comparing asset pricing models: an investment perspective

qWe are grateful for comments by Nick Barberis, Gene Fama, Ravi Jagannathan, Don Keim,David Modest, Toby Moskowitz, William Tyson, an anonymous referee, and workshop participantsat London Business School, Stanford University, University of British Columbia, University ofChicago, University of Maryland, University of Pennsylvania, the Spring 1999 NBER Asset Pricingmeeting, and the 2000 AFA meetings.

*Correspondence address. Finance Department, The Wharton School, University of Pennsyl-vania, Philadelphia, PA 19104-6367, USA. Tel.: 215-8985734; fax: 215-8986200.

E-mail address: [email protected] (R.F. Stambaugh).

Journal of Financial Economics 56 (2000) 335}381

Comparing asset pricing models:an investment perspectiveq

L[ ubos\ PaH stor!, Robert F. Stambaugh",#,*!Graduate School of Business, University of Chicago, Chicago, IL 60637, USA

"Finance Department, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104, USA#The National Bureau of Economic Research, USA

Abstract

We investigate the portfolio choices of mean-variance-optimizing investors who usesample evidence to update prior beliefs centered on either risk-based or characteristic-based pricing models. With dogmatic beliefs in such models and an unconstrained ratioof position size to capital, optimal portfolios can di!er across models to economicallysigni"cant degrees. The di!erences are substantially reduced by modest uncertaintyabout the models' pricing abilities. When the ratio of position size to capital is subject torealistic constraints, the di!erences in portfolios across models become even less impor-tant and are nonexistent in some cases. ( 2000 Elsevier Science S.A. All rights reserved.

JEL classixcation: G11; G12; C11

Keywords: Portfolio selection; Asset pricing models; Investment constraints; Bayesiananalysis

0304-405X/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.PII: S 0 3 0 4 - 4 0 5 X ( 0 0 ) 0 0 0 4 4 - 1

1Examples of risk-based models that contain this implication are the Capital Asset Pricing Model(CAPM) of Sharpe (1964) and Lintner (1965), the intertemporal model of Merton (1973), and theArbitrage Pricing Theory of Ross (1976). For further discussion see Jobson and Korkie (1985),Grinblatt and Titman (1987), and Huberman et al. (1987).

1. Introduction

To an investor seeking a mean-variance e$cient portfolio, a risk-basedpricing model o!ers a powerful insight. If expected excess returns are linearcombinations of exposures (or &betas') to k sources of risk (or &factors'), then therisky portion of any mean-variance e$cient portfolio is a combination ofk benchmark portfolios that mimic those factors.1 In attempting to apply thisinsight, an investor confronts a variety of issues. Alternative pricing models viefor consideration by the investor, who realizes that no model is likely to becompletely accurate. In one class of contenders to risk-based models, assetcharacteristics unrelated to risk exposures enter expected returns due to behav-ioral phenomena such as overreaction, and these &characteristic-based' modelsdo not identify a set of benchmark portfolios for investment. Moreover, theinvestor faces constraints on borrowing and short sales, to at least some degree,whereas a theory's investment implications are often derived in the absence ofsuch constraints.

This study conducts an empirical comparison of asset pricing models. Inorder to provide an economic metric for judging di!erences between models, weanalyze the portfolio-choice problem from the perspective of investors who facethe issues described above. Our investigation reveals the extent to whichalternative pricing models imply di!erent investment choices. An investor viewsa pricing model as providing a way to center prior beliefs, speci"ed with varyingdegrees of con"dence in the model. These prior beliefs about the distribution ofreturns are then updated by the data and used to compute an optimal portfolio,subject to margin requirements ranging from 50% to none at all. Our objectiveis not to choose one pricing model over another but instead to shed some lighton the economic importance of deliberating such a choice. We "nd that, in thepresence of mispricing uncertainty and margin requirements, models with fun-damentally di!erent views about the economic determinants of expected returnsoften imply similar portfolio choices.

Fama and French (1993) and Daniel and Titman (1997) di!er in theirexplanations of the apparent empirical relation between a "rm's expected equityreturn and its market capitalization and book-to-market ratio. In theFama}French model, the latter characteristics are associated with risk expo-sures, whereas they re#ect mispricing in the Daniel}Titman model. From aninvestment perspective, we compare these models to each other and to theSharpe}Lintner CAPM. The investment universes are similar to those analyzed

336 L. Pa& stor, R.F. Stambaugh / Journal of Financial Economics 56 (2000) 335}381

by Daniel and Titman (1997) and Davis, Fama, and French (2000). In particular,portfolios are formed by sorting stocks on total equity capitalization (&size'), theratio of book value to market value of common equity (&book-to-market'), andbetas with respect to the &HML' book-to-market factor of Fama and French(1993). One investment universe consists of the three benchmark positions fromthe Fama}French speci"cation of a factor-based model plus nine spread posi-tions that are long stocks with low HML betas and short stocks with high HMLbetas, holding size and book-to-market constant. These spread positions aredesigned to exploit di!erences between the Fama}French model and theDaniel}Titman model: the expected payo!s are negative under the "rst modelbut zero under the second. Optimal portfolios are computed for various hypo-thetical investors who face this investment universe but believe in di!erentpricing models, with perfect con"dence or with some uncertainty abouta model's pricing ability.

The optimal portfolio for an investor with perfect con"dence in a model'spricing ability can exhibit economically signi"cant di!erences from that of aninvestor with equally strong beliefs in an alternative pricing model. Suppose, forexample, that risk aversion is speci"ed such that all of an investor's wealthis allocated to the value-weighted stock market portfolio if that is the onlyrisky asset available. Then, if an investor with perfect con"dence in theDaniel}Titman characteristic-based model is forced to accept the portfolio of aninvestor with equally strong beliefs in the Fama}French factor-based model, the"rst investor perceives a certainty-equivalent loss of about 8% per year. A sim-ilar loss is perceived by the second investor if forced to accept the portfolio of the"rst. When each investor has some uncertainty about a model's pricing ability,the losses are substantially reduced. For example, if this &mispricing' uncertaintyis such that a spread with long and short positions of one dollar each has anexpected payo!whose deviation from the model has a standard deviation of twocents per year, then the annual certainty-equivalent losses described above dropbelow 2%.

Expected returns are associated with characteristics in both theDaniel}Titman and Fama}French models. Given the latter model's speci"ca-tion of the factors SMB and HML, betas on those factors are correlated withsize and book-to-market (e.g., high book-to-market "rms tend to have highHML betas). Indeed, based on comparisons of investment implications understrict beliefs in both models, each model is closer to the other than to the CAPM,in which characteristics play a weaker role. An investor who believes in thecomplete accuracy of the Daniel}Titman model perceives an annual certainty-equivalent loss of 20% if forced to hold a 100% allocation in the marketportfolio, the choice of an investor with complete con"dence in the CAPM. Thesame loss for the investor with complete con"dence in the Fama}French modelis about 12%. These losses are larger than the corresponding 8% value in theearlier comparison, but mispricing uncertainty again reduces these di!erences

L. Pa& stor, R.F. Stambaugh / Journal of Financial Economics 56 (2000) 335}381 337

substantially. For example, with the same two-cent mispricing uncertaintydescribed previously, the loss for the Daniel}Titman investor drops below 5%,and the loss for the Fama}French investor drops to about 2%.

The above results all describe cases in which an investor is permitted toestablish long and short equity positions of any size. Most investors, however,are likely to face some limit on the aggregate value of risky positions that can beestablished per dollar of invested capital. For example, Regulation T, whichapplies to customers of U.S. broker/dealers, requires 50% margin, or a ratio oftotal position size to invested capital of no more than two. There are practicesby which some investors exceed this limit, such as dealing with non-U.S. brokersor engaging in joint-back-o$ce arrangements, but our understanding is thatRegulation T governs much of the U.S. investment industry at both the indi-vidual and institutional levels. Nevertheless, in addition to a 50% marginrequirement, we also consider margins of only 20% and 10%.

Margin requirements can dramatically reduce, and even eliminate, the cross-model di!erences in investment implications. With margin requirements of 50%and 20%, the Fama}French and Daniel}Titman models yield identical port-folios from the asset universe described above, even for investors whose priorbeliefs preclude any mispricing. Imposing even a 10% margin still has largee!ects. For example, an investor with complete con"dence in the Daniel}Titmanmodel who must hold the portfolio of an equally con"dent Fama}Frenchinvestor perceives an annual certainty-equivalent loss of about 2%, as comparedto 8% when positions are unrestricted. With the two-cent mispricing uncertain-ty described earlier, the loss drops to 65 basis points, compared to about 2% inthe unrestricted case. (Essentially the same statements apply for theFama}French investor who must hold the portfolio of a Daniel}Titmanbeliever.) These results seem especially noteworthy, given that the asset universeis constructed to exploit di!erences between these two models.

We also examine another asset universe that is similar to the "rst, except thatthe three-way-sorted portfolios enter individually instead of being paired inlong-short spreads. Margin requirements give rise to a striking result. Under thetypical 50% margin requirement, an investor with complete con"dence in theaccuracy of the Fama}French model bears substantial risk but allocates nothingto that model's three benchmark positions. Of course, without margin require-ments, those benchmarks constitute the entire risky portion of that investor'sportfolio. This example illustrates a general point that, for an investor facingconstraints, a set of benchmarks can be correct for pricing but not for investing.

With the alternative asset universe, it is less straightforward to representa prior belief in a characteristic-based model, since the simple zero-expectationimplication for the spread positions no longer applies. We develop an alterna-tive approach, applicable more generally, that requires the size and book-to-market characteristics of each portfolio. A comparison of the optimal portfolioschosen by investors with beliefs in di!erent pricing models gives results similar


to those of the preceding analysis, except that the di!erences between thisalternative characteristic-based representation and the Fama}French model areno longer eliminated completely by imposing margin requirements.

The remainder of the study is organized as follows. Section 2 discussesmean-variance portfolio optimization in the presence of margin requirementsand presents examples using the second asset universe discussed above. Themoments of the return distribution for that illustration are taken as known andconstructed assuming the Fama}French model holds exactly, with samplemoments replacing unknown quantities. Section 3 incorporates parameteruncertainty, including uncertainty about a model's pricing ability. The Bayesianapproaches we develop for this purpose are described, and the comparison ofportfolio choices across pricing models is conducted using the "rst asset uni-verse, containing the characteristic-paired spreads. Section 4 reports similarresults for the second universe, after describing the alternative characteristic-based approach developed to accomodate a more general set of non-pairedportfolios. Section 5 discusses brie#y a potential extension to a model-uncertain-ty setting in which an investor assigns probabilities to multiple models inmaking a portfolio choice. Section 6 reviews the study's conclusions.

2. Portfolio choice under investment constraints

De"ne spread position i, established at the end of period t!1, as a purchaseof one asset coupled with a short sale of an equal amount of another. The twoassets are denoted as ¸

iand S

i, and their rates of return in period t are denoted

as RLi ,t

and RSi ,t

. The spread position involves at least one risky asset, which,without loss of generality, is designated as asset ¸

i. Asset S

ican be either risky or

riskless. The investment universe consists of a riskless asset plus n such spreadpositions, and we assume that some amount of margin capital is required toestablish each position. Consider a spread position of size X

iwith a dollar

payo! equal to Xi(R

Li ,t!R

Si ,t), where X

ican be positive or negative. For

a speci"ed c'0, the margin requirements are as follows. If asset Siis risky, then

establishing a spread position of size Xirequires (2/c)DX

iD dollars of capital. If

asset Siis riskless, then establishing a spread position of size X

irequires (1/c)DX

iD

dollars of capital.The total capital required to establish the spread positions is less than the

investor's wealth,=t~1

. That is,

+i|K

(2/c)DXiD#+

ibK(1/c)DX

iD)=

t~1, (1)

where K denotes the set of positions in which Siis risky, or

+i|K

2DwiD#+

ibKDw

iD)c, (2)


where wi,X

i/=

t~1. In other words, c is the maximum permitted total value of

risky long and short positions per dollar of the investor's wealth. A value ofc"2, for example, corresponds to the 50% margin requirement for commonstocks speci"ed by Regulation T.

The amount of total wealth in excess of the margin capital required toestablish the n spread positions is invested in the riskless asset, earning rate R

f,t,

and we assume that the margin capital also earns that rate. The rate of return onthe total portfolio is then given by

Rp,t

"

+ni/1

Xi(R

Li ,t!R

Si ,t)#=

t~1R

f,t=

t~1

, (3)

so the excess portfolio return is simply

Rp,t

!Rf,t

"

n+i/1

wi(R

Li ,t!R

Si ,t). (4)

We also assume a common borrowing and lending rate, so that RSi ,t

"Rf,t

forall of the spread positions in which S

iis riskless (iNK). In essence, the proceeds

from $1 of short sales can be invested in $1 of long positions, but an interest-bearing margin deposit of 2/c dollars is required. Equivalently, each dollar ofa long risky position can be "nanced by borrowing [1!(1/c)] dollars, eachdollar of a short position requires a margin deposit of (1/c) dollars, and interestis earned on margin deposits as well as the proceeds of short sales.

Note that assuming margin capital earns the riskless rate, as stated above,does not imply that the investor's portfolio contains cash (the riskless asset). Thecash position is given by

Xf"=

t~1!+

ibKX

i, (5)

which can be zero or negative (representing borrowing) if there are somepositions in which S

iis riskless, i.e., if the investment universe does not consist

solely of spreads in which both legs are risky. For example, suppose n"2 andthe "rst position has common stocks constituting assets ¸

1and S

1but the

second position has common stocks in only ¸2, so asset S

2is riskless. Let

=t~1

"100, X1"50, and X

2"100. The portfolio contains no cash and has

stock positions of 150 long and 50 short, or 200 in total, thereby meeting exactlya 50% margin requirement (c"2). In essence, 100 in interest-bearing margincapital supporting positions X

1(requiring 50) and X

2(requiring 50) is o!set by

a 100 short position in cash implied by X2.

Let w denote the n-vector with ith element wi. The investor is assumed to

choose w so as to maximize the mean-variance objective function

;"EMRp,t

N!12A VarMR

p,tN, (6)


2Lintner (1965) observes that paying interest on margin deposits and short-sale proceeds givesrise to a case in which (4) holds and allocations obey the restriction +n

i/1Dw

iD)1 (he assumes 100%

margin and all i NK). Lintner allows unlimited borrowing and lending at the riskless rate, however,and therefore maximizes the Sharpe ratio. As he observes, the solution in that case simply amountsto rescaling w in (8) to satisfy the constraint.

subject to the constraint in (2), where A is interpreted as the coe$cient of relativerisk aversion. Let r

tdenote an n-vector with ith element r

i,t,R

Li ,t!R

Si ,t, and

denote the mean vector and variance}covariance matrix of rtas E and <. Then

the optimal portfolio choice w can be rewritten as the solution to

maxw

(w@E!12Aw@<w)

s.t.+i|K

2DwiD#+

ibKDw

iD)c. (7)

When c is in"nite, so that there is no margin requirement, it is well known thatthe solution to (7) is given by

w"

1

A<~1E, (8)

which gives the usual tangent portfolio of risky assets. That is, if each risky&asset' i consists of the zero-investment spread position i plus an investment inthe riskless asset, then w in (8) is proportional to the weights in the riskyportfolio having the maximum Sharpe ratio (expected excess return divided bystandard deviation). When c is "nite, the solution to (7) need not producea portfolio with the maximum Sharpe ratio.2

In the next section, E and < are replaced by moments of Bayesian predictivedistributions corresponding to varying degrees of prior con"dence in alternativepricing models. Before proceeding to that analysis, we use a simpler speci"cationof E and < to illustrate the potential e!ects of investment constraints onportfolio choice. Suppose expected asset returns are known to be given exactlyby, say, the three-factor model of Fama and French. That is, the expected payo!on spread position i is given by

EMritN"b

1,iEMMKT

tN#b

2,iEMSMB

tN#b

3,iEMHML

tN, (9)

where MKTtis the excess return on a value-weighted market index, SMB

tis the

di!erence in returns between small and large "rms, and HMLtis the di!erence

in returns between "rms with high and low book-to-market ratios. Assume that,in the universe of n spread positions, those benchmark positions are included as


the last three, corresponding to the second subset in the following partition:

E"CE

1E

2D, <"C

<11<

12<

21<

22D. (10)

The vector of betas for each of the "rst (n!3) non-benchmark positions,(b

1,ib2,i

b3,i

), is a row of the (n!3)]3 matrix

B"<12<~1

22, (11)

and (9) implies

E1"BE

2. (12)

When c"R, Eqs. (8), (11), and (12) give

w"Cw

1w

2D"

1

AC<

11<

12<

21<

22D

~1

C<

12<~1

22E2

E2

D"1

AC0

<~122

E2D. (13)

That is, the optimal combination involves only the three benchmark positions,and w

2is proportional to the weights in the tangent portfolio corresponding to

those positions. When c is "nite, however, the solution to (7) can yield w1O0

and, as demonstrated below, it can even yield w2"0.

We present here an example with a universe containing n"11 risky posi-tions, the last three of which are the Fama}French (FF) benchmark positions,SMB, HML, and MKT. The eight non-benchmark positions are selected froma larger universe of 27 equity portfolios, constructed in essentially the samemanner as those in Davis et al. (2000). At the end of June of year t, all NYSE,AMEX, and NASDAQ stocks in the intersection of the CRSP and Compustat"les are sorted on market capitalization (&size') and assigned to three categories.The same stocks are also assigned to three categories in an independent sort onthe ratio of book value of equity to market capitalization (&book-to-market').There are equal numbers of NYSE stocks in each of the three size categories aswell as the three book-to-market categories. The intersection of these categoriesproduces nine groups of stocks. The stocks within a group are then sorted byHML beta and assigned to one of three subgroups containing equal numbers ofstocks. Using up to 60 months of data through December of year t!1, the &pre-formation'HML betas are computed in a regression of the stock's excess returnson &"xed-weight' versions of the FF factors, which hold the weights on theconstituent stocks constant at their June-end values of year t. (The latterprocedure, suggested by Daniel and Titman, is designed to increase the disper-sion in the &post-formation' betas of the resulting portfolios.) This three-waygrouping procedure produces 27 value-weighted portfolios, which we identify bya combination of three letters, designating increasing values of size(S, M, B), book-to-market (L, M, H), and HML beta (l, m, h). For example,portfolio SHh contains stocks with the smallest size (S), highest book-to-market


3To obtain the size and book-to-market characteristics of the Fama}French factors required inthe later analysis in Section 4, the factor portfolios are reconstructed following the procedure inFama and French (1993). The returns on those portfolios are used in Section 4 to ensure thecompatibility of factor returns and characteristics, and they are used throughout the rest of the paperfor uniformity. The reconstructed factors are virtually identical to the original Fama}French series,supplied generously by Ken French.

(H), and highest HML beta (h). The eight non-benchmark positions form thesubset of the 27 portfolios with only the high and low (i.e., no medium) values ofsize, book-to-market, and HML beta. Each portfolio is combined with a shortposition in the riskless asset to construct the spread positions consistent with theframework presented earlier.

Values of E and < are constructed to satisfy exact Fama}French three-factorpricing. The values of < and E

2are set equal to sample estimates based on

monthly returns from July 1963 through December 1997, and the 8]1 vectorE1

is then speci"ed using (11) and (12).3 The value of A in (7) is set to 2.83, whichis the value that results in an unconstrained allocation of all wealth to MKTwhen that is the only risky position available, i.e., the investor chooses neither toborrow nor lend. For the sample estimates used here, the optimal marketallocation is actually 101%; the corresponding allocation is exactly 100% in thenext section, which accounts for estimation risk.

Table 1 reports, for di!erent values of c, the optimal position sizes (Xi's) per

$100 of invested wealth (=t~1

"100). The &cash' row reports the overall (net)allocation in (5). In the unconstrained case (c"R), the optimal allocation calls forborrowing 71 and for position sizes of about 64, 358, and 171 in SMB, HML, andMKT. The fact that no allocations are made to any of the non-benchmarkpositions (w

1"0) is consistent with the unconstrained solution in (5). Recall that

a position size of 358 in HML implies that, for each $100 of invested wealth, theinvestor establishes a $358 long position in the FF &high' book-to-market stocksalong with a $358 short position in the &low' book-to-market stocks, whilea position of 171 in MKT implies only a one-way (long) position in a risky asset.The sum of the unconstrained risky positions, long and short, is equal to2(64)#2(358)#171"1,015, a bit more than ten times wealth. Thus, the con-straint on w binds slightly at c"10, as evidenced by the small nonzero allocationsto the non-benchmark positions. The investment constraint in (7) treats then assets as indivisible, which ignores the fact that they are constructed asportfolios of stocks, and a given stock can appear in a non-benchmark portfolio aswell as in each of the FF three benchmarks. A di!erent (more complicated)constraint applies if the latter fact is incorporated. If, for example, all of theunderlying stock positions are held with a single broker, then short and longpositions in the same stock cancel each other, reducing the margin required.

For smaller values of c, properties of the optimal portfolio are a!ectedsigni"cantly by the constraints on w. The standard deviations of the return on


Table 1Optimal allocations under the Fama}French model

The table reports optimal allocations (position sizes) per $100 of wealth for a mean-variance-optimizing investor with relative risk aversion equal to 2.83. The maximum value of risky positionsthat can be established per dollar of wealth is denoted by c. Sample estimates based on monthlyreturns from July 1963 through December 1997 are used to specify the expected payo!s on theFama}French benchmark positions (SMB, HML, and MKT) as well as all betas, variances, andcovariances. The expected payo!s on the "rst eight (non-benchmark) positions obey exactFama}French pricing. The risky components of those positions are a subset of value-weightedportfolios constructed by a three-way sort and identi"ed by a combination of three letters,designating increasing values of size (S, M, B), book-to-market (L, M, H), and HML beta (l, m, h).The &cash' row reports the overall amount invested in the riskless asset, including the amountsimplied by the spread positions in which the second asset is riskless, which are the "rst eightpositions and MKT. (Negative amounts represent borrowing.) Also reported are the annualizedmean and standard deviation (std) of the portfolio's return as well as the correlation (o

c,=) between

the returns on the overall portfolio with and without the constraint.

c

2 5 10 R

SLl !8.1 !0.6 0.0 0.0SLh 0.0 0.0 0.2 0.0SHl 0.0 0.0 1.4 0.0SHh 76.0 64.3 1.9 0.0BLl !37.1 0.0 !0.3 0.0BLh 0.0 0.0 0.0 0.0BHl 6.6 33.5 0.5 0.0BHh 72.1 41.8 0.3 0.0SMB 0.0 0.0 59.9 64.1HML 0.0 179.8 354.6 357.7MKT 0.0 0.0 166.4 170.7Cash !9.6 !39.1 !70.4 !70.7Mean 17.4 25.7 30.8 30.8Std 21.8 28.1 33.0 33.0oc,=

0.86 0.98 1.00 1.00

the optimal portfolio are lower under the constraints: 21.8% per annum withc"2 versus 33.0% with c"R. The correlation between the return on theunconstrained portfolio and the return on the constrained portfolio (denotedoc,=

) is equal to 0.86 for c"2. For c"5, HML is the only FF benchmarkreceiving a nonzero allocation (of 180), and SHh, BHl, and BHh receive alloca-tions of 64, 34, and 42. For c"2, none of the three FF benchmarks enter theoptimal portfolio, while SHh, BHl, and BHh receive allocations of 76, !37, and72. To reiterate, with a 50% margin requirement, the FF benchmark positionsreceive zero weight in this investment universe, even though E and < conformexactly to FF pricing.


4Early applications of Bayesian methods to portfolio choice include Zellner and Chetty (1965),Klein and Bawa (1976), and Brown (1979). Bayesian posterior distributions of measures of portfolioine$ciency are analyzed by Shanken (1987), Kandel et al. (1995), and Wang (1998), who incorporatesshort sale restrictions.

When c"2, substantial long positions are taken in SHh and BHh, whoseHML betas are large (0.80 and 0.84, respectively), since they both contain highbook-to-market stocks with high HML betas. The portfolios of low book-to-market stocks with low HML betas, SLl and BLl, both receive short positions,and their HML betas are !0.42 and !0.65, respectively. In the unconstrainedcase, recall that the optimal portfolio includes a positive exposure of 358 toHML. With c"2, that exposure can be no more than 100, even with all otherexposures set to zero. For the constrained investor who believes inFama}French pricing, going long and short the non-benchmark portfoliosessentially provides an alternative path to high HML exposure that makesbetter use of the permitted overall risky-asset position. That investor largelyavoids the long positions in the stocks with medium and low HML betas presentin the long (H) leg of HML as well as the short positions in the stocks with thehigh and medium HML betas present in the short (L) leg of HML. As a result,the HML beta of the constrained optimal portfolio is 1.51, which exceeds thevalue of 1.00 obtainable by allocating only to HML. This example illustrates thepoint that, for a constrained investor, a set of benchmarks that is correct forpricing need not be correct for investing.

The example presented here simply uses sample estimates of E2

and <. Ofcourse, an investor who relies on a "nite sample of data remains uncertain aboutthe true values of those parameters. Moreover, E

1obeys exact three-factor FF

pricing in this example, whereas an investor might be uncertain about whetherany given pricing model holds exactly. In the next section, we incorporateparameter uncertainty, or &estimation risk', which includes this potentialmispricing uncertainty.

3. Comparing investments under parameter uncertainty

Our objective is to compare pricing models in terms of their implications forportfolio choice. Speci"cally, for a given investment universe containing cashplus n risky positions, we compare the portfolios selected by investors who basetheir prior beliefs on three di!erent pricing models. Two are risk-based, the FFmodel and the CAPM, and relate expected returns to betas on one or more riskfactors. In the Daniel}Titman (DT) characteristic-based model, expected returnsdepend on size and book-to-market, not betas.

To incorporate uncertainty about parameter values, including mispricinguncertainty, we apply Bayesian methods.4 Recall that r

tis the n-vector with ith


element RLi ,t

!RSi ,t

. We assume that rtis drawn independently across t from

a multivariate normal distribution with unknown parameters E and <. Aninvestor has prior beliefs p(E,<), shaped in part by a prior belief about theaccuracy of a given pricing model. The investor forms posterior beliefs p(E,<DR),based on the data MR : r

t, t"1,2,¹N, and forms the predictive distribution for

rT`1

,

p(rT`1

DR)"PEPV

p(rT`1

DR,E,<) p(E,<DR) dEd<. (14)

The investor then solves (7) with E and < replaced by EH and <H, the momentsof the predictive distribution in Eq. (14). As detailed in the Appendix, EH and<H are obtained analytically for the risk-based models and through Gibbssampling for the characteristic-based model. The prior beliefs about E and< arediscussed below. The prior beliefs about < are noninformative, and since themonthly data are fairly informative about second moments, the predictivecovariance matrix<H is quite similar across di!erent pricing models and degreesof mispricing uncertainty.

3.1. Framework

It is useful to cast the problem in a regression setting. Let rt"(r@

1,tr@2,t

)@,following the same partitioning applied to E and < in Eq. (10). That is,r2,t

contains the payo!s on k benchmark positions from a factor-based model,and r

1,tcontains the payo!s on m ("n!k) non-benchmark positions. Con-

sider the multivariate regression,

r1,t

"a#Br2,t

#ut, (15)

where ut

obeys a multivariate normal distribution with mean zero and vari-ance-covariance matrix equal to R. In this regression framework, the set ofparameters (E,<) is replaced by (a, B, R,E

2,<

22), where B is de"ned in Eq. (11),

a"E1!BE

2, (16)

and

R"<11

!B<22

B@. (17)

The factor-based and characteristic-based models imply di!erent restrictionson a, and prior beliefs about a are centered on these restrictions, as will beexplained below. The models impose no restrictions on B, R, E

2, and<

22, so the


prior distributions for these parameters are noninformative. The prior distribu-tion for R is speci"ed as inverted Wishart,

R~1&=(H~1, l), (18)

with degrees of freedom l"15, so that the prior contains only about as muchinformation as a sample of 15 observations (&&' is read &is distributed as'). Fromthe properties of the inverted Wishart distribution (e.g., Anderson, 1984), theprior expectation of R equals H/(l!m!1). We specify H"s2(l!m!1)I

m,

so that E(R)"s2Im. Following an &empirical Bayes' approach, the value of s2 is

set equal to the average of the diagonal elements of the sample estimate of R. Thejoint prior distribution for the remaining parameters (B,E

2,<

22) is assumed to

be di!use and independent of a and R.The CAPM, in which k"1, and the three-factor FF model, in which k"3,

are treated in the same manner. Under the CAPM, the payo!s on the twononmarket positions SMB and HML are simply included in r

1,t, which then has

n!1 elements, whereas those payo!s are included in r2,t

under the FF model,so r

1,tthen has only n!3 elements. The factor-based pricing restriction in Eq.

(12) is equivalent to a"0. To allow for mispricing uncertainty, the priordistribution for a is speci"ed as a normal distribution,

aDR&NA0, p2aA1

s2RBB. (19)

The unconditional prior variance of each element of a, p2a , re#ects the investor'sprior degree of mispricing uncertainty. When pa"0, the investor believesdogmatically in the model, and mispricing is ruled out completely. Whenpa"R, the investor regards the model as useless, since mispricing is com-pletely unrestricted. PaH stor and Stambaugh (1999) introduce this measure ofmispricing uncertainty.

Observe that the conditional prior covariance matrix of a is proportional toR. This speci"cation is motivated by the recognition that there exist portfolioswith high Sharpe ratios if the elements of a are large when the elements of R aresmall. When w is unrestricted (no margin requirements), the maximum squaredSharpe ratio from the universe containing the n positions is given by

S2n"S2

k#a@R~1a, (20)

where S2kis the maximum squared Sharpe ratio from the k benchmark positions.

The importance of bounding a@R~1a as n grows large arises in the arbitrage-pricing literature (e.g., Ingersoll, 1984). For "nite values of n, MacKinlay (1995)demonstrates empirically the importance of a positive association between a andR in reducing the value of S

n. With no mispricing in the factor-based model,

S2n"S2

k. The prior in (19) re#ects a belief that, even with some mispricing, the


risk-based nature of the model makes large values of S2n!S2

kless likely than

under non-risk-based alternatives. For a given pa , large values of S2n!S2

kreceive

lower prior probabilities under (19) than when each element of a has standarddeviation pa but is distributed independently of all other parameters. PaH stor andStambaugh (1999) introduce the same type of prior for a single element of a, andPaH stor (2000) applies the multivariate version in (19) to portfolio-choice prob-lems. MacKinlay (1995) develops a functional relation between a and R whenthe mispricing is entirely risk based, i.e., when a is proportional to sensitivities toa risk factor not included in r

2,t. MacKinlay and PaH stor (2000) use maximum

likelihood procedures to investigate the implications of this relation for estima-tion of expected returns and for portfolio selection.

The investment universe contains n risky positions, and in this section the lastthree are again the FF benchmarks (as in Section 2). Recall that exact FF pricingimplies the restriction on E

1in Eq. (12). The DT model yields a simple

alternative restriction on E1

if the non-benchmark positions are constructed ina particular fashion. For this purpose, we construct a set of nine non-benchmarkpositions as beta spreads within categories of size and book-to-market. Fora given joint classi"cation of size and book-to-market, the payo! on eachspread, per $1 of position size, is produced by going long $1 of the portfolio withthe lowest HML beta and short $1 of the portfolio with the highest HML beta.For example, SH(l-h) denotes the spread position that is long portfolio SHl andshort portfolio SHh. (These portfolios are described in the previous section.) Thekey to such spreads, proposed by Daniel and Titman, is that the characteristicsof the long and short positions are (approximately) the same. Thus, we representthe DT model by the restriction E

1"0, which is equivalent to

a"!BE2. (21)

To allow for mispricing uncertainty, we assume

aDR&N(!BE2, p2aIm). (22)

In contrast to the risk-based speci"cation in (19), R does not appear in (22). Thedi!erence S2

n!S2

kis positive even when (21) holds exactly (using (20) and

assuming BE2O0), and this non-risk-based model provides no reason to limit

this di!erence if (21) is violated (pa'0). Thus, a and R are made independent inthe prior for the DT model. The three elements of E

2are left unrestricted, with

the rationale that at least three unknown parameters would relate those valuesto size and book-to-market (e.g., an intercept and two slopes).

The nine non-benchmark positions are formed as spreads between portfoliosselected from the set produced by a three-way sort on size, book-to-market, andHML beta. As demonstrated earlier, margin requirements can induce an inves-tor to select positions that are extreme in terms of their betas or characteristics.Grouping stocks into 27 portfolios, following Davis et al. (2000), essentiallymaximizes the di!erences among the extreme portfolios, given the limitations of


the data (avoiding portfolios containing very few or no stocks during the earlyyears of the sample). The construction of the investment universe is likely to beimportant to our analysis, especially in the presence of margin requirements,and in Section 4 we explore the robustness of our results to the choice ofinvestment universe.

In each of the nine non-benchmark spreads, the long and short legs areassumed to be matched in terms of their size and book-to-market character-istics. Davis et al. (2000) observe that, despite this objective of the sortingprocedure, the short leg of each spread exhibits a tendency to contain stockswith slightly higher book-to-market ratios than the stocks in the long leg. Thereason is that there remains some positive correlation between book-to-marketratios and HML betas within the nine portfolios formed by a two-way sort onsize and book-to-market. (Recall that the spreads are long low HML betas andshort high HML betas.) Under a DT model in which expected return isincreasing in the book-to-market characteristic, that mismatch implies that theelements of E

1are slightly less than zero, as opposed to the equality implying

(21). The elements of BE2

are likely to be negative (the HML betas are negativeby construction), so under the DT model a slight book-to-market mismatch inthe spreads would require that the prior mean of a ("E

1!BE

2) be somewhat

closer to the zero vector than is !BE2, the prior mean used in (22) correspond-

ing to E1"0. In other words, accounting for the mismatches would move the

prior for the DT model closer to that of the FF model. As detailed below, we "ndthat the di!erences in portfolio choices implied by the DT and FF models areoften made small or nonexistent by incorporating mispricing uncertainty andmargin requirements. Accounting for slight characteristic mismatches in thespreads would tend to further reduce those di!erences.

Black and Litterman (1991, 1992) present a portfolio-selection framework inwhich CAPM-implied expected returns are combined with subjective beliefsabout violations of that model. In that sense, their approach also centers beliefsabout expected returns on a pricing model, and the divergence of the investor'sportfolio from the market benchmark depends on the strengths of the investor'sbeliefs about model violations. Black and Litterman de"ne no role for sampleinformation about expected returns; beliefs about model violations are assignedmeans and variances a priori. In contrast, our approach relies on the strength ofthe sample's information about violations of the pricing model, as well as theinvestor's prior con"dence in the model, to determine how the investor's port-folio departs from the strict implication of the model.

3.2. Results

Table 2 reports optimal allocations per $100 of wealth when prior beliefs arecentered on each of the three pricing models, with varying degrees of mispricinguncertainty (pa ). As mispricing uncertainty increases, optimal allocations must


Tab

le2

Optim

alal

loca

tions

unde

rpar

amet

erun

cert

aint

y

The

table

report

sop

tim

alal

loca

tions(p

ositio

nsize

s)per

$100

ofw

ealth

fora

mea

n-va

rian

ce-o

ptim

izin

gin

vest

orw

ith

rela

tive

risk

aver

sion

equa

lto

2.83

.The

max

imum

valu

eofrisk

ypo

sitions

that

can

be

esta

blis

hed

per

dolla

rofw

ealth

isden

oted

byc.

Misprici

ngunc

erta

inty

,den

oted

byp a,i

sth

eprior

stan

dard

dev

iation

ofth

edi!

eren

cebet

wee

nea

chposition'

san

nua

lized

expec

ted

pay

o!an

dth

eprici

ngm

odel's

exac

tim

plic

atio

n,ex

pres

sed

asa

per

cent

age

ofin

itia

lpos

itio

nsize

.Allo

cation

sar

ere

port

edfo

rpr

ior

belief

sce

nter

edon

thre

edi!er

entprici

ng

mode

ls:t

he

thre

e-fa

ctor

Fam

a}Fre

nch

mode

l(F

F),

the

Dan

iel}

Titm

anch

arac

terist

ic-b

ased

mode

l(D

T),

and

the

CA

PM

(CM

).O

ptim

izat

ion

isba

sed

on

the

pred

ictive

distr

ibution,o

bta

ined

by

updat

ing

the

prio

rbel

iefs

usin

gm

onth

lyre

turn

sfrom

July

1963

thro

ugh

Dec

embe

r19

97.T

heFam

a}Fre

nch

ben

chm

ark

position

sar

ede

not

edby

SMB,

HM

L,a

nd

MK

T.T

heoth

ernin

epositions

aresp

read

sbet

wee

nva

lue-

wei

ghte

dpo

rtfo

liosco

nst

ruct

edby

ath

ree-

way

sort

and

iden

ti"ed

by

aco

mbin

atio

nofth

ree

lett

ers,

des

ignat

ing

incr

easing

valu

esof

size

(S,M

,B),

book-t

o-m

arket

(L,M

,H),

and

HM

Lbet

a(l,

m,h

);th

epor

tfolio

son

each

side

ofa

spre

addi!

erin

HM

Lbet

a(h

igh

vs.l

ow)bu

tar

em

atch

edin

size

and

book

-to-

mar

ket.

The&cas

h'ro

wre

port

sth

eov

eral

lam

oun

tin

vest

edin

the

risk

less

asse

t,w

hic

hin

cludes

theam

ount

sim

plie

dby

the

position

MK

T(w

hic

hsh

orts

the

risk

less

asse

t).N

egat

iveca

sham

ounts

repr

esen

tborr

owin

g.A

lso

repor

ted

are

the

annu

aliz

edm

ean

and

stan

dard

dev

iation

(std

)of

the

port

folio'

sre

turn

with

resp

ect

toth

egi

ven

pred

ictive

distr

ibution.

p a"0

p a"1%

p a"2%

p a"R

(all)

DT

FF

CM

DT

FF

CM

DT

FF

CM

c"2

SL(l-

h)

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

SM(l-

h)0.

00.

00.

00.

00.

00.

00.

00.

00.

00.

0SH

(l-h)

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

ML(l-

h)

0.0

0.0

0.0

0.0

0.0

!12

.70.

00.

0!

12.3

!11

.9M

M(l-

h)0.

00.

00.

00.

00.

00.

00.

00.

00.

00.

0M

H(l-h

)0.

00.

00.

00.

00.

00.

00.

00.

00.

00.

0BL(l-

h)0.

00.

00.

00.

00.

00.

00.

00.

00.

00.

0BM

(l-h)

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

BH

(l-h)

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

SMB

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

HM

L62

.162

.20.

062

.162

.235

.162

.362

.241

.851

.0M

KT

75.8

75.6

100.

075

.875

.610

4.3

75.4

75.6

91.8

74.1

Cas

h24

.224

.40.

024

.224

.4!

4.3

24.6

24.4

8.2

25.9

Mea

n8.

08.

06.

38.

08.

06.

88.

08.

07.

17.

9St

d10

.510

.515

.010

.510

.514

.610

.510

.512

.710

.3


c"5

SL(l-

h)

0.0

0.0

0.0

0.0

0.0

10.0

0.0

0.0

3.0

0.0

SM(l-

h)0.

00.

00.

00.

00.

03.

40.

00.

00.

00.

0SH

(l-h)

0.0

0.0

0.0

0.0

0.0

28.6

0.0

0.0

13.3

0.0

ML(l-

h)

0.0

0.0

0.0

0.0

0.0

!33

.80.

0!

8.2

!46

.1!

50.0

MM

(l-h)

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

MH

(l-h

)0.

00.

00.

00.

00.

00.

40.

00.

00.

00.

0BL(l-

h)0.

00.

00.

00.

00.

017

.00.

00.

00.

00.

0BM

(l-h)

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

BH

(l-h)

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

SMB

0.0

0.0

0.0

0.0

0.0

15.1

0.0

0.0

4.4

0.0

HM

L18

9.2

189.

30.

018

9.2

189.

382

.318

9.5

181.

311

9.4

141.

2M

KT

121.

612

1.4

100.

012

1.6

121.

411

9.0

121.

112

0.9

127.

711

7.6

Cas

h!

21.6

!21

.40.

0!

21.6

!21

.4!

19.0

!21

.1!

20.9

!27

.7!

17.6

Mea

n!

17.4

17.4

6.3

17.4

17.4

8.1

17.4

17.3

12.1

17.2

Std

19.4

19.4

15.0

19.4

19.4

16.4

19.4

19.2

18.5

18.6

c"10

SL(l-

h)

0.0

0.0

0.0

0.0

3.9

15.2

4.2

8.6

22.3

15.0

SM

(l-h)

0.0

0.0

0.0

0.0

0.0

18.4

0.0

0.0

5.8

0.0

SH(l-

h)41

.30.

00.

050

.513

.143

.059

.129

.164

.450

.4M

L(l-

h)

0.0

0.0

0.0

0.0

!24

.3!

45.2

!4.

5!

54.3

!76

.7!

91.9

MM

(l-h)

0.0

0.0

0.0

0.0

0.0

3.7

0.0

0.0

0.0

0.0

MH

(l-h

)0.

00.

00.

00.

00.

07.

40.

00.

00.

30.

0BL(l-

h)57

.60.

00.

047

.82.

034

.631

.54.

337

.28.

3BM

(l-h)

0.0

0.0

0.0

0.0

0.0

!8.

30.

00.

00.

00.

0BH

(l-h)

0.0

0.0

0.0

0.0

0.0

!5.

00.

00.

00.

00.

0SM

B15

.663

.20.

016

.151

.722

.616

.537

.633

.720

.3H

ML

308.

235

2.5

0.0

307.

632

2.3

108.

230

5.8

285.

218

7.3

236.

2M

KT

154.

916

8.2

100.

015

6.0

165.

312

4.9

156.

916

1.9

144.

615

5.5

Cas

h!

54.9

!68

.20.

0!

56.0

!65

.3!

24.9

!56

.9!

61.9

!44

.6!

55.5

Mea

n26

.030

.46.

326

.028

.78.

726

.027

.515

.726

.9St

d25

.232

.815

.025

.330

.817

.625

.628

.621

.926

.3


Tab

le2

(con

tinu

ed)

p a"0

p a"1%

p a"2%

p a"R

(all)

DT

FF

CM

DT

FF

CM

DT

FF

CM

c"R

SL(l-

h)

17.0

0.0

0.0

37.2

16.5

15.2

55.6

37.5

35.8

63.8

SM(l-

h)60

.40.

00.

076

.820

.018

.481

.045

.443

.278

.5SH

(l-h)

178.

60.

00.

017

7.9

46.5

43.0

181.

210

5.4

101.

018

1.6

ML(l-

h)

62.3

0.0

0.0

!26

.8!

48.0

!45

.2!

107.

0!

109.

0!

106.

1!

187.

2M

M(l-

h)!

29.1

0.0

0.0

!1.

34.

13.

714

.69.

38.

815

.6M

H(l-h

)35

.30.

00.

032

.48.

07.

433

.418

.217

.330

.4BL(l-

h)17

9.9

0.0

0.0

179.

537

.834

.616

8.6

85.7

81.2

146.

9BM

(l-h)

3.5

0.0

0.0

!3.

4!

8.9

!8.

3!

12.6

!20

.2!

19.5

!35

.1BH

(l-h)

!0.

80.

00.

0!

2.5

!5.

3!

5.0

!7.

4!

12.0

!11

.8!

20.6

SMB

81.6

63.2

0.0

89.7

71.1

22.6

94.8

81.7

53.1

94.4

HM

L61

3.0

352.

50.

057

9.6

378.

510

8.2

537.

941

3.8

253.

845

5.6

MK

T20

9.4

168.

210

0.0

209.

317

8.3

124.

921

0.1

192.

316

0.1

208.

9C

ash

!10

9.4

!68

.20.

0!

109.

3!

78.3

!24

.9!

110.

1!

92.3

!60

.1!

108.

9

Mea

n46

.830

.46.

345

.431

.58.

745

.835

.919

.146

.0St

d40

.732

.815

.040

.133

.317

.640

.235

.626

.040

.3


approach those based on the sample moments of returns, whatever the pricingmodel. Our aim is to explore the extent to which this behavior occurs atinteresting levels of pa . Results in Table 2 are reported for pa"1% andpa"2% (per annum) as well as the limiting values pa"0 (exact pricing) andpa"R (no use of a pricing model).

Given the true values of B and E2, pa represents the prior volatility of the

errors in E1

obtained from the pricing model. In the absence of a pricing model,one might construct E

1by computing sample means. The volatilities of the

errors in the elements of E1

obtained in that manner provide one benchmark forassessing the magnitude of pa . Of course, such volatilities depend on theinvestment universe. For the universe of spread positions analyzed in thissection, the elements of r

1,thave annualized standard deviations that average

about 10%. Thus, the errors in E1

speci"ed as sample means on average havea 2% volatility in samples of about 25 years and a 1% volatility in samples ofabout 100 years. Sections 2 and 4 analyze an investment universe composedlargely of non-spread positions, whose annualized standard deviations averageabout 20%. The sample sizes corresponding to error volatilities of 2% and 1%are then about 100 and 400 years. When viewed in this context, values ofpa equal to 1%, or even 2%, seem to represent modest degrees of priormispricing uncertainty for these investment universes.

The "rst three columns of Table 2, with pa"0, display the allocationscorresponding to dogmatic beliefs in each of the three pricing models. As inTable 1, the row labeled &cash' includes the overall (net) amount in (5). Recallthat a value for risk aversion of A"2.83 implies that all of the investor's wealthis allocated to the market portfolio when that is the only risky positionavailable. With a dogmatic belief in the CAPM (shortened to &CM' in the table),the market portfolio has the maximum Sharpe ratio, so MKT is again the onlyrisky position with a nonzero allocation for such an investor, and that allocationequals 100 for each value of c considered.

Table 2 presents portfolio allocations for the di!erent levels of investmentconstraints analyzed previously in Table 1 (c"2,5,10,R). The most strikingresult is that, when c"2, the optimal allocations under the FF model arevirtually identical to those under the DT model. This is true even for dogmaticbeliefs in each model (pa"0) and remains true for the nonzero values of pa . Inother words, it makes no di!erence whether a mean-variance investor who mustallocate funds across the 12 risky positions considered here has strong beliefs inthe three-factor model or strong beliefs in the characteristic-based model. Whenconstrained by a 50% margin requirement, the optimal portfolio is the sameunder either model. This result seems noteworthy, since the nine beta-spreadpositions included here are constructed to exploit di!erences between the FFand DT models.

For the three "nite values of pa , the only risky positions receiving non-zeroallocations under the DT and FF models when c"2 are HML and MKT, with


position sizes of 62 and 76, and 24 is placed in cash. The fact that the HML andMKT allocations are virtually identical under the DT and FF models butdi!erent from the allocations under the CAPM can be explained by the predic-tive means of the factors. With pa"0, the predictive mean of MKT is around6.3% per annum across all three models. The predictive mean of HML, how-ever, is 5.1% under the DT and FF models as compared to !1.4% under theCAPM, which prices HML by its MKT beta. With c"2, no funds are allocatedto SMB or any of the nine beta-spread positions; the intuition for this outcomeis deferred to the discussion of risk aversion in the next subsection. The returnon the optimal portfolio has a predictive standard deviation of 10.5% (an-nualized), substantially lower than those of the optimal portfolios under dog-matic beliefs in each model in the unconstrained case (40.7% for the DT modeland 32.8% for the FF model). Note, however, that the investment constraints donot preclude higher standard deviations. For example, a simple two-to-oneleveraging of MKT, permitted under c"2, produces a standard deviation ofabout 30%.

To a large extent, the above observations for c"2 also apply when c"5(corresponding to a 20% margin requirement). The allocations are again vir-tually identical under the FF and DT models when pa"0 and pa"1%. Aswith c"2, HML and MKT are the only risky positions receiving nonzeroallocations, although the HML position is now larger than that of MKT (189versus 121). The optimal portfolios under the DT and FF models divergeslightly at pa"2% only in that ML(l-h) receives a short position of !8 underthe FF model. When pa"R, the optimal allocation again includes a shortposition in ML(l-h).

For margin requirements in the range of 20}50%, the above results reveallittle or no role for mispricing uncertainty (pa) in determining optimal alloca-tions among the opportunities considered. As investment constraints are re-laxed, the degree of mispricing uncertainty exerts more in#uence. When c"R,the unconstrained case, the nine beta spreads receive zero allocations underdogmatic beliefs (pa"0) in the CAPM or FF model, as they must. In contrast,some large nonzero positions in those spreads arise under dogmatic beliefs in theDT model, such as the positions of roughly 180 in both SH(l-h) and BL(l-h). Aspa increases, the allocations change, and the beta-spread positions receivesubstantial nonzero allocations under all three models. With pa"R, where thepricing models are not used at all, the optimal unconstrained portfolio hasa number of large long and short positions, such as 182 in SH(l-h), !187 inML(l-h), 456 in HML, and 209 in MKT. The optimal portfolio also calls forborrowing 109.

Because the payo!s on the 12 risky positions are correlated, large dif-ferences in position-by-position allocations need not produce economicallysigni"cant di!erences in the overall portfolio characteristics. To gauge theeconomic importance of such di!erences across pricing models, we compare


certainty-equivalent returns on the portfolios as follows. Let EH and <H denotethe predictive moments of r

tformed under a given pa and a given pricing model.

For a given c, we "rst compute the certainty-equivalent excess return of theallocation w

Othat is optimal under that predictive distribution,

CEO"w@

OEH!1

2Aw@

O<Hw

O. (23)

Then we compute the certainty-equivalent excess return of a suboptimal alloca-tion w

S,

CES"w@

SEH!1

2Aw@

S<Hw

S, (24)

where wS

is an allocation that is optimal for the same c and pa under thepredictive distribution from a di!erent pricing model. The di!erenceCE

O!CE

Sprovides an economic measure of the di!erence between the two

portfolios. It is the perceived certainty-equivalent loss to an investor witha given degree of belief in one pricing model who is forced to accept the portfolioselection of another investor with the same degree of belief in a di!erent pricingmodel, where both investors face the same constraints. Kandel and Stambaugh(1996) propose the approach wherein a single predictive distribution is used tocompute the certainty equivalents of both portfolios.

Fig. 1 displays the certainty-equivalent losses for an investor who believes inthe characteristic-based model with varying degrees of mispricing uncertainty.For each of four values of c, the "gure displays a plot of the certainty-equivalentloss versus pa . Losses are computed for portfolios from the FF model (solidlines) and from the CAPM (dashed lines). When c"R and pa"0, thecertainty-equivalent loss for the FF allocation is about 8% per annum, aneconomically large magnitude. With mispricing uncertainty, the FF loss dropsto about 5% at pa"1% and to less than 2% at pa"2%. Note that a loss of 2%for the FF model also occurs with no mispricing uncertainty but with a 10%margin requirement (c"10). Although a 2% loss is still economically signi"-cant, it is only one-fourth the size of the loss under the same sets of beliefs but noinvestment constraints. When c"2 and c"5, the certainty-equivalent lossesfor the FF portfolios plot as essentially a #at line at zero, since as observedpreviously, the optimal allocations in the DT and FF models are virtuallyidentical in those cases.

In all cases considered, larger certainty-equivalent losses are associated withthe CAPM portfolios. When c"R and pa"0, the loss for the CAPMportfolio is about 20%. Mispricing uncertainty again reduces that loss to about12% at pa"1% and 4.5% at pa"2%, but those values are considerably largerthan the corresponding losses for the FF model. Unlike the FF losses, theCAPM losses do not completely disappear at the lower values of c. For example,with c"5 and dogmatic beliefs in the models, the CAPM portfolio producesa loss of almost 9% for an investor who believes in the DT model, whereas theFF portfolio produces no loss for such an investor. Nevertheless, a combination


Fig. 1. Certainty-equivalent losses for other models' portfolios from the perspective ofa Daniel}Titman investor. The "gure displays the certainty-equivalent loss (in % per year) fora mean-variance-optimizing investor whose prior beliefs are centered on the Daniel}Titman charac-teristic-based model, with mispricing uncertainty pa , but who is forced to hold portfolios chosen byinvestors with the same degree of belief in either the Fama}French model (solid line) or the CAPM(dashed line). Investor risk aversion is set to A"2.83. The maximum value of risky positions thatcan be established per dollar of wealth is denoted by c. Mispricing uncertainty, denoted by pa , is theprior standard deviation of the di!erence between each position's annualized expected payo! andthe pricing model's exact implication, expressed as a percentage of initial position size.

of realistic investment constraints and modest mispricing uncertainty is stillsu$cient to reduce the losses for the CAPM portfolio to rather low levels. Withc"2 and pa"2%, the annualized loss for the CAPM allocation is only 33basis points.

Fig. 2 displays precisely the same analysis except that the certainty-equivalentlosses are computed for an investor who believes in the FF model instead of theDT model as in Fig. 1. The dashed lines still represent the losses for the CAPMportfolios, but the solid line now represents the losses associated with the DT


Fig. 2. Certainty-equivalent losses for other models' portfolios from the perspective ofa Fama}French investor. The "gure displays the certainty-equivalent loss (in % per year) fora mean-variance-optimizing investor whose prior beliefs are centered on the Fama}French factor-based model, with mispricing uncertainty pa , but who is forced to hold portfolios chosen byinvestors with the same degree of belief in either the Daniel}Titman model (solid line) or the CAPM(dashed line). Investor risk aversion is set to A"2.83. The maximum value of risky positions thatcan be established per dollar of wealth is denoted by c. Mispricing uncertainty, denoted by pa , is theprior standard deviation of the di!erence between each position's annualized expected payo! andthe pricing model's exact implication, expressed as a percentage of initial position size.

portfolios. The losses for the DT portfolios perceived by the investor with FFbeliefs (Fig. 2) are, in all cases, very close to the losses for the FF portfoliosperceived by the investor with DT beliefs (Fig. 1). The magnitudes for some ofthe CAPM losses are somewhat di!erent, most notably at c"R and pa"0,where the loss in Fig. 2 is 12% versus 20% in Fig. 1. This ordering is perhaps notsurprising given that the CAPM and the FF model are both factor-based andhave the MKT factor in common. In general, however, the observations madefor Fig. 1 are unchanged when based on Fig. 2. Again, in terms of investment


implications, the FF and DT models are signi"cantly closer to each other than iseither model to the CAPM.

We also compare portfolios by computing the correlation between theirreturns. As before, let <H denote the predictive covariance matrix of r

tformed

under a given pa and a given pricing model. The predictive correlation betweenthe return on the optimal allocation w

Oand suboptimal allocation w

Sis given by

oOS

"

w@O<Hw

SJw@

O<H

wOJw@

S<Hw

S, (25)

where wS

is again an allocation that is optimal for the same c and pa buta di!erent pricing model. These correlations for the DT predictive distributionare displayed in Fig. 3, which follows precisely the same format used to displaythe certainty-equivalent losses in Fig. 1. (As before, the results based on the FFpredictive distribution are close to those based on the DT predictive distributionand are omitted in the interest of space.) The correlations between the FF andDT portfolios plot as #at lines at 1.0 for c"2 and c"5. For c"10, thecorrelations between the FF and DT allocations are 0.95 or higher. Thus, in thepresence of even weak investment constraints, the FF and DT models implyhighly correlated optimal portfolios. The correlations between either of thoseportfolios and the CAPM portfolio are substantially lower, especially in theabsence of mispricing uncertainty. For c"5 and c"10, the correlations be-tween the CAPM portfolio and the FF or DT portfolios are 0.6 or less. Whenc"R and pa"0, the correlation between the FF and DT portfolios is about0.8, while the correlation of either of those portfolios with the CAPM portfolio isin the vicinity of 0.4. With mispricing uncertainty of pa"2%, those samecorrelations are both around 0.95.

3.3. Risk aversion

The optimal portfolios discussed so far are computed for an investor witha risk aversion coe$cient of A"2.83. Recall that this risk aversion implies thatall of the investor's wealth is allocated to the market portfolio when that is theonly risky position available. Di!erent values of A imply di!erent optimalportfolios and hence di!erent certainty-equivalent losses for allocations fromalternative models. For di!erent levels of risk aversion, Fig. 4 displays certain-ty-equivalent losses analogous to those plotted as solid lines in Fig. 1, whereA"2.83. That is, for an investor with a given degree of belief in the DT model,we plot that investor's certainty-equivalent loss when forced to hold the port-folio of an investor with the same degree of belief in the FF model. The fourgraphs in Fig. 4 display results for risk aversion levels of 1, 5, 10, and 15. Eachgraph plots the certainty-equivalent loss versus mispricing uncertainty (pa) forfour levels of investment constraints, c"2, 5, 10, and R.


Fig. 3. Correlations with other models' portfolios from the perspective of a Daniel}Titman investor.The "gure displays the correlation between the portfolio of a mean-variance-optimizing investorwhose prior beliefs are centered on the Daniel}Titman characteristic-based model, with mispricinguncertainty pa , and the portfolios chosen by investors with the same degree of belief in either theFama}French model (solid line) or the CAPM (dashed line). Investor risk aversion is set to A"2.83.The maximum value of risky positions that can be established per dollar of wealth is denoted by c.Mispricing uncertainty, denoted by pa, is the prior standard deviation of the di!erence between eachposition's annualized expected payo! and the pricing model's exact implication, expressed asa percentage of initial position size.

With no investment constraints (c"R), a large certainty-equivalent loss isperceived by an investor with dogmatic DT beliefs (pa"0) who is forced toaccept the portfolio of a dogmatic FF investor. When risk aversion A"2.83, theannualized loss is about 8%, as observed previously in Fig. 1. Fig. 4 reveals thatthe loss is decreasing in A: the loss is roughly 24% for A"1 but less than 2% forA"15. (Note that the vertical scale is di!erent in the A"1 graph.) As A in-creases, the optimal portfolios from both models involve larger cash positions,


Fig. 4. E!ects of risk aversion on the certainty-equivalent losses for the Fama}French model'sportfolios from the perspective of a Daniel}Titman investor. The "gure displays the certainty-equivalent loss (in % per year) for a mean-variance-optimizing investor whose prior beliefs arecentered on the Daniel}Titman characteristic-based model, with mispricing uncertainty pa , but whois forced to hold portfolios chosen by investors with the same degree of belief in the Fama}Frenchmodel. Investor risk aversion is denoted by A. Mispricing uncertainty, denoted by pa , is the priorstandard deviation of the di!erence between each position's annualized expected payo! and thepricing model's exact implication, expressed as a percentage of initial position size. The maximumvalue of risky positions that can be established per dollar of wealth is denoted by c, and each plotdisplays results for c"2 (solid), c"5 (dash), c"10 (dash-dot), and c"R (dots).

and this e!ect reduces the certainty-equivalent loss from holding the alternativeportfolio.

In Fig. 4, a 50% margin requirement (c"2) eliminates the loss for the "rstthree levels of risk aversion (A"1, 5, and 10), duplicating the result obtainedearlier with A"2.83 in Fig. 1. As before, the portfolio choices under theconstraint are virtually identical for investors with dogmatic beliefs in the FFand DT models. As in Table 2, those optimal portfolios call for zero allocationsto SMB and the nine beta-spread portfolios. (In the absence of constraints, a DT


investor chooses substantial nonzero allocations to those positions, and a FFinvestor chooses a positive allocation to SMB.) When A"1, Fig. 4 revealsa zero certainty-equivalent loss with only a 10% margin, whereas the corre-sponding loss with A"2.83 is about 2%. Dogmatic DT and FF investors withA"1 continue to allocate nothing to SMB and the nine beta spreads. WithA"2.83, both investors choose positive (but di!erent) allocations to SMB, andthe DT investor allocates funds to some of the beta spreads as well (Table 2). Thefact that these positions receive zero allocations under the lower A can beunderstood by considering their expected payo!s. Under dogmatic DT beliefs,the nine beta spreads have zero expected payo!s (E

1"0). The posterior mean

of SMB is positive but lower than that of HML and the highest-mean payo!,MKT. The positions with low means o!er diversi"cation, but capturing thatdiversi"cation requires allocating capital to those positions. With margin re-quirements and su$ciently low risk aversion, that required capital is better usedin positions with higher expected payo!s.

With investment constraints, the certainty-equivalent loss can increase withrisk aversion. For example, with c"2 and pa"0, the loss for the FF portfolio iszero for A"1 but about 45 basis points for A"15. Although the optimalportfolios under both models indeed involve larger cash positions for A"15,the constrained optimal portfolios also include some positions that are notincluded for A"1. In particular, the optimal portfolios from both modelsinclude only HML and MKT when A"1, as observed above. In contrast, forA"15, the optimal portfolio from the FF model also includes a $12 position inSMB (per $100 invested), while the optimal portfolio from the DT modelincludes nonzero allocations (between 10 cents and $12) to all 11 risky positions.With this higher level of risk aversion, the DT investor "nds the zero-meanHML-beta spreads attractive for diversi"cation reasons, whereas the dogmaticFF investor allocates nothing to those positions, whether constrained or not.

Fig. 5 reports losses for an investor who believes in the DT model but is forcedto hold the portfolio of an investor with CAPM beliefs. This "gure con"rms theobservations made for Fig. 4. Certainty-equivalent losses decrease as riskaversion increases as well as when investment constraints and mispricing uncer-tainty are introduced. Note that the scales for the vertical axes in Fig. 5 aredi!erent from those in Fig. 4 since, to an investor with DT beliefs, the losses fromholding a CAPM investor's portfolio are larger than those from holding theportfolio of an FF investor. This result is observed for all levels of risk aversionand is consistent with the evidence for A"2.83 presented in the earlier "gures.With A"1, however, the DT portfolio is quite close to the CAPM portfolio,with the latter producing less than a 50 basis point loss for all values of pa . Forrisk aversion so low, the highest-mean position MKT constitutes the bulkof each portfolio's allocation. In general, though, the FF and DT modelsare substantially closer to each other than to the CAPM in terms of theirimplications for portfolio choice. With either realistic investment constraints or


Fig. 5. E!ects of risk aversion on the certainty-equivalent losses for the CAPM's portfolios from theperspective of a Daniel}Titman investor. The "gure displays the certainty-equivalent loss (in % peryear) for a mean-variance-optimizing investor whose prior beliefs are centered on theDaniel}Titman characteristic-based model, with mispricing uncertainty pa , but who is forced tohold portfolios chosen by investors with the same degree of belief in the CAPM. Investor riskaversion is denoted by A. Mispricing uncertainty, denoted by pa , is the prior standard deviation ofthe di!erence between each position's annualized expected payo! and the pricing model's exactimplication, expressed as a percentage of initial position size. The maximum value of risky positionsthat can be established per dollar of wealth is denoted by c, and each plot displays results for c"2(solid), c"5 (dash), c"10 (dash-dot), and c"R (dots).

modest mispricing uncertainty, the portfolio implications from the FF and DTmodels are very similar.

4. An alternative characteristic-based model

The representation of the characteristic-based model in the previous sectionrequires positions to be constructed as spreads between assets with matched


characteristics. This section proposes an alternative representation that can, inprinciple, be applied to any set of equity positions. At the same time, thisalternative model makes stronger assumptions about the relation betweenexpected returns and characteristics. Our main goal in pursuing this secondapproach is to explore the robustness of the empirical results in the previoussection to changes in the speci"cation of the investment universe.

In this second characteristic-based model (&CB2'), the expected excess returnon a (positive-cost) equity portfolio is assumed to be a linear function of (known)characteristics. Let C denote an n]¸ matrix in which each of columns2 through ¸ contains the values of a characteristic, such as book-to-market. Inthe empirical application presented here, the two characteristics are size andbook-to-market (¸"3). Then CB2 can be represented by the restriction

E"Cc, (26)

where c is an ¸]1 vector. For a position in which asset Siis risky, such as SMB

and HML, the corresponding element in the "rst column of C is equal to zero.For the other positions, the "rst column of C contains the value one, to includean intercept in the linear relation between expected excess returns and charac-teristics. Note that this model restricts all n elements of E, whereas the modelsexamined in the previous section imply restrictions only on the n!k elements ofE1, the expected payo!s on the non-benchmark positions.To allow for mispricing uncertainty in this characteristic-based model, the

prior distribution for E is speci"ed as

E D c&N(Cc, p2aIn). (27)

As in the previous models, pa represents the degree of mispricing uncertainty.The lower the value of pa , the higher the prior con"dence in the model's pricingrestriction in (26). As before, r

tis assumed to be normally distributed with mean

E and covariance matrix <. The prior distributions for < and c are speci"ed asnoninformative. Optimal portfolios are computed as in (7) using the predictivemean and covariance matrix, which are obtained through Gibbs sampling (asexplained in the Appendix).

The investment universe analyzed here consists of cash plus the same set ofn"11 positions used in the example in Section 2. The second and third columnsof C contain values of each position's size and book-to-market, which are thesame two characteristics used in the DT model analyzed in Section 3 (i.e., thesame as used to construct the characteristic-matched spreads). The character-istics in C are computed as follows. For each stock, size is the natural logarithmof total equity capitalization, and book-to-market is the ratio of book value tomarket value of common equity. In each month, the values of size and book-to-market for each three-way-sorted portfolio (described in Section 2) are com-puted as the value-weighted averages of those characteristics for the stocks inthe portfolio. This computation is also performed for the portfolios &S' and &B'


used to construct the FF factor SMB, and similar calculations are performed forthe portfolios used to construct the FF factors HML and MKT. In the positionsconstructed as spreads between equity portfolios (SMB and HML), the charac-teristic of one portfolio is subtracted from that of another. The formulation in(26) treats the characteristics as known constants, which are obtained from themonthly time series as follows. In each month, the characteristic of each positionis divided by the cross-sectional average of that characteristic across the n posi-tions. The time-series averages of these standardized series are then used as thevalues in C. Plots of the standardized characteristics exhibit no apparent trendand exhibit much less time-series variation than do plots of the raw character-istics.

Table 3 reports optimal allocations per $100 of wealth when prior beliefs arecentered on each of three pricing models, CB2, FF, and CAPM (CM). (Theformat is otherwise identical to Table 2.) Consider "rst the optimal allocationsobtained with a dogmatic belief in the FF model (pa"0). Recall that theseallocations are reported in Table 1 in a simpli"ed setting that does not accountfor estimation risk. Not surprisingly, the optimal portfolio in the unconstrainedcase (c"R) includes only the three FF positions as before. With a 10% marginrequirement (c"10), the optimal portfolio in Table 1 includes some smallnon-benchmark positions, whereas the portfolio in Table 3 is still una!ected bythe constraint. With a 50% margin requirement (c"2), the striking result seenearlier in Table 1 remains after accounting for estimation risk: an investor whobelieves dogmatically in the FF model invests only in non-benchmark positions.

In Table 3, the optimal portfolios from the characteristic-based and FFmodels for pa"0 and c"2 are no longer identical, as in Table 2. For example,SHl receives a zero allocation under the FF model but the largest allocation (of114) under CB2. The latter allocation is not surprising. The posterior distribu-tions for the elements of c in (26) are such that expected returns are decreasing insize and increasing in book-to-market. In particular, nearly 95% of the posteriormass of the size coe$cient lies below zero, while virtually all of the posteriormass of the book-to-market coe$cient lies above zero. Thus, SHl and SHh havethe highest expected returns. Although SHl has a lower expected return thanSHh (14.2% versus 14.8% per year), it also has a lower standard deviation(19.9% versus 21.4% per year) and somewhat smaller covariances with the otherpositions, so it is more attractive to a CB2 investor. Nevertheless, SHh receivesthe second largest allocation (of 28) under CB2. Although the compositions ofthe optimal portfolios under CB2 and FF di!er, neither contains allocations inthe FF benchmark positions. Also, the annual means and standard deviations ofthe two portfolios are quite similar: 18.1% and 21.0% for the CB2 portfolio, and17.1% and 21.6% for the FF portfolio. The two portfolios become even moresimilar when dogmatic beliefs in both models are relaxed. As in Table 2, relaxingthe investment constraints leads to more borrowing and hence to optimalportfolios with higher means and standard deviations. For c"R and pa"R,


Tab

le3

Optim

alal

loca

tions

unde

rpar

amet

erun

cert

aint

y(w

ith

the

alte

rnat

ive

char

acte

rist

ic-b

ased

mod

el)

The

tabl

ere

por

tsop

tim

alal

loca

tion

s(p

osition

size

s)per

$100

ofw

ealth

fora

mea

n-va

rian

ce-o

ptim

izin

gin

vest

orw

ith

rela

tive

risk

aver

sion

equa

lto

2.83

.The

max

imum

valu

eof

risk

ypositions

that

can

be

esta

blis

hed

per

dolla

rofw

ealth

isden

oted

byc.

Misprici

ng

unc

erta

inty

,den

ote

dby

p a,is

the

prior

stan

dard

dev

iation

of

the

di!

eren

cebet

wee

nea

chposition'

san

nua

lized

expec

ted

pay

o!an

dth

eprici

ngm

odel's

exac

tim

plic

atio

n,ex

pres

sed

asa

per

cent

age

ofin

itia

lposition

size

.Alloc

atio

ns

are

repo

rted

for

prio

rbel

iefs

cent

ered

onth

ree

di!er

entpr

icin

gm

ode

ls:t

he

thre

e-fa

ctor

Fam

a}Fre

nch

model

(FF

),th

eC

APM

(CM

),an

dth

eal

tern

ativ

ech

arac

terist

ic-b

ased

mod

el(C

B2)

.O

ptim

izat

ion

isbas

edon

the

pred

ictive

distr

ibution,o

bta

ined

by

updat

ing

the

prior

bel

iefs

usin

gm

ont

hly

retu

rnsfrom

July

1963

thro

ugh

Dec

embe

r19

97.T

heFam

a}Fre

nch

ben

chm

ark

pos

itio

nsar

ede

not

edby

SMB,

HM

L,a

nd

MK

T.T

herisk

yco

mpon

ents

oft

he"rs

tei

ghtposition

sar

ea

subs

etofv

alue

-wei

ghte

dpor

tfolio

sco

nstr

uct

edby

ath

ree-

way

sort

and

iden

ti"ed

by

aco

mbi

nat

ion

ofth

ree

letter

s,des

ignat

ing

incr

easing

valu

esofs

ize

(S,M

,B),

book

-to-m

arket

(L,M

,H),

and

HM

Lbe

ta(l,

m,h

).The&cas

h'ro

wre

port

sth

eov

eral

lam

ount

inve

sted

inth

erisk

less

asse

tan

din

clud

esth

eam

ount

sim

plie

dby

the

pos

itio

nsth

atsh

ortth

erisk

less

asse

t(a

llpo

sition

sex

cept

SMB

and

HM

L).

Neg

ativ

eca

sham

oun

tsre

pre

sent

borr

ow

ing.

Also

report

edar

eth

ean

nual

ized

mea

nan

dst

andar

ddev

iation

(std

)oft

hepo

rtfo

lio's

retu

rnw

ith

resp

ect

toth

egi

ven

pre

dict

ive

distr

ibution.

p a"0

p a"1%

p a"2%

p a"R

(all)

CB

2FF

CM

CB

2F

FC

MC

B2

FF

CM

c"2

SLl

!29

.0!

10.9

0.0

!26

.6!

35.0

!14

.2!

25.3

!32

.8!

12.5

!25

.4SLh

!13

.10.

00.

0!

17.0

0.0

!31

.7!

20.8

!11

.0!

36.7

!22

.4SH

l11

3.5

0.0

0.0

125.

134

.167

.212

3.3

85.8

107.

112

1.6

SHh

28.4

77.5

0.0

1.0

65.6

6.7

0.0

27.8

2.3

0.0

BLl

0.0

!34

.30.

00.

00.

015

.10.

00.

00.

00.

0BLh

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

BH

l0.

06.

60.

00.

00.

011

.70.

00.

00.

50.

0BH

h16

.070

.60.

030

.353

.520

.030

.742

.740

.930

.6SM

B0.

00.

00.

00.

00.

00.

00.

00.

00.

00.

0H

ML

0.0

0.0

0.0

0.0

5.9

0.0

0.0

0.0

0.0

0.0

MK

T0.

00.

010

0.0

0.0

0.0

33.5

0.0

0.0

0.0

0.0

Cas

h!

15.8

!9.

60.

0!

12.8

!18

.1!

8.2

!7.

9!

12.5

!1.

7!

4.3

Mea

n18

.117

.16.

316

.516

.07.

515

.815

.410

.515

.5St

d21

.021

.615

.019

.220

.515

.718

.218

.716

.717

.6


Tab

le3

(con

tinu

ed)

p a"0

p a"1%

p a"2%

p a"R

(all)

CB

2FF

CM

CB

2FF

CM

CB

2F

FC

M

c"5

SLl

!44

.6!

0.6

0.0

!52

.5!

30.7

!14

.7!

62.4

!54

.5!

34.9

!62

.6SLh

!79

.30.

00.

0!

97.4

!39

.8!

40.0

!11

3.7

!89

.2!

84.1

!11

5.2

SHl

212.

30.

00.

024

4.5

108.

267

.426

3.1

202.

917

8.8

261.

6SH

h23

.963

.50.

06.

240

.7!

15.3

11.4

31.6

2.2

11.1

BLl

0.0

0.0

0.0

0.0

0.0

41.3

0.0

0.0

44.8

0.0

BLh

0.0

0.0

0.0

0.0

0.0

7.6

0.0

0.0

0.0

0.0

BH

l20

.433

.90.

07.

214

.2!

4.2

0.0

0.0

3.5

0.0

BH

h15

.141

.00.

040

.348

.90.

749

.454

.727

.349

.4SM

B0.

00.

00.

00.

00.

033

.70.

00.

00.

00.

0H

ML

52.2

180.

50.

025

.910

8.8

80.9

0.0

33.6

62.1

0.0

MK

T0.

00.

010

0.0

0.0

0.0

79.5

0.0

0.0

0.0

0.0

Cas

h!

47.8

!37

.80.

0!

48.3

!41

.5!

22.5

!47

.7!

45.5

!37

.8!

44.3

Mea

n28

.625

.56.

328

.024

.38.

428

.526

.115

.628

.6St

d27

.028

.015

.026

.526

.117

.126

.726

.122

.026

.2

c"10

SLl

!40

.80.

00.

0!

47.8

!23

.0!

15.5

!58

.5!

47.4

!35

.9!

69.5

SLh

!11

8.1

0.0

0.0

!13

8.7

!66

.2!

42.0

!15

6.9

!12

3.3

!97

.9!

170.

8SH

l27

8.9

0.0

0.0

296.

515

5.8

61.1

322.

926

5.8

170.

834

5.1

SH

h0.

00.

00.

00.

00.

0!

30.9

0.0

0.0

!28

.80.

0BLl

74.1

0.0

0.0

79.8

63.9

46.9

74.0

78.1

99.0

69.3

BLh

!27

.10.

00.

00.

012

.39.

50.

00.

017

.80.

0BH

l21

.20.

00.

00.

00.

0!

9.9

0.0

0.0

!7.

00.

0BH

h0.

00.

00.

00.

05.

5!

0.7

8.0

14.7

2.7

16.9

SMB

0.0

63.2

0.0

0.0

13.2

63.6

0.0

0.0

65.9

0.0

HM

L22

0.0

352.

50.

021

8.6

307.

110

4.8

189.

823

5.3

186.

516

4.2

MK

T0.

016

8.2

100.

00.

032

.810

6.0

0.0

0.0

35.3

0.0

Cas

h!

88.2

!68

.20.

0!

89.9

!81

.0!

24.5

!89

.4!

87.8

!56

.1!

91.0

Mea

n40

.630

.46.

340

.532

.58.

541

.437

.218

.342

.5St

d35

.232

.815

.035

.033

.317

.335

.534

.525

.136

.0


c"R

SLl

!45

.20.

00.

0!

54.2

!29

.0!

15.5

!63

.5!

53.4

!38

.5!

72.7

SLh

!14

2.8

0.0

0.0

!16

8.9

!81

.7!

42.0

!18

9.3

!15

0.5

!10

4.4

!20

4.7

SHl

256.

00.

00.

024

4.1

106.

661

.125

9.8

196.

315

1.9

272.

7SH

h!

106.

40.

00.

0!

190.

2!

77.4

!30

.9!

199.

1!

142.

6!

76.6

!20

0.4

BLl

163.

30.

00.

024

7.0

102.

146

.925

5.1

188.

111

6.5

261.

5BLh

!10

1.6

0.0

0.0

8.2

23.3

9.5

41.3

42.9

23.5

62.9

BH

l!

12.2

0.0

0.0

!22

.9!

23.4

!9.

9!

41.3

!43

.1!

24.6

!59

.4BH

h!

46.7

0.0

0.0

!24

.9!

4.5

!0.

7!

16.2

!8.

3!

1.6

!10

.3SM

B19

1.6

63.2

0.0

350.

219

5.7

63.6

379.

730

8.1

158.

039

8.2

HM

L49

0.5

352.

50.

059

6.6

444.

810

4.8

597.

052

7.0

260.

459

5.0

MK

T26

2.6

168.

210

0.0

190.

717

5.1

106.

018

1.3

183.

111

6.6

179.

5C

ash

!12

6.9

!68

.20.

0!

128.

8!

91.0

!24

.5!

128.

3!

112.

5!

62.6

!12

9.1

Mea

n49

.830

.46.

351

.734

.38.

553

.043

.519

.254

.7St

d42

.032

.815

.042

.734

.817

.343

.339

.226

.144

.0


the optimal portfolio has a mean of 54.7% and a standard deviation of 44.0%; itinvolves borrowing 129 and its largest position (of 595) is HML.

As in the previous section, the position-by-position analysis of the optimalallocations under the factor-based and characteristic-based models is comp-lemented by an analysis of certainty-equivalent losses. Fig. 6 is the equivalent ofFig. 1, except that the characteristic-based model is now CB2 (instead of DT)and the investment universe is di!erent (as explained earlier). With no invest-ment restrictions (c"R) and a dogmatic belief in each model (pa"0), thecertainty-equivalent loss for the FF portfolio is about 8% per year, quite similarto the corresponding loss in Fig. 1. As in Fig. 1, the loss for the CAPM portfoliois much higher, almost 21%. For pa"3%, the FF loss is only 17 basis points,whereas the CAPM loss is 2.2%. CB2 is also closer to the FF model than to theCAPM when investment constraints are present (c(R). When c"2 and pa is2% or more, both the CAPM and the FF model are very close to CB2 (the lossesare no more than 19 basis points).

Comparisons of CB2 and the CAPM under the predictive distribution fromthe FF model (an equivalent of Fig. 2, not reported to save space) lead toobservations very similar to those made for Fig. 6. In general, optimal portfoliosfrom the FF and CB2 models are quite similar when realistic investmentconstraints and modest mispricing uncertainty are incorporated. This result nodoubt re#ects an association between characteristics and expected returnspresent in the characteristic-based model as well as the FF model, as notedearlier, whereas that association is weaker under the CAPM due to the lowercorrelation between characteristics and market betas.

5. Incorporating model uncertainty

In Sections 3 and 4, optimal allocations are computed by combining data withprior beliefs centered at a particular pricing model. Such an approach isconsistent with a common practice of "rst choosing the &best' model, usingjudgment or model selection criteria, and then proceeding as if the selectedmodel were the only one relevant. If the investor entertains several modelsa priori, this practice does not account for &model uncertainty' in the selectionprocess. For example, an investor who is uncertain as to whether expectedreturns are modeled better as risk-related or characteristic-related can poten-tially bene"t from combining the implications of both models. It seems conceiv-able, given available empirical evidence, that one might be less than comfortablediscarding one of the two models entirely. For example, Davis et al. (2000) "ndthat a t-test rejects the null hypothesis that the FF model prices an equallyweighted combination of characteristic-matched HML-beta spreads in the1973}93 period, but the hypothesis cannot be rejected in the longer 1929}97period. At the same time, the point estimate underlying the latter result does


Fig. 6. Certainty-equivalent losses for other models' portfolios from the perspective of an investorwho believes in the alternative characteristic-based model. The "gure displays the certainty-equivalent loss (in % per year) for a mean-variance-optimizing investor whose prior beliefs arecentered on the alternative characteristic-based model, with mispricing uncertainty pa , but who isforced to hold portfolios chosen by investors with the same degree of belief in either theFama}French model (solid line) or the CAPM (dashed line). Investor risk aversion is set to A"2.83.The maximum value of risky positions that can be established per dollar of wealth is denoted by c.Mispricing uncertainty, denoted by pa, is the prior standard deviation of the di!erence between eachposition's annualized expected payo! and the pricing model's exact implication, expressed asa percentage of initial position size.

have its sign in the direction of a characteristic-based alternative. Moreover, theauthors report that the FF model, viewed as a null hypothesis, is formallyrejected within a characteristic-sorted universe based on an F-test. Fortunately,investors need not limit their attention to only one of the two models.

Suppose the investor considers a universe of J models M1,2, M

Jof expected

returns. Let p(rT`1

DMj, R) denote the predictive distribution of returns from

model j, and let P(MjDR) denote the posterior probability of model j. The


predictive distribution that accounts for model uncertainty is

p(rT`1

DR)"J+j/1

p(rT`1

DMj,R) P(M

jDR). (28)

Optimal allocations are again computed from (7), but E and < are no longerreplaced by EH

jand <H

j, the mean and covariance matrix of the predictive

distribution from model j, but rather by EHM

and <HM

, the moments of thepredictive distribution that accounts for model uncertainty. These moments arecomputed as follows (see Leamer, 1978):

EHM"

J+j/1

EHjP(M

jDR) (29)

<HM"

J+j/1

<HjP(M

jDR)#

J+j/1

(EHj!EH

M)(EH

j!EH

M)@P(M

jDR). (30)

The predictive mean is an average of the predictive means from the J models,weighted by the model probabilities. The predictive covariance matrix has twocomponents. The "rst is a weighted average of the predictive covariancematrices from the J models, and the second is the covariance matrix of thepredictive means across models. The posterior model probabilities P(M

jD R) are

computed as

P(MjDR)"

P(Mj)p(R DM

j)

+Jj/1

P(Mj)p(R D M

j), (31)

where P(Mj) denotes the prior probability assigned to model j before observing

the data, and p(R DMj) denotes the so-called marginal likelihood of model j. The

marginal likelihood is computed as

p(R D Mj)"Pp(h

jDM

j) p(R D h

j, M

j) dh

j, (32)

where hj

denotes the parameters, p(hjDM

j) denotes the prior distribution, and

p(RDhj, M

j) denotes the likelihood function, all from model j.

The calculation of posterior model probabilities is beyond the scope of thisstudy. Such a task would have to address several issues. First, recall that thepriors for parameters not involving a pricing restriction are speci"ed as nonin-formative or even di!use, which are in some sense completely noninformative.Di!use priors are improper in that they are not integrable over the parameterspace. With improper priors, posterior model probabilities can be computedonly in some special cases, as discussed in Kass and Raftery (1995). Second, the


marginal likelihood is often quite sensitive to the choice of a noninformativeprior distribution, more so than is the posterior distribution typically used inestimation. A prior distribution can be made essentially noninformative but stillproper by specifying a large prior variance, but the latter can generally assumea wide range of values. For example, doubling an already large variance keepsthe prior noninformative, hence having little e!ect on the posterior distribution,but the change can greatly a!ect the marginal likelihood. Therefore, a study thatcomputes marginal likelihoods should analyze their sensitivity to speci"cationsof noninformative priors. Finally, prior model probabilities should also bethoughtfully speci"ed. For example, suppose the universe of models includesone with strong theoretical motivation (e.g., the CAPM) as well as one that ispartially motivated by observing data that either overlap or are correlated withthe sample R. It might be reasonable to assign a higher prior probability to theformer model.

In Sections 3 and 4, we assume that an investor's prior beliefs center ona particular pricing model, say the DT model, and this assumption correspondsto assigning a posterior probability of one to that model in the context describedabove. In those sections, the e!ect of model uncertainty is assessed by comput-ing the loss to an investor who is forced to hold the allocation that would beoptimal if that investor assigned a probability of one to a di!erent model, say theFF model. Such a scenario measures the maximum e!ect of model uncertaintyin that two-model universe. The loss from accepting the FF allocation is smallerwhen the probability assigned to the DT model is less than one. Recall that eventhe maximum e!ect of model uncertainty is quite small with modest mispricinguncertainty and realistic investment constraints.

6. Conclusions

This study compares asset pricing models from the perspective of investorswho center their prior beliefs on the models and then update those beliefs withdata for the 1963}97 period. The pricing models considered include theSharpe}Lintner CAPM, the three-factor model of Fama and French (1993), andthe characteristic-based model of Daniel and Titman (1997). An investor whohas dogmatic beliefs in a given pricing model perceives a large certainty-equivalent loss if forced to hold the portfolio chosen by an investor with equallystrong beliefs in another model. The largest such losses occur when the secondportfolio is chosen by a CAPM believer, but the di!erences between theFama}French and Daniel}Titman models are also large when judged by thismetric.

The di!erences described above are reduced by a consideration of two issuesconfronting investors. First, even an investor who prefers a given model isunlikely to believe it to be completely accurate. The investor's prior beliefs more


likely include some degree of mispricing uncertainty. Second, most investors facemargin requirements to at least some degree. Both of these issues, especially thesecond, diminish the importance of di!erences among the pricing models froman investment perspective. In fact, we "nd that considering such issues canvirtually eliminate any di!erences between the Fama}French and Daniel}Tit-man models, even though these models re#ect fundamentally di!erent viewsabout the economic determinants of expected returns. It is noteworthy thatthese models lead to similar portfolio choices within investment universesconstructed to exploit di!erences between the models.

This study is not intended to assist investors in choosing one pricingmodel over another. One might instead view our results as questioning theeconomic importance of deliberating such a choice. Moreover, a rationalBayesian investor who is uncertain about which model to use will generally usethem all, weighted by posterior model probabilities. Computing such probabilit-ies is beyond the intended scope of this study but o!ers a direction for futureresearch.

Finally, the single-period mean-variance framework provides only one ofmany investment perspectives from which pricing models might be comparedempirically. For example, the di!erences between risk-based and characteristic-based models could be more important for investors who optimize multiperiodobjective functions. Alternatives to the i.i.d. stochastic setting, adopted here fortractability, could further enrich an empirical comparison of models froma multiperiod investment perspective. Such a perspective would be more consis-tent with that of the representative investor in Merton's (1973) intertemporalversion of a risk-based model. (An investor in the present study is not assumedto be representative, insofar as di!erent assumptions about behavior and objec-tives might be required to derive a particular pricing model.) Such issues presentadditional directions for future research.

Appendix

We provide here the methods for obtaining the "rst two moments, EH and<H,of the predictive distribution of returns, p(r

T`1DR), for each of the pricing models

in Sections 3 and 4. As shown below, these predictive moments can be computeddirectly from the "rst and second posterior moments of each model's para-meters.

A.1. Risk-based models

De"ne >"(r1,1

,2, r1,T

)@, X"(r2,1

,2, r2,T

)@, and Z"(nT

X), where nT

de-notes a ¹-vector of ones. Also de"ne the (k#1)]m matrix A"(a B)@, and leta"vec(A). For the ¹ observations t"1,2,¹, the regression model in Eq. (15)


can be written as

>"ZA#;, vec(;)&N(0, R?IT), (A.1)

where ;"(u1,2, u

T)@. The matrix R"(> X) contains the entire sample.

De"ne the statistics AK "(Z@Z)~1Z@>, a("vec(AK ), RK "(>!ZAK )@(>!ZAK )/¹,EK2"X@n

T/¹, and <K

22"(X!n

TEK @2)@(X!n

TEK @

2)/¹. (Recall that h denotes all of

the parameters in a given model.) The likelihood function can be factored as

p(RDh)"p(>Dh,X) p(XDh), (A.2)

where

p(>Dh,X)JDRD~T@2expM!12tr(>!ZA)@(>!ZA)R~1N

JDRD~T@2expG!¹

2trRK R~1!

1

2tr(A!AK )@Z@Z(A!AK )R~1H

JDRD~T@2expG!¹

2trRK R~1!

1

2(a!a( )@(R~1?Z@Z)(a!a( )H

(A.3)

and

p(XDh)JD<22

D~T@2expM!12tr(X!n

TE@

2)@(X!n

TE@

2)<~1

22N

JD<22

D~T@2expG!¹

2tr<K

22<~1

22!

¹

2tr(E

2!EK

2)(E

2!EK

2)@<~1

22 H.(A.4)

The joint prior distribution of all parameters is

p(h)"p(aDR) p(R) p(B) p(E2) p(<

22), (A.5)

where

p(aDR)JDRD~1@2expG!1

2a@A

p2as2

RB~1

aH, (A.6)

p(R)JDRD~(l`m`1)@2expM!12trHR~1N, (A.7)

p(B)J1, (A.8)

p(E2)J1, (A.9)

p(<22

)JD<22

D~(k`1)@2. (A.10)


The priors of B, E2, and <

22are di!use. The prior of R is inverted Wishart with

a small number of degrees of freedom, so that it is essentially noninformative.The prior on a given R is normal and centered at the pricing restriction. Notethat

a@Ap2as2

RB~1

a"a@(R~1?D)a, (A.11)

where D is a (k#1)](k#1) matrix whose (1, 1) element is s2/p2a and all otherelements are zero.

Combining the likelihood in Eqs. (A.2)}(A.4) with the prior in Eqs.(A.5)}(A.10) yields the posterior distribution of h:

p(hDR)Jp(RDh) p(h). (A.12)

Both the likelihood and the prior can be factored into two parts, one thatinvolves the regression parameters (a, R) and another that involves the bench-mark moments (E

2,<

22). As a result, the posterior distribution also splits into

two parts. The joint posterior of the regression parameters is

p(a, RDR)JDRD~(k`1)@2expM!12[a@(R~1?D)a#(a!a( )@

](R~1?Z@Z)(a!a( )]NDRD~(T`l`m~k`1)@2

]expM!12tr(H#¹RK )R~1N. (A.13)

Let F"D#Z@Z, Q"Z@(IT!ZF~1Z@)Z. Completing the square on a yields

p(a, RDR)JDRD~(k`1)@2expM!12(a!a8 )@(R~1?F)(a!a8 )N

]DRD~(T`l`m~k`1)@2expM!12tr(H#¹RK #AK @QAK )R~1N, (A.14)

where

a8 "(Im?F~1Z@Z)a( . (A.15)

It follows from (A.14) that

aDR,R&N(a8 , R?F~1), (A.16)

R~1DR&=(¹#l!k, (H#¹RK #AK @QAK )~1). (A.17)

Therefore,

a8 "E(aDR)"(Im?F~1Z@Z)a( , (A.18)

RI "E(RDR)"1

¹#l!m!k!1(H#¹RK #AK @QAK ), (A.19)

Var(aDR)"RI ?F~1. (A.20)

We denote posterior means using tildes for the remainder of the Appendix.


The joint posterior of the benchmark moments is

p(E2,<

22DX)JD<

22D~(T`k`1)@2

]expG!¹

2tr<K

22<~1

22!

¹

2tr(E

2!EK

2)(E

2!EK

2)@<~1

22 H. (A.21)

It follows that

E2D<

22, R&NAEK 2 ,

1

¹

<22B, (A.22)

<~122

DR&=(¹!1, (¹<K22

)~1). (A.23)

Therefore,

EI2"E(E

2DR)"EK

2, (A.24)

<I22

"E(<22

DR)"¹

¹!k!2<K

22, (A.25)

Var(E2DR)"

1

¹!k!2<K

22. (A.26)

Note that the predictive mean obeys the relation,

EH"E(rT`1

DR)"E(E(rT`1

Dh,R)DR)"E(EDR)"EI . (A.27)

Since B and E2

are independent in the posterior, the mean of the predictivedistribution is

EH"EI "EAa#BE

2E

2KRB"A

a8 #BI EI2

EI2

B, (A.28)

where a8 and BI are obtained from Eq. (A.18) using a8 "vec((a8 BI )@).Partition the predictive covariance matrix as

<H"C<H

11<H

12<H

21<H

22D. (A.29)

The "rst submatrix,<H11

, can be represented in terms of its (i, j) element. Denotethe i-th element of r

1,T`1as y

i,T`1, the ith element of a as a

i, the ith element of

uT`1

as ui,T`1

, the ith row of B as b@i, the ith column of A as a

i, and the (i, j)


element of R as pi,j

. Note that

yi,T`1

"ai#b@

ir2,T`1

#ui,T`1

(A.30)

"[1 r@2,T`1

]ai#u

i,T`1, (A.31)

since ai"(a

ib@i)@. The predictive covariance between y

i,T`1and y

j,T`1, the (i, j)

element of <H11

, can be obtained using the decomposition

Cov(yi,T`1

, yj,T`1

DR)"E(Cov(yi,T`1

, yj,T`1

Da,R)DR)

#Cov(E(yi,T`1

Da,R), E(yj,T`1

Da, R)DR). (A.32)

To compute the "rst term in (A.32), observe using (A.30) that

Cov(yi,T`1

, yj,T`1

Da,R)"b@i<H

22bj#p8

i,j, (A.33)

since, given the assumed properties of ui,T`1

, Cov(r2,T`1

, ui,T`1

DR)"0 for alli and

Cov(ui,T`1

, uj,T`1

DR)"E(Cov(ui,T`1

, uj,T`1

Dh, R)DR)"p8i,j

. (A.34)

Taking the expectation of (A.33) with respect to a gives

E(Cov(yi,T`1

, yj,T`1

Da,R)DR)"bI @i<H

22bIj#tr[<H

22Cov(b

i, b@

jDR)]#p8

i,j.

(A.35)

To compute the second term in (A.32), observe using (A.31) that

E(yi,T`1

Da, R)"[1 EI @2]a

i, (A.36)

so

Cov(E(yi,T`1

Da,R), E(yj,T`1

Da, R)DR)"[1 EI @2]Cov(a

i, a@

jDR)[1 EI @

2]@. (A.37)

Note that Cov(bi, b@

jDR) and Cov(a

i, a@

jDR) are submatrices of Var(aDR) in (A.20).

To compute the remaining submatrices in (A.29), observe that

<H"Var(rT`1

DR)"E(<DR)#Var(EDR)"<I #Var(EDR), (A.38)

which follows from the law of iterated expectations and the variance decomposi-tion rule. Applying this decomposition to the lower-right submatrix gives

<H22

"<I22

#Var(E2DR), (A.39)

and applying it to the o!-diagonal submatrices gives

<H12

"<H{21

"E(B<22

DR)#Cov(a#BE2,E@

2DR)

"BI <I22

#BI Var(E2DR). (A.40)


A.2. The xrst characteristic-based model

The likelihood function and the prior on (B, R,E2,<

22) are the same as in the

factor-based model presented in Section A.1. The only di!erence from thefactor-based model is in the prior for a:

p(aDB,E2)JexpG!

1

2p2a(a#BE

2)@(a#BE

2)H. (A.41)

The conditional prior on a is normal and centered at the pricing restriction.Note that

1

p2a(a#BE

2)@(a#BE

2)"

1

p2a(1 E@

2)A

a@

B@B(a B)A1

E2B

"

1

p2atr(1 E@

2)AA@A

1

E2B

"tr A@WAA1

p2aImB

"a@A1

p2aIm?WBa, (A.42)

where

W"A1 E@

2E2

E2E@2B. (A.43)

The full conditional posterior distribution of a is

p(aD ) )JexpG!1

2Ca@A1

p2aIm?WBa#(a!a( )@(R~1?Z@Z)(a!a( )DH

JexpM!12(a!a6 )@G(a!a6 )N, (A.44)

where

G"A1

p2aIm?WB#(R~1?Z@Z), (A.45)

a6 "G~1(R~1?Z@Z)a( . (A.46)


Hence, the full conditional posterior of a is a normal distribution:

aD ) &N(a6 , G~1). (A.47)

The full conditional posterior distribution of R is

p(RD ) )JDRD~(T`l`m`1)@2expM!12tr[(>!ZA)@(>!ZA)#H]R~1N.

(A.48)

Hence, the full conditional posterior of R is an inverted Wishart distribution:

R~1D ) &=(¹#l, [(>!ZA)@(>!ZA)#H]~1). (A.49)

The full conditional posterior distribution of E2

is

p(E2D ) )JexpM!1

2[E@

2B@BE

2#2E@

2B@a#trE

2n@TnTE@

2<~1

22

!2E@2<~1

22X@n

T]N

JexpM!12(E

2!EM

2)@P(E

2!EM

2)N, (A.50)

where

P"B@B#¹<~122

, (A.51)

EM2"P~1(¹<~1

22EK

2!B@a). (A.52)

Hence, the full conditional posterior of E2

is a normal distribution:

E2D )&N(EM

2,P~1). (A.53)

The full conditional posterior distribution of <22

is

p(<22

D ) )JD<22

D~(T`k`1)@2expM!12tr(X!n

TE@

2)@(X!n

TE@

2)<~1

22N. (A.54)

Hence, the full conditional posterior of <22

is an inverted Wishart distribution:

<~122

D )&=(¹, [(X!nTE@

2)@(X!n

TE@

2)]~1). (A.55)

Posterior draws of (a, R, E2,<

22) can be obtained using Gibbs sampling (see

Casella and George, 1992). A chain of draws is constructed by making repeateddraws from the full conditional distributions in (A.47), (A.49), (A.53), and (A.55).After an initial burn-in stage, these draws are taken from the joint posteriordistribution p(a, R,E

2,<

22DR). The predictive moments are computed as in Eqs.

(A.28) through (A.40), where the required posterior moments in those equationsare computed using the draws from the Gibbs chain.


A.3. The alternative characteristic-based model

As before, rt

is assumed to be normally distributed with mean E andcovariance matrix <. Unlike the previous models, however, this model is notrecast in a regression framework. The likelihood function is therefore

p(RDh)JD<D~T@2expM!12tr(R!n

TE@)@(R!n

TE@)<~1N. (A.56)

The set of parameters h is now rede"ned as h"(E,<, c), where c is de"nedbelow.

The prior on h is

p(E,<, c)"p(EDc)p(<)p(c), (A.57)

where

p(EDc)JexpG!1

2p2a(E!Cc)@(E!Cc)H, (A.58)

p(<)JD<D~(n`1)@2, (A.59)

p(c)JexpG!1

2p2c(c!c6 )@(c!c6 )H. (A.60)

The prior on < is di!use. The prior on c is normal with a large variance p2c , sothat it is noninformative. The normal prior on E given c is centered at the pricingrestriction.

The full conditional posterior distribution of E is

p(ED ) )JexpG!1

2 C(E!Cc)@(p2aIn)~1(E!Cc)

#(E!EK )@A1

¹

<B~1

(E!EK )DHJexpM!1

2(E!Ec)@(<c

E)~1(E!Ec)N, (A.61)

where

EK "R@nt/¹, (A.62)

<cE"A

1

p2aIn#¹<~1B

~1, (A.63)

Ec"<cEA

1

p2aCc#¹<~1EK B. (A.64)


Hence, the full conditional posterior of E is a normal distribution:

ED )&N(Ec,<cE). (A.65)

The full conditional posterior distribution of < is

p(<D ) )JD<D~(T`n`1)@2expM!12tr(R!n

TE@)@(R!n

TE@)<~1N. (A.66)

Hence, the full conditional posterior of < is an inverted Wishart distribution:

<~ 1D )&=(¹, [(R!nTE@)@(R!n

TE@)]~1). (A.67)

The full conditional posterior distribution of c is

p(cD ) )JexpM!12[(E!Cc)@(p2aIn)~1(E!Cc)#(c!c6 )@(p2cIL )~1(c!c6 )]N

JexpM!12(c!cc)@(<cc )~1(c!cc)N, (A.68)

where

<cc"A1

p2aC@C#

1

p2cILB

~1, (A.69)

cc"<ccA1

p2aC@E#

1

p2cc6 B. (A.70)

Hence, the full conditional posterior of c is a normal distribution:

cD )&N(cc,<cc). (A.71)

Posterior draws of the parameters are again obtained using Gibbs sampling.A chain of draws is constructed by making repeated draws from the fullconditional distributions in (A.65), (A.67), and (A.71). After an initial burn-instage, these draws are taken from the joint posterior distribution p(E,<, cDR).The posterior mean of < and the posterior mean and variance of E arecomputed using the posterior draws. The predictive moments EH and <H arethen computed using (A.27) and (A.38).

When pa"0, E"Cc. In this special case, the full conditional posteriors arederived for a reduced set of parameters (c,<). Whereas < is drawn in the samemanner as before, c is now drawn from the following normal distribution:

cD )&N(c0,<0c ), (A.72)

where

<0c"AC@A<

¹B~1

C#

1

p2cILB

~1, (A.73)

c0"<0cCC@A<

¹B~1

EK #1

p2cc6 D. (A.74)


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