A Theoretial Extension of the Consumption-based CAPM Model Jingyuan Li Shool of Management Huazhong University of Siene and Tehnology Wuhan 430074, China Email: jingyuanht@yahoo.om.n Georges Dionne Canada Researh Chair in Risk Management, HEC Montréal, CIRRELT, and CIRPÉE, Canada Email: georges.dionne@he.a January 25, 2011 Abstrat We extend the Consumption-based CAPM (C-CAPM) model for representative agents with different risk attitudes. We introdue the onept of expetation dependene and show that for a risk averse representative agent, it is the first-degree expetation dependene rather than the ovariane that determines C-CAPM s riskiness. We extend the assumption of risk aversion to prudene and provide another dependene ondition to obtain the values of asset prie and equity premium. Results are generalized to higher-degree risk hanges and higherorder risk averse representative agents, and are linked to the equity premium puzzle. Key words: Consumption-based CAPM; Risk premium; Equity premium puzzle; Expetation dependene JEL lassifiation: D51; D80; G12 This work is supported by the National Siene Foundation of China (General Program) 70602012 and by the Soial Siene Researh Counil of Canada (SSRCC).
1 Introdution Consumption-based apital asset priing model (C-CAPM), developed in Rubinstein (1976), Luas (1978), Breeden (1979) and Grossman and Shiller (1981), relates the risk premium on eah asset to the ovariane between the asset return and the intertemporal marginal rate of substitution of a deision maker. The most important omparative statis results for C-CAPM is how an asset s prie or equity premium hanges as the quantity of risk and the prie of risk hange. The results of omparative statis analysis thus form the basis for muh of our understanding of the soures of hanges in onsumption (maroeonomi) risk and risk aversion that drive asset pries and equity premia. The two objetives of this study are to propose a new theoretial framework for C-CAPM and to extend its omparative statis. We use general utility funtions and probability distributions to investigate C-CAPM. Our model provides insight into the basi onepts that determine asset pries and equity premia. The C-CAPM priing rule is sometimes interpreted as implying that the prie of an asset with a random payoff falls short of its expeted payoff if and only if the random payoff positively orrelates with onsumption. Liu and Wang (2007) show that this interpretation of C-CAPM is not generally orret by presenting a ounterexample. We introdue more powerful statistial tools to obtain the appropriate dependene between asset payoff and onsumption. We first disuss the onept of expetation dependene developed by Wright (1987) and Li (2011). We show that, for a risk averse representative agent, it is the first-degree expetation dependene between the asset s payoff and onsumption rather than the ovariane that determines C- CAPM s riskiness. Our result also reinterprets the ovariane between an asset s payoff and the marginal utility of onsumption in terms of the expetation dependene between the payoff and onsumption itself. We extend the assumption of risk aversion to prudene and provide another dependene ondition. Finally, we interpret C-CAPM in a general setting: for the i th -degree risk averse representative agent, 1 with i = 2,.., N + 1, it is the N th -order expetation dependene that determines C-CAPM s riskiness. 1 Risk aversion in the traditional sense of a onave utility funtion is indiated by i = 2, whereas i = 3 gives downside risk aversion in the sense of Menezes, Geiss and Tressler (1980). i th -degree risk aversion is equivalent to preferenes satisfying risk apportionment of order i. See Ekern (1980) and Eekhoudt and Shlesinger (2006) for more disussions. 1
Our study relates to Gollier and Shlesinger (2002) who examine asset pries in a representativeagent model of general equilibrium with two differenes. First, we study asset prie and equity premium driven by maroeonomi risk as in the traditional C-CAPM model while Gollier and Shlesinger s model onsiders the relationship between the riskiness of the market portfolio and its expeted return. Seond, Gollier and Shlesinger (2002) s model is a stati model whereas our results rest on a dynami framework. Our study also extends the literature that examines the effets of higher-degree risk hanges on the maroeonomy. Eekhoudt and Shlesinger (2006) investigate neessary and suffiient onditions on preferenes for a higher-degree hange in risk to inrease saving. Our study provides neessary and suffiient onditions on preferenes for a higher-degree hange in risk to set asset pries and equity premia. The paper proeeds as follows. Setion 2 introdues several onepts of dependene. Setion 3 provides a reinterpretation of C-CAPM for risk averse representative agents. Setion 4 extends the results of Setion 3 to prudent and higher-order risk averse agents respetively. Setion 5 disusses the results in relation to loal indexes of risk aversion and Setion 6 interprets the results in terms of the equity premium puzzle and onludes the paper. 2 Conepts of dependene The onept of orrelation oined by Galton (1886) had served as the only measure of dependene during the first 70 years of the 20th entury. However orrelation is too weak to obtain meaningful onlusions in many eonomi and finanial appliations. For example, ovariane is a poor tool for desribing dependene for non-normal distributions. Sine Lehmann s introdution of the onept of quadrant dependene in 1966, stronger measures of dependene have reeived muh attention in the statistial literature 2. Suppose x ỹ [a, b] [d, e], where a, b, d and e are finite. Let F (x, y) denote the joint and F X (x) and F Y (y) the marginal distributions of x and ỹ. Lehmann (1966) introdues the following onept to investigate positive dependene. Definition 2.1 (Lehmann, 1966) ( x, ỹ) is positively quadrant dependent, written P QD( x, ỹ), if F (x, y) F X (x)f Y (y) for all (x, y) [a, b] [d, e]. (1) 2 For surveys of the literature, we refer to Joe (1997), Mari and Kotz (2001) and Embrehts (2009). 2
The above inequality an be rewritten as F X (x ỹ y) F X (x) (2) and an interpretation of definition (2.1) is provided by Lehmann as follows: knowledge of ỹ being small inreases the probability of x being small. PQD has its interest in modeling dependent risks beause it an take into aount the simultaneous downside (upside) evolution of risks. The marginal and the onditional CDFs an be hanged simultaneously 3. Wright (1987) introdued the following related onept of dependene into the eonomis literature. Definition 2.2 If F ED( x y) = [E x E( x ỹ y)] 0 for all y [d, e], (3) and there is at least some y 0 in some set S with prob(s) > 0, for whih a strong inequality holds, then x is positive f irst degree expetation dependent on ỹ. The family of all distributions F satisfying (3) will be denoted by F 1. Similarly, x is negative first-degree expetation dependent on ỹ if (3) holds with the inequality sign reversed. The totality of negative first-degree expetation dependent distributions will be denoted by G 1. Wright (1987, page 113) interprets negative first-degree expetation dependene as follows: when we disover ỹ is small, in the preise sense that we are given the trunation ỹ y, our expetation of x is revised upward. Having x positively (negatively) first-degree expetation dependent on ỹ is a stronger ondition than positive (negative) quadrant dependene between x and ỹ, but a weaker ondition than orrelation (see Wright (1987) and Li (2011) for disussions of these onepts and examples). Li (2011) proposes the following weaker dependene measure: Definition 2.3 If SED( x y) = = y d y d [E x E( x ỹ t)]f Y (t)dt (4) F ED( x t)f Y (t)dt 0 for all y [d, e], then x is positive seond-degree expetation dependent on ỹ. 3 Portfolio seletion problems with positive quadrant dependeny have been explored by Pellerey and Semeraro (2005) and Dahraoui and Dionne (2007), among others.. Pellerey and Semeraro (2005) assert that a large subset of the elliptial distributions lass is PQD. For more examples, see Joe (1997). 3
The family of all distributions F satisfying (4) will be denoted by F 2. Similarly, x is negative seond-degree expetation dependent on ỹ if (4) holds with the inequality sign reversed, and the totality of negative seond-degree expetation dependent distributions will be denoted by G 2. It is obvious that F 1 F 2 and G 1 G 2 but the onverse is not true. Beause x and ỹ are positively orrelated when (see Lehmann 1966, lemma 2) ov( x, ỹ) = b e a d [F (x, y) F X (x)f Y (y)]dxdy = e d F ED( x t)f Y (t)dt 0, (5) then ov( x, ỹ) 0 is only a neessary ondition for ( x, ỹ) F 2 but the onverse is not true. Comparing (4) and (5), we know that ov( x, ỹ) is the 2nd entral ross moment of x and ỹ, while SED( x y) is related to 2nd entral ross lower partial moment of x and ỹ whih an be explained as a measure of downside risk omputed as the average of the squared deviations below a target. Rewriting 1 th ED( x y) = F ED( x y), 2 th ED( x y) = SED( x y) = y d F ED( x t)f Y (t)dt, repeated integrals yield: N th ED( x y) = y d (N 1) th ED( x t)dt, for N 3. (6) Definition 2.4 (Li 2011) If k th ED( x e) 0, for k = 2,..., N 1 and N th ED( x y) 0 for all y [d, e], (7) then x is positive N th -order expetation dependent on ỹ (N th ED( x y)). The family of all distributions F satisfying (7) will be denoted by F N. Similarly, x is negative N th -order expetation dependent on ỹ if (7) holds with the inequality sign reversed, and the totality of negative N th -order expetation dependent distributions will be denoted by G N. From this definition, we know that F N 1 F N and G N 1 G N but the onverse is not true. N th - order expetation dependene is related to N th -order entral ross lower partial moment of x and ỹ (See, Li (2011) for more details). Several reent researhes in finane have foused on estimators of higher-order moments and omoments of the return distribution (i.e. oskewness and okurtosis) and showed that these estimates generate a better explanation of investors portfolios. (See, Martellini and Ziemann (2010) for more details). For our purpose, omparative expetation dependene has to be defined. Sibuya (1960) introdues the onept of dependene funtion Ω F : Ω F = F (x, y) F X (x)f Y (y). (8) 4
We propose the following definition, whih generalizes Sibuya s (1960) definition, to quantify omparative expetation dependene. Definition 2.5 Define i th ED F and i th ED H, for i = 1,.., N, as the i th expetation dependene under distribution F (x, y) and H(x, y) respetively. Distribution F (x, y) is more first-degree expetation dependent than H(x, y), if and only if F ED F ( x y)f Y (y) F ED H ( x y)h Y (y) for all y [d, e]. Distribution F (x, y) is more N th -order expetation dependent than H(x, y) for N 2, if k th ED F ( x e) k th ED H ( x e), for k = 2,..., N 1 and N th ED F ( x y) N th ED H ( x y) for all y [d, e]. (9) When N = 1, F (x, y) is more first-degree expetation dependent than H(x, y) if F (x, y) F X (x)f Y (y) H(x, y) H X (x)h Y (y). (10) Hene Ω F Ω H is a suffiient ondition for F (x, y) having more first-degree expetation dependent than H(x, y). 3 C-CAPM for a risk averse representative agent 3.1 Consumption-based asset priing model The well known onsumption-based asset priing model an be expressed as the following two equations (see e.g. Cohrane 2005, page 13-14) and p t = E t x t+1 R f + β ov t[u ( t+1 ), x t+1 ] u, (11) ( t ) E t Rt+1 R f = ov t[u ( t+1 ), R t+1 ] E t u ( t+1 ) where p t is the prie in period t of an asset with random payoff x t+1 and gross return R t+1 in period t + 1, β is the subjetive disount fator, R f is the gross return of the risk-free asset, u ( ) is the marginal utility funtion, t is the onsumption in period t, and t+1 is the onsumption in period t + 1. E R t+1 R f is the asset s risk premium. When the representative agent s utility funtion is the power funtion, u( t ) = 1 γ t 1 1 γ (12) where γ is the oeffiient of relative risk aversion and t+1 and x t+1 are onditional lognormally distributed, (12) beomes (Campbell 2003, page 821) E t r t+1 r f + var t( r t+1 ) 2 = γov t (log t+1, r t+1 ), (13) 5
where r t+1 = log(1 + R t+1 ) and r f = log(1 + R f ). The first term on the right-hand side of (11) is the standard disounted present-value formula. This is the asset s prie for a risk-neutral representative agent or for a representative agent when asset payoff and onsumption are independent. The seond term is a risk aversion adjustment. (11) states that an asset with random future payoff x t+1 is worth less than its expeted payoff disounted at the risk-free rate if and only if ov[u ( t+1 ), x t+1 ] 0. (12) shows that all assets have an expeted return equal to the risk-free rate plus a risk adjustment under risk aversion. (13) states that the log risk premium is equal to the produt of the oeffiient of relative risk aversion and the ovariane of the log asset return with onsumption growth. We now provide a generalization of these results. Suppose ( x t+1, R t+1, t+1 ) [x, x] [R, R] [, ]. From theorem 1 in Cuadras (2002), we know that ovariane an always be written as ov t [u ( t+1 ), x t+1 ] = Beause we an write x x x x [F ( t+1, x t+1 ) F Ct+1 ( t+1 )F Xt+1 (x t+1 )]u ( t+1 )dx t+1 d t+1. (14) [F Xt+1 (x t+1 t+1 t+1 ) F Xt+1 (x t+1 )]dx t+1 = E x t+1 E( x t+1 t+1 t+1 ), (15) (see, e.g., Tesfatsion (1976), Lemma 1), hene, we have = = ov t [u ( t+1 ), x t+1 ] (16) [E x t+1 E( x t+1 t+1 t+1 )]F Ct+1 ( t+1 )u ( t+1 )d t+1 (by (15)) F ED( x t+1 t+1 )u ( t+1 )F Ct+1 ( t+1 )d t+1. Using (16), (11) an be rewritten as p t = E t x t+1 } R{{ f } disounted present value effet = E t x t+1 R f β β F ED( x t+1 t+1 )F Ct+1 ( t+1 )[ u ( t+1 ) u ( t ) ]d t+1 } {{ } first degree expetation dependene effet F ED( x t+1 t+1 )F Ct+1 ( t+1 )AR( t+1 )MRS t+1, t d t+1, where AR(x) = u (x) u (x) is the Arrow-Pratt absolute risk aversion oeffiient, and MRS x,y = u (x) u (y) is the marginal rate of substitution between x and y. We an also rewrite (12) as E t Rt+1 R f = (17) F ED( R t+1 t+1 )F Ct+1 ( t+1 ) [ u ( t+1 ) }{{} E t u ( t+1 ) ] d t+1 (18) onsumption risk effet }{{} prie of risk effet 6
Beause R f = 1 β u ( t) E t u ( t+1 ) (see e.g. Cohrane 2005, page 11), we also have E t Rt+1 R f = βr f F ED( R t+1 t+1 )F Ct+1 ( t+1 )AR( t+1 )MRS t+1, t d t+1 (19) (17) shows that an asset s prie involves two terms. The effet, measured by the first term on the right-hand side of (17), is the disounted present value effet. This effet depends on the expeted return of the asset and the risk-free interest rate. The sign of the disounted present value effet is the same as the sign of the expeted return. This term aptures the diret effet of the disounted expeted return, whih haraterizes the asset s prie for a risk-neutral representative agent. The seond term on the right-hand side of (17) is alled first-degree expetation dependene effet. This term involves β, the expetation dependene between the random payoff and onsumption, the Arrow-Pratt risk aversion oeffiient and the intertemporal marginal rate of substitution. The sign of the first-degree expetation dependene indiates whether the movements on onsumption tend to reinfore (positive first-degree expetation dependene) or to ounterat (negative first-degree expetation dependene) the movements on an asset s payoff. (18) states that the expeted exess return on any risky asset over the risk-free interest rate an be explained as an integral of a number represented by the quantity of onsumption risk times the prie of this risk. The quantity of onsumption risk is measured by the first-degree expetation dependene of the exess stok return with onsumption, while the prie of risk is the Arrow-Pratt risk aversion oeffiient times the intertemporal marginal rate of substitution. We obtain the following proposition from (17) and (18). Proposition 3.1 The following statements hold: F 1 ; G 1 ; (i) p t E t x t+1 R f (ii) p t E t x t+1 R f for any risk averse representative agent (u 0) if and only if ( x t+1, t+1 ) for any risk averse representative agent (u 0) if and only if ( x t+1, t+1 ) (iii) E t Rt+1 R f for all risk averse representative agent (u 0) if and only if ( R t+1, t+1 ) F 1 ; (iv) E t Rt+1 R f for all risk averse representative agent (u 0) if and only if ( R t+1, t+1 ) G 1. Proof See Appendix A. 7
Proposition 3.1 states that, for a risk averse representative agent, an asset s prie is lowered (or equity premium is positive) if and only if its payoff is positively first-degree expetation dependent with onsumption. Conversely, an asset s prie is raised (or equity premium is negative) if and only if its payoff is negatively first-degree expetation dependent with onsumption. Therefore, for a risk averse representative agent, it is the first-degree expetation dependene rather than the ovariane that determines its riskiness. Beause ( x t+1, t+1 ) F 1 (G 1 ) ov t ( x t+1, t+1 ) 0( 0) and the onverse is not true, we onlude that a positive (negative) ovariane is only a neessary ondition for risk averse agent paying a lower (higher) asset prie (or having a positive (negative) equity premium). 3.2 Comparative risk aversion The assumption of risk aversion has long been a ornerstone of modern eonomis and finane. Ross (1981) provides the following strong measure for omparative risk reversion: Definition 3.2 (Ross 1981) u is more Ross risk averse than v if and only if there exists λ > 0 suh that for all x, y u (x) v (x) λ u (y) v (y). (20) More risk averse in the sense of Ross guarantees that the more risk averse deision-maker is willing to pay more to benefit from a mean preserving ontration. Under whih ondition does a hange in the representative agent s risk preferenes redue the asset prie? To answer this question let us onsider a hange of the utility funtion from u to v. From (17) and (18), for agent v, we have p t = E t x t+1 R f β F ED( x t+1 t+1 )F Ct+1 ( t+1 )[ v ( t+1 ) v ( t ) ]d t+1 (21) and E t Rt+1 R f = F ED( R t+1 t+1 )F Ct+1 ( t+1 )[ v ( t+1 ) E t v ( t+1 ) ]d t+1. (22) Intuition suggests that if asset return and onsumption are positive dependent and agent u is more risk averse than agent v, agent u should have a larger risk premium than agent v. This intuition an be reinfored by Ross risk aversion and first-degree expetation dependene, as stated in the following proposition. 8
Proposition 3.3 Let p u t and p v t denote the asset s pries orresponding to u and v respetively. The following statements hold: (i) p u t p v t for all ( x t+1, t+1 ) F 1 if and only if v is more Ross risk averse than u; (ii) p u t p v t for all ( x t+1, t+1 ) G 1 if and only if u is more Ross risk averse than v; (iii) u has a larger risk premium than agent v for all ( R t+1, t+1 ) F 1 if and only if u is more Ross risk averse than v; (iv) u has a larger risk premium than agent v for all ( R t+1, t+1 ) G 1 if and only if v is more Ross risk averse than u. Proof See Appendix A. Proposition 3.3 indiates that, when an asset s prie first-degree positively (negatively) expetation depends on onsumption, an inrease in risk aversion in the sense of Ross dereases (inreases) the asset prie. Proposition 3.3 also shows that, for all risk averse representative agents, assets whose gross returns are positively first-degree expetation dependent with onsumption must promise higher expeted returns to indue agents to hold them. Conversely, assets that negatively first-degree expetation depend on onsumption, suh as insurane, an offer expeted rates of return that are lower than the risk-free rate, or even negative (net) expeted returns. The results of Proposition 3.3 annot be obtained with the Arrow-Pratt relative risk aversion measure. We onsider the onvenient power utility form u() = 1 γu 1 1 γ u and v() = 1 γv 1 1 γ v. γ u and γ v are u and v s relative risk aversion oeffiients respetively. Intuition would suggest that, when an asset s gross return and onsumption are positively dependent, γ u γ v implies that u s risk premium will be higher. However, the following ounter example shows that, in the range of aeptable values of parameters, the Arrow-Pratt relative risk aversion oeffiient is neither a neessary nor a suffiient ondition to obtain higher risk premium when asset s gross return and onsumption are positively first-degree expetation dependent. Counter Example Suppose u() = 1 γ 1 1 γ, t+1 [1, 3] almost surely and ( R t+1, t+1 ) F 1 (note that in this ase ov t ( R t+1, t+1 ) 0), from (18) we obtain E t Rt+1 R f = = F ED( R t+1 t+1 )F Ct+1 ( t+1 )[ u ( t+1 ) E t u ( t+1 ) ]d t+1 (23) F ED( R t+1 t+1 )F Ct+1 ( t+1 ) γ γ 1 t+1 E t γ d t+1, t+1 9
hene, when an asset s gross return and onsumption are positively first-degree expetation 0 if and only if d γ γ 1 t+1 E t γ t+1 is not always true beause it ontains the variations of the marginal rate of substitution. Sine dependent, d[et R t+1 R f ] dγ d γ γ 1 t+1 E t γ /dγ = t+1 γ 1 t+1 (E t γ {[1 γ(γ + 1) 1 t+1 )2 /dγ 0. We now show that d γ γ 1 t+1 /dγ 0 E t γ t+1 t+1 ]E t γ t+1 + γ2 E t γ 1 t+1 }, (24) we obtain d[e t R t+1 R f ] dγ 0 if and only if [1 γ(γ + 1) 1 t+1 ]E t γ t+1 + γ2 E t γ 1 t+1 0. Beause [1 γ(γ + 1) 1 t+1 ]E t γ t+1 + γ2 E t γ 1 t+1 (25) [1 1 3 γ(γ + 1)]E t γ t+1 + γ2 E t γ 1 t+1 (sine t+1 3 almost surely), then for γ = 2 and t+1 suh that E t γ t+1 = 1 5 and E t γ 1 t+1 = 1 21, we have [1 1 3 γ(γ + 1)]E t γ t+1 + γ2 E t γ 1 t+1 = 1 5 + 4 < 0. (26) 21 Therefore, a higher Arrow-Pratt relative risk aversion oeffiient is neither a neessary nor a suffiient ondition to obtain higher risk premium. 3.3 Changes in joint distributions The question dual to the hange in risk aversion examined above is as follows: Under whih ondition does a hange in the joint distribution of random payoff and onsumption inrease the asset s prie? We may also ask the same question for the risk premium by using the joint distribution of an asset s gross return and onsumption. To address these questions, let us denote Et H and F ED H as the expetation and first order expetation dependeny under distribution H(x, y). Let p F t and p H t denote the orresponding pries under distributions F (x, y) and H(x, y) respetively. From (17), we have p H t = EH t x t+1 R f β F ED H ( x t+1 t+1 )H Ct+1 ( t+1 )[ u ( t+1 ) u ( t ) ]d t+1. (27) Similarly, from (18) we have E H t R t+1 R f = F ED H ( R t+1 t+1 )H Ct+1 ( t+1 )[ u ( t+1 ) E H t u ( t+1 ) ]d t+1. (28) From (17), (18), (27) and (28), we obtain the following result. Proposition 3.4 The following statements hold: (i) Suppose E F t x t+1 = E H t x t+1, then p F t p H t for all risk averse representative agents if and only if F (x, y) is more first-degree expetation dependent than H(x, y); 10
(ii) For all risk averse representative agents, F (x, y) is more first-degree expetation dependent than H(x, y) if and only if the risk premium under F (x, y) is greater than under H(x, y). Proof See Appendix A. Part (i) of Proposition 3.4 shows that a pure inrease in first-degree expetation dependene represents an inrease in asset riskiness for all risk averse investors. The next orollary onsiders a simultaneous variation in expeted return. Corollary 3.5 For all risk averse representative agents, E F t x t+1 E H t x t+1 and F (x, y) is more first-degree expetation dependent than H(x, y) imply p F t p H t ; Proof The suffiient onditions are diretly obtained from (17) and (27). Corollary 3.5 states that, for all risk averse representative agents, a derease in the expeted return and the first-degree expetation dependene between return and onsumption will derease the asset s prie. Again, the key available onept for predition is omparative first-degree expetation dependene. 4 C-CAPM for a higher-order risk averse representative agent 4.1 C-CAPM for a risk averse and prudent representative agent The onept of prudene and its relationship to preautionary savings was introdued by Kimball (1990). Sine then, prudene has beome a ommon and aepted assumption in the eonomis literature (Gollier 2001). All prudent agents dislike any inrease in downside risk in the sense of Menezes et al. (1980) (See also Chiu, 2005.). Dek and Shlesinger (2010) provide a laboratory experiment to determine whether preferenes are prudent and show behavioural evidene for prudene. In this setion, we will demonstrate that we an get weaker dependene onditions for asset prie and equity premium than first-degree expetation dependene, when the representative agent is risk averse and prudent. We an integrate the right-hand term of (16) by parts and obtain: ov t [u ( t+1 ), x t+1 ] = = F ED( x t+1 t+1 )u ( t+1 )F Ct+1 ( t+1 )d t+1 (29) t+1 u ( t+1 )d( [E x t+1 E( x t+1 t+1 s)]f Ct+1 (s)ds) 11
t+1 = u ( t+1 ) [E x t+1 E( x t+1 t+1 s)]f Ct+1 (s)ds t+1 = u () t+1 E x t+1 E( x t+1 t+1 s)]f Ct+1 (s)dsu ( t+1 )d t+1 [E x t+1 E( x t+1 t+1 s)]f Ct+1 (s)ds = u ()ov t ( x t+1, t+1 ) E x t+1 E( x t+1 t+1 s)]f Ct+1 (s)dsu ( t+1 )d t+1 SED( x t+1 t+1 )u ( t+1 )d t+1. From equation (5), we know that a positive SED implies a positive ov( x t+1, t+1 ) but the onverse is not true. Hene, we have from (29) that ov t ( x t+1, t+1 ) 0 is only a neessary ondition for ov t [u ( t+1 ), x t+1 ] 0 for all u 0 and u 0. With a positive SED funtion, prudene is also neessary. (11) and (12) an be rewritten as: or p t = E t x t+1 } R{{ f βov t ( x t+1, t+1 )[ u () } u ( t ) ] }{{} disounted present value effet ovariane effet β SED( x t+1 t+1 )[ u ( t+1 ) u ]d t+1 ( t ) }{{} seond degree expetation dependene effet (30) p t = E t x t+1 R f βov t ( x t+1, t+1 )AR()MRS,t (31) β SED( x t+1 t+1 )AP ( t+1 )MRS t+1, t d t+1, where AP (x) = u (x) u (x) is the index of absolute prudene 4, and or E t Rt+1 R f = ov t ( R t+1, t+1 )[ u () E t u ( t+1 ) ] + }{{} onsumption ovariane effet SED( R t+1 t+1 ) u ( t+1 ) E t u ( t+1 ) d t+1 }{{} onsumption seond degree expetation dependene effet (32) E t Rt+1 R f = βr f ov t ( R t+1, t+1 )AR()MRS,t + βr f SED( R t+1 t+1 )AP ( t+1 )MRS t+1, t d t+1. 4 Modia and Sarsini (2005), Crainih and Eekhoudt (2008) and Denuit and Eekhoudt (2010) propose u (x) u (x) instead of u (x) u (x) (Kimball, 1990) as an alternative andidate to evaluate the intensity of prudene. 12 (33)
Condition (31) inludes three terms. The first one is the same as in ondition (17). The seond term on the right-hand side of (31) is alled the ovariane effet. This term involves β, the ovariane of asset return and onsumption, the Arrow-Pratt risk aversion oeffiient and the marginal rates of substitution. The third term on the right-hand side of (31) is alled seonddegree expetation dependene effet, whih reflets the way in whih seond-degree expetation dependene of risk affets asset s prie through the intensity of downside risk aversion. Again (31) affirms that positive orrelation is only a neessary ondition for all risk averse and prudent agents to pay a lower prie. Equation (32) shows that a positive SED reinfores the positive ovariane effet to obtain a positive risk premium. We state the following propositions without proof (The proofs of these propositions are similar to the proofs of Propositions in Setion 3, and are therefore skipped. They are however available from the authors.). Proposition 4.1 The following statements hold: (i) p t E t x t+1 R f and only if ( x t+1, t+1 ) F 2 ; (ii) p t Et x t+1 R f and only if ( x t+1, t+1 ) G 2 ; for any risk averse and prudent representative agent (u 0 and u 0) if for any risk averse and prudent representative agent (u 0 and u 0) if (iii) E t Rt+1 R f for all risk averse and prudent representative agents (u 0 and u 0) if and only if ( x t+1, t+1 ) F 2 ; (iv) E t Rt+1 R f for all risk averse and prudent representative agents (u 0 and u 0) if and only if ( x t+1, t+1 ) G 2. Modia and Sarsini (2005) provide a omparative statis riterion for downside risk in the spirit of Ross (1981). Definition 4.2 (Modia and Sarsini 2005) u is more downside risk averse than v if and only if there exists λ > 0 suh that for all x, y u (x) v (x) λ u (y) v (y). (34) More downside risk aversion an guarantee that the deision-maker with a utility funtion that has more downside risk aversion is willing to pay more to avoid the downside risk inrease as defined by Menezes et al. (1980). We an therefore extend Proposition 3.3 as follows: 13
Proposition 4.3 The following statements hold: (i) p u t p v t for all ( x t+1, t+1 ) F 2 if and only if v is more Ross and downside risk averse than u; (ii) p u t p v t for all ( x t+1, t+1 ) G 2 if and only if u is more Ross and downside risk averse than v; (iii) u has a larger risk premium than agent v for all ( x t+1, t+1 ) F 2 if and only if u is more Ross and downside risk averse than v; (iv) u has a larger risk premium than agent v for all ( x t+1, t+1 ) G 2 if and only if v is more Ross and downside risk averse than u. We also obtain the following results for hanges in joint distributions. Proposition 4.4 The following statements hold: (i) Suppose Et F x t+1 = Et H x t+1, then p F t p H t for all risk averse and prudent representative agents if and only if F (x, y) is more seond-degree expetation dependent than H(x, y); (ii) For all risk averse and prudent representative agents, F (x, y) is more seond-degree expetation dependent than H(x, y) if and only if the risk premium under F (x, y) is greater than under H(x, y). Corollary 4.5 For all risk averse and prudent representative agents, E F t x t+1 E H t x t+1 and F (x, y) is more seond-degree expetation dependent than H(x, y) implies p F t p H t ; 4.2 C-CAPM for a higher-order representative agent Ekern (1980) provides the following definition to sign the higher-order risk attitude. Definition 4.6 (Ekern 1980) An agent u is N th degree risk averse, if and only if ( 1) N u (N) (x) 0 for all x, (35) where u (N) ( ) denotes the N th derivative of u(x). Ekern (1980) shows that all agents having utility funtion with N th degree risk aversion dislike a probability hange if and only if it produes an inrease in N th degree risk. Risk aversion in the traditional sense of a onave utility funtion is indiated by N = 2. When N = 3, we obtain u 0 whih means that marginal utility is onvex, or implies prudene. Eekhoudt and 14
Shlesinger (2006) derive a lass of lottery pairs to show that lottery preferenes are ompatible with Ekern s N th degree risk aversion. Jindapon and Neilson (2007) generalize Ross risk aversion to higher-order risk aversion. Definition 4.7 (Jindapon and Neilson 2007) u is more N th -degree Ross risk averse than v if and only if there exists λ > 0 suh that for all x, y u (N) (x) v (N) (x) λ u (y) v (y). (36) Li (2009) and Denuit and Eekhoudt (2010) provide ontext-free explanations for higher-order Ross risk aversion. In Appendix B, we generalize the results of setion 3 and 4.1 to higher-degree risks and higher order representative agents. 5 Asset pries and two loal absolute indexes of risk attitude If we assume that t and t+1 are lose enough, then we an use the loal oeffiient of risk aversion and loal downside risk aversion (see Modia and Sarsini, 2005) 5 to obtain the following first-order approximation formulas for (17) and (30): and p t E t x t+1 R f + β u ( t ) u F ED( x t+1 t+1 )F Ct+1 ( t+1 )d t+1 (37) ( t ) = E t x t+1 R f βar( t )ov t ( x t+1, t+1 ) p t E t x t+1 R f + β u ( t ) u ( t ) ov t( x t+1, t+1 ) β u ( t ) u ( t ) = E t x t+1 R f βar( t )ov t ( x t+1, t+1 ) βap ( t ) SED( x t+1 t+1 )d t+1 (38) SED( x t+1 t+1 )d t+1. When the variation of onsumption is small, (37) implies that absolute risk aversion and ovariane determine asset pries while (38) implies that absolute risk aversion, absolute prudene, ovariane and SED determine asset pries. We mentioned before that SED( x y) is related to 2nd entral ross lower partial moment of x and ỹ, hene (38) provides a theoretial explanation of the importane of higher-order risk preferenes, higher-order moments and omoments in 5 See Denuit and Eekhoudt (2010) for higher order risk attitudes. 15
finane 6. We must emphasize that we obtain only approximations of asset pries when we use Arrow-Pratt measures of risk aversion and prudene. 6 Conluding remarks on the equity premium puzzle We disuss the impliations of our results on the equity premium puzzle. The major disrepany between the C-CAPM model preditions and empirial reality is identified as the equity premium puzzle in the literature. As mentioned in Setion 3, the key empirial observations of the equity premium puzzle based on (13) an be summarized as follows: When the representative agent s utility funtion is the power funtion, and t+1 and x t+1 are onditional lognormally distributed, observed equity premium an be explained only by assuming a very high oeffiient of relative risk aversion. In other words, it is diffiult to explain the existene of observed high risk premia with the ovariane beause of the smoothness of onsumption over time. However, the equity premium puzzle onlusion is built on speifi utility funtions and return distributions. Our results show that, for general utility funtions and distributions, ovariane is not the key element of equity premium predition. It is very easy to find ounter intuitive results. For example, given positively orrelated gross return and onsumption distributions, a lower Arrow-Pratt oeffiient of relative risk aversion may result in higher equity premium. Alternatively, given representative agent s preferene, a lower ovariane between gross return and onsumption may result in a higher equity premium. Therefore, (13) is not a robust theoretial predition of equity premia. Our results prove that asset priing and equity premium settings and their omparative statis imply the following robust preditions: (a) it is expetation dependene between gross return and onsumption that determines asset riskiness rather than ovariane; (b) when gross return and onsumption are positive expetation dependent, higher risk aversion in the sense of Ross is equivalent to a higher equity premium; () when a representative agent s risk preferene satisfies higher-order risk aversion, more expetation dependene between gross return and onsumption is equivalent to higher equity 6 For the empirial studies of higher-order risk preferenes, higher-order moments and omoments in finane, we refer to Harvey and Siddique (2000); Dittmar (2002); Mitton and Vorkink (2007) and Martellini and Ziemann (2010) 16
premium. Beause the omparative Ross risk aversion is fairly restritive upon preferene, some readers may regard (b) as negative, beause no standard utility funtions satisfy suh ondition on the whole domain. However, there are utility funtions satisfying omparative Ross risk aversion on some domain. For example, Crainih and Eekhoudt (2008) and Denuit and Eekhoudt (2010) assert that ( 1) N+1 u(n) u is an appropriate loal index of N th order risk attitude. On the other hand, some readers may think that the fat that no standard utility funtions satisfy these onditions would undersore the need to develop experimental methods to identify these onditions. Ross (1981), Modia and Sarsini (2005), Li (2009) and Denuit and Eekhoudt (2010) provide ontext-free experiments for omparative Ross risk aversion. More researh is needed in both diretions to develop the theoretial foundations for C-CAPM. This paper takes a first step in that diretion. We have proposed a new unified interpretation to C-CAPM, whih we have related to the equity premium puzzle problem. Our results are important beause C- CAPM shares the positive versus normative tensions that are present in finane and eonomis to explain asset pries and equity premia. 7 Appendix A: Proofs of propositions 7.1 Proof of Proposition 3.1 (i): The suffiient onditions are diretly obtained from (17) and (18). We prove the neessity by a ontradition. Suppose that F ED( x t+1 t+1 ) < 0 for 0 t+1. Beause of the ontinuity of F ED( x y), we have F ED( x t+1 0 t+1 ) < 0 in interval [a,b]. Choose the following utility funtion: αx e a x < a ū(x) = αx e x a x b (39) αx e b x > b, where α > 0. Then ū (x) = α α + e x x < a a x b (40) α x > b 17
and ū (x) = 0 x < a e x a x b (41) 0 x > b. Therefore, p t = E t x t+1 R f β 1 b u F ED( x t+1 t+1 )F Ct+1 ( t+1 )e t+1 d t+1 > E t x t+1 ( t ) a R f. (42) This is a ontradition. (ii) (iii) and (iv): We an prove them by the same approah used in (i). 7.2 Proof of Proposition 3.3 (i): The suffiient onditions are diretly obtained from (17), (18), (21) and (22). We prove the neessity by a ontradition. Suppose that there exists some t+1 and t suh that u ( t+1 ) v ( t+1 ) > u ( t) v ( t ). Beause u, v, u and v are ontinuous, we have hene u ( t+1 ) v ( t+1 ) > u ( t ) v ( t ) u ( t+1 ) v ( t+1 ) > u ( t ) v ( t ) for all ( t+1, t ) [γ 1, γ 2 ], (43) for all ( t+1, t ) [γ 1, γ 2 ], (44) and u ( t+1 ) u ( t ) > v ( t+1 ) v ( t ) for all ( t+1, t ) [γ 1, γ 2 ]. (45) If F (x, y) is a distribution funtion suh that F ED( x t+1 t+1 )F Y (y) is stritly positive on interval [γ 1, γ 2 ] and is equal to zero on other intervals, then we have This is a ontradition. γ2 p u t p v t = β F ED( x t+1 t+1 )F Y (y)[ u ( t+1 ) γ 1 u ( t ) (ii) (iii) and (iv): We an prove them by the same approah used in (i). v ( t+1 ) v ] < 0. (46) ( t ) 7.3 Proof of Proposition 3.4 (i): The suffiient onditions are diretly obtained from (17), (18), (27) and (28). We prove the neessity by a ontradition. Suppose F ED F ( x t+1 t+1 )F Ct+1 ( t+1 ) < F ED H ( x t+1 t+1 )H Ct+1 ( t+1 ) for 0 t+1. Owing the ontinuity of F ED F ( x t+1 t+1 )F Ct+1 ( t+1 ) F ED H ( x t+1 t+1 )H Ct+1 ( t+1 ), 18
we have F ED F ( x t+1 0 t+1 )F C t+1 ( 0 t+1 ) < F ED H( x t+1 0 t+1 )H C t+1 ( 0 t+1 ) in interval [a,b]. Choose the following utility funtion: ū(x) = αx e a x < a αx e x a x b αx e b x > b, (47) where α > 0. Then ū (x) = α α + e x x < a a x b (48) α x > b and ū (x) = 0 x < a e x a x b (49) 0 x > b. Therefore, p F t p H t = β 1 u ( t ) b a [F ED H ( x t+1 t+1 )F Ct+1 ( t+1 ) F ED F ( x t+1 y)f Ct+1 ( t+1 )]e t+1 d t+1 > 0. (ii): We an prove the seond part of the proposition by the same approah used in (i). (50) 8 Appendix B: Higher-order risks and higher order representative agents We integrate the right-hand term of (16) by parts repeatedly until we obtain: ov[u ( t+1 ), x t+1 ] = N ( 1) k u (k) ()k th ED( x t+1 ) (51) k=2 + ( 1) N+1 u (N+1) ( t+1 )N th ED( x t+1 t+1 )d t+1, for n 2. Then (11) and (12) an be rewritten as: p t = E t x t+1 R f }{{} disounted present value effet β N k th ED( x t+1 )[( 1) k+1 u(k) () u ( k=2 t ) ] }{{} higher order ross moments effet 19 (52)
β N th ED( x t+1 t+1 )[( 1) N+2 u(n+1) ( t+1 ) u ]d t+1 ( t ) }{{} N th order expetation dependene effet = E t x t+1 N R f β k th ED( x t+1 )AR (k) ()MRS,t k=2 β N th ED( x t+1 t+1 )AR (k+1) ( t+1 )MRS t+1, t d t+1 where AR (k) (x) = ( 1) k+1 u (k) (x) u (x) is the absolute index of k th order risk aversion, and = E t Rt+1 R f (53) N k th ED( R t+1 )[( 1) k+1 u (k) () E k=2 t u ( t+1 ) ] }{{} onsumption ross moments effet + N th ED( R t+1 t+1 )[( 1) N+2 u(n+1) ( t+1 ) E t u ( t+1 ) ] }{{} N th degree expetation dependene effet N = βr f k th ED( R t+1 )AR (k) ()MRS,t k=2 d t+1 +βr f N th ED( R t+1 t+1 )AR (k+1) ( t+1 )MRS t+1, t d t+1. Condition (52) inludes three terms. The first one is the same as in ondition (17). The seond term on the right-hand side of (52) is alled higher-order ross moments effet. This term involves β, the intensity of higher-order risk aversion, the marginal rates of substitution and the higher-order ross moments of asset return and onsumption. The third term on the right-hand side of (52) is alled N th degree expetation dependene effet, whih reflets the way in whih N th -degree expetation dependene of risks affet asset prie through the intensity of absolute N th risk aversion and the marginal rates of substitution. We state the following propositions without proof (The proofs of these propositions are similar to the proofs of Propositions in Setion 3, and are therefore skipped. They are however available from the authors.). Proposition 8.1 The following statements hold: (i) p t E t x t+1 R f if ( x t+1, t+1 ) F N ; (ii) p t Et x t+1 R f if ( x t+1, t+1 ) G N ; for any i th risk averse representative agent with i = 2,..., N + 1 if and only for any i th risk averse representative agent with i = 2,..., N + 1 if and only 20
(iii) E t Rt+1 R f for all i th risk averse representative agents with i = 2,..., N + 1 if and only if ( x t+1, t+1 ) F N ; (iv) E t Rt+1 R f for all i th risk averse representative agents with i = 2,..., N + 1 if and only if ( x t+1, t+1 ) G N. Proposition 8.1 suggests that, for a i th -degree risk averse representative agent with i = 1,.., n = 1, an asset s prie is lowered if and only if its payoff N th -order positively expetation depends on onsumption. Conversely, an asset s prie is raised if and only if it N th -order negatively expetation depends on onsumption. Therefore, for i th -degree representative agents with i = 1,.., N + 1, it is the N th -order expetation dependene that determines its riskiness. The next two propositions and Corollary 8.4 have a similar general intuition when ompared with those in Setion 3. Proposition 8.2 The following statements hold: (i) p u t p v t for all ( x t+1, t+1 ) F N if and only if v is more i th risk averse than u for i = 2,..., N + 1; (ii) p u t p v t for all ( x t+1, t+1 ) G N if and only if u is more i th risk averse than v for i = 2,..., N + 1; (iii) u has a larger risk premium than agent v for all ( x t+1, t+1 ) F N if and only if u is more Ross i th risk averse than v for i = 2,..., N + 1; (iv) u has a larger risk premium than agent v for all ( x t+1, t+1 ) G N if and only if v is Ross i th risk averse than v for i = 2,..., N + 1. Proposition 8.3 The following statements hold: (i) Suppose Et F x t+1 = Et H x t+1, then p F t p H t for all i th risk averse representative agents with i = 2,..., N + 1 if and only if F (x, y) is N th more expetation dependent than H(x, y); (ii) For all i th risk averse representative agents with i = 2,..., N + 1,, F (x, y) is more i th - degree expetation dependent than H(x, y) if and only if the risk premium under F (x, y) is greater than H(x, y). Corollary 8.4 For all i th risk averse representative agents with i = 2,..., N + 1, E F t x t+1 E H t x t+1 and F (x, y) is more N th expetation dependent than H(x, y) implies p F t p H t ; 21
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