The CAPM versus the risk neutral pricing model

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The CAPM versus the risk neutral pricing model

Dominique Pepin

To cite this version:

Dominique Pepin. The CAPM versus the risk neutral pricing model. 2002. �hal-00966459�

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: dominique.pepin@univ-poitiers.fr

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:

( ) ( )

mt 1 t

mt t 1 t ft

mt 1 t ft

t 1

t V R

R , R , Cov

R R E R

R E

−

− −

− = ι + β − β = (1)

The notations used are as follows: R_t,β and

ι

are N-vectors. R contains the random returns of the N _t risky assets and ι=^t

(

^{1 1}  ¹

)

. Rmt is the market portfolio return, and R is the riskless asset return. ft

1

Et₋ is the conditional expectations operator conditioning on the information available to investors at the end of the period t-1. Covt₋1

(

Rt,Rmt

)

is the conditional covariance of the N-vector R with _t R_mt, and

mt 1

t R

V₋ 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:

ι

−1 t = ft

t R R

E (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 (e.g., Ross, 1978, Harrison and Kreps, 1979 and Kreps, 1981). In t

the absence of arbitrage opportunities, the basic equation of asset pricing can be written as follows:

( )

^{= ι}

−1 t t

t m R

E (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

2

complete markets. Let M≠ ∅ denote the set of all random variables with finite second moments that satisfy Equation (3): M⁼

{

mt :Et−1

(

mtRt

)

^{= 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:

(

t t

)

²

1 M t mt 2

yt minE y m

D = ₋ −

∈ . The least-squares distance D²y_t between yt and M provides a way to assess the usefulness of an asset pricing model when it is misspecified. Hansen and Jagannathan (1997) showed that:

( )

[

^{− ι}

] [ ( ) ] ^{[ ( )}

^{− ι}

^]

=^t _{t−1 t t t−1 t}^t _t ⁻¹ _{t−1 t t}

2

yt E y R E R R E y R

D (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. D²_yt is equal to the

maximum pricing error ^Et₋1

(

^mt^Rpt

)

− ¹, where R is a portfolio return such that pt ^Et−¹

( )

^{R2pt = 1}. 3. Comparing the CAPM with the RNPM

Let y_1t and y be the following candidate SDF : _2t

ft t

1 R

y = 1 (5)

(

mt t 1 mt

)

mt 1 t ft

ft mt 1 t ft t

2 R E R

R V R

R R E R

y 1 ₋

−

− − −

−

= (6)

y_1t and y are the RNPM et CAPM’s SDF respectively. By substituting equations (5) and (6) _2t successively in (3), y_1t and y taking the place of _2t m , one gets equations (1) and (2)._t

We suppose that y_1t∉ M and y_2t ∉ 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 ?

By using Hansen and Jagannathan’s measure of model misspecification, we can assess that the CAPM

is not better than the RNPM if D²_y2t ≥ D²_y1t. When this condition is met, the CAPM’s maximal pricing error is larger than that of the RNPM.

By noting that ^tyAy− ^txAx=^t(y− x)A(y− x)+ 2 ^t(y− x)Ax for any symmetrical square matrix A,

the difference of D²y_2t and D²y_1t is :

( )

[ ] [ ( ) ] ^{[ ( ) ]}

( )

[

⁻

] [ ( ) ] [ ( )

^{− ι}

]

+

−

−

=

−

−

−

−

−

−

−

−

−

t t 1 1 t 1 t t t 1 t t t 1 t 2 1 t t

t t 1 t 2 1 t 1 t t t 1 t t t 1 t 2 1 t 2 t

t y1 2

t y2

R y E R R E R ) y y ( E 2

R ) y y ( E R R E R ) y y ( E D

D

(7)

By taking account of (5) and (6), equation (7) can be rewritten:

( )

[ ]

( )

[ ]

_^{ −} _^

 β

 

 −

−

β

 β

 

 −

=

−

−

ι

−

− −

−

− −

t ft 1 t 1 t t t 1 t t ft

ft mt 1 t

1 t t t 1 t t 2

ft ft mt 1 t 2

t y1 2

t y2

R R E 1 R R R E

R R 2 E

R R R E

R R D E

D

(8)

Let us notice that D²y_2t and D²y_1t are not different when E_t−1R_mt − R_ft = 0. So we suppose thereafter that E_t−1R_mt − R_ft > 0. It is supposed in addition that β is not a zero vector, so that

( )

[

^{Et 1 Rt tRt}

]

^{1 0}

tβ ₋ ⁻ β > . From equation (8), one gets then the essential result of the paper:

( )

[ ]

( )

[ ]

^β

β



 − ι

β

≥

−

⇔

>

− ₋

−

−

−

−

− 1

t t t 1 t t

t ft 1 t 1 t t t 1 t t

ft ft

mt 1 t 2

t y1 2

t y2

R R E

R R E 1 R R E R

2 R R E 0 D

D (9)

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.

4

References

Campbell, J.Y. [2000], « Asset pricing at the millennium », Journal of Finance, 55, 1515-1567.

Hansen, L.P., J. Heaton and E.G.J. Luttmer [1995], « Econometric evaluation of asset pricing models », Review of Financial Studies, 8, 237-274.

Hansen, L.P. and R. Jagannathan [1997], « Assessing specification errors in stochastic discount factor models », Journal of Finance, 52, 557-590.

Harrison, J.M. and D. Kreps [1979], « Martingales and arbitrage in multiperiod securities markets », Journal of Economic Theory, 20, 381-408.

Kreps, D.M. [1981], « Arbitrage and equilibrium in economies with infinitely many commodities », Journal of mathematical Economics, 8, 15-35.

Lintner, J. [1965], « The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets », Review of Economics and Statistics, 47, 13-37.

Mossin, J. [1966], « Equilibrium in a capital asset market », Econometrica, 34, 768-783.

Ross, S.A. [1978], « A simple approach to the valuation of risky streams », Journal of Business, 51, 453-475.

Sharpe, W.F. [1964], « Capital asset prices: A theory of market equilibrium under conditions of risk », Journal of Finance, 19, 425-442.