Constraints and Time Consistency in Dynamic Mean–Variance Optimization
Summary
The document presents an infinite-horizon portfolio-control problem with time-varying covariance risk, quadratic trading costs, a discount factor, and a constraint that portfolio weights sum to one. The question asks how to use a quadratic value-function guess and verify it when the budget constraint is present. The replies caution that a sum-to-one condition alone does not specify long-only investing, and that nonnegative or box constraints turn mean–variance allocation into a quadratic programming problem without a general closed-form solution.
The discussion notes that some long-short position constraints can be expressed with norms; an L2 penalty can be incorporated into the covariance-like quadratic term, while an L1 restriction does not yield a closed form. A separate reply raises time inconsistency in standard dynamic mean–variance problems and points to related literature. These are brief answers rather than a derivation of the Bellman equation, and they do not solve the stated control problem or specify all assumptions required for a numerical implementation.
Key ideas
- A sum-to-one constraint does not itself imply nonnegative asset weights.
- Nonnegative or box-constrained mean–variance allocation is generally handled as a quadratic program.
- An L2 position penalty can be incorporated into a quadratic risk term.
- The reply flags time inconsistency as a concern in dynamic mean–variance optimization.
- The document offers conceptual cautions rather than a full Bellman solution.
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Full text
# Help guessing the solution to an optimal control problem
# Help guessing the solution to an optimal control problem
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?
## Answer by Michael Isichenko (score 1)
https://quant.stackexchange.com/a/68234
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.
## Answer by phdstudent (score 0)
https://quant.stackexchange.com/a/71474
I thought that the standard mean-variance dynamic problem is actually time inconsistent. See for example: Basak and Chabakauri (2010)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.