The Consumption Capital Asset Pricing Model (C-CAPM)
Consider a representative agent solving the joint consumption and portfolio allocation problem:
v ( m t ) = max u ( c t ) + E t [ ∑ n = 1 ∞ β n u ( c t + n ) ] s.t. m t + 1 = ( m t − c t ) R ~ t + 1 + y t + 1 R ~ t + 1 = ∑ i = 1 m ω t , i R t + 1 , i + ( 1 − ∑ i = 1 m ω t , i ) R \begin{gathered}\begin{aligned}
\vFunc(\mRat_{t}) & = \max \uFunc(c_{t}) + \Ex_{t}\left[\sum_{n=1}^{\infty} \Discount^{n} \uFunc(c_{t+n}) \right]
\\ & \text{s.t.}
\\ \mRat_{t+1} & = (\mRat_{t}-c_{t})\Rport_{t+1} + y_{t+1}
\\ \Rport_{t+1} & = \sum_{i=1}^{m}\omega_{t,i}\Risky_{t+1,i}+\left(1-\sum_{i=1}^{m} \omega_{t,i}\right)\Rfree
\end{aligned}\end{gathered} v ( m t ) m t + 1 R ~ t + 1 = max u ( c t ) + E t [ n = 1 ∑ ∞ β n u ( c t + n ) ] s.t. = ( m t − c t ) R ~ t + 1 + y t + 1 = i = 1 ∑ m ω t , i R t + 1 , i + ( 1 − i = 1 ∑ m ω t , i ) R where R \Rfree R R t + 1 , i \Risky_{t+1,i} R t + 1 , i i i i t t t t + 1 t+1 t + 1 ω t , i \omega_{t,i} ω t , i i i i R ~ t + 1 \Rport_{t+1} R ~ t + 1 y t + 1 y_{t+1} y t + 1 t + 1 t+1 t + 1
As usual, the objective can be rewritten in recursive form:
v ( m t ) = max { c t , ω 1 , t , ω 2 , t , … } u ( c t ) + β E t [ v ( R ~ t + 1 ( m t − c t ) + y t + 1 ) ] \begin{gathered}\begin{aligned}
\vFunc(\mRat_{t}) & = \max_{\{c_{t},\omega_{1,t},\omega_{2,t},\ldots\}} \uFunc(c_{t}) +\beta \Ex_{t}\left[\vFunc(\Rport_{t+1}(\mRat_{t}-c_{t})+{y}_{t+1})\right]
\end{aligned}\end{gathered} v ( m t ) = { c t , ω 1 , t , ω 2 , t , … } max u ( c t ) + β E t [ v ( R ~ t + 1 ( m t − c t ) + y t + 1 ) ] The first order condition with respect to c t c_{t} c t
u ′ ( c t ) = β E t [ R ~ t + 1 v ′ ( m t + 1 ) ] . \begin{gathered}\begin{aligned}
\uFunc^{\prime}(c_{t}) & = \beta \Ex_{t}[\Rport_{t+1}\vFunc^{\prime}({m}_{t+1})].
\end{aligned}\end{gathered} u ′ ( c t ) = β E t [ R ~ t + 1 v ′ ( m t + 1 )] . and the FOC with respect to ω t , i \omega_{t,i} ω t , i
E t [ ( R t + 1 , i − R ) v ′ ( m t + 1 ) ] = 0 \begin{gathered}\begin{aligned}
\Ex_{t}[(\Risky_{t+1,i}-\Rfree)\vFunc^{\prime}({m}_{t+1})] & = 0
\end{aligned}\end{gathered} E t [( R t + 1 , i − R ) v ′ ( m t + 1 )] = 0 But the usual logic of the Envelope theorem tells us that
u ′ ( c t + 1 ) = v ′ ( m t + 1 ) , \begin{gathered}\begin{aligned}
\uFunc^{\prime}(c_{t+1}) & = \vFunc^{\prime}(\mRat_{t+1}),
\end{aligned}\end{gathered} u ′ ( c t + 1 ) = v ′ ( m t + 1 ) , so, substituting (5) (3) (4)
u ′ ( c t ) = E t [ β R ~ t + 1 u ′ ( c t + 1 ) ] E t [ ( R t + 1 , i − R ) u ′ ( c t + 1 ) ] = 0. \begin{gathered}\begin{aligned}
\uFunc^{\prime}(c_{t}) & = \Ex_{t}\left[\beta \Rport_{t+1} \uFunc^{\prime}({c}_{t+1})\right]
\\ \Ex_{t}[(\Risky_{t+1,i}-\Rfree)\uFunc^{\prime}({c}_{t+1})] & = 0
.
\end{aligned}\end{gathered} u ′ ( c t ) E t [( R t + 1 , i − R ) u ′ ( c t + 1 )] = E t [ β R ~ t + 1 u ′ ( c t + 1 ) ] = 0. Now assume CRRA utility, u ( c ) ≡ c 1 − ρ / ( 1 − ρ ) \uFunc(c) \equiv c^{1-\rho}/(1-\rho) u ( c ) ≡ c 1 − ρ / ( 1 − ρ ) (6) c t − ρ c_{t}^{-\rho} c t − ρ
E t [ ( c t + 1 / c t ) − ρ ( R t + 1 , i − R ) ] = 0 \begin{gathered}\begin{aligned}
\Ex_{t}[(c_{t+1}/c_{t})^{-\rho}(\Risky_{t+1,i}-\Rfree)] & = 0
\end{aligned}\end{gathered} E t [( c t + 1 / c t ) − ρ ( R t + 1 , i − R )] = 0 We can now follow the same steps as in the “Equity Premium Puzzle” section to obtain the relation that for every asset i i i
E t [ R t + 1 , i ] − R ≈ ρ cov t ( Δ log c t + 1 , R t + 1 , i ) 1 − ρ E t [ Δ log c t + 1 ] ≈ ρ cov t ( Δ log c t + 1 , R t + 1 , i ) \begin{gathered}\begin{aligned}
\Ex_{t}[\Risky_{t+1,i}]-\Rfree & \approx \frac{\rho \text{cov}_{t}(\Delta \log {c}_{t+1},\Risky_{t+1,i})}{1-\rho \Ex_{t}[ \Delta \log {c}_{t+1}]}
\\ & \approx \rho \text{cov}_{t}(\Delta \log {c}_{t+1},\Risky_{t+1,i})
\end{aligned}\end{gathered} E t [ R t + 1 , i ] − R ≈ 1 − ρ E t [ Δ log c t + 1 ] ρ cov t ( Δ log c t + 1 , R t + 1 , i ) ≈ ρ cov t ( Δ log c t + 1 , R t + 1 , i ) What does this imply about asset pricing?
Consider an asset for which the return covaries positively with consumption, cov ( R t + 1 , i , Δ log c t + 1 ) > 0 \mbox{cov}(\Risky_{t+1,i},\Delta \log c_{t+1}) > 0 cov ( R t + 1 , i , Δ log c t + 1 ) > 0 marginal utility will negatively covary with the return. Thus the expected return must be higher for an asset that “does well” when consumption is high. But for a given average stream of dividends or payouts, if the average return is high, the average price must be low. Thus this indicates that prices should be low for assets whose payoffs are procyclical, and high for assets whose payoffs are countercyclical.