Stage 4 — From selected FMPs to a final portfolio

The final stage of factor research combines selected factor-mimicking portfolios into holdings that satisfy an investment mandate. It uses their expected contributions and shared risks to choose factor allocations, then translates those allocations into security-level decisions.

The animation and calculations below follow that process through candidate selection, allocation, and final portfolio constraints. A further forward test measures how the complete construction behaves with turnover and costs.

What the animation shows

The left panel starts with FMPs that passed the acceptance rules. Correlation and family checks prevent several versions of the same idea from occupying the whole allocation. Their training-window return histories form a matrix from which expected returns and cross-portfolio covariance are estimated.

In the center, the signal-space optimizer assigns weights to those FMPs. A high-scoring FMP can receive less weight when it duplicates another FMP; a modest but diversifying FMP can remain useful. The weighted FMPs then map back to a target vector of security holdings.

The right panel applies the same asset risk model used during FMP construction, adds the mandate's position and exposure constraints, and produces the final portfolio. The last step runs the entire chain—including FMP selection and estimation—through unseen periods.

Estimate the FMP opportunity set

For the selected set \(\mathcal{S}\), let \(F\) contain the training-window returns of the selected FMPs. Estimate the expected-return vector \(\widehat{\mu}_F\) and the annualized covariance matrix \(\widehat{\Omega}_F\) from that window only:

\[\widehat{\mu}_F = \frac{1}{T}\sum_{t=1}^{T}F[t,:]^{\mathsf T},\]

\[\widehat{\Omega}_F = \frac{A}{T-1} \sum_{t=1}^{T} \left(F[t,:]^{\mathsf T}-\widehat{\mu}_F\right) \left(F[t,:]^{\mathsf T}-\widehat{\mu}_F\right)^{\mathsf T}.\]

Some configurations map ranking evidence such as ICIR into an allocation score. That mapping must be declared and calibrated: ICIR is dimensionless, and the calibration supplies the expected-return scale.

Combine the FMPs in signal space

The nominal FMP-allocation problem is

\[\underset{\omega}{\operatorname{maximize}} \quad \widehat{\mu}_F^{\mathsf T}\omega -\lambda_\omega \omega^{\mathsf T}\widehat{\Omega}_F\omega.\]

Here \(\omega\) contains the allocations across FMPs. Covariance brings shared risk and diversification into the allocation. The unconstrained first-order solution is

\[\omega^* = \left(2\lambda_\omega\widehat{\Omega}_F\right)^{-1} \widehat{\mu}_F.\]

If \(H_t\) contains the current security weights of the selected pure FMPs, the combined target portfolio is

\[\boxed{h^*=H_t\omega^*.}\]

Map the target to asset-level alpha

The target is connected to the downstream asset optimizer through the same covariance model \(Q_t\) used to purify the individual FMPs:

\[\boxed{\alpha^*=Q_t h^*.}\]

This consistency has a useful consequence. In the unconstrained problem,

\[\underset{w}{\operatorname{maximize}} \quad (\alpha^*)^{\mathsf T}w -\lambda_h w^{\mathsf T}Q_t w,\]

the solution follows the direction of the FMP target,

\[w^*=\frac{1}{2\lambda_h}h^*.\]

The final optimizer can therefore add real-world constraints while preserving the meaning of the target it received.

Apply the mandate and test the whole chain

The production solve may add a benchmark, long-only requirement, position bounds, sector and factor budgets, active-risk limits, liquidity, borrow, turnover, and transaction costs. These constraints decide which parts of the idealized target are feasible for a particular portfolio.

At every historical rebalance, StrategyNet freezes the selected FMP set, expected-return estimate, covariance estimate, FMP allocation, implied alpha, and final positions before observing the next return. Net performance is

\[r_{p,t+1}^{\mathrm{net}} = w_t^{\mathsf T}r_{t+1} -\operatorname{Cost}(w_t-w_{t-1}).\]

The output is the complete portfolio record: positions, FMP contributions, risk exposures, turnover, costs, return, and drawdown. That record can be traced backward through the selected FMPs to the original versioned signals.