Optimisation

This page is a technical deep dive into our optimisation implementation of our various blocks, layers and paths. We recommend reading our introduction and our model pages to fully understand where this deep dive fits in the PathLit ecosystem. These are quick reads and well spent time!

Allocation strategies

l3ewp - Equally-Weighted Portfolios

takes the covariances between the assets and computes the equally-weighted portfolio weights.

What is the math PathLit uses?

Objective Function:

Let $w_i$ be the weight of the $i$th asset of $N$ total assets in a portfolio, the allocation is:

$$w_i = \frac{1}{N}, \quad i=\{1,...,N\}$$

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l3ivp - Inverse-Volatility Portfolios

takes the covariances between the assets and computes the inverse-volatility weighted portfolio weights.

What is the math PathLit uses?

Objective Function: Let $w_i$ be the weight of the $i$th asset of $N$ total assets in a portfolio and $\Sigma$ be the estimated covariance matrix, the allocation is:

$$\mathbf{w} = \frac{\sigma^{-1}}{\mathbf{1}^T \sigma^{-1}}, \quad \hbox{where,} \quad \sigma^2=diag(\Sigma)$$

Sources:
Sorensen, E.H. & Hua, R. & Qian, E.. (2005). Contextual fundamentals, models, and active management - Improving on one-size-fits-all. Journal of Portfolio Management. 32. 23-36+4. 10.3905/jpm.2005.599493.

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l3rpp - Risk-Parity Portfolios

takes the covariances between the assets and computes the risk-parity weighted portfolio weights.

What is the math PathLit uses?

Objective Function: Let $w_i$ be the weight of the $i$th asset of $N$ total assets in a portfolio and $\Sigma$ be the estimated covariance matrix, the allocation is:

$$\begin{aligned} \min_{\mathbf{w}} \quad & \sum_{i,j=1}^N (w_i(\Sigma \mathbf{w})_i - w_j(\Sigma \mathbf{w})_j)^2\\ \textrm{s.t.} \quad & \mathbf{1}^T\mathbf{w}=1, \\ &\mathbf{w}\geq0 \\ \end{aligned}$$

Sources:
Sorensen, E.H. & Hua, R. & Qian, E.. (2005). Contextual fundamentals, models, and active management - Improving on one-size-fits-all. Journal of Portfolio Management. 32. 23-36+4. 10.3905/jpm.2005.599493.

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l3mdp - Maximum Diversification Portfolios

takes the covariances between the assets and computes the maximum diversification weighted portfolio weights.

What is the math PathLit uses?

Objective Function:

Let $w_i$ be the weight of the $i$th asset of $N$ total assets in a portfolio and $\Sigma$ be the estimated covariance matrix, the allocation is:

$$\begin{aligned} \min_{\mathbf{w}} \quad & \frac{\mathbf{w}^T \sigma}{\sqrt{\mathbf{w}^T \Sigma \mathbf{w}}} \quad \hbox{where,} \quad \sigma^2 = diag(\Sigma)\\ \textrm{s.t.} \quad & \mathbf{1}^T\mathbf{w}=1 \\ \end{aligned}$$

Sources:
Choueifaty, Yves & Coignard, Yves. (2008). Toward Maximum Diversification. Journal of Portfolio Management - J PORTFOLIO MANAGE. 35. 40-51. 10.3905/JPM.2008.35.1.40.

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l3mdcp - Maximum Decorrelation Portfolios

takes the covariances between the assets and computes the maximum decorrelation weighted portfolio weights.

What is the math PathLit uses?

Let $w_i$ be the weight of the $i$th asset of $N$ total assets in a portfolio and $\Sigma$ be the estimated covariance matrix, the allocation is:

Objective Function:

$$\begin{aligned} \min_{\mathbf{w}} \quad & \mathbf{w}^T \mathbf{C} \mathbf{w} \quad \hbox{where,} \quad \mathbf{C} = diag(\Sigma)^{-\frac{1}{2}} \ \Sigma \ diag(\Sigma)^{-\frac{1}{2}}\\ \textrm{s.t.} \quad & \mathbf{1}^T\mathbf{w}=1 \\ \end{aligned}$$

Sources:
Christoffersen, Peter & Errunza, Vihang & Jacobs, Kris & Langlois, Hugues. (2012). Is the Potential for International Diversification Disappearing? A Dynamic Copula Approach. Review of Financial Studies. 25.10.2139/ssrn.2066076.

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l3gmvp - Global Minimum Variance Portfolios

takes the covariances between the assets and computes the global minimum-variance weighted portfolio weights.

What is the math PathLit uses?

Let $w_i$ be the weight of the $i$th asset of $N$ total assets in a portfolio and $\Sigma$ be the estimated covariance matrix, the allocation is:

Objective Function:

$$\begin{aligned} \min_{\mathbf{w}} \quad & \mathbf{w}^T \Sigma \mathbf{w} \\ \textrm{s.t.} \quad & \mathbf{1}^T\mathbf{w}=1 \\ \end{aligned}$$

Sources:
Markowitz, Harry. (1952). Portfolio Selection. The Journal of Finance. 7. 77-91. 10.1111/j.1540-6261.1952.tb01525.x.

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l3hrp - Hierarchical Risk-Parity Portfolios

takes the covariances between the assets and computes the hierarchical risk-parity weighted portfolio weights.

As per Prado's seminal work that introduced HRP:

HRP applies modern mathematics (graph theory and machine learning techniques) to build a diversified portfolio based on the information contained in the covariance matrix. However, unlike quadratic optimizers, HRP does not require the invertibility of the covariance matrix. In fact, HRP can compute a portfolio on an ill-degenerated or even a singular covariance matrix, an impossible feat for quadratic optimizers. Monte Carlo experiments show that HRP delivers lower out-of-sample variance than CLA, even though minimum-variance is CLA’s optimization objective. HRP also produces less risky portfolios out-of-sample compared to traditional risk parity methods.

The algorithm can be broken down into 3 major steps:

1. Hierarchical Tree Clustering
2. Matrix Seriation
3. Recursive Bisection

Refer to Prado's paper for in depth analysis of the algorithm:

Source: Lopez de Prado, Marcos. (2016). Hierarchical Portfolio Construction: Dispelling Markowitz's Curse. SSRN Electronic Journal. 10.2139/ssrn.2708678.

Framework

The framework used is an open-source highly performant, nonlinear optimisation framework library known as NLopt. Within the framework, the Sequential Least Squares Quadratic Programming (SLSQP) method is used to derive the solutions. Specified in NLopt as NLOPT_LD_SLSQP, this is a sequential quadratic programming (SQP) algorithm for nonlinearly constrained gradient-based optimisation which supports both inequality and equality constraints), based on the implementation by Dieter Kraft and described in:

• Dieter Kraft, "A software package for sequential quadratic programming", Technical Report DFVLR-FB 88-28, Institut für Dynamik der Flugsysteme, Oberpfaffenhofen, July 1988.
• Dieter Kraft, "Algorithm 733: TOMP–Fortran modules for optimal control calculations," ACM Transactions on Mathematical Software, vol. 20, no. 3, pp. 262-281 (1994).

The problem is treated as a sequence of constrained least-squares problems, but such a least-squares problem is equivalent to a quadratic programme. The algorithm optimizes successive second-order quadratic / least-squares approximations of the objective function, with first-order, affine, approximations of the constraints.

Reference: https://nlopt.readthedocs.io/en/latest/