The Best-Beta CAPM (BCAPM)

(Literature: Paper Zou 1)(Literature: Paper Zou 1)

3-22006 by Liang Zou. All rights reserved.

Outline

• The Variation-Around-Target Preferences

• Re-examination of Modern Portfolio Theory

• The Separation Theorem

• Derivation of the BCAPM

• Comparing the BCAPM with the CAPM

• Applications

3-32006 by Liang Zou. All rights reserved.

Introduction• Mean-variance (MV) CAPM has been with us for 40

years. Despite the lengthy debate over its theoretical and empirical validity it remains a major model for estimating cost of capital, discussing market efficiency, and so on.

• Empirical evidence suggests, however, that the CAPM may entail pricing errors and that the market index may not be mean-variance efficient.

• Although many multifactor models are proposed, these are empirically motivated models and are likely to be subject to data mining. More seriously, the factors that are reportedly useful for improving the explanatory and predictive power of the CAPM often lack theoretical insights.

3-42006 by Liang Zou. All rights reserved.

Motivating the BCAPM• Why improving the accuracy of

expected returns so important? Consider the dividend-growth model for evaluating a company’s equity value:

First period dividend

Required rate of return

Dividend growth rate

E

DV

k g

D

k

g

3-52006 by Liang Zou. All rights reserved.

What a small difference can do

• A critical problem is how to estimate the required rate of return k. One solution is to use the CAPM and estimate the beta of the risky cash flows.

• Accuracy of k is highly important, however.

• For instance, based on a current dividend yield of 2%, a 1% difference in k could lead to 50% difference in the estimated firm equity value!

• The reason is that the market capitalizes the firm’s future earnings. Since there are infinitely many periods ahead, any small improvement in the precision of expected return (or required rate of return) per period could add up to substantial improvement in the estimated present value of the firm.

3-62006 by Liang Zou. All rights reserved.

What is Best-Beta?

• The best-beta is defined as

return) excess s(market'

return) excess s(asset'

)(

)(

0

0

2

rrx

rrx

xE

xxE

mm

m

mB

3-72006 by Liang Zou. All rights reserved.

What is the Best-Beta CAPM?

• The best-beta CAPM predicts the same expected return – beta relation as the CAPM, except that the beta is different.

2

: ( ) ( )

: ( ) ( )

( , ) ( ),

( ) ( )

MV MVm

B Bm

MV Bm m

m m

CAPM E x E x

BCAPM E x E x

Cov x x E xx

Var x E x

3-82006 by Liang Zou. All rights reserved.

Compare BCAPM with CAPM• Let the true expected return and beta

be given (which we do not know):

• The pricing error of the CAPM and the BCAPM are the difference between the true and predicted expected return, denoted by alpha:

( ) ( )mE x E x

)()(

)()(

mBB

mMVMV

xExE

xExE

3-92006 by Liang Zou. All rights reserved.

Theorem• The relation between the MV beta and

the best beta is given by

2 2

22

2

2

(1 )

[ ( )]where (0,1)

( )

We call the "eta ratio", which is

a variation of the Sharpe ratio.

B MVm m

mm

m

m

E x

E x

3-102006 by Liang Zou. All rights reserved.

Implication of the theorem• The preceding theorem implies that

the best beta is always closer to the true beta than the MV-beta, that is, one of the following relations must hold for all situations and all assets:

MV B

MV B

MV B

3-112006 by Liang Zou. All rights reserved.

Proof of the theorem• By definition,

2 2

2

2 2

2 2

( ) ( ) ( ) ( )

( ) ( )

( ) [ ( )]

( ) ( )

(1 )

B m m m

m m

MVm m

m m

MVm m

E xx Cov xx E x E x

E x E x

Var x E x

E x E x

3-122006 by Liang Zou. All rights reserved.

Theorem• The preceding theorem implies

2 2

2

( ) (1 ) ( ) ( )

(1 )

thus either 0

or 0

or 0

B MVm m

B MVm

B MV

MV B

MV B

E x E x E x

3-132006 by Liang Zou. All rights reserved.

Empirical Example• The U.S. equity annual returns on the

NYSE, AMEX, and NASDAQ have an average risk premium of 7.67% (with standard deviation of 16.03%) over the period of 1952 to 1999 and 10.61% (with standard deviation of 15.15%) over 1983 to1999.

• We then derive the eta ratio to be

• 0.18629 (over 1952-1999)

• 0.32907 (over 1983-1999)

3-142006 by Liang Zou. All rights reserved.

The eta ratio as a function of market risk premium and volatility

00.10.20.30.40.5 E(xm)

00.1

0.20.3

m

0

0.2

0.4

0.6

0.8

1

2

3-152006 by Liang Zou. All rights reserved.

Derivation of the BCAPM

• Assume each investor has the objective

With two individual parameters: degree of risk aversion l and a target return .

• This objective function can be seen as an approximation of expected utility.

2max ( ) ( )2p p

pE r E r

3-162006 by Liang Zou. All rights reserved.

Separation Theorem

• Then we can show that every one wants to maximize the eta ratio:

• In equilibrium, the common optimal risky portfolio P must be the market portfolio and we can then derive the BCAPM (see paper Zou 1).

2

22

( )

( )P

PP

E x

E x

3-172006 by Liang Zou. All rights reserved.

From SDF to BCAPM• Alternatively, the BCAPM can also be derived

by specifying the stochastic discount factor (SDF) as follows:

• Question: What is the difference between this derivation and the previous derivation?

2

2

, ( ) 0

( ) ( )

( ) ( )

( )( ) ( ) (BCAPM)

( )

m

m

m m

mm

m

a bx E x

aE x bE xx

aE x bE x

E xxE x E x

E x

3-182006 by Liang Zou. All rights reserved.

Insights

• Consider a zero-price investment strategy with a long position in any asset i and some short position in market and cash:

• What b gives the best hedge of market risk?

• Answer: the best beta if we use the least-squares criterion.

0(1 )i mr br b r

3-192006 by Liang Zou. All rights reserved.

The least squares solution

20min ( (1 ) )b i mE r br b r

The necessary and sufficient condition yields

2

( )

( )Bm

m

E xxb

E x

3-202006 by Liang Zou. All rights reserved.

Further Insights• Without loss of generality, we can

always write out the equation:

• The question is how we should interpret alpha: Do we assume that it belongs to a correctly specified linear model, or do we assume that it represents a source of pricing error due to model uncertainty?

• The BCAPM treats alpha as source of error.

with ( ) 0i i i mx b x E

3-212006 by Liang Zou. All rights reserved.

How many masters do you serve?• CAPM attempts to serve two masters:

Alpha wants to be zero, and Beta wants to

• These two masters ask you to do the same thing if and only if “happens” to be zero when you choose the optimal b.

• Instead, BCAPM serves only one master, Beta, who wants you to

2min ( )b i mE x bx

2min ( )b i mE x bx

3-222006 by Liang Zou. All rights reserved.

The Econometrics View

• Equivalently, the MV alpha and beta are the solution to minimizing expected residual squares:

• The best beta and the corresponding alpha are derived by minimizing the combined errors:

2,min ( ) ,MV MVb iE b

2,min ( ) ,B Bb i iE b

3-232006 by Liang Zou. All rights reserved.

The Econometrics of BCAPM• When we use the historical data to estimate the

best beta and alpha, it can be shown that

• So that the BCAPM also gives more accurate estimates of the alpha and beta.

• Question: How would you compare the t-statistics of the estimated alphas of the two models?

2

2 2

( ) (1 ) ( )

( ) (1 ) ( )

B MVm

B MVm

Var Var

Var Var

3-242006 by Liang Zou. All rights reserved.

Concluding Remarks• MPT can be entirely re-written without much

ado. Given that investors may indeed have personal target returns, however, the assumption of the BCAPM could be considered to be more realistic.

• The best-beta approach could be useful for designing more powerful (or more sensible) tests of the CAPM-like models.

• The BCAPM can be readily extended to multifactor models, and its improvement over the MV models become increasingly more significant as the factor number increases.