Asset Pricing Implications

In this subsection I derive several spillover effects of the Business Cycle theory into the Asset Pricing theory, by using consumption-based preferences. The analyze is quite common and follows the main literature (Campbell 2003; Cochrane 2001).

Major parts of this analyze is following, e.g. by Lettau and Uhlig (2002) and Uhlig (2004a). In the following, I review these findings and their derivations for all prefer-ences used in this paper and figure out the importance of the respective preference by evaluating stylized business cycle facts like the Frisch elasticity and the EIS as well as asset pricing facts like the Sharpe ratio.

At first let me decompose the logarithmic returns and the logarithmic pricing kernel into their expected values and innovations:

r_i,t+1 =E_tr_i,t+1+ε_R,t+1 and m_t+1 =E_tm_t+1+ε_M,t+1, (2.19) where r_i = log (R_i), m = log (M), while the innovations of the log-returns of asset i and the logarithmic pricing kernel are assumed to be normally distributed with ε_Ri,t ∼ N0, σ_R²_i,t and ε_M,t ∼ N0, σ²_M,t.

Assuming that the pricing kernel and asset returns are log-normal distributed and homoscedastic, the Euler equation (2.4) for any asset i can be written as:

0 = E_tr_i,t+1+E_tm_t+1+ 1 2

σ_R²

i+σ²_m+ 2σ_Rim, (2.20) where σ_Rim is the unconditional covariance of the innovations in returns and the pricing kernel. Obviously, given this equation the expected return of a risky asset is given by,

E_tr_i,t+1+ σ²_R

i

2 =−E_tm_t+1− 1 2

σ_m² + 2σ_Rim, (2.21) where the second term on the left hand side is the Jensen’s inequality adjustment, because of using expectations of log returns. For risk-less asset the variance and covariance are zero and (2.21) reduces to:

Etrf,t+1 =−Etmt+1− σ_m²

2 (2.22)

Using both, (2.21) and (2.22), its obviously that the risk premium of a risky asset i

over a riskless asset is given by:

E_t[r_i,t+1−r_f,t+1] +σ_R²

i

2 =−σ_Rim. (2.23)

Given this general derivation, it is easy to resolve these steps given the preference-based pricing kernel, which depends on leisure and consumption. Similarly, we can decompose log-consumption and log-leisure into,

c_t+1 =E_tc_t+1+ε_c,t+1 and l_t+1 =E_tl_t+1+ε_l,t+1, (2.24) and the pricing kernel M can now be decomposed into:

mt+1 =Etmt+1+ηmcεc,t+1+ηmlεl,t+1, (2.25) whereη_mcandη_mlare the elasticities of the pricing kernel with respect to innovations in consumption and leisure. The innovations in consumption and log-leisure are assumed to be independently normally distributed withεc,t∼ N0, σ²_c,t and ε_l,t∼ N 0, σ_l,t² .

Given these assumptions the corresponding risk premium can be calculated as:

E_t[r_i,t+1−r_f,t+1] + σ²_R

i

2 =−(η_mcσ_Ric+η_mlσ_Ril) . (2.26) As shown by Hansen and Jagannathan (1991), the first-order condition for excess returns, 0 =E_t^hM_t+1R^e_t+1ⁱ, implies that the Sharpe ratio (SR) for any asset is given by:

SRt = max

{all assets}

E_tR^e_t+1

σ_t(R^e_t+1) =−ρt

Mt+1, R^e_t+1 σt(M_t+1)

E_t(M_t+1) ≤ σt(M_t+1)

E_t(M_t+1) , (2.27) where ρ_t is the conditional correlation between the excess return and the pricing kernel. Using the log-normality characteristics of M, the largest possible Sharpe ratio is given by (see Campbell and Cochrane 1999):

SR_t= σt(exp (M_t+1))

E_t(exp (M_t+1)) =expσ_m,t² −1^1/2 (2.28) For the nonseparable preferences used in this paper this implies that the highest

possible Sharpe ratio is given by:

SR_t=expη_mc² σ²_c,t+η²_mlσ_l,t² −1^1/2 (2.29) This representation of the Sharpe ratio implies a perfect correlation between the pricing kernel and asset returns. Within CBAPM the pricing kernel is driven by consumption, and also leisure in my case, where consumption has an smaller corre-lation than one and next to that leisure is negative correcorre-lation with asset returns.

A more accurate representation for the Sharpe ratio can be found by assuming that the risk premium is measured as R^e_t =R_t/R_t^f. Using (eq. 2.26), the Sharpe ratio is given by:

SRt =−ηmcσc,tρcR−ηmlρlRσl,t , (2.30) where ρcR, ρlR are the correlation of consumption and leisure with asset returns.

By equations 2.26 2.29, the risk premium and the Sharpe ratio depend only on the elasticities η_mc and η_ml, which are both denoted by the preference parameters.

To illustrate the spillover effects between the Business Cycle literature and asset pricing literature by using consumption-based asset pricing models, let investigate these parameter in more detail (e.g. Lettau and Uhlig 2002; Uhlig 2004a).

For all power utility models e.g. CRRA, habit formation, "Catching up with the Joneses", or Campbell-Cochrane as widely discussed in the literature, and also for the extensions used in this paper holds that the elasticity ηmc is related to the elasticity of intertemporal consumption substitution (EIS) since:

η_mc =− 1

EIS . (2.31)

To elicit the elasticities η_mc and η_ml for the preferences used in this paper, first, I investigate the internal and external habit model. Recall, that log-consumption and log-leisure each follow an auto-regressive processes described in (eq. 2.24) and note that the pricing kernel (eq. 2.11) depends on conditional expectations. As shown by Lettau and Uhlig (2002) the elasticity η_mc can be derived as:

η_mc= a 1− H¯^c

C¯

· 1 +τ_c

H¯^c C¯ βνe^agc

1−τ_cβνe^agc , (2.32)

where gc is the trend of log-consumption and ¯H^c/C¯ =νe^−gc the steady-state value

of the fraction of habit of total consumption. As mentioned before, τ_c controls for internal vs. external habit formation. In the case of external habit formation the formula reduces to the first factor of the right hand side of 2.32, represents the effect that the elasticity of changes in consumption depends on the effective consumption, which is a proportion of consumption net of the corresponding habit (Lettau and Uhlig 2002). The second factor of the right hand side of 2.32 only occur for internal habit formation (τ_c = 1) and represents that within time-nonseparable preferences an increase in consumption today also has an effect on the marginal utility tomorrow.

Recapturing the formulation for a in eq. 2.9, if the parameter χ decreases, the proportion of consumption with respect to leisure in the utility decreases, and the elasticityη_mcdecreases as well, and finally the willingness to substitute consumption between today and tomorrow, as captured by the EIS increases. A similar result can be found for the second elasticity,

ηml = b 1−H¯^l

L¯

·1 +τ_cβψνe^agc

1−τ_cβνe^agc , (2.33)

where the steady state value of the fraction of habit of total leisure is ¯H^l/L¯ = ψ because leisure is modeled as a stationary AR(1)-process. As shown by Uhlig (2004a), this elasticity is the cross-derivative of consumption and leisure.

Note that the elasticities η_mc and η_ml are, up to here, both constant over time.

Evaluating these elasticities for the preferences used in theexternal-ARmodel yields a time-varying elasticity:

η_mc,t=− 1

EIS_t =a·(1 +λ(s^c_t)) . (2.34) The time-variation of η_mc is an important feature of the model by Campbell and Cochrane (1999), because it allows to model a time-varying Sharpe ratio as observed in the data. Therefore, the EIS is also changing over time. For η_ml the findings are similar:

η_ml,t =b·1 +λs^l_t (2.35) To compare the EISs and Sharpe ratios implied by the different models, I refer to averages over time for the elasticities, EIS, or the Sharpe ratio for these preferences.

As shown, a CBAPM prices assets by the elasticities η_mc and η_ml, which are de-noted only by the preference parameters. These parameters also specify individual behavior as illustrated by EIS. Furthermore, the Business Cycle literature uses the

marginal utilities of leisure and consumption to resolve for wages, which the agents demand for a reduction of his leisure. In absence of any real or nominal rigidities, such frictionless wages are given by:

w_t^f = M U_t^l M U_t^c ,

where M U_t^c is the marginal utility regarding consumption and M U_t^l the marginal utility regarding leisure. Obviously, the frictionless wage, w^f, is the pricein addi-tional consumption units the agent demands for reducing leisure by one unit.

As motivated by Uhlig (2007), the Frisch elasticity (FE) seems to be a good measure to capture this additional feature of the preferences. Moreover, the Frisch elasticity is a well investigated measure in the empirical literature (see section 2.4.1). It is defined as the elasticity of labor supply with respect to frictionless wages, holding the marginal rate of consumption constant,

FE = dn dw^f

w^f n

_U

c

. (2.36)

For the preferences used in this paper the Frisch elasticity can be evaluated as:

FE = Un

¯ n

"

U_nn− U_nc² U_cc

# (2.37)

Under the common assumption that in the steady state, leisure is given by two-third of the total time endowment (¯l = 2/3), the Frisch elasticity can be written depending on preference parameters:

FE = (1−ψ) ¯l 1−¯l

· (1−χ(1−γ)) (1−τ βψ) 1 +τ βν²

γ+τ β(1−χ)χ^h(ν−ψ)²(1 +γ²) + 4νψ+ 2ψ²+ 2−^{hν2+(1+β)ψ}_(1−χ)χ ²ⁱⁱ (2.38) The equation looks quite scary. This is due to the fact that both consumption and leisure decision today have an effect on the marginal utilities of leisure and of consumption tomorrow. Furthermore, the equation nicely illustrates, what happens if the number of former periods within the habits is increasing. For I = J % ∞ the Frisch elasticity reduces to the well-known solution for external habit formation, namely:

FE = (1−ψ) ¯l

1−¯l · 1−χ(1−γ)

γ (2.39)

This confirms the discussion above, that including lags into the internal habit, re-moves the differences between internal and external habit formation.

Solving for the Frisch elasticity for the external-AR model, the extended Campbell-Cochrane preferences, the following equation obtains:

FE_t=S_t^l l¯_t

1−¯lt

· 1−χ(1−γ)

γ . (2.40)

Similar to EIS and Sharpe ratio the Frisch elasticity is time-varying for this utility function. The second term of the right hand side of this equation is the Frisch elasticity if no habit in leisure would exist (φ = 0). This part of the equation is constant over time. If the leisure of the individual is only a little bit larger than his habit of leisure, S_t^l is small and the Frisch elasticity decreases. This means that the individual is less willing to increase working time if wages are increasing. By contrast when S_t^l is large, the agent would increase his working time also for small increases in wages.

Im Dokument Essays on asset pricing and the macroeconomy (Seite 39-44)