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I am considering an investor facing a discrete-time multi-period minimization problem $$ \min_{\{v_t\}_{t=0}^\infty}\Bigg[\sum_{t=0}^\infty(1-\rho)^{t+1}\bigg(\frac{1}{2}v_{t}\Omega_{t+1}v_{t}'\bigg)+\frac{(1-\rho)^t}{2}\bigg(\frac{1}{2}\Delta v_t'\Lambda\Delta v_t\bigg) \Bigg] \quad \text{s.t.} \quad v_t'\textbf{1}=1 $$ Let $v_t$ be a vector of weights attached to each asset, $\Omega_t$ be the time-varying covariance matrix and $\Lambda$ be a symmetric matrix of trading cost. Finally, $\rho\in(0,1)$ be the discount factor and $\textbf{1}$ being a vector of 1's. This problem has a corresponding value function $$ V(v_{t-1})=\min_{v_t}\Bigg[\frac{1}{2}\Delta v_t'\Lambda\Delta v_t+(1-\rho)\bigg(\frac{1}{2}v_{t}\Omega_{t+1} v_{t}' +\mathbb{E}_t[V(v_{t})] \bigg)\Bigg] - \lambda(v_{t-1}'\textbf{1}-1) $$

I am looking to find the Bellman equation via the 'guess and verify' method (similar to Gârleanu and Pedersen, 2013).

Without the constraint ($v_{t-1}'\textbf{1}=1$), I can verify that $$ V(v_t)=v_t'A_{vv}v_t+A_0 $$ is a solution with $A_{vv}$ a symmetric matrix of parameters. But with the constraint, I have been unable to find a guess that solves the problem. Can you find a suitable guess that includes the constraint?

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2 Answers 2

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Your constraint of positions summing up to one would be strange unless you imply a long-only constraint as well. There is no closed-form solution to mean-variance optimization with non-negativity (or box) constraints. This is a standard quadratic programming problem, which can be efficiently handled numerically. In a long/short context, position constraints are imposed in terms of $L_1$ or $L_2$ norm. The former case wouldn't allow a closed-form solution either; the latter just adds to your $\Omega$ matrix. Further generalizations, including non-local impact costs, slippage, continuous time, etc., are described in my recent book.

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I thought that the standard mean-variance dynamic problem is actually time inconsistent. See for example: Basak and Chabakauri (2010)

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