from one period were with the Betas for a second period. Table 7.3 presents a typical result
showing how highly correlated the Betas are for the period 7154—6161 and 7/61m6/68. It is apparent from this table that, whiie Betas on very large portfolios contain a great
deal of information about future Betas on these portfolios, Betas on individual securities
contain much iess information about the future Betas on securities. Why might observed
Betas in one period differ from Betas in a second period? One reason is that the risk (Beta)
of the security or portfolio might change. ‘A second reason is that the Beta in each period
is measured with a random error, and the larger the random error, the less predictive power
Betas from one period will have for Betas in the next period. Changes in security Betas will differ from security to security. Some will go up, some
will go down. These changes will tend to cancel out in a portfolio, and we observe less
change in the actual Beta on portfolios than on securities. Table 7.3 Association of Betas Over Time Number of Securities in Correlation Coefﬁcient of
the Portfolio Coefﬁcient Determination l 0.60 0.36 2 0.73 0.53 4 0.84 0.71 7 0.83 0.77 10 0.92 0. 35 20 0.97 0.95 35 0.97 0.95 50 0.98 0.96 M l2In fact, the analysis will produce an estimate of the standard error in both or; and 3;. This can be used to make
interval estimates of future Alphas and Betas under the assumption of stetionarity. cHAE'TEﬂ 7 THE CORREMTlON STRUCTURE OF SECURITY RETURNS 1 Likewise, one would expect that the errors in estimating Beta for individual securities
would tend to cancel out when securities are combined, and therefore, there would be less
eﬁor in measuring a portfolio’s Beta.E3 Since portfolio Betas are measured with less error
and since Betas on portfolios change less than Betas on securities, historical Betas on port:
folios are better predictors of future Betas than are historical Betas on securities. Adjusting Historical Estimates Can we further improve the predictive ability of Betas on securities and portfolios? To aid
in answaring this question, let us examine a simpie hypothetical distribution of Betas Assume the true Betas on all stocks are'really one. If we estimate Betas for all stocks some
of our estimated Betas will be one, but some will be above or below one due to sampling
error in the estimate. Estimated Betas above into would be above one simply due to positive
sampling errors. Estimated Betas beiow oneplwould be below one due to negative sampling
errors. Furthermore, since there is no reason to suspect that a positive sampling error for a
stock will be foliowed by a positive sampling error for the same stock, we would ﬁnd that
historical Beta did a worse job of predicting future Beta than did a Beta of one for all stocks Now, assume we have different Betas for different stocks. The Beta we calculate for any
stock Will be, in part, a function of the true underlying Beta and, in part, a function of sam
pling error. If we, compute a very high estimate of Beta for a stock, we have an increased
probability that we have a positive sampling error, while if we compute a very low estimate
of Beta, we have an increased chance that we have a negative sampling error. If this sce—
nario is correct. we should find that Betas, on the average, tend to converge to one in suc
ceSStve time periods. Estimated Betas that are a lot larger than one should tend to be
followed by estimated Betas that are cioser to one (lower), and estimated Betas below one
should tend to be foliowed by higher Betas. Evidence that this does, in fact happen has been
presented by Blume (1975) and Levy (1971). Biume’s results are rcproddced in Table 7 4 The reader should examine the table and conﬁrm the tendency of Betas in the forecast
period to be closer to one than the estimates of these Betas obtained from historical data.14 13    
hAssumIng that the reiationshlp between R;, and Rm, is described by a stationary bivariate normal distribution
t en the standard error m the measurement of Beta for a security is given by ’ Get = Get 10m
The standard error for the {i on a portfolio is given by 0'5}: = Gap/Gm Where where N is the number of securities in the portfolio and T is the number of time periods. 10:93:: cjtlent tlgsth the residuals for different stocks are not perfectly correlated, averaging them across stocks will Singie‘gnde: HBO: lt e residuals and, hence, the value of 0%,, on the portfoiio. In particular, if the assumptions of the WW” m e are met and if stocks are held In equal proportions, the standard error of the Beta on the portfolio
equal the average standard error on all stocks times the reciprocal of the number of stocks in the portfolio. 1 . .
rtThrough this section when we speak of Betas, we are referring to estimates of Betas. 1 44 PART 2 PORTFOLiO WLYSES r Two Successive Periods rooster 7161—6 7 0.393 0.620
0.612 0.707
0.810 0.861
0.987 0.914
[.133 0.995
1.337 1.169 Source: Blurne, Matched. “On the Assessment of Risk," Journal of Finance, VI, No. 1 (March 1971). Measuring the Tendency of Bates to Regress Toward One—
Blume’s Technique Since Betas in the forecast period tend to be closer to one than the estimate obtained from
historical data, the next obvious step is to try to modify past Betas to capture this tendency.
B1ume{1975) was the first to propose a scheme for doing so. He corrected past Betas by
directly measuring this adjustment toward one and assuming that the adjustment in one
period is a good estimate of the adjustment in the next. Let us see how this could work. We could caicuiate the Betas for all stocks for the period
1948—1954. We could then caiculate the Betas for these same stocks for the period
195 5—1961. We could then regress the Betas for the later period against the Betas for the
earlier period as shown in Figure 7.2. Note that each observation is the Beta on the same
stocic for the period 1948w1954 and 1955—1961. Following this procedure, we would
obtain a line that measures the tendency of the forecaster] Betas to be closer to one than
the estimates from historical data. When Biume did this for the period mentioned, he obtained
like a 0.343 + 0.6770“ where 13,; stands for the Beta on stock 1’ in the later period {1955—1961) and B“ stands for the [3 for stock i for the earlier period (1948—1954). The relationship implies that the Beta
in the later period is 0.343 + 0.677 times the Beta in the earlier period. Assume We wish Beta
i955—61 0.677 0.343 Beta
“34354 Figue "' CWER 7 THE CORREL/id’lON STRUCTURE OF SECURiTY RETURNS 1 45 to forecast the Beta for any stock for the period 1962—1968. We then compute (via regres
sion analysis) its Beta for the years 1955m196i. To determine how this Beta should be
inadiiied, we substitute it for B” in the equation. We then compute 0;; from the foregoing
equation and use it as our forecast. . Notice the effect of this on the Beta for any stock. If [3,1 were 2.0, then our forecast
would be 0.343 + 0.677(2)» m 1.697, rather than 2.0. If B“ were 0.5, our forecast wooid
be. 0.343 + 0.677(05) = 0.682 rather than 0.5. The equation lowers high values of Beta
and raises low vaiues. One more characteristic of this equation should be noted. It modi—
ﬁes the average level of Betas for the population of stocks. Since it measures the relation»
ship between Betas over two periods, if the average Beta increased ovar these two periods,
it assumes that average Betas will increase over the next period. Unless there is reason to
suspect a continuous drift in Beta, this/will he an undesirable property. If there is no rea~
son to expect this trend in the average Betazto continue, then the estimates can be improved
by adjusting the forecasted Betas so that their mean is the same as the historical mean. To make this point more concrete, let us examine an example. Assume that in estimao
ing the equation, Blame found the average Beta in i948—i95ti was one and the average
Beta in 1955—1961 was 1.02. These numbers are consistent with his results, though there
are other sets of numbers that woaici also be consistent with his results. Now, to determine
what the average forecasted Beta should be for the period 1962—4968. we simply substi
tute 1.02 in the righthand side of the estimating equation. The answer is i033. As dis—
cussed earlier, Blume’s technique results in a continued extrapolation of the upward trend
in Betas observed in the earlier periods. If there is no reason to believe that the next period’s average Beta will be more than this
period’s, then the forecasts should be improved by adjusting the forecast Beta to have the
same mean as the historical mean. This involves subtracting a constant from all Betas after
adjusting them toward their mean. In our example, this is achieved by subtracting 1.033
from each forecast of Beta and adding 1.02. Measuring the Tendency of Betas to Regress Toward Unem
Vasicek’s Technique Recall that the actual Beta in the forecast period tends to be closer to the average Beta than
is the estimate obtained from historical data. A straightforward way to adjust for this rem
dency is to simpiy adjust each Beta toward the average Beta. For example, taking onehalf
of the historical Beta and adding it to onehalf of the average Beta moves each historical
Beta halfway toward the average. This technique is widely used.15 ' It would be desirable not to adjust all stocks the same amount toward the average but
rather to have the adjustment depend on the size of the uncertainty (sampling error) about
Beta. The larger the sampling error, the greater the chance of large differences from the
average, being due to sampling error, and the greater the adjustment. Vasicek (1973) has
suggested the following scheme that incorporates these properties: If we lot {31 equal the
average Beta across the sample of stocks in the historical period, then the Vasicek procedure
Involves taking a weighted average of B1 and the historical Beta for security i. Let (1%,; stand
for the variance of the disrribation of the historical estimates of Beta over the sample of
stocks. This is a measure of the variation of Beta across the sample of stocks under consid
eration. Let 02g,“ stand for the square of the standard error of the estimate of Beta for rs . . , . .
For example, Merrill Lynch has used a Simple weighting technique like this to adjust its Betas. I PART 2 PORTFOLIO ANALYSIS security i measured in time period 1. This is a measure of the uncertainty associated with
the measurement of the individual securities Beta. Vasiceit (1973) suggested weights of 2 0— (52.
W m 2 for B“ and MW ‘3”
of” + 53,, (5—51 + om; Note that these weights add up to 1 and that the more the uncertainty about either estimate
of Beta, the lower the weight that is placed on it. The forecast of Beta for security i is 2 for lit a 2
5.2 =__E9il__ﬁi +W39ngq
! 2 g 2 l
6E1 +6311 o5i +613“ This weighting procedure adjusts observations with large standard errors further toward
the mean than it adjusts observations with small standard errors. As Vasicelt has shown,
this is a Bayesian estimation technique.16 While the Bayesian technique does not forecast a trend in Betas as does the Blume tech
nique, it suffers from its own potential source of bias. In the Bayesian technique, the
weight placed on a stock’s Beta, relative to the weight on the average Beta in the sampie,
is inversely related to the stock’s standard error of Beta. High Beta stocks have larger stan
dard errors associated with their Betas than do iow Beta stocks. This means that high Beta
stocks will have their Betas lowered by a bigger percentage of the distance from the aver~
age Beta for the sampie than low Beta stocks will have their Betas raised. Hence, the esti—
mate of the average future Beta will tend to be lower than the average Beta in the sample
of stocks over which Betas are estimated. Unless there is reason to believe that Betas will continually decrease, the estimate of
Beta can be further improved by adjusting all Betas upward so that they have the same
mean as they had in the historical period. Accuracy of Adjusted Beta Let us examine how well the Biume and the Bayesian adjustment techniques worked as
forecasters, compared to unadjusted Betas. Klemkosicy and Martin ( 1975) tested the abil
ity of these techniques to forecast over three 5—year periods for both oneustocir and ten—
stock portfolios. As would be suspected, in ali cases both the Biume and Bayesian
adjustment techniques led to more accurate forecasts of future Betas than did the unad—
justed Betas. The average squared error in forecasting Beta was often cut in half when one
of the adjustment techniques was used. K1emkosicy and Martin used an interesting decom
position technique to search for the source of the forecast error. Speciﬁcally, the source of
error was decomposed into that part of the error due to a misestimate of the average level
of Beta, that part due to the tendency to overestimate high Betas and underestimate iow
Beta, and that part that is unexplained by either of the first two influences. As might be
expected, when the Blame and Bayesian techniques were compared with the unadjusted
Betas, almost ail of the decrease in error came from the reductions in the tendency to over—
estimate high Betas and underestimate low Betas. This is not surprising because this is
exactly what the two techniques were designed to achieve. Klemltosicy and Martin found
that the Bayesian technique had a slight tendency to outperform the Blurne technique. “The reader should note that this is just one of an inﬁnite number of ways of forming prior distributions. For
example, priors could have been set equal to l (the average for all stocks market weighted) or to an average Beta
for the industry to which the stock belongs, and so on. CW5]? 7 THE CORRELATiON STRUCl'URE OF SECURiTY RETURNS ] 47 However, the differences were small and the ordering of the techniques varied across dif
ferent periods of time. , "Most of the literature dealing with Betas has evaluated Beta adjustment techniques by
their ability to better forecast Betas. However, there is another, and perhaps more important,
cg'rerion by which the performance of alternative Betas can be judged. At the beginning of
this chapter we disauSScd the fact that the necessary inputs to portfolio analysis were
expected returns, variances, and correlations. We believe that analysts can be asked to pro
yids estimates of expected returns and variances, but that correlations will probably con~
[inuc to be generated from some sort of historical model.” One way Betas can be used is
to generate estimates of the correlation between securities. The correlations between stocks
(given the assumptions of the singie—lndeit modei) can be expressed as a function of Beta. '2 p, .. “a r .. grﬁqu 1 "Jul—' _ m
J 65.0} {5ij Another way to test the usefulness of Betas, as well as the performance of alternative forew
casts of Betas, is to see how well Betas forecast the correlation sttucture between secun'ties. Betas as Forecasters of Correlation Coefficients Elton, Gruber, arid Urich (1978) have compared the ability of the following models to fore
cast the correlation structure between securities: 1. The historical correlation matrix itself 2. Forecasts of the correlation matrix prepared by estimating Betas from the prior his
torical period 3. Forecasts of the correlation matrix prepared by estimating Betas from the prior two
periods and updating via the Blunts technique 4. Forecasts prepared as in the third model but where the updating is done via the Vasicek
Bayesian technique One of the most striking results of the study was that the historical correlation matrix
itself was the poorest of all techniques. In most cases it was outperformed by all of the Beta
forecasting techniques at a statistically significant level. This indicates that a large part of
the observed correlation structure between securities, not captured by the singleindex
model, represents random noise with respect to forecasting. The point to note is that the
singleindex model, developed to simplify the inputs to portfolio analysis and thought to
lose information due to the simpliﬁcation involved, actually does a better job of forecast
ing than the full set of historical data. The compalison of the three Beta techniques is more ambiguous. In each of two ﬁve
ycar samples tested, the Blame adjustment technique outperformed both the unadjusted
Betas and the Betas adjusted via the Bayesian technique. The difference it: the techniques
was statistically signiﬁcant. However, the Bayesian adjustment technique performed bet‘
ter than the unadjusted Beta in one period and worse in a second. in both cases, the results
were statistically significant. This calls for some further analysis. The performance of any
forecasting technique is, in part, a function of its forecast of the average correlation
between all stocks and, in part, a function of its forecast of previous differences from the u  . . . . . . . .
I! IS pOSSIbIe that analysts Will be used to subjectively modify historical esumates of Beta to improve their at:ch
may. Several ﬁrms currently use analysts’ modiﬁed estimates of Beta. 1 48 PART 2 PORTFOIJO ANALYSIS mean. We might stop for a moment and see why each of the Beta techniques might pro
duce forecasts of the average correlation coefﬁcient between all stocks that are different
from the average correlation coefﬁcient in the data to which the techniqae is ﬁtted. Let us start with the unadjusted Betas. This model assumes that the only correlation
between stocks is one due to common correiation with the market. It ignores all other
sources of correlation such as industry effects. To the extent that there are other sources of
correlation that are, on the whole, positive, this technique wili underestimate the average
correlation coefﬁcient in the data to which it is ﬁtted. This is exactly what Elton, Gruber,
and Urich (3978) showed happened in both periods over which the modei was ﬁtted. The Biume technique suffers from the same bias, but it has two additional sources of
bias. One is that the Blume technique adjusts a1} Betas toward one. This tends to raise die
average correlation coefﬁcient estimated from the Biume technique. The correlation coef
ﬁcient is the product of two Betas. To the extent that Betas are reduced to one symmetrh
caliy {with no change in mean), the cross products between them will tend to be larger. For
example, the product of 1.1 and 0.9 is larger than the product of 1.2 and 0.8. There is
another source of potential problems in the Blume technique. Remember that the Blume
technique adjusts the Betas in period two for the changes in Betas between period one and
two. if the average change in Beta between periods one and two is positive (negative), the
Blume technique will adjust the average Beta for period two up (down).15 In the Elton,
Gruber, and Urich study there was an upWard drift in Betas over the period studied and
this, combined with the tendency of the Blame technique to shrink ail Betas toward one,
resuited in forecasts of an average correlation coefﬁcient well above the average correia—
tion coefﬁcient for the sampie to which the model was fitted. The Bayesian adjustment to Betas, like the Blume adjustment, has some upward fore
cast bias because of its tendency to shrink Betas toward one, but it does not continue to
project a trend in Betas and, hence, correlation coefﬁcients as the Blume technique does.
However, as pointed out earlier, it has a new source of bias—none that tends to pull Betas
and correlation coefﬁcients in a downward direction. This occurs because hi gh Beta stocks
are adjusted more toward the mean than low Beta stocks. Short of empirical tests, it is difficult to say whether, given any set of data, the alterna«
tive sources of bias, which work in different directions, will increase or decrease the fore—
cast acouracy of the result. We do know that unless there are predictable trends in average
correlation coefﬁcient, the effect of these biases on forecast accuracy will be random from
period to period. 'ihis source of randomness can be eliminated. One way to do it is to force
the average correlation coefficient, estimated by each technique, to be the same and to be
equal to the average coireiation coefficient that existed in the period over which the model
was titted. If correlation coefﬁcients do not have stable trends, this wili be an efﬁcient fore—
cast procedure. It uses only available data and is also easy to do. When the adjustments were made, the Bayesian adjustment produced the most accurate
forecasts of the future correlation matrix. Its difference from the Blame technique, the unad—
justed Beta, and the historical matrix was statistically signiﬁcant in all periods tested. The
second—ranked technique varied through time with the Binnie adjustment, outperforming the
unadjusted Beta in one period and the unadjusted Beta outperforming Blume in one period.19 isThis would be a desirable property if bends in average correlation coefﬁcients were expected to persist ovor
time. but we see no reason to expect them to do so.
‘9In addition, tests were made that forted the average correlation coefﬁcient from each technique to be the same and equal to the average correlation coefﬁcient that occurred in the forecaet period. This is equivalent to perfect
foresight with respect to the average correlation coefficient. The rankings were the same as those discussed above when this was done. warren 7 THE CORRELATION STRUCTURE OF SECURiTY RETURNS g 4.9 The forecasts from the three Beta techniques were compared with the forecasts from a
fqpﬂh Beta estimate, Beta eqnais one for all stocks, as welt as with the historical correla
tion matrix, as a forecast of the future. The mean forecast was adjusted to be the same for
ail techniques. The performance of the historical correlation matrix and the Beta~equals»one
model were inferior to the performance of all other models at a statisticaily signiﬁcant level. Let us stop a minute and review the work on estimating Betas. There are two reasons for
estimating Betas: The ﬁrst is in order to forecast future Betas. The second is to generate corw
relation coefﬁcients as input to the portfolio problem. Empirical evidence strongly suogests
that to forecast future Betas one should use either the Bayesian adjustment or the Blame
adjustment rather than unadjusted Betas. The evidence on the choice between the Blume and
Bayesian adjustment is mixed, but the Bayesian adjustment seems to work slightly better. If the goal is estimating the future correlation matrix as an input to the portfolio problems
things get more compiex. Unadjusted Betas and adjusted Betas, both by the Bayesian and the:
Blame techniques, all contain potential bias as forecasters of future correlation matrices.20
The forecasts from all of these techniques can be examined directly or the forecasts can be
adjusted to remove bias in the forecast of the average correlation coefﬁcient. The first fact to
note is that each of these three estimates of Beta outperfonns the historical correlation matrix
as a forecast of the future correlation matrix. Second, note that when compared to a Beta of
one, all produce better forecasts. The ranking among these three techniques is a function of
whether we malts the adjustment to the average forecast. Since we believe it is appropriate
to do so, we ﬁnd'that the Bayesian adjustment technique performs best. In Chapter 8 we wili
discuss forecasting future correlation coefficients using a combination of past Betas and
other forecasts derived from historical data. I Recently, attempts have been made to incorporate more data than past return informa—
tion into the forecasts of Betas. We will now take a brief iook at some of the work that has
been done in this area. Fundamental Betas Beta is a risk measure that arises from the relationship between the return on a stock and
the return on the market. However, we know that the risk of a ﬁrm should be determined
by some combination of the iirm’s fundamentals and the market characteristics of the farm’s stock. If these relationships could be determined, they would help us both to better
understand Betas and to better forecast Betas. One of the earliest attempts to relate the Beta of a stock to ﬁmdamental ﬁrm variables was
performed by Beaver, Kettler, and Scholcs (1970). They examined the relationship between
seven ﬁrm variables and the Beta on a company’s stock. The seven variables they used were
1 Dividend payout (dividends divided by earnings). 2 Asset growth (annual change in total assets). 3. Leverage (senior securities divided by total assets). 4 Liquidity (current assets divided by current iiabilities). 5 Asset size (total assets). 6. Earning variability (standard deviation of the earnings price ratio). '7. Accounting Beta (the Beta that arises from a time series regression of the earnings of
the tinn against average earnings for the economy, often called the earnings Beta). 2r: . .
As discussed earlier. a smaller set of potential biases is present when Betas are estimated. a 5 D FAR? 2 Foursome ANALYSES An examination of these variables would lead us to expect a negative relationship
between dividend payout and Beta under one of two arguments: I. Since management is more reloctant to cut dividends than raise them, high payout is
indicative of conﬁdence on the part of management concerning the ievei of future, earnings.
2. Dividend payrnents are less risky than capitai gains; hence, the company that pays on;
more of its earnings in dividends is less risky. Growth is usually thought of as positively associated with Beta. Highgrowth ﬁrms are
thought of as more risky than low—growth firms. Leverage tends to increase the volatility of the earnings stream, hence to increase risk and
Beta. A firm with high liquidity is thought to be less risky than one with low liquidity and, hence,
iiquidity should be negatively reiated to market Beta. Large firms are often thought to be less risky than small ﬁrms, if for no other reason than
that they have better access to the capital markets. Hence, they should have lower Betas Finaiiy, the more variable a company’s earning stream and the more highly correlated it is
with the market, the higher its Beta shouid be. Table 7.5 reports some of the results from the Beaver ct al. {1970) study. Note all variabies had the sign that we expected. The next logical step in developing fundamental Betas is to incorporate the effects of
relevant fundamental variables simultaneously into the analysis. This is usually done by
relating Beta to several fundamental variables via multiple regression analysis. An equation of the following form is estimated: g!zag+a[X1+d2X2+m+€1NXN+€5 where each X. is one of the N variables hypothesized as affecting Beta. Several studies have
been perforrned that link Beta to a set of fundamental variabies, such as that studied by
Beaver et a1. (1970).” The list of variables that has been studied and linked to Betas is too Table 7.5 Correlation between Accounting MeaSures of Risk and Market Beta Period 1 Period 2 19474956 19574965
OneSteak FiveStocic One~Stock FiveStock
Variable Portfolio Portfolio i’ortfolio Portfolio
Payout #050 ~03? “ 0.24 "0.45
Growth 0.23 0.51 0.03 0.07
Leverage 0.23 945 0.25 0.56
Liquidity —0.13 "0.44 —{).02 “0.0%
Size 0.07 0. i3 "0.26 ~0.30
Earnings Variability 0.58 0.7"! {1.36 0,62
Earnings Beta 0.39 0.67 0.23 (145 2‘For examples of the use of fundamental data to estimate Betas, see Cohen at at. {1978), Francis 0975).
Hawawini and Vera (1980), Blurne 0975). and Hill and Stone {1980). The ability of fundamental data to aid in
the prediction of future Betas has been mixed. Some studies find large improvements in forecasting ability willie others do not. CWTER 7 THE CORRELATION STRUCTURE OF SECURITY RETURNS ‘! 51 long t0 review here. For example, Thompson (1978} reviews43 variables while Rosenberg
and Marathe {1975) reviews 101. Rather than discuss the long list of variables that has been used to generate fundamental Betas, let us review the relative strengths and weak
nesses of fundamental and historical Betas as well as. one system, proposed by Barr I Rosenberg (i976, 1975, 3973), that has been put forth to combine both types of Betas. The advantage of Betas based on historical return data is that they measure the response
gfeach stock to market m0vernents. The disadvantage of this type of Beta is that it reﬂects
changes in the size or importance of company characteristics oniy after a long period of
time has passed. For example, assume a company increased its debtto—equity ratio. We
would expect its Beta to increase. However, if we are using 50 months of return date to
estimate Beta, one month after the company increased its debt—to—equity ratio, only one of
the 60 data points will reﬂect the new information. Thus, the change in dcbtto—equity ratio
would have only a very minor impact onlhe‘Beta computed from historical return data.
Similarly, one fuli year after the event only 12 of the 60 data points used to measure Beta
wiil reﬂect the event. On the other hand, fundamental Betas respond quickly to a change in the companies’
characteristics since they are computed directly from these characteristics. However, the
weakness of fundamental Betas is that they are computed under the assnmption that the
responsiveness of all Betas to an underlying fundamentai variable is the same. For exam—
ple, they assume that the Beta for {BM wili change in exactly the same way with a given
change in its dedi—to—cquity ratio as wili the Beta of Generai Motors (GM).22 By combining the techniques of historical Betas and fundamental Betas into one system,
Barr Rosenberg hopes to gain the advantages of each without being subject to the disad»
vantages of either. In addition, because Rosenberg and MoKibben (1973) found that there
were persistent differences between the Betas of different industries, Rosenberg and
Marathe (1975) introduced a set of industry dummy variables into the analysis to capture
these differences. Rosenberg’s system can be described as follows:23 [3, 2 no +alx1+a2x2 +a3x3 +~+a7x7 «twang "inn+a46x46 (7.7) x; represents 14 descriptions of market variability. These 14 descriptions include his
torical values of Beta as well as other market characteristics of the stock such as
share trading, volume, and stock price range. x2 represents seven descriptors of earnings variability. These descriptors include
measures of earnings variability, earnings Betas, and measures of the unpre
dictability of earnings such as the amount of extraordinary earnings reported. X3 represents eight descriptors of unsuccess and low valuation. These descriptors
include growth in earnings, the ratio of book value to stock price, relative strength,
and other indicators of perceived success. x4 represents nine descriptors of immaturity and smallness. These descriptors include
total assets, market share, and other indicators of size and age. :55 represents nine descriptors of growth orientation. These descriptors include dividend
yield, earnings price ratios, and other measures of historical and perceived growth. 22Each of the regression coefﬁcients of Equation {7.6) (cg, (2) has only one value for all ﬁrms. This means that
“hang: of 1 unit in X. will change the Beta ofevery firm by a; units. 23Resenherg changes the variables in his system over time. This description is based on his system as it existed
at a point in time as described in (1975). 1 52 PART 2 PORTFOLIO ANALYer x5 represents nine descriptors of ﬁnancial risk. These include measures of leverage,
interest coverage, and liquidity. X7 represents six descriptors of firm characteristics. These include indicators of stock
listings and broad types of business. 253 through 1:45 are industry dummy variables. These variables allow the fact that dif»
ferent industries tend to have different Betas, all other variables held constant, to be taken into account. While conceptually the Rosenberg technique is easy to grasp, the multitude of variables
(101) makes it difficult to grasp the meaning of the parameterized model. The reason for
moving to this complex model is to improve forecasting ability. While the model is too
new for extensive testing to have been performed, Rosenberg and Marathe’s (1975) initial
testing indicates that the model involving both fundamental data and historical Betas leads
to better estimates of future Betas than the use of either type of estimate in isolation. Before ending this chapter, we should mention one more type of model that is beginning
to attract attention. The Rosenberg system quickly reﬂects changes in Beta that have
occurred because it uses data that reflect present conditions {fundamental ﬁrm variables}
to modify historical Betas as forecasts of the future. A more ideal system would employ
forecasts of future fundamental ﬁrm variables to modify historical estimates of Betawin
other words, substitute estimates of future values on the right~hand side of Equation (7.7)
rather than concurrent values. Now it seems unlikely that analysts can do this for the 101
variables used in Rosenberg’s system. However, simpler systems employing a much
smaller number of variables are being used in this way. THE MARKET MODEL Although the singledndex model was developed to aid in portfolio management, a less
restrictive form~known as the market modelwhas found increased usage in finance. The
market model is identical to the singlewindex model except that the assumption that
cov(e.eJ) = {l is not made?“ The model starts with the simpler linear relationship of returns and the market R2 = at ‘l' 55R»: + er and produces an expected value for any stock of Rt = 0h +§3i§m Since it does not make the assumption that all covariances among stocks are doc to a
common covariance with the market, however, it does not lead to the simple expressions
of portfolio risk that arise under the single—index model. We will meet the market model again as we progress through this book. It is used exten
sively in Chapter 17 on the efficient market. The point to keep in mind is that the discus
sion of estimating Beta is equally as applicable whether we are talking about the market
model or the single—index model. 2“Actually. although the singleindex model can he deﬁned in terms of any inﬂuence (e.g.. the rate of return on
liverwurst). we usually think of the index as the rate of return on some market portfolio. The market model is always deﬁned in terms of a market portfolio. cHAPTER 7 THE CORREMTiON STRUCTURE OF SECURITY RETURNS 1 53 AN EXAMPLE Amanager of a large pension fund will often utilize several domestic stock managers. T he
Pension fund sponsor (manager) can view the asset allocation problem as equivalent to
selecting among various stock mutual funds. The data for the portfolios being considered
by a large pension fund are as follows: NAME 0&1 [31’ 03(
1, Small stock 6 1.4 65
2. Value 4 0.8 20
3. Growth 4.5 1,3 45
4. Large capitalization 0.8 ‘ ,‘0.90 24 5, Special sitnation 6.2 1_.1 45 The Alphas, Betas, and residual risks were initially computed by running a regression of
each fund’s return on the return of the 385? index using live years of monthly returns.
These estimates were then modiﬁed by the plan sponsor to reflect their beliefs.
Management projected that the S&P index at this point had an expected return of 12.5%
and an estimated standard deviation of return of 14.9%. The expected returns, standard
deviations of return, and covariance using the singleindex model are s E e 6 +1.4(12s): 23.5
1?; m 4 «l— 0.8(12.5)m14
it} = 4.5 +1.3(12.5) 2 20.75
it; = 0.3 + e.9(12.5)=12.05
as = 0.2 +2.1(12.5} = 23.95 0: =[(t4)1(149)2 “55]” a 22.36
62 a [defeat9): +20]"2 a 12.73
63 : [(1.3)g(14.9)2 + 45]V2 = 205 2
64 = [(99l2(14.9)2 mil” =14.2s 05 = [(1.1)2u49)2 445]”2 217.71 on = (1.4)(0.8)(14.9}2 :29
033 a (1.4)(1.3}{14.9)2 a 404
em = (1.4)(0.9)(14.9)2 = 280
615 2 (1.4)(1. i)(14.9)2 a 342 623 = 231
(524 = 625 $195
034 = (535 3 317
045 = 220 ; 54, PART 2 Port/wont) metros These estimates for portfolio inputs are not necessarily the same as would be obtained
from historical data. However, the Betas for fund 1 and 2. were the historical betas using
the prior five years of data. Thus if the covariance between the residuals f0r fund i and 2
were zero, the estimate oi the covariance using the singlenindex model and the historical
estimate would be the same. The covariance between assets 1 and 2 computed directly
from the historical data was 271. The estimate from the singleindex model was 249. The
difference arose because there was a small positive correlation between residuals for fund
i and fund 2. The justiﬁcation for using the singiewindex model to estimate inputs is a
belief that this positive residual resulted by chance, and zero is a better estimate of its
future value than the actual past value. The optimum proportions using these inputs, a risk—
less rate of 5%. and the procedures discussed in Chapter 6 are FUND A wrrn SHORT sates no such? SALES
1 692694: 78%
2 5797% 0
3 4218922 22%
4 “10.14370 0
5 —5797% 0 The solution with short sales is of course unreasonable both because pension managers
cannot short sell and because of the magnitude of the numbers. The large numbers come
about because mutual funds are very highly correlated with one another and small differ
ences result in large positions being taken. In Chapter 9 we will analyze the problem when
We have developed the tools for a simpler analysis. QUESTIONS AND PROBLEMS 1. Monthly return data are presented below for each of three stocks and the 8&1D index
(corrected for dividends) for a 12month period. Calculate the following quantities: A. Aipha for each stock B. Beta for each stocic C. The standard deviation of the residuals from each regression D. The correlation coefﬁcient between each security anti the market
t. The average return on the market F. The variance of the market Month A B C S&P
1 12.05 25.2.0 31.67 12.28
2 55.27 2.85 15.32 5.99
3 “4.12 5.45 30.58 2.41
4 1.57 4.56 —14.43 4.48
5 3.16 3.72 31.98 4.4%
6 ""2579 10.79 ~0.72 4.43
7 ~89”? 5.38 ~19.64 6.77
8 “1.18 2.97 “20.00 M211
9 1.07 1.52 11.51 3.46
30 12.75 10.75 5.63 6.16
11 7.48 3.79 ~46? 2.47
1'2. “0.94 2.32 7.94 "1.15 CHAPTER 7 THE CORREMTiON S‘E’RUCTURE OF SECURFTY RETURNS I 55 A. Compute the mean return and variance of return for each stock in Problem 1
using ' (1) The singleindex model.
(2} The historical data. 3, Compute the covariance between each possible pair of stocks using
{l} The singleindex model.
{2) The historical data. C. Compute the return and standard deviation of a portfolio constructed by placing
one—third of your funds in each “stock, using
(1) The Singleindex modei.
(2) The historical data. D. Explain why the answers to parts Phi and A2 were the same, whiie the answers
to parts 133, B2 and (3.1, (2.2 were different. Show that the Vasicek technique leads to a simple proportional weighting of the mar
ket Beta and the stock’s Beta if the standard error of all Betas is the same. A. If the Blame adjustment equation is ﬁt and the appropriate equation is
135;“ 7 + ; what is your best forecast of Beta for each of the stocks in Question I?
B. If the parameters of the Vasicek technique are ﬁt, and they are 2 m. 2 __
egress, amass; a. =1.er
0123“; = 0.36, 03.“; a an what is your best forecast of Beta for each of the stocks in Question I? A s c o
u 3 t 4
a 1.5 1.3 .8 . .9
(I e:' 3 i 2 4 Given the data above and the fact that lim = 8 and om = 5, calculate the foilowing:
(a) The mean return for each security. (13) The variance of each security’s return. (c) The covariance of returns between each security. Using the data in Problem 5 and assuming an equally weighted portfolio, calculate the
foiiowing: (a) [3,.
{5) up
(0) Up
(6) Rp I 56 PART 2 Using Biume’s technique where {3,2 m 0.343 + 0.6778,, calculate 6,3 for the securi
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