$©$ February 8, 2021, Christopher D. Carroll Portfolio-Multi-CRRA

CRRA Portfolio Choice with Two Risky Assets

Merton (1969) and Samuelson (1969) study optimal portfolio allocation for a consumer with Constant Relative Risk Aversion utility $u (c) = (1 − ρ )− 1c1 − ρ$ who can choose among many risky investment options.

Using their framework, here we study a consumer who has wealth $a t$ at the end of period $t$ , and is deciding how much to invest in two risky assets with lognormally distributed return factors $R = (R ,R )′ t+1 1,t+1 2,t+1$ , $( )′ log Rt+1 = rt+1 = (r1,t+1, r2,t+1 )′ ∼ 𝒩 (r1, σ21), (𝒩 (r2, σ22)$ , with covariance matrix

( 2 ) σ 1 σ12 . σ12 σ22

If the period- $t$ consumer invests proportion $ςi$ of $at$ in risky asset $i$ , $i = 1,2$ (so that $ς1 = (1 − ς2)$ and vice-versa), spending all available resources in the last period of life¹ $t + 1$ will yield:

c = (ς ⋅ R )a t+1 ◟---◝◜t+1◞ t ≡Rt+1

where $Rt+1$ is the portfolio-weighted return factor.

Campbell and Viceira (2002) point out that a good approximation to the portfolio rate of return is obtained by

rt+1 = r1,t+1 + ς2(r2,t+1 − r1,t+1 ) + ς2(1 − ς2)η∕2

(1)

where

2 2 η = (σ 1 + σ 2 − 2 σ12).

Using this approximation, the expectation as of date $t$ of utility at date $t + 1$ is:

[( )1− ρ] − 1 r1,t+1 ς2(r2,t+1− r1,t+1)+ ς2(1− ς2)η∕2 𝔼t[u (ct+1 )] ≈ (1 − ρ ) 𝔼t ate e [ ] ≈ (1 − ρ )− 1a1− ρe (1− ρ)ς2(1− ς2)η∕2 𝔼 e (r1,t+1+ ς2(r2,t+1− r1,t+1))(1− ρ) ◟ -----◝◜----t-◞ -----------------t----- ----------------------- constant < 0 ◟ ◝ ◜ ◞ excess return utility factor

where the ﬁrst term is a negative constant under the usual assumption that relative risk aversion $ρ > 1.$

Our foregoing assumptions imply that

(1− ρ )(ς r + ς r ) ∼ 𝒩 ((1 − ρ)(ς r + ς r ),(1 − ρ)2(ς 2σ2+ ς 2σ2+2 ς ς σ )) 1 1,t+1 2 2,t+1 1 1 2 2 1 1 2 2 1 2 12

(using $[LogELogNormTimes ]$ ). With a couple of extra lines of derivation we can show that the log of the expectation in (2) is

[ (r1,t+1+ς2(r2,t+1− r1,t+1))(1− ρ)] 2 2 2 2 2 log 𝔼t e = (1 − ρ)(ς1r1 + ς2r2) + (1 − ρ ) (ς1σ 1 + ς2σ 2 + 2 ς1ς2σ12))∕2

Substituting from (2) for the log of the expectation in (2), the log of the ‘excess return utility factor’ in (2) is

2 2 2 2 (1− ρ )ς2(1− ς2) η∕2+ (1− ρ )(r1+ ς2(r2− r1))+ (ρ − 1) (σ 1+ ς2η+2 ς2(σ12 − σ2))∕2.

The $ς$ that minimizes this log will also minimize the level; the FOC for minimizing this expression is

(1 − 2ς )η ∕2 + r − r + (1 − ρ)(ς η + (σ − σ2 )) = 0 2 2 1 2 12 1 η- 2 (r2 − r1 + 2 ) + (1 − ρ)(σ12 − σ 1) = ρ ης2.

(2)

( ) r2 − r1 + η∕2 + (1 − ρ)(σ12 − σ21) ς2 = --------------------------------------- ρη

(3)

and note that if the ﬁrst asset is riskfree so that $σ1 = σ12 = 0$ then this reduces to

( 2 ) r2-−--r1-+-σ-2∕2- ς2 = ρ σ2 2

(4)

but the log of the expected return premium (in levels) on the risky over the safe asset in this case is $2 φ ≡ log R2 ∕R1 = r2 − r1 + σ 2∕2$ (recalling that we have assumed $σ = σ2 = 0 12 1$ ), so (4) becomes

( ) φ ς2 = ----- ρ σ22

(5)

which corresponds to the solution obtained for the case of a single risky asset in Portfolio-CRRA.

References

Campbell, John Y., and Luis M. Viceira (2002): Appendix to ‘Strategic Asset Allocation: Portfolio Choice for Long-Term Investors’. Oxford University Press, USA, Available on 2011/01/22 at http://kuznets.fas.harvard.edu/ $∼$ campbell/papers/bookapp.pdf.

Merton, Robert C. (1969): “Lifetime Portfolio Selection under Uncertainty: The Continuous Time Case,” Review of Economics and Statistics, 51, 247–257.

Samuelson, Paul A (1963): “Risk and Uncertainty: A Fallacy of Large Numbers,” Scientia, 98(4-5), 108–113.

Samuelson, Paul A. (1969): “Lifetime Portfolio Selection by Dynamic Stochastic Programming,” Review of Economics and Statistics, 51, 239–46.

__________ (1989): “The Judgment of Economic Science on Rational Portfolio Management: Indexing, Timing, and Long-Horizon Eﬀects,” The Journal of Portfolio Management, 16(1), 4–12.