Quantitative Researcher | Mustafa MAJJI
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The Markowitz model is a classical framework for portfolio optimization, introduced by Harry Markowitz in 1952.
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It is based on the principle that investors seek to maximize expected returns while minimizing portfolio volatility, measured by variance.
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Since the model uses variance as a risk metric, it implicitly assumes that asset returns are normally distributed—an assumption that simplifies analysis but may not always reflect real market behavior.
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However, estimating the expected return vector and covariance matrix from historical data introduces estimation errors, which can lead to poor out-of-sample performance. To address this, various regularization techniques have been developed to stabilize portfolio weights and improve robustness.
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Weight Constraints: Imposing constraints on portfolio weights (e.g., long-only, maximum position size) is itself a form of regularization that helps reduce overfitting.
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Resampling Methods: These methods aim to mitigate estimation error by generating multiple alternative scenarios from the original data:
- Monte Carlo Simulation:Asset returns are simulated from a multivariate normal distribution using the sample mean and covariance matrix.
- Bootstrap Resampling:Returns are randomly drawn with replacement from the original dataset to create new samples.
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L1-Constrained Portfolio (Lasso Regularization): Adds an L1 penalty on the portfolio weights, promoting sparsity.
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L2-Constrained Portfolio (Ridge Regularization):Adds an L2 penalty on the weights, encouraging smaller and more evenly distributed allocations.
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