Formulating Tactical Asset Tilts as a Quadratic Optimization

Summary

The document frames tactical portfolio tilts, such as overweighting or underweighting a stock-bond pair trade relative to a benchmark, as a constrained optimization problem. The decision vector represents trade sizes, the signal vector represents desired direction or strength, and a covariance matrix describes risk across the trades. Additional restrictions on the tilts can encode implementation or portfolio requirements.

The proposed formulation is a quadratically constrained quadratic program: maximize a signal-based linear objective with a risk-related term, subject to a quadratic equality constraint on portfolio risk. The response suggests solving the constraint first and then considering the resulting linear objective, while further constraints may simplify or shape the feasible set. The document provides a mathematical framing rather than an implementation in R, numerical example, or guidance on choosing signals, penalty parameters, risk targets, or a solver. Those choices would be needed to turn the sketch into a usable allocation process.

Key ideas

• Represent tactical tilts with a vector of trade sizes and score them using signals.
• Use the covariance matrix to express risk across candidate pair trades.
• A quadratic risk equality constraint makes the setup a quadratically constrained quadratic program.
• Additional bounds or sign constraints can encode portfolio restrictions.
• The formulation does not specify parameter calibration or solver implementation.

Tags

# Optimal portfolio construction for tactical asset allocation


# Optimal portfolio construction for tactical asset allocation












This is the first time I post question here so if there is anything that does not follow the rule, please bear with me and let me know.

I am trying to solve this optimization question but I don't know yet how to do it in R.

What I try to do here is to find the way to optimally overweight/underweight a pair trade (i.e: stock-bond) compared to the benchmark.

x: vector of bet size/ tilt (%)

Score: vector of signal score (i.e -1,0,1,...)

Q: co-variance matrix of each pair trade.

I also intend to add constraints on values of x (nonnegative, etc).

Looking forward to learning from you and have a good day!

## Answer by Attack68 (score 1)

https://quant.stackexchange.com/a/40085

This a Quadratically Constrained Quadratic Program (QCQP) (try searching for that) albeit the usual inequality constraint has been replaced by your equality constraint.

maximise over $x_i$ $$x_i'S_i - 0.01^2\lambda_i$$ s.t. $$x_i'Qx_i=0.01^2$$

You may have some success if you investigate techniques for solving the constraint in the first place and then the resultant linear objective function may be easier after that, particularly if you further restrict $x_i$ with additional constraints.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.