Why the No-Short-Sale Maximum-Return Portfolio Favors Top Assets

Summary

The document formulates a maximum expected return portfolio as minimizing negative expected return, subject to fully invested weights and nonnegative positions. It asks how to construct the Lagrangian and derive a closed-form solution for the portfolio weights. The stated setup is a linear objective with simplex constraints, rather than a risk-adjusted portfolio optimization that trades off expected return against variance.

The document itself stops at the question and contains no derivation or answer. From the formulation, the optimum places all weight on an asset with the highest expected return; if multiple assets tie, any fully invested nonnegative allocation among the tied assets is optimal. Thus, without additional constraints or a risk penalty, the problem does not produce a diversified allocation. The expected returns are inputs and no estimation method, empirical evidence, or treatment of uncertainty is provided, so the result is an optimization implication rather than an investment recommendation.

Key ideas

• The objective maximizes expected return under a full-investment constraint and prohibits short positions.
• With only these constraints, an optimal portfolio allocates entirely to an asset with the highest expected return.
• If multiple assets share the maximum expected return, any allocation across those tied assets is optimal.
• The formulation includes no risk penalty, so it does not inherently favor diversification.
• The document poses the derivation question but does not supply its own solution.

Tags

# Maximum expected return portfolio: Lagrangean derivation of closed-form analytical solution


# Maximum expected return portfolio: Lagrangean derivation of closed-form analytical solution












\begin{align} \arg \min_w \enspace & -w^\top \mu \\ \mathrm{s.t.} \enspace & 1_N^\top w = 1 \\ & w_i \geq 0 \enspace \forall i=1,\dots, N \end{align}

is the optimization problem for solving the maximum expected return portfolio's weight vector $w$, where $\mu$ is the vector of asset expected returns, and the two constraints ensure the weights sum to 1 and are non-negative (the constrained no short-sales portfolio).

The Lagrangean of the above is

$$ L(w,\lambda_1,\lambda_2) = -w^\top \mu - \lambda_1( 1_N^\top w - 1)- \lambda_2(?)$$ where the $\lambda$s are the Lagrangean multipliers.

How can I take the first order conditions (derivatives) of the Lagrangean $L$ to get the closed-form analytical solution of $w$?

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