The CAPM versus the risk neutral pricing model
Dominique Pepin
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The CAPM versus the Risk Neutral Pricing Model
Dominique Pépin*
University of Poitiers, Centre de Recherche sur l’Intégration Economique et Financière, France
Abstract
We compare the risk neutral pricing model with the CAPM when it is understood that both models are
incorrect. We show that the former is better than the latter when a condition that we give is satisfied.
Keywords: CAPM; stochastic discount factor; pricing errors
JEL classification: G12
1. Introduction
The Capital Asset Pricing Model (CAPM) of Sharpe (1964), Lintner (1965) and Mossin (1966) is one
of the most popular asset pricing models. Since the beginning of the 1980s, so many deviations from the
CAPM or "anomalies" have been discovered in stock returns that financial economists had to
acknowledge the empirical failure of the model. Following Campbell (2000), p.1557, we think that "it is
unrealistic to hope for a fully rational, risk-based explanation of all the empirical patterns that have been
discovered in stock returns". It must be accepted that very asset pricing model may present some
imperfections. Therefore, the relevant question to be asked is the following: to what extent are these
imperfections acceptable?
The substance of every asset pricing model involves the impact of risk on asset returns. If such a model
is not better than one which ignores the trade-off between risk and return, we can consider it as too
imperfect to be used in applications. We therefore compare the CAPM with the Risk Neutral Pricing
Model (RNPM) when it is understood that both models do not price all portfolios correctly. By using
Hansen and Jagannathan’s measure of model misspecification (e.g., Hansen et Jagannathan, 1997), we
show that the CAPM is a worse asset pricing model than the RNPM when a specific condition is satisfied.
*
Department of Economics, CRIEF, 93 Recteur Pineau Avenue, 86022 Poitiers Cedex, France. Tel.: +33-5-49-28-75-51; fax:
+33-5-49-28-14-49
E-mail address: [email protected]
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2. Measuring the misspecification of an asset pricing model
The CAPM rests on the assumption that all investors are single-period mean-variance optimizers. As a
result, the market portfolio is mean-variance efficient, which implies a beta pricing relation between all
assets and the market portfolio:
E t − 1 R t = R ft ι + β ( E t − 1 R mt − R ft ) ,
β =
Cov t − 1 ( R t , R mt )
Vt − 1 R mt
(1)
The notations used are as follows: R t , β and ι are N-vectors. R t contains the random returns of the N
risky assets and ι = t (1 1  1) . R mt is the market portfolio return, and R ft is the riskless asset return.
E t − 1 is the conditional expectations operator conditioning on the information available to investors at the
end of the period t-1. Cov t − 1 ( R t , R mt ) is the conditional covariance of the N-vector R t with R mt , and
Vt − 1 R mt is the conditional variance of R mt .
The RNPM implies that the asset valuation is risk neutral. As all investors are risk neutral, the
equilibrium expected asset returns equal the riskless interest rate:
E t − 1 R t = R ft ι
(2)
We admit that both valuation equations (1) and (2) do not correctly price all portfolios and we wonder
what the requirement is so that the CAPM be more imperfect than the RNPM.
Models of asset pricing with frictionless markets imply that asset pricing can be represented by a
stochastic discount factor or SDF m t (e.g., Ross, 1978, Harrison and Kreps, 1979 and Kreps, 1981). In
the absence of arbitrage opportunities, the basic equation of asset pricing can be written as follows:
E t− 1 ( m t R t ) = ι
(3)
Every asset pricing model like (1) or (2) can be obtained by specifying the SDF in (3). Except possibly
when there are arbitrage opportunities present in the data set used in the empirical investigation, the set of
correctly specified discount factors is nonempty and typically large. The SDF is unique only in the case of
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complete markets. Let M ≠ ∅
denote the set of all random variables with finite second moments that
satisfy Equation (3): M = { m t : E t − 1 ( m t R t ) = ι } . Let y t ∉ M denote some « proxy » variable for a SDF
that does not satisfy Equation (3). Following Hansen, Heaton and Luttmer (1995) and Hansen and
Jagannathan (1997), we consider the following least-squares measure of misspecification:
D 2y t = min E t − 1 ( y t − m t ) 2 . The least-squares distance D 2y between yt and M provides a way to assess
t
m ∈M
t
the usefulness of an asset pricing model when it is misspecified. Hansen and Jagannathan (1997) showed
that:
[ (
D 2y t = t [ E t − 1 ( y t R t ) − ι ] E t − 1 R t t R t
)]
−1
[ E t− 1 ( y t R t ) − ι ]
(4)
Hansen et Jagannathan (1997) in addition showed that their measure of specification error focus on the
most mispriced portfolios, while correcting for portfolio size in a particular way.
D 2y t
is equal to the
( )
2
maximum pricing error E t − 1 ( m t R pt ) − 1 , where R pt is a portfolio return such that E t − 1 R pt = 1 .
3. Comparing the CAPM with the RNPM
Let y1t and y 2t be the following candidate SDF :
y1t =
1
R ft
(5)
y 2t =
E R − R ft
1
( R mt − E t − 1R mt )
− t − 1 mt
R ft
R ft Vt − 1 R mt
(6)
y1t and y 2t are the RNPM et CAPM’s SDF respectively. By substituting equations (5) and (6)
successively in (3), y1t and y 2t taking the place of m t , one gets equations (1) and (2).
We suppose that y1t ∉ M
and y 2 t ∉ M . So neither model can perfectly describe the asset returns.
What is the condition for the CAPM to be a worse asset pricing model than the RNPM ?
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By using Hansen and Jagannathan’s measure of model misspecification, we can assess that the CAPM
2
2
is not better than the RNPM if D y 2 t ≥ D y1t . When this condition is met, the CAPM’s maximal pricing
error is larger than that of the RNPM.
By noting that t yAy− t xAx = t ( y − x )A( y − x ) + 2 t ( y − x )Ax for any symmetrical square matrix A,
2
2
the difference of D y 2 t and D y1t is :
[ (
)] [ E ( ( y
) ] [E ( R R )] [ E ( y
D 2y 2 t − D 2y1t = t [ E t − 1 ( ( y 2 t − y1t )R t ) ] E t − 1 R t t R t
+ 2 [ E t − 1 ( ( y 2 t − y1t )R t
t
t− 1
−1
t− 1
2t
−1
t
t
t− 1
t
− y1t )R t ) ]
1t R t
) − ι]
(7)
By taking account of (5) and (6), equation (7) can be rewritten:
D 2y 2 t
−
D 2y1t
 E R − R ft 

=  t − 1 mt
R ft


 E R − R ft
− 2 t − 1 mt
R ft

2
t
[ (
β E t− 1 R t t R t
[ (
)]
 t
 β E t − 1 R t t R t

−1
)]
β
−1
 1

E t − 1 
R t − ι 
 R ft

(8)
2
2
Let us notice that D y 2 t and D y1t are not different when E t − 1 R mt − R ft = 0 . So we suppose thereafter
that E t − 1 R mt − R ft > 0 . It is supposed in addition that β
t
[ (
β E t− 1 R t t R t
)]
−1
is not a zero vector, so that
β > 0 . From equation (8), one gets then the essential result of the paper:
t
D 2y 2 t − D 2y1t > 0 ⇔ E t − 1 R mt − R ft ≥ 2R ft
[ (
β E t− 1 R t t R t
t
)]
[ (
−1
 1

E t − 1 
R t − ι 
 R ft

β E t− 1 R t t R t
)]
−1
(9)
β
The inequality (9) defines the required condition for the CAPM to be, within the meaning of Hansen
and Jagannathan (1997), a worse valuation model than the RNPM. When the condition (9) is met, the
CAPM produces a maximum pricing error larger than that of the basic model which ignores the trade-off
between risk and return. It is thus so bad that it is not better than the simplest pricing model. This
inequality can be presented like a sufficient condition to reject the CAPM, not because it would be
statistically inadequate with the data of asset returns, but because it leads to too significant pricing errors.
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References
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