Convex Portfolio Optimisation
Structure, Stability, and the Mathematics of Capital Allocation
Portfolio construction is fundamentally an optimisation problem.
Given a set of assets, each with uncertain returns and associated risks, the objective is to allocate capital in a way that satisfies specific criteria. These criteria may include maximising expected return, minimising risk, or achieving a balance between the two under a defined set of constraints.
The challenge lies in the nature of the problem. Financial markets are complex, uncertain, and high-dimensional. The optimisation landscape can be irregular, with multiple competing solutions and sensitivity to assumptions.
Convex portfolio optimisation provides a structured approach to this problem. By formulating the objective and constraints in a convex framework, it ensures that the optimisation process is stable, tractable, and yields solutions that are globally optimal within the defined model.
The Concept of Convexity
Convexity is a property of mathematical functions and sets.
A function is convex if, for any two points within its domain, the value of the function at a weighted average of those points is less than or equal to the weighted average of the function values at those points. In intuitive terms, a convex function has a single “bowl-shaped” structure, this property has important implications for optimisation.
In a convex problem:
any local minimum is also a global minimum
the solution is unique or lies within a well-defined set
optimisation algorithms converge reliably
This contrasts with non-convex problems, where multiple local minima may exist and finding the global optimum becomes significantly more difficult.
Portfolio Optimisation as a Convex Problem
Many portfolio optimisation problems can be formulated as convex.
A common example is the mean–variance framework, where the objective is to minimise portfolio variance subject to a target return. Variance is a convex function of portfolio weights, and linear constraints on weights preserve convexity. This structure allows the optimisation to be solved efficiently and reliably.
Convex formulations can incorporate a range of constraints, including:
budget constraints
limits on position size
requirements for diversification
As long as these constraints are convex, the overall problem remains convex.
Stability and Robustness
One of the key advantages of convex optimisation is stability, solutions are less sensitive to initial conditions and algorithmic choices.
This is particularly important in financial contexts, where:
data may be noisy
estimates may be uncertain
models may be imperfect
A convex framework provides a degree of robustness. While it does not eliminate sensitivity to inputs, it ensures that the optimisation process itself does not introduce additional instability.
Constraints and Real-World Implementation
In practice, portfolio optimisation is subject to constraints.
These constraints reflect real-world considerations such as:
liquidity limitations
regulatory requirements
risk management policies
Convex optimisation allows these constraints to be incorporated directly into the model. This ensures that the resulting portfolio is not only theoretically optimal, but also feasible in practice. The inclusion of constraints transforms the optimisation from a purely mathematical exercise into a structured decision-making process.
Regularisation and Sparsity
Convex frameworks can incorporate regularisation techniques. Regularisation introduces additional terms into the objective function to control the structure of the solution.
For example, penalties may be applied to:
large positions
rapid changes in allocation
concentration of risk
These techniques help prevent overfitting to historical data and improve out-of-sample performance. They also contribute to interpretability, producing portfolios that are more stable and manageable.
Risk Measures and Convexity
The choice of risk measure influences the formulation of the optimisation problem, variance is commonly used due to its convexity.
However, alternative measures such as:
conditional value at risk
drawdown-based metrics
Can also be incorporated within convex frameworks under certain conditions, this flexibility allows for a broader representation of risk.
Limitations of Convex Approaches
Despite its advantages, convex optimisation is not a complete solution.
The framework relies on assumptions about:
expected returns
covariance structures
stability of relationships
These inputs are uncertain and may change over time.
Additionally, not all relevant aspects of financial markets are easily expressed in convex form. Some features, such as discrete decisions or complex non-linear relationships, may require non-convex formulations.
Convex optimisation provides clarity and tractability, but it does not eliminate model risk.
The MorMag Perspective
At MorMag, convex portfolio optimisation is used as a foundational tool for capital allocation. It provides a structured framework for balancing risk and return under constraints. However, its outputs are interpreted within a broader context.
This includes:
probabilistic assessment of inputs
awareness of model limitations
integration with dynamic market analysis
The objective is not to rely solely on optimisation, but to use it as part of a comprehensive decision-making process.
From Mathematics to Allocation
Convex optimisation bridges the gap between mathematical theory and practical allocation. It translates abstract objectives into concrete portfolios, ensuring that decisions are consistent with defined criteria; this translation is essential, as without it, insights remain theoretical.
Conclusion
Convex portfolio optimisation provides a powerful framework for structuring capital allocation in financial markets.
By ensuring that optimisation problems are well-defined and tractable, it enables reliable identification of optimal solutions within the constraints of the model. While subject to limitations related to inputs and assumptions, it offers stability, clarity, and flexibility.
At MorMag, this framework forms part of a disciplined approach to portfolio construction, integrating mathematical rigour with contextual understanding.
In financial markets, the challenge is not only to identify opportunities, it is to allocate capital effectively. Convex optimisation provides a foundation for meeting that challenge with structure and precision.
