! N -{

IndexModels THE MARKOWITZPROCEDURE introduced in the preceding chapter suffers from two d r a w b a c k s .F i r s t ,t h e m o d e l r e q u i r e sa h u g e

returns.Once its properties are analyzed,we proceed to an extensiveexample of estimation of the single-indexmodel. We reviewthe

number of estimates to fill the covariance statisticalproperties of these estimates and matrix. Second, the model does not oro- show how they relate to the practical issues v i d e a n y g u i d e l i n et o t h e f o r e c a s t i n go f t h e facing portfolio managers. s e c u r i t yr i s k p r e m i u m st h a t a r e e s s e n t i a tl o Despitethe simplificationthey offer, index construct the efficient frontier of risky as- models remain true to the concepts of the sets. Since past returns are a poor guide to expected future returns, this drawback can

efficient frontier and portfolio optimization. Empirically,index models are as valid as the

be telling.

assumption of normality of the rates of reIn this chapter we introduce index mod- turn on availablesecurities.To the extent that els that simplifyestimationof the covariance short-term returns are well approximated by matrix and greatly enhance the analysisof normal distributions, index models can be security risk premiums.By allowing us to ex- used to selectoptimal portfolios nearlyas acplicitly decompose risk into systematicand curately as the Markowitz algorithm. Finally, firm-specificcomponents,these models also shed considerablelight on both the power

we examine optimal risky portfolios constructed using the index model. While the

and limits of diversification.They also allow us to measurethese components of risk for

principles are the same as those employed in the previous chapter, the properties of the portfolio are easier to derive and interpret in this context. We illustratehow to use

particularsecuritiesand portfolios. We begin the chapter by describing a single-factorsecurity market and show how it can justify a single-indexmodel of security

the index model by constructingan optimal risky portfolio using a small sample of firms.

I

I I I

i

This portfolio is compared to the correspond- i ing portfolio constructed from the Markowitz i model. We concludewith a discussionof several

practicalissuesthat arisewhen implementing the indexmodel.

MARKET A SINGLE-FACTOR SECURITY The Input List of the lvtarkowitzModel The successof a portfolio selection rule dependson the quality of the input list, that is, the estimates of expected security returns and the covariance matrix. In the long run, efficient portfolios will beat porfolios with less reliable input lists and consequentlyinferior reward-to-risk trade-offs. Supposeyour security analystscan thoroughly analyze50 stocks.This meansthat your input list will include the following: n : 50 estimatesof expectedretums n : 50 estimatesof variances -n\12:1.225 h2 estimates of covariances t,ZZSestimates This is a formidable task, particularly in light of the fact that a 5O-security porrfolio is relatively small. Doubling n to 100 will nearly quadruple the number of estimates to 5,150. If n : 3,000, roughly the number of NYSE stocks, we need more than 4.5 million estimates. Another difficulty in applying the Markowitz model to portfolio optimization is that errors in the assessmentor estimation of correlation coefficients can lead to nonsensical results. This can happenbecausesome setsof correlation coefficients are mutually inconsistent, as the following example demonstrates:r

Asset

Standard Deviation (7d

A B c

20 20 20

Correlation Matrix A

1.00 0.90 0.90

B

C

0.90 1.00 0.00

0.90 0.00 1.00

Supposethat you construct a portfolio with weights: - 1.00; 1.00; 1.00, for assetsA; B; C, respectively, and calculate the portfolio variance. You will find that the portfolio variance appearsto be negative (-200). This of course is not possible becauseportfolio variances cannot be negative: we conclude that the inputs in the estimated correlatiqt matrix must be mutually inconsistent.Of course, true conçlation coefficients are always consistent.2But we do not know theseffue correlationsand can only estimatethem with some imprecision. Unfortunately, it is difficult to determine whether a correlation matrix is inconsistent, providing another motivation to seek a model that is easier to implement. tWe are grateful to Andrew Kaplin and Ravi Jagannathan, Kellogg Graduate School of Management, Northwestern University, for this example. 2The mathematical term for a correlation matrix that cannot generate negative portfolio variance is "positive definite."

258

CHAPTERI

IndexModels

Introducing a model that simplifies the way we describe the sources of security risk allows us to use a smaller,consistentset of estimatesof risk parametersand risk premiums. The simplification emergesbecausepositive covariancesamong security returns arise from common economic forces that affect the fortunes of most firms. Someexamplesof common economic factors are businesscycles, interest rates, and the cost of natural resources.The unexpected changesin these variables cause, simultaneously,unexpectedchangesin the rates of return on the entire stock market. By decomposinguncertainty into fhese systemwide versusfirm-specific sources,we vastly simplify the problem of estimating covariance and correlation.

Normalityof Returnsand SystematicRisk We can always decomposethe rate of return on any security, l, into the sum of its expected plus unanticipatedcomponents: ri:

E(r) -l e,

(8.1)

where the unexpectedreturn, er,has a mean of zero and a standard deviation of o; that measuresthe uncertainty about the security return. When security returns can be well approximated by normal distributions that are correlated acrosssecurities,it is said that securities arejoint normally distributed. This assumption alone implies that, at any time, security returns are driven by one or more common variables.When more than one variable drives normally distributed security returns, these returns are said to have a multivariate normal distribution. We begin with the simpler case where only one variable drives the joint normally distributed returns, resulting in a single-factor security market. Extension to the multivariate case is straiehtforward and is discussedin later chapters. Supposethe common factor that drives variation in security returns is some macroeconomic variable, which we will call m, that affects all firms. Then we can decomposethe sourcesof uncertainty into uncertainty about the economy as a whole, which is captured by m, and uncertainty about the firm in particular, which is capturedby ei. rnthis case,we amend Equation 8.1 to accommodatetwo sourcesof variation in return:

r,:E(r,)*m*e,

(8.2)

The macroeconomic factor, m, measuresunanticipatedmacro surprises.As such, it has a mean of zero (over time, surpriseswill averageout to zero), with standarddeviation of o,. In contrast, er measuresonly the firm-specific surprise.Notice that rn has no subscript because the same corlmon factor affects all securities. Most important is the fact that m and e, are uncorrelated, that is, becausee; is firm-specific, it is independentof shocks to the common factor that affect the entire economy.The variance of r,thus arisesfrom two uncorrelatedsources,systematicand firm specific. Therefore,

o?: o,_+o2(e,)

(s.3)

The common factor, m, generatescorrelation acrosssecurities,since all securitieswill respond to the same macroeconomicnews, while the firm-specific surprises,capturedby e,, are assumedto be uncorrelatedacrossfirms. Since lrr is also uncorrelatedwith any of the firm-specific surprises,the covariancebetween any two securitiesI andj is C o v ( r , ,r r ) : C o v ( m * e , , m * e , ) : o 2 ^

(S.4)

Finally, we recognize that some securitieswill be more sensitivethan others to macroeconomic shocks.For example, auto firms might respondmore dramatically to changesin generaleconomic conditions than pharmaceuticalfirms. We can capturethis refinementby

259

260

PARTll

Portfolio Iheory and Practice assigning each firm a sensitivity coefficient to macro conditions. Therefore, if we denote the sensitivity coefficient for firm I by the Greek letter beta, P,, we modify Equation 8.2 as follows to obtain the single-factor model: r,:

E(r,) * S,m * e;

(8.5)

Equation 8.5 tells us the systematicrisk of security i is determinedby its beta coefficient. "Cyclical" trms have greater sensitivity to the market and therefore higher systematic risk. The systematicrisk of security i is Blo'^, and its total risk is

"?:g?"I+oz(e,)

(s.6)

The covariancebetween any pair of securitiesalso is determinedby their betas:

Cov(4,r,) : Cov(\imi e,,Bim + e) : F,9p'*

(8.7)

In terms of systematicrisk and market exposure,this equaiion tells us that firms are close substitutes.Equivalent beta securitiesgive equivalent market positions. Up to this point we have used only statistical implications from the joint normality of security returns. Normality of security returns alone guarantees that portfolio returns are also normal (from the "stability" of the normal distribution discussedin Chapter5) and that there is a linear relationship between security returns and the common factor. This greatly simplifies portfolio analysis. Statistical analysis, however, does not identify the common factor, nor does it specify how that factor might operate over a longer investment period. However, it seemsplausible (and can be empirically verified) that the variance of the common factor changesrelatively slowly through time, as do the variances of individual securities and the covariances among them. We seek a variable that can proxy for this common factor. To be useful, this variable must be observable,so that we will be able to estimateits volatilitv as well as the sensitivity ofindividual securitiesreturns to variation in its value.

T H ES I N G L E . I N D EMXO D E L A reasonable approach to making the single-factor model operational is to assert that the rate of return on a broad index of securities such as the S&P 500 is a valid proxy for the common macroeconomic factor. This approach leads to an equation similar to the singlefactor model, which is called a single-index model becauseit uses the market index to proxy for the common factor.

TheRegressionEquationof the Single-lndexModel Since the S&P 500 is a portfolio of stockswhose prices and ratesofreturn can be observed, we have a considerable amount of past data with which to estimate systematic risk. We denote the market index by M,with excessreturn of Ry: ru - tf, and standarddeviation of oy. Becausethe index model is linear, we can estimate the sensitivity (or beta) coefficient of a security on the index using a single-variable linear regression. Vfu regress the excessreturn of a security, Ri: ri - rf, oî the excessrefurn of the index, Rr. To estimate the regression,we collect a historical sample of paired observations,.R(r)and Rlt), where t denotesthe date ofeach pair ofobservations (e.g., the excessreturns on the stock and the index in a particular month),3The regression equation is (8.8) Ri(r) : a; + B,.Ry(/) 'f e,(t) 3Practitionersoften use a "modifred" index model that is similar to Equation8.8 but that usestotal rather than excessreturns.Thispracticeis mostcommonwhendailydataareused.In this casetherateofreturnon bills is on per day,sototal andexcessretumsarealmostindistinguishable. theorderof only about.Ùl%o

CHr\ZfER8

IndexModels

The intercept of this equation (denoted by the Greek letter alpha, or cr) is the security's expectedexcessreturn when the market excessreturn is zero. The slope coefficient, B,, is the security beta. Beta is the security's sensitivity to the index: it is the amount by which the security return tends to increaseor decreasefor every lVo increaseor decreasein the return on the index. er is the zero-mean, firm-specific surprise in the security return in time r, also called the residual.

TheExpectedReturn-BetaRelationship BecauseE(ei) : 0, if we take the expectedvalue of E(R) in Equation 8.8, we obtain the expectedreturn-beta relationship of the single-index model:

E(Rr): a,* B,E(R*)

(E.e)

The secondterm in Equation 8.9 tells us that part of a security's risk premium is due to the risk premium of the index. The market risk premium is multiplied by the relative sensitiv'ity, or beta, of the individual security.We call this the systematicrisk premium becauseit derives from the risk premium that characterizesthe entire market, which proxies for the condition of the full economyor economicsystem. The remainder of the risk premium is given by the first term in the equation,a. Alpha is a nonmarket premium. For example, cr may be large if you think a security is underpriced and therefore offers an attractive expected return. Later on, we will see that when security prices are in equilibrium, such attractive opportunities ought to be competed away, in which case a will be driven to zero. But for now, let's assumethat each security analyst comes up with his or her own estimatesof alpha. If managersbelieve that they can do a superiorjob of security analysis,then they will be confident in their ability to find stocks with nonzero values of alpha. V/e will see shortly that the index model decomposition of an individual security's risk premium to market and nonmarket componentsgreatly clarifies and simplifies the operation of macroeconomicand security analysiswithin an investmentcompany.

Risk and Covariancein the Single-lndexModel Rememberthat one of the problems with the Markowitz model is the overwhelming number of parameterestimatesrequired to implement it. Now we will seethat the index model simplification vastly reducesthe number of parametersthat must be estimated.Equation 8.8 yields the systematicand firm-specific componentsof the overall risk of each security,and the covariance between any pair of securities.Both variances and covariancesare determined by the security betas and the properties of the market index:

Totalrisk : Systematic risk * Firm-specificrisk ol : glo2, + o2(e,) Covariance: Productof betasX Marketindexrisk Cov(r,,r.):gigfzu Conelation: Productof correlations with themarketindex ^- "'

Corr(ri, ,j): "

-:9'9io" oioj

9g"9 io" i--!----!!-!--!!t!-: Con(4, rr) X Con(r,. rr) oioMojoM

(8.10)

261

262

PARTll

PortfolioTheoryand Practice Equations 8.9 and 8.10 imply that the set of parameter estimatesneeded for the singleindex model consistsof only cr, B, and o(e) for the individual securities,plus the risk premium and variance of the market index.

TheSet of EstimatesNeededfor the Single-lndexModel We summarizethe results for the sinele-index model in the table below. Symbol 1 . Thestock'sexpectedreturnif the marketis neutral,that is, if the market'sexceis return,rM- rr, is zero The componentof returndue to movementsin the overallmarket;B;is the security'sresponsiveness to marketmovements eventsthat 3. The unexpectedcomponentof reilurndue.to.unexpected are relevantonlyto this security(firmspecific) 4. The variance attributable to the uncertainty of the common

cL; gitu- rà

macroeconomicfactor

5 . The variance attributable to firm-specific uncertaintv

êi

9?"'" étl

drét The data below describe a three-stockfinancialmarket that satisfiesthe single-indexmodel. Mean Standard Stock Capitalization Beta ExceggReturn Devlatlon A B c

$3,000 $1,940 $1,360

1.0 0.2 1.7

10Y" 2 17

The standard deviation of the market index portfolio is 25o/". (a) What is the mean excess return of the index porrfolio? (b) What is the covariancebetween Stock A and Stock B? (c) What is the covariancebetween Stock I and the index? (d) Break down the variance of Stock I into its systematic and firm-specific components.

These calculations show that if we have: .

n estimates of the extra-market expectedexcessreturns, q.t

.

n estimatesof the sensitiviw coefficients, B;

.

n estimatesof the firm-specific variances,oz(e,)

.

1 estimate for the market risk premium

.

1 estimate for the variance of the (common) macroeconomic factor, o2*

then these (3n+2) estimateswill enable us to prepare the entire input list for this single-index security universe. Thus for a 50-security portfolio we will need 152estimates rather than 1,325; for the entire New York Stock Exchange, about 3,000 securities,we will need 9,002 estimatesrather than approximately 4.5 million! It is easyto seewhy the index model is such a useful abstraction.For large universesof securities,the number of estimatesrequired for the Markowitz procedure using the index model is only a small fraction of what otherwise would be needed. Another advantageis less obvious but equally important. The index model abstractionis crucial for specializationof effort in security analysis.If a covarianceterm had to be calculated directly for each security pair, then security analystscould not specializeby industry. For example, if one group were to specialize in the computer industry and another in the auto industry, who would have the common background to estimate the covaiance betweenlBM and GM? Neither group would have the deepunderstandingof other industries necessaryto make an informed judgment of co-movementsamong indusffies. In contrast,

CHAPTER8

IndexModels

265

the index model suggestsa simple way to compute covariances.Covariancesamong securities are due to the influence of the single common factor, representedby the market index return, and can be easily estimatedusing the regressionEquation 8.8. The simplification derived from the index model assumption is, however, not without cost. The "cost" of the model lies in the restrictions it places on the structure of assetreturn uncertainty. The classification of uncertainty into a simple dichotomy-macro versus micro risk----oversimplifiessourcesof real-world uncertainty and misses some important sources of dependencein stock returns. For example, this dichotomy rules out industry events,eventsthat may affect many firms within an industry without substantiallyaffecting the broad macroeconomy. This last point is actually fairly subtle. Imagine that the single-index model is perfectly accurate, except that the residuals of two stocks, say British Petroleum (Bp) and Royal Dutch Shell are correlated.The index model will ignore this correlation (it will assumei1 is zero), while the Markowitz algorithm (which accountsfor the full covariancebetween every pair of stocks) will automatically take the residual correlation into account when minimizing portfolio variance.If the universe of securitiesfrom which we must construct the optimal portfolio is small, the two models will yield substantively different optimal portfolios. The portfolio of the Markowitz algorithm will place a smaller weight on both BP and Shell (becausetheir mutual covariancereducestheir diversification value), resulting in a portfolio with lower variance. Conversely,when correlation among residuals is negative,the index model will ignore the potential diversification value of these securities. The resulting inferior "optimal" portfolio will place too little weight on these securities, resulting in an unnecessarilyhigh variance. The optimal portfolio derived from the single-index model therefore can be signitcantly inferior to that of the full- (Markowitz) covariance model when stocks with correlated residuals have large alpha values and account for a large fraction of the Supposethat the index model for the excessreturns of portfolio. If many pairs of the covered stocks.A and B is estimated with the following results: stocks exhibit residual correlation. it ' RA:1.0o/o + .9Ry + e4 is possible that a multi-index model, -l 1.1R1y which includes additional factors to Rs : - 2.Oo/o * es capture those extra sourcesof crossom: 2Oo/o security correlation, would be better o(ei - 30% suited for portfolio analysis and cono(er) : 197o struction. We will demonstrate the Find the standard deviation of each stock and the covarieffect of correlated residuals in the ance between them. spreadsheetexample in this chapter, and discuss multi-index models in later chapters.

The lndex Model and Diversification The index model, first suggestedby Sharpe,aalso offers insight into portfolio diversification. Supposethat we choosean equally weighted portfolio ofn securities.The excessrate ofreturn on each security is given by Ri:o'i*BiRn,*ei Similarly, we can write the excessreturn on the portfolio of stocks as Rp : crr -f BrRy * ep awilliam F. Sharpe,'A Simplified Model of Portfolio Analysisl' Management science, January 1963.

(S.11)

264

PARTfl

Portfolio Theoryand Practice We now show that, as the number of stocks included in this portfolio increases,the part of the portfolio risk attributable to nonmarket factors becomes ever smaller. This part of the risk is diversified away. In contrast,market risk remains,regardlessof the number of firms combined into the portfolio. To understandtheseresults, note that the excessrate of return on this equally weighted portfolio, for which each portfolio weight w,: lln, is

* e,) n,:fw,n,: If ^,: * É,*,*B,R* (

.

,

(8.12)

:*I",*l*IB, ln,+)L", a

n

i=r

\i=r

\

,

,

i=l

/

Comparing Equations 8.11 and 8.12, we see that the portfolio has a sensitivity to the market given by

,r:+àr,

(8.13)

which is the averageof the individual B,s. It has a nonmarket return component of

o r : n13+o ,

(8.r4)

which is the average of the individual alphas, plus the zero mean variable ep:

1+ iLei

(8.rs)

t=l

which is the averageof the firm-specific components.Hence the portfolio's variance is o2, : B2ro2*+ o2 (er)

(8.16)

The systematicrisk component of the portfolio variance,which we defined as the component that dependson marketwide movements, is B'ro'* and dependson the sensitivity coefficients of the individual securities.This part of the risk dependson portfolio beta and o2, and will persist regardlessof the extent of portfolio diversification. No matter how many stocks are held, their common exposure to the market will be reflected in portfolio systematicrisk.s In contrast, the nonsystematiccomponent of the portfolio variance is o2(ep) and is attributable to firm-speciûc components,e,. Becausethese ersare independent,and all have zero expectedvalue, the law of averagescan be applied to conclude that as more and more stocks are added to the portfolio, the firm-specific componentstend to cancel out, resulting in ever-smallernonmarket risk. Such risk is thus termed diversifiable. To seethis more "portfolio" of rigorously, examine the formula for the variance of the equally weighted firm-specific components.Becausethe e;s are uncorrelated, o2 (ep)

:Z(il'u(e):Loz'"'

(8.17)

where o2 (e) is the averageof the firm-specific variances.Becausethis averageis independent of n, when n gets large; o2(ep)becomesnegligible. sOfcourse, riskby mixingnegative onecanconstruct a portfoliowithzerosystematic B andpositiveB assets. The point of our discussionis that the vast majority of securitieshave a positive B, implying that well-diversified portfolios with small holdings in large numbers of assetswill indeed have positive systematicrisk.

CHAPTLR8

IndexModels

2ô5

o"

DiversifiableRisk

o'(e") =a2(e)/n

F7a'"

I SystematicRisk

Fl G U RE 8.1 The varianceof a portfolio with risk coefficientBpin the single-factoreconomy

To summarize, as diversification increases,the total variance of a portfolio approaches the systematic variance, defined as the variance of the market factor multiplied by the squareof the portfolio sensitivity coefficient, B!. fnis is shown in Figure 8. l. Figure 8.1 shows that as more and more securities are combined into a portfolio, the portfolio variance decreasesbecauseof the diversification of firm-specific risk. However, the po.wer of diversification is limited. Even for very large n, part of the risk remains becauseof the exposureof virtually all assetsto the common, or market, factor. Therefore, this systematicrisk is said to be nondiversifiable. This analysis is borne out by empirical evidence. We saw the effect Reconsider the two stocks in Concept Check 2. Supof porffolio diversification on portfopose we form an equally weighted portfolio of A and lio standarddeviations in Figure 7.2. B. What will be the nonsysterhaticstandard deviation of These empirical results are similar to that portfolio? the theoretical graph presented here in Figure 8.1.

E S T I M A T I NTGH ES I N G L E . I N D EMXO D E L Armed with the theoretical underpinnings of the single-index model, we now provide an extendedexample that begins with estimation of the regressionequation 8.8 and continues through to the estimation of the full covariance matrix of security returns. To keep.the presentation manageable,we focus on only six large U.S. corporations: Hewlett-Packard and Dell from the information technology (IT) sector of the S&P 500,

2ôô

PI\RTll

PortfolioTheoryand Practice

Target and Wal-Mart from the retailing sector, and British Petroleum and Royal Dutch Shell from the energy sector. We work with monthly observationsof rates of return for the six stocks, the S&P 500 portfolio, and T-bills over the period April 2001 to March 2006 (60 observations).As a first step, the excessreturns on the sevenrisky assetsare computed. We start with a detailed look at the preparationof the input list for Hewlett-Packard(HP), and then proceedto display the entireinput list. Later in the chapter,we will showhow theseestimatescan be used to construct the optimal risky portfolio.

The SecurityCharacteristicLine for Hewlett-Packard The index model regressionequation(8.8), restatedfor Hewlett-Packard(HP) is Rup(/) : cr", * Bn.R5a,soo(/)t e"r(t) The equationdescribesthe (linear) dependenceof HP's excessreturn on changesin the stateof the economyas representedby the excessreturnsof the S&P 500 index portfolio. The regressionestimatesdescribe a straight line with intercept cr* and slope Bsp,which we call the security characteristic line (SCL) for HP. Figure 8.2 showsa graph of the excessreturns on HP and the S&P 500 portfolio over the 60-month period from April 2001 to March 2006. The graph shows that HP returns generally follow thoseof the index, but with much larger swings.Indeed,the annualizedstandard deviation of the excessreturn on the S&P 500 portfolio over the period was 13.58Va,whlle that of HP was 38.I7Vo.The swings in HP's excessreturns suggesta larger-than-average sensitivity to the index, that is, a beta greaterthan 1.0. The relationshipbetweenthe returns of HP and the S&P 500 is made clearerby the scatter diagram in Figure 8.3, wherethe regressionline is drawn throughthe scatter.The vertical distanceof eachpoint from the regressionline is the value of HP's residual,esp(r), correspondingto that particular date. The rates in Figures 8.2 and 8.3 are not annualized, and the scatter diagram shows monthly swings of over +30Vo for HP, but returnsin the rangeof -ll7o to 8.5Vofor the S&P 500. The regression analysisoutput obtainedby using Excel is shownin Table 8.1.

.4000 .3000 àe .2000 c .1000 OJ .0000 É. C,

x u.l

TheExplanatory Powerof the SCLfor HP

-.1000 -.2000 -.3000 -.4000 r

-

E

N

N

&

C

i

D

(

à

n

â

$

r

B

)

l

ç

o

.

Ë

O

=

Month/Year

FIGURE 8.2 Excess returnson HPandS&P500for April2001 to March2006

{

Considering the top panel of Table 8.1 first, we see that the correlation of HP with the S&P 500 is quite high (.1238),telling us that HP tracks changes in the returns of the S&P 500 fairly closely.The R-square (.5239) tells us that variation in the S&P 500 excess returns explains aboû 52Voof the variation in the HP series.The adjustedR-square(which is slightly smaller) corrects for an

CHAPTLRI upward bias in R-square that arises because we use the fitted values of two parameters,6the slope (beta) and intercept (alpha), rather than their true, but unobservablevalues.V/ith 60 observations,this bias is small. The standard error of the regression is the standard deviation of the residual, which we discussin more detail shortly. This is a measureof the slippagein the averagerelationship between the stock and the index due to the impact of firmspecific factors. This measure, of course,is basedon in-sampledata.A more severe test is to look at returns from periods later than the one covered by the regressionsample and test the power of the independentvariable (the S&P 500) to predict the dependent variable (the return on HP). Correlation between regression forecasts and realizations of out-of-sample data is almost always considerably lower than in-sample correlation.

I i

\

OJ

6

o I x

_.10.".:%.,

s

IJJ

.:oK'

t

rr

-'/

-.2

t

ExcessReturns.S&P500

FIGURE 8.3 Scatterdiagramof HB the S&P500,and the line (SCL)for HP securitycharac'teristic

TABTE8.1 Excel output: Regression statistics for the SCL of Hewlett-Packard

.7238 .5239 .5157 .0767 60

ANOVA

Regression Residual Total

lntercept s&P500

df

55

MS

I 58 59

.3752 .3410 .7162

.3752 .0059

Coefficients

Standard Error

t-Stat

VValue

.0099 .2547

0.8719 7.9888

.3868 .0000

0.0086 2.0348

267

.2

RegressionStatistics MultipleR R-square AdjustedR-square Standarderror Observations

IndexModels

6In general, the adjustedR-square (Rl) is derived from the unadjustedby R::

1 - (1 - R')

, where k

isthenumberofindependentvariables(here,k:1).Anadditionaldegreeoffreedomislosttotheestimateof the interceDt.

268

PARTfl

PortfolioTheoryand Practice

Analysisof Variance The next panel of Thble 8. I showsthe analysisof variance(ANOVA) for the SCL. The sum of squares(SS) of the regression (.3752) is the ponion of the variance of the dependent variable (HP's return) that is explained by the independentvariable (the S&P 500 return); it is equal to Bilo3*r*. The MS column for the residual (.0059) shows the variance of the unexplained portion of HP's return, that is, the portion of return that is independentof. the market index. The square root of this value is the standard error (SE) of the regression (.0767) reported in the first panel. If you divide the total SS of the regression(.7162) by 59, you will obtain the estimate of the variance of the dependentvariable (Hp), .012 per month, equivalent to a monthly standarddeviation of ll7o. When annualized,Twe obtain an annualized standarddeviation of 38.17Vo,as reported earlier. Notice that the R-square (the ratio of explained to total variance) equals the explained (regression)SS divided by the total SS.8

TheEstimateof Alpha Moving to the bottom panel, the intercept (.0086 : .867oper month) is the estimateof HP's alpha for the sample period. Although this is an economically large value (10.327oon an annual basis), it is statistically insignificant. This can be seenfrom the three statisticsnext to the estimated coefficient. The first is the standarderror of the estimate (0.0099).eThis is a measureof the imprecision of the estimate.If the standarderror is large, the range of likely estimation error is correspondingly large. The t-statistic reported in the bottom panel is the ratio of the regressionparameterto its standarderror. This statistic equals the number of standarderrors by which our estimate exceedszero, and thereforecan be usedto assessthe likelihood that the true but unobserved value might actually equal zero rather than the estimate derived from the data.rOThe intuition is that if the true value were zero, we would be unlikely to observe estimated values far away (many standarderrors) from zero. So large r-statisticsimply low probabilities that the true value is zero. In the caseof alpha, we a.reinterestedin the averagevalue of HP's return net of the impact of market movements.Supposewe define the nonmarket componentof HP's return as ?When annualizing monthly data, average retum and variance are multiplied by 12. However, because variance is

murtipriedbv 12'standarddev'iï':"1':,"iil;1-

FiroÉorr** o'(e*)

:.3752 = szzs .7162

Equivalently,R-squareequalsI minusthefractionof variancethat is norexplainedby marketretums,i.e., I mihus the ratio of firm-specificrisk to total risk. For HP,this is

'-s;;Sntt:r-ffi:

szto

eWe can relate the standard error of the alpha estimate to the standard error of the residuals

as follows:

l . (AveS&P500)' 5s(s*)=o(e*;./1 * -\ln Var(S&P500)x(n-l)

roTher-statisticis basedon the assumptionthat retumsarenormally distributed. general, ln if we standardizethe estimateof a normally distributedvariableby computingits differencefrom a hypothesizedvalue and dividin! by the standarderror of the estimate(to expressthe differenceasa numberof standardenors), the resultingvariablewill havea r-distribution.With a largenumberof observations,the bell-shapedr-distributionapproachesthe normaldistribution.

CHAPTERI

lndexModels

its actual return minus the return attributable to market movements during any period. Call this HP's firm-specific return, which we abbreviateas R6. - R/" : RHpRfim-specific BnpRsapsoo If Ry"were normally distributed with a mean of zero, the ratio of its estimate to its standard error would have a r-distribution. From a table of the l-distribution (or using Excel's TINV function) we can find the probability that the true alpha is actually zero or even lower given the positive estimateof its value and the standarderror of the estimate.This is called the level of significance oL as in Table 8.1, the probability or p-value. The conventional cut-off for statistical significance is a probability of less than 5Vo, which requires a /-statistic of about 2.0. The regression output shows the r-statistic for HP's alpha to be .8719, indicating that the estimateis not significantly different from zero. That is, we cannot reject the hypothesisthat the true value of alpha equalszero with an acceptablelevel of. confidence.The p-value for the alpha estimate (.3868) indicatesthat if the true alpha were zero, the probability of obtaining an estimate as high as .0086 (given the large standard error of .0099) would be .3868, which is not so unlikely. We concludethat the sample average of R6 is too low to reject the hypothesisthat the true value of alpha is zero. But even if the alpha value were both economically and statistically significant within the sample, we still would not use that alpha as a forecastfor a future period. Overwhelming empirical evidenceshowsthat 5-year alpha valuesdo not persistover time, that is, there seemsto be vinually no correlation between estimatesfrom one sampleperiod to the next. In other words, while the alpha estimated from the regressiontells us the averagereturn on the security when the market was flat during that estimation period, it does not forecast what the firm's performance will be in future periods. This is why security analysis is so hard. The past does not readily foretell the future. We elaborateon this issue in Chapter 11 on market efficiency.

Thc Estimateof Beta The regressionoutput in Table 8.1 showsthe beta estimatefor HP to be 2.0348, more than twice that of the S&P 500. Such high market sensitivity is not unusual for technology stocks.The standardenor (SE) of the estimateis .2547.11 The value ofbeta and its SE produce a large /-statistic (7.9888), and ap-value ofpractically zero. We can confidently reject the hypothesis that HP's ffue beta is zero. A more interesting /-statistic might test a null hypothesisthat HP's beta is greaterthan the marketwide averagebeta of 1.0.This /-statistic would measurehow many standarderrors separate the estimatedbeta from a hypothesizedvalue of l. Here too, the difference is easily large enough to achievestatistical significance: Estmated value - Hypothesizedvalue _ Standard error

2.03-l : 4 . 0 0 .2547

However, we should bear in mind that even here, precision is not what we might like it to be. For example, if we wanted to consffuct a confidenceinterval that includes the true but unobserved value of beta with 95Vo probability, we would take the estimated value as the center of the interval and then add and subtract about two standarderrors. This producesa range between 1.43 and 2.53, which is quite wide. "

5g1g1:

o("o) ' Var(R*)x(n-l)

269

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Portfolio Theoryand Practice

Firm-SpecificRisk Themonthlystandarddeviationof HP'sresidualis7.67Vo,or 26.6Vo annually.This is quite large,on top of HP'shigh-levelsystematic risk.Thestandarddeviationof systematic risk is B x o(S&P 500) : 2.03x 13.58: 27.57vo.NoticerhatHP's firm-specificrisk is aslarge asits systematicrisk, a commonresultfor individualstocks.

Correlationand Covariance lv{atrix Figure 8.4 graphs the excess returns of the pairs of securities from each of the three sectors with the S&P 500 index on the same scale. We see that the IT sector is the most variable, followed by the retail sector, and then the energy sectoç which has the lowest volatility. Panel 1 in Spreadsheet8.1 shows the estimatesof the risk parametersof the S&P 500 portfolio and the six analyzed securities. You can seefrom tlie high residual standard errors how important diversification is. These securitieshave tremendousfirm-specific risk. Portfolios concentratedin these (or other) securitieswould have unnecessarilyhigh volatility and inferior Sharperatios. Panel 2 shows the correlation matrix of the residuals from the regressions of excess returns on the S&P 500. The shadedcells show correlations of same-sectorstocks, which are as high as .7 for the two oil stocks (BP and Shell). This is in contrastto the assumption of the index model that all residuals are uncorrelated. Of course, these correlations are, to a great extent, high by design, because we selected pairs of firms from the same industry. Cross-industry correlations are typically far smaller, and the empirical estimates of correlations of residuals for industry indexes (rather than individual stocks in the same industry) would be far more in accord with the model. In fact, a few of the stocks in this sample actually seem to have negatively correlated residuals. Of course,correlation also is subject to statistical sampling error, and this may be a fluke. Panel 3 produces covariancesderived from Equation 8.10 of the single-index model. Variancesof the S&P 500 index and the individual covered stocks appearon the diagonal. The variance estimates for the individual stocks equal glof,+o21e). The off-diagonal terms are covariancevalues and equal B, B,o| .

PORTFOLIO CONSTRUCTION AND THE S I N G L E . I N D EMXO D E L In this section,we look at the implications of the index model for portfolio construction.12 V/e will see that the model offers several advantages,not only in terms of parameter estimation, but also for the analytic simplification and organizational decentralizationthat it makespossible.

Alpha and SecurityAnalysis Perhaps the most important advantage of the single-index model is the framework it provides for macroeconomic.andsecurity analysis in the preparation of the input list that r2The use of the index model to construct optimal risky portfolios was originally developed in Jack Treynor and Fischer Black, "How to Use Security Analysis to Improve Portfolio Selection," Joumal of Business, January 1973.

CHAPTLRI

IndexModels

s . 2 -

o , r G

A

= O

>

s&P500

-HP DELL -. 11 -.2

Mar.03 Mar.04 Month/Year

Mar.06

8 . 2 - s&P500 -wMT

G

É.

-

TARGET

-

s&P500

c

Mar.03 Mar.04 Month/Year

8 . 2 9o . 1

-BP SHELL

^ à . 0 c o

_ 1

Mar.04 Mar.03 Month/Year

FIGURE

Mar.05

Mar.06

Excessreturnson portfolio assets

is so critical to the efficiency of the optimal portfolio. The Markowitz model requires estimates of risk premiums for each security. The estimate of expected return depends on both macroeconomic and individual-firm forecasts. But if many different analysts perform security analysis for a large organization such as a mutual fund company, a likely result is inconsistency in the macroeconomic forecasts that partly underlie expectations

271

eXcel

A

c

B

D

E

F

G

P a n e l1 : R i rik Parameters of the Investable UnivéEèlânn"ati

Pleasevisit us at m.mhhe.com/bkm

tl

I

BP

SHELL

J

at-

2 SD ot excess return

3 4 5 6 7 I I

s&P 500 HP DELL

'10

WMT TARGET BP SHELL

Beta 1.00 203

0 1358 0.3817 0.290 0.1935 0 261

90rretaItQ

SD of Residual

with the s&P 500

0 1358

0

o 2762 o 1672

0 2656

o72

o 2392

o.62

0 0841

058 0.43 066

127

o.172(

o 1757 0 . 19 8 1

o.1822

o47

035

067

0 0634 0.0914

o 1722

0.1988

0 1780

046

11

1 2 Panel 2: Correlation of Residuals t3 14

't5 t6 17

t8 t9 N

HP HP DELL WMT TABGET BP SHELL

DELL

o08 -0 34 -0 10

WMT

TABGET

BP

1

-o.20 -0 06

'I

o17 o12 o28

-0 19

-0.19

-o.24

q,l$l

-0 13

I

o.7

21 22 Panel 3: Th( Index Model Covariance Matrix I

24 25 26 27 28 29 30 31

32 33

s&P 500

'1 00

Beta

s&P 500

1.00

HP DELL WMT TARGET BP SHELL

203

HP 203 0.0375

0.0375

o 0227

DELL

o.0227 o.0462

0.0462

0.62

0 . 0 11 4

o.0232

1.27

o 0234

o.47

0.0086

067

o.o124

o 0475 0 0175 0.0253

WMT TARGET 0.62 1.27 0 0't t4 0.0234 o 0232 o 0475 00141 0 0288

0.0141 0 0288 0 0106 0 0153

o47 0 0086 0.0175 0.0106 0 0053 0 0109

0.0145 0 0145

0.0053 o.oo77

0.0109 0.0157

;j;

0.0058

o67 o 0124 0 0253 0.0153

o oo77 0 0157 0.005€ 't.æt3g

u

Cells on the diagonal(shadowed eoual to variance 35 formulain cell C26 =84 2

36 37

Off-diagonal cells equal to covariance

formulain cell C27 = c$25 $827-$B$4"2 multrpliesbetatrom row and columnbv index'

38

39

& Panel 4: Macro Forecast and Forecasts of Alpha Values 41

42 43

s&P500 HP DELL WMT -0.0100 -0 0050 44 Alpha 0 00150 rl5 Riskoremir 0 0600 0 0750 o 1'121 0 0689 6 4l Panel 5: Computation of the Optimal Risky portfolio

TARGET

0.0075 o.0447

BP

o o12 0 0880

SHELL 0 0024 0.0305

Æ 49

s&P500

|ctive

Pf A

50 o'(e)

0 070a

u/o'(e)

0 5505

52 wo(i) 53 lwo(i)l' al d ^

1 0000|

51

55

o'(e^)

56

wi

DELL

o 0572

o 2126 -o 1748 0.3863 -0 3176 0.1492

0.1009

WMT

TARGET

SHELL 0.0317

0 1911

0 4045

0 0789

0 3472 0 1205

0.7349

0 1433

0.5400

0.0205

0 0596

o.1262

o.0246

o.0447

0 0880

0 0305

0 0392

-o 2941 0.0865

BP

o 0297

0.0309| -0 1619

o.0222 0 0404 0 1691

57 W'(Riskvoor0l 58 Beta 59 RiskpremiumJ

o 4282

0 1718

1.0922

o ooos| -o osao

60 SD 6t Sharpe Ratio

0.06|

o 135s I

0.0878 0 2497 035

0 . 0 7 5 0 1 0 . 1 1 2I1

o44

S P R E A D S H E E8T. 1 lmplementing the index model

272

HP

-0.0505 0.0689

Ovêrall Pf

CHAPTLR8

IndexModels

of returns across securities. Moreover, the underlying assumptionsfor market-index risk and return often are not explicit in the analysis ofindividual securities. The single-index model creates a framework that separatesthese two quite different sourcesof return variation and makes it easier to ensure consistencyacross analysts.We can lay down a hierarchy of the preparation of the input list using the framework of the single-index model. 1. Macroeconomic analysis is used to estimatethe risk premium and risk of the market index. 2. Statistical analysis is used to estimatethe beta coefficients of all securitiesand their residual variances,o2(er). 3. The portfolio managerusesthe estimatesfor the market-index risk premium and the beta coefficient of a security to establishthe expectedreturn of that security absent any contribution from security analysis.The market-driven expectedreturn is conditional on information common to all securities,not on information gleaned from security analysis of particular firms. This market-driven expectedreturn can be used as a benchmark. 4. Security-specificexpectedreturn forecasts(specifically, security alphas)are derived from various security-valuationmodels (such as those discussedin Part Five). Thus, the alpha value distills the incremental risk premium attributable to private information developedfrom security analysis. In the context of Equation 8.9, the risk premium on a security not subject to security analysis would be 9,8(R,). In other words, the risk premium would derive solely from the security's tendency to follow the market index. Any expectedreturn beyond this benchmark risk premium (the security alpha) would be due to some nonmarket factor that would be uncoveredthrough security analysis. The end result of security analysis is the list of alpha values. Statistical methods of estimating beta coefficients are widely known and standardized;hence,we would not expect this portion of the input list to differ greatly acrossportfolio managers.In contrast, macro and security analysis are far less of an exact science and therefore provide an arena for distinguished performance.Using the index model to disentanglethe premiums due to market and nonmarket factors, a portfolio managercan be confident that macro analystscompiling estimatesof the market-index risk premium and security analystscompiling alpha values are using consistentestimatesfor the overall market. In the context of portfolio construction, alpha is more than just one of the components of expected return. It is the key variable that tells us whether a security is a good or a bad buy. Consider an individual stock for which we have a beta estimate from statistical considerations and an alpha value from security analysis. We easily can find many other securities with identical betas and therefore identical systematic componentsof their risk premiums. Therefore, what really makes a security attractive or unattractive to a portfolio manager is its alpha value. In fact, we've suggestedthat a security with a positive alpha is providing a premium over and above the premium it derives from its tendency to track the market index. This security is a bargain and therefore should be overweighted in the overall portfolio compared to the passive alternative of using the market-index portfolio as the risky vehicle. Conversely, a negative-alphasecurity is overpriced and, other things equal, its portfolio weight should be reduced. In more extreme cases,the desired portfolio weight might even be negative, that is, a short position (if permitted) would be desirable.

275

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Portfolio Theoryand Practice

The IndexPortfolio as an InvestmentAsset The processofcharting the efficient frontier using the single-index model can be pursued much like the procedure we used in Chapter 7, where we used the Markowitz model to find the optimal risky portfolio. Here, however, we can benefit from the simplification the index model offers for deriving the input list. Moreover, portfolio optimization highlights another advantageof the single-index model, namely, a simple and intuitively revealing representationof the optimal risky portfolio. Before we get into the mechanicsof optimization in this setting, however, we start by considering the role of the index portfolio in the optimal portfolio. Supposethe prospectusof an investment company limits the universe of investable assetsto only stocks included in the S&P 500 portfolio. In this case,the S&P 500 index captures the impact of the economy on the large stocks the firm may include in its portfolio. Supposethat the resourcesof the company allow coverageof only a relatively small subset of this so-called investable universe.If these analyzedfirms are the only ones allowed in the portfolio, the portfolio manager may well be worried about limited diversification. A simple way to avoid inadequatediversification is to include the S&P 500 portfolio as one of the assetsof the portfolio. Examination of Equations 8.8 and 8.9 reveals that if we treat the S&P 500 portfolio as the market index, it will have a beta of 1.0 (its sensitivity to itself), no firm-specific risk, and an alpha of zero-there is no nonmarket component in its expectedreturn. Equation 8.10 shows that the covarianceof any security, i with the index is B,o"r. To distinguish the s&P 500 from the n securities covered by the firm, we will designateit the (n * l)th asset.Vy'ecan think of the S&P 500 as apassiveportfolio thar the managerwould select in the absenceof security analysis.It gives broad market exposure without the need for expensivesecurity analysis.However, if the manageris willing to engage in such research,she may devise an active portfolio that canbe mixed with the index to provide an even better risk-return trade-off.

TheSingle-lndex-Model lnput List If the portfolio manager plans to compile a portfolio from a list of n actively researched firms and a passive market index portfolio, the input list will include the following estimates: 1. Risk premium on the S&P 500 portfolio. 2. Estimate of the standarddeviation of the S&p 500 portfolio. 3. n setsofestimates of(a) beta coefficients, (b) stock residual variances,and (c) alpha values. (The alpha values for each security,together with the risk premium of the S&P 500 and the beta of each security, will allow for determination of the expectedreturn on each security.)

TheOptimalRiskyPortfolioof the Single-lndex Model The single-index model allows us to solve for the optimal risky portfolio directly and to gain insight into the nature of the solution. First we confirm that we easily can set up the optimization processto chart the eftcient frontier in this framework along the lines of the Markowitz model. With the estimates of the beta and alpha coefficients, plus the risk premium of the index portfolio, we can generatethe n * I expectedreturns using Equation 8.9. With the

CHAPTLRI

lndexModels

estimatesof the beta coefficients and residual variances,together with the variance of the index portfolio, we can construct the covariancematrix using Equation 8.10. Given a column of risk premiums and the covariancematrix, we can conduct the optimization program describedin Chapter 7. Vy'ecan take the description of how diversification works in the single-index framework of Section 8.2 a step further. We showed earlier that the alpha, beta, and residual variance of an equally weighted portfolio are the simple averagesof thoseparametersacrosscomponent securities.Moreover, this result is not limited to equally weighted portfolios. It applies "simple average" with "weighted average," to any portfolio, where we need only replace using the portfolio weights. Specifically, n+l dP:

s

;

).wia;

for theindex cr,*,:crr:0

n*l

F": Ivv,F,

: for the index 9,*r: 9,, 1

i

(8.18)

n+l

o2(ep):lwloz1e,)

;

f o r t h ei n d e x o 2 ( e n * ) :

o2(eu):0

The objective is to maximize the Sharperatio of the portfolio by using portfolio weights, wr, . . ., w,*r. With this set of weights, the expectedreturn, standarddeviation, and Sharpe ratio ofthe portfolio are n1'l

E(Rp): a, + E(Ry)}r:2rg,+

o r:lB2ro2, * o

n+l

E(Rr)),w,F,

'(e,)t't' :1",(x,,u, * ) i,7,",r)

(8.1e)

. _ -E(RP)

'

ù o :

oP

At this point, as in the standardMarkowitz procedure,we could use Excel's optimization program to maximize the Sharperatio subject to the adding-up constraint that the portfolio weights sum to 1.0. However, this is not necessarybecausethe optimal portfolio can be derived explicitly using the index model, and has considerableintuitive appeal.It is instructive to outline the logical thread of the solution. We will not show every algebraic step,but will insteadpresentthe major results and interpretationsof the procedure. Before delving into the results, let us first explain the basic trade-off the model reveals. If we were interestedonly in diversification, we would just hold the market index. Security analysis gives us the chanceto uncover securitieswith a nonzero alpha and to take a differential position in those securities.The cost ofthat differential position is a departurefrom efficient diversification, in other words, the assumption of unnecessaryfirm-specific risk. The model shows us that the optimal risky portfolio tradesoff the searchfor alpha against the departurefrom efficient diversification. The optimal risky portfolio turns out to be a combination of two component portfolios: (1) an active portfolio, denotedbyA, comprised of the n analyzedsecurities(which we call the acTiveportfolio becauseit follows from active security analysis), and (2) the

275

276

PARTll

Portfolio Theoryand Practice market-index portfolio, the (n * 1)th assetwe include to aid in diversification, which we call the passlveportfolio and denoteby M. Assume first that the active portfolio has a beta of 1.0. In that case,the optimal weight in the active portfolio would be proportional to the ratio ao/o2(e). This ratio balancesthe contribution ofthe active portfolio (its alpha) againstits contribution to the portfolio variance(residual variance).The analogousratio for the index portfolio is E(Rr)lo], and hence the initial position in the active portfolio (i.e., if its beta were 1) is ct, n

w;

o'o E(RM)

(8.20)

o'*

Next, we amend this position to account for the actual beta of the active portfolio. For any level of o2a,the correlation between the active and passiveportfolios is greater when the beta of the active portfolio is higher. This implies less diversification benefit from the passive portfolio and a lower position in it. Correspondingly, the position in the active portfolio increases.The precise modification for the position in the active portfolio is as follows:13

(8.21)

w\

'rfA:

t + ( 1 - 9 r )w Ï

TheInformationRatio Equations 8.20 and 8.21 yield the optimal position in the active portfolio once we know its alpha, beta, and residual variance. v/ith wj in the active portfolio and I - wj invested in the index portfolio, we can compute the expected return, standard deviation, and Sharperatio of the optimal risky portfolio. The Sharperatio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy). The exact relationship is e

-

)

I a. l5 , , : 5 , , +l------a--| 1o(eo)J

(8.22)

Equation 8.22 shows us that the contribution of the active portfolio (when held in its optimal weight, wi; to the Sharpe ratio of the overall risky portfolio is determined by the ratio of its alpha to its residual standard deviation. This important ratio is called the information ratio. This ratio measures the extra return we can obtain from security analysis compared to the firm-specific risk we incur when we over- or underweight securities relative to the passive market index. Equation 8.22 therefore implies that r3With a little algebraic manipulation, beta can be shown to equal the product of correlation between the index and the active portfolio and the ratio of SD(index)iSD(active portfolio). If Êa > 1, then correlation is higher than envisioned in Equation 8.20, so the diversification value ofthe index is lower. This requires the modification in Eouation8.21.

CHAPTER8

IndexModels

to maximize the overall Sharpe ratio, we must maximize the information ratio of the active portfolio. It turns out that the information ratio of the active portfolio will be maximized if we invest in each security in proportion to its ratio of u,/o2(e,). Scaling this ratio so that the total position in the active portfolio adds up to w*o, the weight in each security is cIi

(8.23)

,i: ro S a i 2,=, o2 {e')

With this set of weights, we find that the contribution of each security to the information ratio of the active portfolio dependsonits own information ratio, that is,

[ "^ ]':f [_n_-l'

(8.24)

Lo("o)l i1l"@,))

The model thus reveals the central role of the information ratio in efficiently taking advantageof security analysis. The positive contribution of a security to the portfolio is made by its addition to the nonmarket risk premium (its alpha). Its negative impact is to increasethe portfolio variance through its firm-specific risk (residual variance). In contrast to alpha, notice that the market (systematic) component of the risk premium, $,E(Rr),is offset by the security's nondiversifiable (market) 1rsk, B!o2r, and both are driven by the same beta. This trade-off is not unique to any security, as any security with the same beta makes the same balanced contribution to both risk and return. Put differently, the beta of a security is neither vice nor virtue. It is a property that simultaneously affects the risk and risk premium of a security. Hence we are concerned only with the aggregatebeta ofthe active portfolio, rather than the beta ofeach individual security. We see from Equation 8.23 that if a security's alpha is negative, the security will assume a short position in the optimal risky portfolio. If short positions are prohibited, a negative-alpha security would simply be taken out of the optimization program and assigned a portfolio weight of zero. As the number of securities with nonzero alpha values (or the number with positive alphas if short positions are prohibited) increases,the active portfolio will itself be better diversified and its weight in the overall risky portfolio will increaseat the expenseof the passiveindex portfolio. Finally, we note that the index portfolio is an efficient portfolio only if all alpha values are zero. This makes intuitive sense.Unless security analysis reveals that a security has a nonzero alpha, including it in the active portfolio would make the portfolio less attractive. In addition to the security's systematicrisk, which is compensatedfor by the market risk premium (through beta), the security would add its firm-specific risk to portfolio variance. With a zero alpha,however,the latter is not compensatedby an addition to the nonmarket risk premium. Hence, if all securities have zero alphas, the optimal weight in the active portfolio will be zero, and the weight in the index portfolio will be 1.0. However, when security analysis uncovers securitieswith nonmarket risk premiums (nonzero alphas),the index portfolio is no longer efficient.

277

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Portfolio fheory and Practice

Summary of OptimizationProcedure To summarize,once security analysisis complete, and the index-model estimatesof security and market index parametersare established,the optimal risky portfolio can be formed using thesesteps: l. Compute the initial position of each security in the active portfolio as wl:a'/o2(e,). 2. Scale those initial positions to force portfolio weights to sum to I by dividing by 'i their sum. that is. ,,:

t,? i=l

3. Computethe alpha of the activeportfolio: oo:\i_r*,o,. 4. Computethe residualvarianceof the activeportfolio: o2 (e):

Il=, wl o21e,7.

| "o/^ 5. Compute theinitialposition in theactive portfolio: ,2:l##l |

/"'"

l )

6. Computethe betaof the activeporrfolio:Êo: )1

, w,B,. 7. Adjust the initial position in the active portfolio: - - " - - - - ',.o ' A - - .-'"o^ 1+(1-F)roo' 8. Note: the optimal risky portfolio now has weights: wr:

| - wi;wi : w*Aw i.

9. Calculate the risk premium of the optimal risky portfolio from the risk premium of the index portfolio and the alpha of the active portfolio: E(Rp) : (wu * wo B)E(R4) I w'oao. Norice that the beta of the risky portfolio is wi + wi B, becausethe beta of the index portfolio is 1.0. 10. Compute the variance of the optimal risky portfolio from the variance of the index portfolio and the residual variance of the active portfolio: o2, : (w-* + w*og o)2o2, + fw.oo(e))2 .

An Example We can illustrate the implementation of the index model by constructing an optimal portfolio from the S&P 500 index and the six stocks for which we analyzedrisk parametersin Section8.3. This example entails only six analyzedstocks, but by virtue of selecting ûree pairs of firms from the same industry it is designed to produce relatively high residual correlations. This should put the index model to a severetest, as the model ignores the correlation between residuals when producing estimatesfor the covariance matrix. Therefore, comparison of results from the index model with the full-blown covariance(Markowitz) model should be instructive

Risk Premium Forecasts Panel4 of Spreadsheet 8.1 containsestimatesof alphaand the risk premiumfor eachstock.Thesealphasordinarily would be the most important

CHAPTERI

IndexModels

279

production of the investment company in a real-life procedure. Statistics plays a small role here; in this arena, macro/security analysis is king. In this example, we simply use illustrative values to demonstratethe portfolio construction process and possible results. You may wonder why we have chosen such small, forecast alpha values. The reason is that even when security analysis uncovers a large apparentmispricing, that is, large alpha values, these forecasts must be substantially trimmed to account for the fact that such forecasts are subject to large estimation error. We discuss the important procedure of adjusting actual forecastsin Chapter 27.

Panel 5 of Spreadsheet8.1 displays calculations for the The Optimal Risky Portfolio optimal risky portfolio. They follow the summary procedureof Section 8.4 (you should try to replicate these calculations in your own spreadsheet).In this example we allow short sales.Notice that the weight of each security in the active portfolio (see row 52) has the same sign as the alpha value. Allowing short sales,the positions in the active portfolio are quite large (e.g., the position in BP is .7349); this is an aggressiveportfolio. As a result, the alpha of the active portfolio (2.227o)is larger than that of any of the individual alpha forecasts.However, this aggressivestancealso results in a large residual variance (.0404, which correspondsto a residual standarddeviation of 2OVo).Therefore, the position in the active portfolio is scaleddown (seeEquation 8.20) and ends up quite modest (.1718; cell C57), reinforcing the notion that diversification considerationsilre paramount in the optimal risky portfolio. The optimal risky portfolio has a risk premium of 6.48Vo,standarddeviation of 14.227o, and a Sharpe ratio of .46 (cells J58-J61). By comparison, the Sharperatio of the index portfolio is .06/.1358: .44 (cell861), which is quite close to that of the optimal risky portfolio. The small improvement is a result of the modest alpha forecaststhat we used. In Chapter 11 on market efficiency and Chapter 24 on performanceevaluationwe demonstrate that such results are common in the mutual fund industry. Of course, some portfolio managerscan and do produce portfolios with better performance. The interesting question here is the extent to which the index model produces results that are inferior to that of the .14 model. full-covariance (Markowitz) .12 Figure 8.5 shows the efficient frontiers fromthe two modelswiththe exampledata. - -lu È We find that the difference is quite negligi'Ë .oa o ble. Table 8.2 comparesthe compositions and expected performance of the global i.9 .oo * Efficientfrontier (full covariance) É .04 minimum variance (G) and the optimal - Efficientfrontier (indexmodel) risky portfolios derived from the two ^ s&P500 .02 models. The significant difference be,00 tween the two portfolios is limited to the .40 .00 .05 .10 .15 .20 .25 .30 portfolios that are minimum-variance Deviation Standard driven only by considerations of variance. As we move up the efficient frontier, the required expectedreturns obviate FI G U R E .8.5 Efficientfrontiers with the index model and full-covariancematrix the impact of the differences in covariance and the portfolios become similar in performance.

280

PARTll

TABTE8.2 Comparisonof portfolios from the single-indexand full-covariance models

PortfolioTheoryand Practic€

Global MinimumVariancePortfolio Full-Covariance Model Mean .0371 5D .1089 Sharperatio .3409 Portfolio Weights 5&P500 .88 - .11 HP -.01 DELL WMT .25 -.18 TARGET BP .22 -.02 SHELL

Index Model

.0354 .1052 .3370 .83 -.17 -.05 .'t4 -.08 .20 .12

Optimal Portfolio Full-Covariance Model Index Model .0677 .1471 .4605 .75 .10 -.04 -.03 .1.0 .25 -.12

.0649 .1423 .4558 .83 .07 -.06 -.05 .06 .13 .03

8.5 PRACTICAL ASPECTS OF PORTFOLO MANAGEMENT W I T HT H EI N D E XM O D E L

The tone of our discussionsin this chapter indicates that the index model is the preferred one for practical portfolio management.Switching from the Markowitz to an index model is an important decision and hence the first question is whether the index model is really inferior to the Markowitz full-covariance model.

ls the IndexModel lnferiorto the Full-Covariance Model? This question is partly related to a more general question of the value of parsimonious models.As an analogy,considerthe questionof adding additional explanatoryvariables in a regressionequation. We know that adding explanatory variables will in most cases increaseR-square,and in no casewill R-squarefall. But this does not necessarilyimply a better regressionequation.raA better criterion is contribution to the predictive power of the regression.The appropriatequestion is whether inclusion of a variable that contributes to in-sample explanatory power is likety to contribute to out-of-sample forecast precision. Adding variables, even ones that may appear significant, sometimescan be hazardousto forecast precision. Put differently, a parsimonious model that is stingy about inclusion of independentvariables is often superior. Predicting the value of the dependentvariable dependson two factors, the precision of the coefficient estimatesand the precision of the forecasts of the independent variables. When we add variables, we introduce erïors on both counts. This problem applies as well to replacing the single-index with the full-blown Markowitz model, or even a multi-index model of security returns. To add another index, we need both a forecast of the risk premium of the additional index portfolio and estimates of security betas with respect to that additional factor. The Markowitz model allows far more flexibility in our modeling of asset covariance structure compared to the single-index model. But that advantagemay be illusory if we can't estimate those raln fact, the adjustedR-square may fall if the additional variable does not contribute enough explanatory power to compensatefor the extra degreeof freedom it uses.

CHAPTLR8

IndexModels

covariances with any degree of confidence. Using the full covariance matrix invokes estimation risk of thousandsof terms. Even if the full Markowitz model would be better in principle, it is very possible that cumulative effect of so many estimation errors will result in a portfolio that is actually inferior to that derived from the single-index model. Against the potential superiority of the full-covariance model, we have the clear practical advantageof the single-index framework. Its aid in decentralizing macro and security analysis is anotherdecisive advantage.

ThelndustryVersionof the IndexModel Not surprisingly, the index model has attractedthe attention of practitioners.To the extent that it is approximately valid, it provides a convenientbenchmark for security analysis. A portfolio managerwho has no special information about a security nor insight that is unavailableto the generalpublic will take the security's alpha value as zero, and, according to Equation 8.9, will forecast a risk premium for the security equal to p,nr. If we restate this forecast in terms of total returns, one would expect

E(rnp): ? + pnp[E(ry) - ry)

(8.2s)

A portfolio manager who has a forecast for the market index, E(rr), and observesthe risk-free T-bill rate, ?, can use the model to determine the benchmark expectedreturn for any stock. The beta coefficient, the market t'rsk, qi2r,and the firm-specific risk, o2(e), can be estimated from historical SCLs, that is, from regressionsof security excessreturns on market index excessreturns. There are many sourcesfor suchregressionresults. One widely used sourceis Research Computer ServicesDepartmentof Merrill Lynch, which publishesa monthly Security Risk "beta book." The Web sites listed at the end of the Evaluation book, commonly called the provide security betas. chapter also Security Risk Evaluation usesthe S&P 500 as the proxy for the market portfolio. It relies on the 60 most recentmonthly observationsto calculateregressionparameters.Merrill Lynch and most seryiceslsusetotal retums,rather than excessreturns (deviationsfrom T-bill rates), in the regressions.In this way they estimate a variant of our index model, which is r:a*br**e*

(8.26)

r-rt:cr*B(4a-r7)*e

(8.27)

instead of

To seethe effect of this departure,we can rewrite Equation 8.27 as

r : rs* o.'t $ry- 9rrl- e : otI rr(l - B) + Bry + e

(8.28)

Comparing Equations 8.26 and 8.28, you can see thatif r7 is constant over the sample period, both equationshave the same independentvariable, ry, znd residual, e. Therefore, the slope coefficient will be the samein the two regressions.16 However, the intercept that Menill Lynch calls alpha is really an estimate of cr* ri(l - B).Theapparentjustificationforthisprocedureisthat,onamonthlybasis,ry(1B) is smallandisapttobe swampedbythevolatilityof actualstockreturns.Butitis worthnoting that for I + l, the regressionintercept in Equation 8.26 will not equal the index model alpha as it does when excessreturns are used as in Equation8.27. rsValueLine is another common source of security betas.Value Line uses weekly rather than monthly data and usesthe New York Stock Exchange index instead of the S&P 500 as the market proxy. r6Actually, rTdoesvary over time and so should not be grouped casually with the constant term in the regression. Howevet variations in rJare tiny compared with the swings in the market return. The actual volatility in the T-bill rate has only a small impact on the estimated value of B.

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PortfolioTheoryand Practice Another way the Merrill Lynch proceduredepartsfrom the index model is in its use of percentagechangesin price insteadof total ratesof return.This meansthat the index model variantof Merrill Lynch ignoresthe dividendcomponentof stockreturns. Table 8.3 illustratesa page from the beta book which includesestimatesfor HewlettPackard.The third column, Close Price, shows the stock price at the end of the sample period.The next two columnsshow the betaand alphacoefficients.Rememberthat Merrilt Lynch's alphais actuallyan esrimateof o + rr( I - 9). Much of the output that Merrill Lynch reportsis similar to the Excel output (Table8.1) that we discussedwhen estimatingthe index model for Hewlett-Packard.The R-square statisticis the ratio of systematicvarianceto total variance,the fraction of total volatility attributableto market movements.Merrill Lynch actually reportsadjustedR-squares(see footnote6), which accountsfor the instancesof negativevalues.For most firms, R-square is substantiallybelow .5, indicating that undiversifiedstockshave far more firm-specific than systematicrisk. This highlightsthe practicalimportanceof diversification. The ResidStd Dev-n columnis the standarddeviationof the monthly regressionresiduals, also sometimescalled the standarderror of the regression.Like Excel, Menill Lynch also reportsthe standarderrorsof the alpha and betaestimatesso we can evaluatethe precisionof the estimates.Notice that the estimatesof betaarefar more precisethanthoseof alpha. The next-to-lastcolumn is calledAdjustedBeta.The motivationfor adjustingbetaestimatesis that, on average,the betacoeff,cientsof stocksseemto move toward I over time. One explanationfor this phenomenonis intuitive. A businessenterpriseusually is establishedto producea specificproductor service,and a new firm may be more unconventional than an older one in many ways, from technologyto managementstyle.As it grows,however,a firm often diversifies,first expandingto similar productsand later to more diverse operations.As the firm becomesmore conventional,it startsto resemblethe rest of the economyevenmore.Thus its betacoefficientwill tend to changein the directionof 1. Another explanationfor this phenomenonis statistical.We know that the averagebeta over all securitiesis 1. Thus,beforeestimatingthe betaof a security,our bestforecastof the beta would be that it is l. When we estimatethis beta coefficientover a particularsample period,we sustainsomeunknown samplingerror of the estimatedbeta.The greaterthe differencebetweenour betaestimateand 1, the greateris the chancethat we incurreda large estimationerror and that betain a subsequentsampleperiod will be closerto l.

E-lnvestments

Beta Estimates Go to http://finance.yahoo.com and clickon the /nvestinglink.Thiswill open the Market overview page. click on the Stockstab and then the stock screenerlink. Selectthe link for the Java Yahoo!FinanceScreenerthat letsyou createyour own screens.ln the Criteriabox, scrolldown to TradingandVolume on the menu and choose8eta. In the Conditionsbox, choose