Resampled Portfolio Stacking

A high-level presentation of Resampled Portfolio Stacking for portfolio optimization with fully general parameter uncertainty.

The fully general Resampled Portfolio Stacking equations.

This post gives a high-level introduction to Resampled Portfolio Stacking, which is a method for portfolio optimization with fully general parameter uncertainty introduced in Chapter 6 of the Portfolio Construction and Risk Management book¹.

The fundamental perspectives for the Resampled Portfolio Stacking approach were originally introduced in the Portfolio Optimization and Parameter Uncertainty article², which presented the Exposure Stacking method.

Resampled Portfolio Stacking is basically a generalization of Exposure Stacking, which allows us to use other targets than the resampled portfolio exposures.

Resampled efficiency history

Resampled portfolio efficiency was originally introduced in the Efficient Asset Management book³.

The original idea was to optimize mean-variance efficient frontiers B times with resampled parameter estimates for the mean vector and covariance matrix.

Resampled efficient portfolios are then computed by averaging the exposures over the individual portfolio indices.

For example, if an efficient frontier is computed over N=10 portfolios, we use each of the 10 indices to average over the B samples:

\(w_{resampled, n}= \frac{1}{B} \sum_{b=1}^{B} w_{b, n} \quad n \in \{1,2,...,N\}\)

While the original resampled efficient optimization method outperforms the usual mean-variance optimization⁴, several critiques remain:

1. It was not well-understood why the method performed well in practice.

2. It is limited to just efficient frontier optimization.

3. It was only defined for mean-variance.

The Resampled Portfolio Stacking method solves all of the above issues.

Resampled portfolio optimization foundation

To understand the Resampled Portfolio Stacking approach, we have to understand what we are actually doing when we perform resampled portfolio optimization.

The perspectives are thoroughly presented in the Portfolio Construction and Risk Management book, so they are only briefly summarized here.

When we are doing resampled portfolio optimization, we are essentially treating the parameters, or more generally the next generation market model

\((R, p),\)

as data. Here, R is the usual matrix of joint P&L and risk factor simulations, while p is the joint scenario probability vector⁵.

From this perspective, the mean-risk estimators such as mean-variance or mean-CVaR⁶ are unbiased but have a high variance.

In summary, mean-risk estimators give us the correct optimal exposures for the particular parameter values or market model, but they are very sensitive to this input.

When we do resampled portfolio optimization, we are purposefully introducing market model uncertainty to get B samples of the optimal portfolio exposures:

\(e_{b}^{\star}= \underset{e}{\text{argmax}} \ f\left(R_{b},p_{b}, \mathcal{E},e\right),\)

where f is the risk-adjusted return objective such as mean-variance or, even better, mean-CVaR.

We can then assign different weights to the B resampled portfolio exposures to get the final resampled portfolio estimate:

\(e_{w}^{\star}=\sum_{b=1}^{B}w_{b}e_{b}^{\star}.\)

Resampled Portfolio Stacking objective

In the original resampled efficiency suggestion, the sample weights are simply set to be uniform

\(w_{b} = \frac{1}{B} \quad \text{for all} \ b \in \{1,2,...,B\}.\)

However, we might be able to achieve a better trade-off between bias and variance by optimizing some properly designed objective.

As shown in the Exposure Stacking article, the original suggestion corresponds to optimizing a naive multivariate stacking objective that uses the optimal exposures as targets.

Resampled Portfolio Stacking instead choose the sampled weights that minimize a proper multivariate regression stacking objective including L-fold cross-validation:

\(w^{\star}=\underset{w_{1},w_{2},\ldots,w_{b}}{\mbox{argmin}}\ \frac{1}{L}\sum_{l=1}^{L}\left(\frac{1}{\left|\mathcal{K}_{l}\right|}\sum_{k\in\mathcal{K}_{l}}\left\Vert t\left(e_{k}^{\star}\right)-\sum_{b\notin\mathcal{K}_{l}}w_{b}t\left(e_{b}^{\star}\right)\right\Vert _{2}^{2}\right).\)

The Resampled Portfolio Stacking objective might look very complex at first, but it can be formulated as a quadratic programming problem as shown in the appendix of the Exposure Stacking article.

In the most elementary Exposure Stacking case, the targets are simply the optimal resampled exposures.

In the more general Resampled Portfolio Stacking case, we can, for example, use the marginal risk contributions or the marginal risk-adjusted return contributions. These objectives also allow us to account for diversification interactions and even perform intelligent rebalancing tests⁷.

Resampled Portfolio Stacking benefits

The most important benefit is that it is possible to achieve better out-of-sample risk-adjusted returns, as shown both in the original article and the Portfolio Construction and Risk Management book.

Another benefit is that the approach allows us to handle resampled derivatives portfolio optimization⁸ using Entropy Pooling⁹.

Since the Resampled Portfolio Stacking approach is still quite new, extensive practical testing is needed to assess the magnitude of the risk-adjusted gains, in addition to gaining experience with which objectives work well.

You can easily test out the method on your own data and portfolios by using the fortitudo.tech Python package¹⁰.

1. Portfolio Construction and Risk Management Book: https://antonvorobets.substack.com/p/pcrm-book

2. Portfolio Optimization and Parameter Uncertainty SSRN article: https://ssrn.com/abstract=4709317

3. Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation, Oxford University Press

4. Modern Portfolio Theory’s Evolutionary Road Institutional Investor article: https://www.institutionalinvestor.com/article/2btg68hlbj7j7hfikxam8/home/modern-portfolio-theorys-evolutionary-road

5. Time- and State-Dependent Resampling: https://antonvorobets.substack.com/p/time-and-state-dependent-resampling

6. Variance for Intuition, CVaR for Optimization: https://antonvorobets.substack.com/p/variance-for-intuition-cvar-for-optimization

7. Intelligent Portfolio Rebalancing: https://antonvorobets.substack.com/p/intelligent-portfolio-rebalancing

8. Derivatives Portfolio Optimization and Parameter Uncertainty: https://antonvorobets.substack.com/p/10-derivatives-portfolio-optimization-parameter-uncertainty

9. Entropy Pooling Collection: https://antonvorobets.substack.com/p/entropy-pooling-collection

10. fortitudo.tech open-source Python package: https://github.com/fortitudo-tech/fortitudo.tech