CRRA Portfolio Choice with Two Risky Assets
Merton (1969) Samuelson (1969) u ( c ) = ( 1 − ρ ) − 1 c 1 − ρ \uFunc(c) = (1-\CRRA)^{-1}c^{1-\CRRA} u ( c ) = ( 1 − ρ ) − 1 c 1 − ρ
Using their framework, here we study a consumer who has wealth a t \aRat_{t} a t t t t R t + 1 = ( R 1 , t + 1 , R 2 , t + 1 ) ′ \Risky_{t+1}=(\Risky_{1,t+1}, \Risky_{2,t+1})' R t + 1 = ( R 1 , t + 1 , R 2 , t + 1 ) ′ log R t + 1 = r t + 1 = ( r 1 , t + 1 , r 2 , t + 1 ) ′ ∼ ( N ( r 1 , σ 1 2 ) , N ( r 2 , σ 2 2 ) ) ′ \log \Risky_{t+1} = \risky_{t+1}=(\risky_{1,t+1}, \risky_{2,t+1})' \sim \left ( \mathcal{N}(\risky_1,\sigma^{2}_{1}), \mathcal{N}(\risky_2,\sigma^{2}_{2}) \right)' log R t + 1 = r t + 1 = ( r 1 , t + 1 , r 2 , t + 1 ) ′ ∼ ( N ( r 1 , σ 1 2 ) , N ( r 2 , σ 2 2 ) ) ′
( σ 1 2 σ 12 σ 12 σ 2 2 ) . \left(\begin{array}{cc}\sigma_1^2 & \sigma_{12} \\ \sigma_{12}& \sigma_2^2\end{array}\right). ( σ 1 2 σ 12 σ 12 σ 2 2 ) . If the period-t t t ς i \riskyshare_i ς i a t \aRat_{t} a t i i i i = 1 , 2 i=1,2 i = 1 , 2 ς 1 = ( 1 − ς 2 ) \riskyshare_{1}=(1-\riskyshare_{2}) ς 1 = ( 1 − ς 2 ) [1] t + 1 t+1 t + 1
c t + 1 = ( ς ⋅ R t + 1 ) ⏟ ≡ R ~ t + 1 a t c_{t+1} = \underbrace{(\riskyshare \cdot \Risky_{t+1} )}_{\equiv \Rport_{t+1}} \aRat_{t} c t + 1 = ≡ R ~ t + 1 ( ς ⋅ R t + 1 ) a t where R ~ t + 1 \Rport_{t+1} R ~ t + 1
Campbell & Viceira (2002)
r ~ t + 1 = r 1 , t + 1 + ς 2 ( r 2 , t + 1 − r 1 , t + 1 ) + ς 2 ( 1 − ς 2 ) η / 2 \rport_{t+1} = \risky_{1,t+1}+\riskyshare_{2} (\risky_{2,t+1}-\risky_{1,t+1})+ \riskyshare_2 (1-\riskyshare_2) \eta/2 r ~ t + 1 = r 1 , t + 1 + ς 2 ( r 2 , t + 1 − r 1 , t + 1 ) + ς 2 ( 1 − ς 2 ) η /2 where
η = ( σ 1 2 + σ 2 2 − 2 σ 12 ) . \eta=(\sigma_1^2+\sigma_2^2-2\sigma_{12}). η = ( σ 1 2 + σ 2 2 − 2 σ 12 ) . Using this approximation, the expectation as of date t t t t + 1 t+1 t + 1
E t [ u ( c t + 1 ) ] ≈ ( 1 − ρ ) − 1 E t [ ( a t e r 1 , t + 1 e ς 2 ( r 2 , t + 1 − r 1 , t + 1 ) + ς 2 ( 1 − ς 2 ) η / 2 ) 1 − ρ ] ≈ ( 1 − ρ ) − 1 a t 1 − ρ ⏟ constant < 0 e ( 1 − ρ ) ς 2 ( 1 − ς 2 ) η / 2 E t [ e ( r 1 , t + 1 + ς 2 ( r 2 , t + 1 − r 1 , t + 1 ) ) ( 1 − ρ ) ] ⏟ excess return utility factor \begin{aligned}
\Ex_{t}[\uFunc(c_{t+1})] & \approx (1-\CRRA)^{-1}\Ex_{t}\left[\left(\aRat_{t}e^{\risky_{1,t+1}}e^{\riskyshare_2 (\risky_{2,t+1}-\risky_{1,t+1})+\riskyshare_2(1-\riskyshare_2)\eta /2}\right)^{1-\CRRA}\right]
\\ & \approx \underbrace{ (1-\CRRA)^{-1}\aRat_{t}^{1-\CRRA}}_{\text{constant $< 0$}}\underbrace{e^{ (1-\CRRA)\riskyshare_2(1-\riskyshare_2)\eta/2}\Ex_{t}\left[e^{(\risky_{1,t+1}+\riskyshare_2 (\risky_{2,t+1}-\risky_{1,t+1})) (1-\CRRA)}\right]}_{\text{excess return utility factor}}
\end{aligned} E t [ u ( c t + 1 )] ≈ ( 1 − ρ ) − 1 E t [ ( a t e r 1 , t + 1 e ς 2 ( r 2 , t + 1 − r 1 , t + 1 ) + ς 2 ( 1 − ς 2 ) η /2 ) 1 − ρ ] ≈ constant < 0 ( 1 − ρ ) − 1 a t 1 − ρ excess return utility factor e ( 1 − ρ ) ς 2 ( 1 − ς 2 ) η /2 E t [ e ( r 1 , t + 1 + ς 2 ( r 2 , t + 1 − r 1 , t + 1 )) ( 1 − ρ ) ] where the first term is a negative constant under the usual assumption that relative risk aversion ρ > 1. \CRRA>1. ρ > 1.
Our foregoing assumptions imply that
( 1 − ρ ) ( ς 1 r 1 , t + 1 + ς 2 r 2 , t + 1 ) ∼ N ( ( 1 − ρ ) ( ς 1 r 1 + ς 2 r 2 ) , ( 1 − ρ ) 2 ( ς 1 2 σ 1 2 + ς 2 2 σ 2 2 + 2 ς 1 ς 2 σ 12 ) ) (1-\CRRA) (\riskyshare_1 \risky_{1,t+1}+\riskyshare_2 \risky_{2,t+1})
\sim \mathcal{N}( (1-\CRRA)(\riskyshare_1\risky_1+\riskyshare_2 \risky_2), (1-\CRRA)^2 ( \riskyshare_1^2\sigma_1^2+\riskyshare_2^2\sigma_2^2+2\riskyshare_1\riskyshare_2\sigma_{12})) ( 1 − ρ ) ( ς 1 r 1 , t + 1 + ς 2 r 2 , t + 1 ) ∼ N (( 1 − ρ ) ( ς 1 r 1 + ς 2 r 2 ) , ( 1 − ρ ) 2 ( ς 1 2 σ 1 2 + ς 2 2 σ 2 2 + 2 ς 1 ς 2 σ 12 )) (using LogELogNormTimes (5)
log E t [ e ( r 1 , t + 1 + ς 2 ( r 2 , t + 1 − r 1 , t + 1 ) ) ( 1 − ρ ) ] = ( 1 − ρ ) ( ς 1 r 1 + ς 2 r 2 ) + ( 1 − ρ ) 2 ( ς 1 2 σ 1 2 + ς 2 2 σ 2 2 + 2 ς 1 ς 2 σ 12 ) / 2 \log \Ex_{t}\left[e^{(\risky_{1,t+1}+\riskyshare_2 (\risky_{2,t+1}-\risky_{1,t+1})) (1-\CRRA)}\right] = {(1-\CRRA)(\riskyshare_1 \risky_1+\riskyshare_2 \risky_2) + (1-\CRRA)^2 (\riskyshare_1^2\sigma_1^2+\riskyshare_2^2\sigma_2^2+2\riskyshare_1\riskyshare_2\sigma_{12})/2} log E t [ e ( r 1 , t + 1 + ς 2 ( r 2 , t + 1 − r 1 , t + 1 )) ( 1 − ρ ) ] = ( 1 − ρ ) ( ς 1 r 1 + ς 2 r 2 ) + ( 1 − ρ ) 2 ( ς 1 2 σ 1 2 + ς 2 2 σ 2 2 + 2 ς 1 ς 2 σ 12 ) /2 Substituting from (7) (5) (5)
( 1 − ρ ) ς 2 ( 1 − ς 2 ) η / 2 + ( 1 − ρ ) ( r 1 + ς 2 ( r 2 − r 1 ) ) + ( ρ − 1 ) 2 ( σ 1 2 + ς 2 2 η + 2 ς 2 ( σ 12 − σ 2 2 ) ) / 2. (1-\CRRA)\riskyshare_2(1-\riskyshare_2)\eta/2+(1-\CRRA) (\risky_1+\riskyshare_2(\risky_2-\risky_1))+(\CRRA-1)^2 (\sigma_1^2+\riskyshare_2^2 \eta+2\riskyshare_2(\sigma_{12}-\sigma_2^2))/2
. ( 1 − ρ ) ς 2 ( 1 − ς 2 ) η /2 + ( 1 − ρ ) ( r 1 + ς 2 ( r 2 − r 1 )) + ( ρ − 1 ) 2 ( σ 1 2 + ς 2 2 η + 2 ς 2 ( σ 12 − σ 2 2 )) /2. The ς \riskyshare ς FOC for minimizing this expression is
( 1 − 2 ς 2 ) η / 2 + r 2 − r 1 + ( 1 − ρ ) ( ς 2 η + ( σ 12 − σ 1 2 ) ) = 0 ( r 2 − r 1 + η 2 ) + ( 1 − ρ ) ( σ 12 − σ 1 2 ) = ρ η ς 2 . \begin{aligned}
(1-2\riskyshare_2)\eta/2+ \risky_2-\risky_1+(1-\CRRA) (\riskyshare_2 \eta+(\sigma_{12}-\sigma_1^2)) & = 0 \\
\quad(\risky_2-\risky_1+\frac{\eta}{2})+(1-\CRRA) (\sigma_{12}-\sigma_1^2) & = \CRRA \eta \riskyshare_2.
\end{aligned} ( 1 − 2 ς 2 ) η /2 + r 2 − r 1 + ( 1 − ρ ) ( ς 2 η + ( σ 12 − σ 1 2 )) ( r 2 − r 1 + 2 η ) + ( 1 − ρ ) ( σ 12 − σ 1 2 ) = 0 = ρ η ς 2 . So
ς 2 = ( r 2 − r 1 + η / 2 + ( 1 − ρ ) ( σ 12 − σ 1 2 ) ρ η ) \riskyshare_2 = \left(\frac{\risky_2-\risky_1+\eta/2+(1-\CRRA)(\sigma_{12}-\sigma_1^2)}{\CRRA\eta}\right) ς 2 = ( ρ η r 2 − r 1 + η /2 + ( 1 − ρ ) ( σ 12 − σ 1 2 ) ) and note that if the first asset is riskfree so that σ 1 = σ 12 = 0 \sigma_{1}=\sigma_{12}=0 σ 1 = σ 12 = 0
ς 2 = ( r 2 − r 1 + σ 2 2 / 2 ρ σ 2 2 ) \riskyshare_2 = \left(\frac{\risky_2-\risky_1+\sigma^{2}_{2}/2}{\CRRA\sigma^{2}_{2}}\right) ς 2 = ( ρ σ 2 2 r 2 − r 1 + σ 2 2 /2 ) but the log of the expected return premium (in levels) on the risky over the safe asset in this case is φ ≡ log R 2 / R 1 = r 2 − r 1 + σ 2 2 / 2 \EpremLog \equiv \log \Risky_{2}/\Risky_{1} = \risky_{2}-\risky_{1}+\sigma^{2}_{2}/2 φ ≡ log R 2 / R 1 = r 2 − r 1 + σ 2 2 /2 σ 12 = σ 1 2 = 0 \sigma_{12}=\sigma^{2}_{1}=0 σ 12 = σ 1 2 = 0 (11)
ς 2 = ( φ ρ σ 2 2 ) \riskyshare_2 = \left(\frac{\EpremLog}{\CRRA\sigma^{2}_{2}}\right) ς 2 = ( ρ σ 2 2 φ ) which corresponds to the solution obtained for the case of a single risky asset in the Portfolio Choice with CRRA Utility (Merton-Samuelson) .
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