\begin{algorithm} 
\caption{Multi-Period Transition Framework}\label{alg:MPTF}
\begin{algorithmic}[H]
\Require $P_0, \mathcal{T}, Y_{-L:0}, \mathbf{X}_{-L:0}, V_0, \mathbf{T}, C_0$, OPT, Forecaster, MarketObserver
\Ensure $Y_0^T\mathcal{T} +  C_0 \leq V_0$
\State $t \gets 0$
\State $P_t \gets P_0$
\State $C_t \gets C_0$
\State $Y \gets Y_{-L:t}$
\While{$t \leq \mathbf{T}$}
    \State $V_t \gets Y_t^TP_t + C_t$ \Comment{Calculate portfolio value at current timestep}
    \State $\hat{Y}_{t+1:\mathbf{T}}\gets \text{Forecaster}(Y_{-L:t}, \mathbf{X}_{-L:t})$ \Comment{Predict unknown time steps with known information}
    \State $Z_{t:\mathbf{T}}, W_{t:\mathbf{T}} \gets \text{OPT}([Y_t\ \hat{Y}_{t+1:\mathbf{T}}], P_t, C_t, \mathcal{T})$ \Comment{Run Optimization Policy}
    \State $P_{t+1} \gets P_t + Z_t$ \Comment{Execute trades at current timestep}
    \State $C_{t+1} \gets C_t - Y_t\cdot Z_t - F\Vec{1} \cdot W_t$    \Comment{Calculate cash}
    \State $Y_{t+1} \gets \text{MarketObserver}()$ \Comment{Observe the market for new price information}
    \State $t \gets t + 1$
\EndWhile
\Ensure $P_\mathbf{T} \geq \mathcal{T}$
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{Multi-Period Transition Framework}\begin{algorithmic}
[H]
\Require $P_0, \mathcal{T}, Y_{-L:0}, \mathbf{X}_{-L:0}, V_0, \mathbf{T}, C_0$, OPT, Forecaster, MarketObserver
\Ensure $Y_0^T\mathcal{T} + C_0 \leq V_0$
\State $t \gets 0$
\State $P_t \gets P_0$
\State $C_t \gets C_0$
\State $Y \gets Y_{-L:t}$
\While{$t \leq \mathbf{T}$}
\State $V_t \gets Y_t^TP_t + C_t$ \Comment{Calculate portfolio value at current timestep}
\State $\hat{Y}_{t+1:\mathbf{T}}\gets \text{Forecaster}(Y_{-L:t}, \mathbf{X}_{-L:t})$ \Comment{Predict unknown time steps with known information}
\State $Z_{t:\mathbf{T}}, W_{t:\mathbf{T}} \gets \text{OPT}([Y_t\ \hat{Y}_{t+1:\mathbf{T}}], P_t, C_t, \mathcal{T})$ \Comment{Run Optimization Policy}
\State $P_{t+1} \gets P_t + Z_t$ \Comment{Execute trades at current timestep}
\State $C_{t+1} \gets C_t - Y_t\cdot Z_t - F\Vec{1} \cdot W_t$ \Comment{Calculate cash}
\State $Y_{t+1} \gets \text{MarketObserver}()$ \Comment{Observe the market for new price information}
\State $t \gets t + 1$
\EndWhile
\Ensure $P_\mathbf{T} \geq \mathcal{T}$
\end{algorithmic}
\end{algorithm}
{
    "title": "optimizing transition strategies for small to medium sized portfolios",
    "id": "2401.13126",
    "abstract": "this work discusses the benefits of constrained portfolio turnover strategies for small to medium-sized portfolios. we propose a dynamic multi-period model that aims to minimize transaction costs and maximize terminal wealth levels whilst adhering to strict portfolio turnover constraints. our results demonstrate that using our framework in combination with a reasonable forecast, can lead to higher portfolio values and lower transaction costs on average when compared to a naive, single-period model. such results were maintained given different problem cases, such as, trading horizon, assets under management, wealth levels, etc. in addition, the proposed model lends itself to a reformulation that makes use of the column generation algorithm which can be strategically leveraged to reduce complexity and solving times.",
    "categories": "q-fin.cp",
    "doi": "",
    "created": "2024-01-23",
    "updated": "2024-01-24",
    "authors": [
        "nakul upadhya",
        "alexandre granzer-guay"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.13126"
}
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