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 $\uFunc(\cRat) = (1-\CRRA)^{-1}\cRat^{1-\CRRA}$ who can choose among many risky investment options.

Using their framework, here we study a consumer who has wealth $\aRat_{t}$ at the end of period $t$ , and is deciding how much to invest in two risky assets with lognormally distributed return factors $\Risky_{t+1}=(\Risky_{1,t+1}, \Risky_{2,t+1})'$ . As in Portfolio Choice with CRRA Utility, $\risky_{i} \equiv \log \Ex_{t}[\Risky_{i,t+1}]$ is the log of asset $i$ ’s expected return factor, so the log returns are distributed $\log \Risky_{t+1} = \risky_{t+1}=(\risky_{1,t+1}, \risky_{2,t+1})' \sim \left ( \mathcal{N}(\risky_1-\sigma^{2}_{1}/2,\sigma^{2}_{1}), \mathcal{N}(\risky_2-\sigma^{2}_{2}/2,\sigma^{2}_{2}) \right)'$ , with covariance matrix

\left(\begin{array}{cc}\sigma_1^2 & \sigma_{12} \\ \sigma_{12}& \sigma_2^2\end{array}\right).

(1)

If the period- $t$ consumer invests proportion $\riskyshare_i$ of $\aRat_{t}$ in risky asset $i$ , $i=1,2$ (so that $\riskyshare_{1}=(1-\riskyshare_{2})$ and vice-versa), spending all available resources in the last period of life^[1] $t+1$ will yield:

\cRat_{t+1} = \underbrace{(\riskyshare \cdot \Risky_{t+1} )}_{\equiv \Rport_{t+1}} \aRat_{t}

(2)

where $\Rport_{t+1}$ is the portfolio-weighted return factor.

Campbell & Viceira (2002) point out that a good approximation to the log of the portfolio’s return factor is obtained by

\rport_{t+1} = \risky_{1,t+1}+\riskyshare_{2} (\risky_{2,t+1}-\risky_{1,t+1})+ \riskyshare_2 (1-\riskyshare_2) \eta/2

(3)

where

\eta=(\sigma_1^2+\sigma_2^2-2\sigma_{12}).

(4)

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

\begin{aligned} \Ex_{t}[\uFunc(\cRat_{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}

(5)

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

Our foregoing assumptions imply that

(1-\CRRA) (\riskyshare_1 \risky_{1,t+1}+\riskyshare_2 \risky_{2,t+1}) \sim \mathcal{N}( (1-\CRRA)(\riskyshare_1(\risky_1-\sigma_1^2/2)+\riskyshare_2 (\risky_2-\sigma_2^2/2)), (1-\CRRA)^2 ( \riskyshare_1^2\sigma_1^2+\riskyshare_2^2\sigma_2^2+2\riskyshare_1\riskyshare_2\sigma_{12}))

(6)

(using NormTimes and SumNormsIsNorm). With a couple of extra lines of derivation we can show that the log of the expectation in (5) is

\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-\sigma_1^2/2)+\riskyshare_2 (\risky_2-\sigma_2^2/2)) + (1-\CRRA)^2 (\riskyshare_1^2\sigma_1^2+\riskyshare_2^2\sigma_2^2+2\riskyshare_1\riskyshare_2\sigma_{12})/2}

(7)

Substituting from (7) for the log of the expectation in (5), the log of the “excess return utility factor” in (5) is

(1-\CRRA)\riskyshare_2(1-\riskyshare_2)\eta/2+(1-\CRRA) (\risky_1-\sigma_1^2/2+\riskyshare_2(\risky_2-\risky_1+(\sigma_1^2-\sigma_2^2)/2))+(\CRRA-1)^2 (\sigma_1^2+\riskyshare_2^2 \eta+2\riskyshare_2(\sigma_{12}-\sigma_1^2))/2 .

(8)

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

\begin{aligned} (1-2\riskyshare_2)\eta/2+ \risky_2-\risky_1+(\sigma_1^2-\sigma_2^2)/2+(1-\CRRA) (\riskyshare_2 \eta+(\sigma_{12}-\sigma_1^2)) & = 0 \\ \risky_2-\risky_1+\CRRA(\sigma_1^2-\sigma_{12}) & = \CRRA \eta \riskyshare_2, \end{aligned}

(9)

since $\eta/2+(\sigma_1^2-\sigma_2^2)/2 = \sigma_1^2-\sigma_{12}$ . So

\riskyshare_2 = \frac{\risky_2-\risky_1}{\CRRA\eta}+\frac{\sigma_1^2-\sigma_{12}}{\eta}.

(10)

The second term is asset 2’s share in the portfolio of the two assets with the smallest variance; the first adds a speculative demand that rises with asset 2’s expected return over asset 1’s and falls with risk aversion and with the variance $\eta$ of the difference between their log returns.

If the first asset is riskfree, so that $\sigma_{1}=\sigma_{12}=0$ and $\eta=\sigma^{2}_{2}$ , this reduces to

\riskyshare_2 = \left(\frac{\risky_2-\risky_1}{\CRRA\sigma^{2}_{2}}\right) = \left(\frac{\EpremLog}{\CRRA\sigma^{2}_{2}}\right),

(11)

where $\EpremLog \equiv \log \EPrem = \risky_{2}-\risky_{1}$ is the log of the expected excess return factor $\EPrem \equiv \Risky_{2}/\Risky_{1}$ . This is the solution for a single risky asset in Portfolio Choice with CRRA Utility.

Footnotes¶

The portfolio allocation solution obtained below induces back to earlier periods of life, as Samuelson (1963) and Samuelson (1989) famously emphasized.
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References¶

Merton, R. C. (1969). Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case. Review of Economics and Statistics, 51(3), 247–257. 10.2307/1926560
Samuelson, P. A. (1969). Lifetime Portfolio Selection by Dynamic Stochastic Programming. Review of Economics and Statistics, 51, 239–246.
Campbell, J. Y., & Viceira, L. M. (2002). Appendix to Strategic Asset Allocation: Portfolio Choice for Long-Term Investors. Oxford University Press. https://scholar.harvard.edu/files/campbell/files/bookapp.pdf
Samuelson, P. A. (1963). Risk and Uncertainty: A Fallacy of Large Numbers. Scientia, 98(4–5), 108–113.
Samuelson, P. A. (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.