CARA-Normal Portfolio Choice with Risky and Risk-Free Assets

Summary

The document asks whether a proposed optimal holding for a CARA investor with normally distributed risky-asset payoffs is correct. Its setup compares expected payoff net of the risk-free investment cost with payoff variance, and suggests scaling expected excess payoff by risk aversion and variance to obtain the position. For multiple risky assets, this is a vector portfolio problem: the covariance matrix, rather than a single variance, determines how positions interact.

The post presents a derivation but no independent answer or numerical evidence. Its notation shifts between dividends, prices, and returns, and the stated covariance or variance objects are ambiguous. The result depends on defining wealth consistently, specifying the risk-free payoff over the same horizon, and using the full covariance matrix for multiple assets. As written, the formula should be treated as a question about the derivation rather than a confirmed result.

Key ideas

• CARA utility with normally distributed wealth leads to a mean-variance objective.
• The optimal risky position depends on expected excess payoff and payoff covariance.
• For multiple risky assets, the covariance matrix replaces a scalar variance.
• Payoff, price, and return units must be aligned before applying the formula.

Tags

# Is this equation correct for portfolio optimization for CARA normal with N risky and one riskless asset?


# Is this equation correct for portfolio optimization for CARA normal with N risky and one riskless asset?












Suppose the consumer Solves $\max -e^{-\gamma W}$ where $W=X^T D -X^Tp R_f$ where $X$ is the vector invested in a risky asset and $D\sim N(E[D],\Sigma^2_D)$ and $R=\sim N(E[R],\Sigma^2_R)$. Then ${ X=(\gamma \Sigma_R)^{-1}(E[R-R_f])}$. Is this formula correct?

My reasoning is as follows: $e^{-\gamma W}=e^{-\gamma X(E[D]-p R_f)+\frac{1}{2}\gamma^2 X \Sigma X^T}$ Hence $ X=(\gamma \Sigma_D)^{-1}(E[D-p R_f])$ Hence $ X=(\gamma \Sigma_R)^{-1}(E[R-R_f])$

Here $\Sigma_D$ and $\Sigma_R$ refer to variance vector for dividend and returns.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.