Interpreting the UMVE Portfolio Weights Formula

Summary

The document examines a formula for the weights of a conditionally optimal portfolio, called the unconstrained minimum-variance efficient portfolio, in a setting with multiple risky assets and no riskless asset. The weights depend on the conditional mean excess-return vector and its covariance matrix: the covariance-adjusted mean is scaled by a factor involving the squared risk-adjusted mean. The author is seeking a derivation or source because the cited studies do not appear to state the formula in the same form.

No derivation or answer is provided, so the formula’s assumptions and connection to the cited results remain unresolved in the document. The discussion is useful as a pointer to dynamic portfolio construction with conditional moments, but it is not a complete optimization treatment. In particular, implementation would require clarifying the return normalization, admissible strategies, and conditions under which the covariance matrix is invertible.

Key ideas

• The portfolio weights use conditional expected excess returns and their covariance matrix.
• The formula scales covariance-adjusted expected returns by a risk-dependent factor.
• The document situates the result in a setting without a riskless asset.
• The requested derivation and source are not supplied, leaving assumptions to be verified.

Tags

# Explaination and Reference of Formula for Portfolio Optimization


# Explaination and Reference of Formula for Portfolio Optimization












I stumbled upon Pricing Currency Risks by Chernov and Dahlquist (Jrl Fin, 2023). They state on page 698-699: "Suppose that we have $N$ basis assets with an $N × 1$ vector of excess returns $R^e_{t+1}$. The conditional mean of this vector is $\mu_t = E_t (R^e_{t+1})$ and its conditional covariance matrix is $\Omega_t = V_t (R^e_{t+1})$. An admissible trading strategy $p$ in the basis assets has an $N × 1$ vector of weights $w_{pt}$ that are determined only by information available up until time t. The resulting excess portfolio return is then $R_{p,t+1} = w^T_{pt} R^e_{t+1}$. The UMVE portfolio is a dynamic trading strategy in these assets and obtains the MSR, both conditionally and unconditionally. Ferson and Siegel (2001) and Jagannathan (1996) show that the UMVE portfolio weights are $$w_t^* = \frac{1}{1 + \mu_t^\top \Omega_t^{-1} \mu_t} \Omega_t^{-1} \mu_t \text{.}$$"

However, I couldn't find this exact formula in either of the referenced papers. While it may be a derivation or simplification from their results, the formula as presented does not appear explicitly (with different notation of course). I specifically checked Ferson and Siegel (2001), under the chapter Multiple Risky Assets in the section No Riskless Asset.

I would appreciate any insights, a derivation of the formula, or a reference to a proof.

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