Economics Letters 125 (2014) 219–222

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Economics Letters

journal homepage: www.elsevier.com/locate/ecolet

A characterization of the – pricing model Kerry Back ∗ Jones Graduate School of Business and Department of Economics, Rice University, Houston, TX 77005, USA h i g h l i g h t s

• If there are no ‘‘extremely attractive returns’’, the coskewness–cokurtosis pricing model holds. • The converse also holds. • An extremely attractive return has a positive alpha. • An extremely attractive return also has residual risk with desirable coskewness and cokurtosis. article info a b s t r a c t

Article history: The coskewness–cokurtosis pricing model is equivalent to absence of any positive-alpha return for which Received 8 July 2014 the residual risk has positive coskewness and negative cokurtosis with the market. This parallels the CAPM Received in revised form and also the fundamental theorem of asset pricing. 3 September 2014 © 2014 Published by Elsevier B.V. Accepted 7 September 2014 Available online 16 September 2014

JEL classification: G12

Keywords: Coskewness Cokurtosis Stochastic discount factor

¯ 1. Introduction where Rf is the risk-free return, Rm is the market return, Rm is the expected market return, and λ0, λ1, λ2 > 0. The signs of the coeffi- The coskewness–cokurtosis pricing model of Kraus and Litzen- cients in (1) are based on the assumption that investors dislike vari- berger(1976) and Dittmar(2002) extends the capital asset pricing ance, prefer positive skewness to negative skewness, and dislike model (CAPM) of Sharpe(1964), Treynor(1999), Lintner(1969), kurtosis.1 Thus, high /low coskewness/high cokurtosis and Mossin(1966) by including coskewness and cokurtosis in ad- assets are undesirable and consequently sell at low prices, produc- dition to covariance with the market return as priced risks. The mo- ing high expected returns. tivation for the model is that investors may care about skewness Consider the projection of an excess return R−Rf on the market and kurtosis in addition to mean and . If so, then investors excess return Rm − Rf : who hold the market portfolio would evaluate a marginal change   in the holding of an asset in terms of its effect on variance, skew- R − Rf = α + β Rm − Rf + ε, (2) ness, and kurtosis, and these marginal effects are captured by co- = variance, coskewness, and cokurtosis. The model asserts that each where β cov (R, Rm) /var (Rm) and where ε has mean zero. It is return R and the corresponding expected return R¯ satisfy easy to see that the coskewness–cokurtosis pricing model implies that there cannot be any returns possessing all of the following ¯   ¯ 2 R − Rf = λ1cov (R, Rm) − λ2cov R, Rm − Rm properties:

  ¯ 3 + λ3cov R, Rm − Rm , (1)

1 This is consistent with utility functions being concave with positive third ∗ derivatives and negative fourth derivatives, conditions that follow from risk Tel.: +1 9792044248. aversion, decreasing absolute risk aversion, and decreasing absolute prudence E-mail addresses: [email protected], [email protected]. (Arditti, 1967; Kimball, 1993; Haas, 2007). http://dx.doi.org/10.1016/j.econlet.2014.09.008 0165-1765/© 2014 Published by Elsevier B.V. 220 K. Back / Economics Letters 125 (2014) 219–222

(i) α > 0, 2. Main result  ¯ 2  (ii) E Rm − Rm ε > 0, and Suppose there is a risk-free asset with return Rf and N risky  ¯ 3  (iii) E Rm − Rm ε < 0. assets with returns Ri. By ‘‘return’’, we mean gross return, that is, one plus the rate of return. The set of returns is the set of random If such a return were to exist, it would be a positive alpha return variables with a residual risk having positive coskewness and negative N cokurtosis with the market. Such a return would be extremely  R = R + w (R − R ), (4) attractive to investors with mean–variance–skewness–kurtosis f i i f i=1 preferences who hold the market portfolio, and the coskewness– N cokurtosis pricing model implies that such extremely attractive for w = (w1, . . . , wN ) ∈ R . Assume that each of the returns Ri returns cannot exist. This is parallel to the CAPM, which implies has a finite fourth , so all of the returns (4) also have finite that positive alpha returns cannot exist. fourth moments. As previously remarked, the equivalence of (b) The contribution of this note is to show that the absence of re- and (c) below is a standard result, but we include it for the sake of turns with properties (i)–(iii) is a sufficient as well as necessary completeness. condition for the coskewness–cokurtosis pricing model. Thus, we derive the model from the absence of returns that are extremely at- Theorem. The following statements are equivalent: tractive to investors with mean–variance–skewness–kurtosis pref- (a) There does not exist a return R satisfying erences who hold the market portfolio. We do not need to assume (i) α ≥ 0, that there is a representative investor whose utility function can    − ¯ 2 ≥ be well approximated by a Taylor series expansion as do Kraus and (ii) E Rm Rm ε 0, and Litzenberger(1976) and Dittmar(2002), though of course our mo-  ¯ 3  (iii) E Rm − Rm ε ≤ 0, tivation for assuming the absence of returns satisfying (i)–(iii) is not far from their representative investor hypothesis. with at least one strict inequality, where α and ε are defined from To put the result in perspective, it is useful to consider three R by the projection (2). (b) There exist 0 such that the coskewness–cokurtosis separate classes of securities markets: λ0, λ1, λ2 > pricing model (1) holds for all returns R. M1: markets in which there is no return satisfying (i), (c) There exist a0 and a1, a2, a3 > 0 such that M defined by (3) is an M2: markets in which there is no return satisfying both (i) and SDF. (ii),

M3: markets in which there is no return satisfying all of (i)–(iii). 3. Proof of the theorem Clearly, M ⊂ M ⊂ M . The CAPM holds in the smallest set of 1 2 3 For any w ∈ N , the return R defined by (4) has an alpha equal markets M . The coskewness pricing model holds in the middle set R 1 to M2 (the coskewness model is the coskewness–cokurtosis model with a zero coefficient on cokurtosis). The coskewness–cokurtosis N α = wiαi, pricing model holds in the largest set M3. = As with all factor pricing models, the coskewness–cokurtosis i 1 pricing model is equivalent to an affine representation of a where αi is the alpha of the return Ri in the projection (2) for stochastic discount factor (SDF). An SDF is a M i = 1,..., N. Furthermore, the residual risk of R is [ ] = such that E MR 1 for all (gross) returns R. The coskewness– N cokurtosis pricing model holds if and only if there is an SDF of the  ε = wiεi, form i=1  2  3 M = a − a R + a R − R¯ − a R − R¯ , (3) where εi is the residual risk of the return Ri in the projec- 0 1 m 2 m m 3 m m ′ tion (2). Let A1 = (α1 ··· αN ) . Let A2 denote the column vec- for constants a with a , a , a > 0. Thus, the result of this note can  2  i 1 2 3 tor formed by stacking the numbers E R − R¯  ε for i = be expressed as: there are no returns satisfying (i)–(iii) if and only m m i if there is an SDF of the form (3). This is analogous to what Dybvig 1,..., N. Let A3 denote the vector formed by stacking the numbers  ¯ 3  and Ross(1989) call the fundamental theorem of asset pricing. The E Rm − Rm εi for i = 1,..., N. Now, let A denote the N×3 ma- fundamental theorem of asset pricing states that the following are trix that has A , A , and −A as its columns. Given w ∈ N , we have equivalent: (1) absence of arbitrage opportunities, (2) existence 1 2 3 R of a strictly positive SDF, and (3) existence of an optimum for an  α  investor with monotone utility. The absence of returns satisfying  ¯ 2  ′  E Rm − Rm ε  (i)–(iii) imposes more structure on the space of returns than does A w =   ,   ¯ 3  the absence of arbitrage opportunities. Consequently, the absence −E Rm − Rm ε of returns satisfying (i)–(iii) has strong implications for the nature of the SDF.2 The key step in the proof of the theorem below and where R is the return (4) with α and ε defined by the projection (2). ′ N the key step in the proof that (1) implies (2) in the fundamental Assume (a) holds. Then, the set {A w | w ∈ R } inter- 3 theorem of asset pricing are the same, namely, they invoke a sects the positive orthant of R only at the origin. By Tucker’s version of the separating hyperplane theorem. Complementarity Theorem (Rockafellar, 1970, Theorem 22.7), it ′ ′ follows that there exist η1, η2, η3 > 0 such that η A w ≤ 0 for all N ′ ′ N w ∈ R , where η = (η1 η2 η3) . Because the set {A w | w ∈ R } is a linear subspace of 3, this implies η′A′w = 0 for all w ∈ N . Thus, 2 R R Note, however, that, whereas the absence of arbitrage opportunities implies the for any w ∈ N with corresponding return (4) and projection (2), existence of a strictly positive SDF, the SDF (3) cannot be strictly positive unless the R market return is bounded. See Dybvig and Ingersoll(1982) for a discussion of this  ¯ 2   ¯ 3  point in connection with the CAPM. η1α + η2E Rm − Rm ε − η3E Rm − Rm ε = 0. (5) Download English Version: https://daneshyari.com/en/article/5059223

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