Factor models are typically linear multiple regressions defining risk (to keep it simple, take volatility) and return of factor portfolios...

\$ r_{a} = B \cdot r_{f} \$

\$ r_{p} = w_{a} \cdot r'_{a} = w_{a} \cdot B \cdot r'_{f} = w_{f} \cdot r'_{f} \$

With $ r_{a} $ as a vector of asset returns, $ B $ as a matrix of factor exposures of all assets, $ r_{f} $ as a vector of factor returns, $ r_{p} $ as portfolio return and $ w_{a} $ as a vector of asset weights in the portfolio.

We assume that the factor model is "complete" in the sense that it fully explains the variability in the investment universe without residuals (i.e. idiosyncratic risk). An example of a complete model is a statistical factor model derived from a Principal Component Analysis (PCA) on the correlation matrix of asset returns.

The factor portfolio selection problem is an entirely unrestricted mean-variance problem in factor space...

\$ \max_{w_{f}} w_{f} \cdot r'_{f} - \frac{1}{2 \cdot \lambda} \cdot w_{f} \cdot \Omega_{f} \cdot w'_{f} \$

As factor exposures are not directly investable, the big question is how to convert optimal factor allocations into investable asset allocations. Formally, this is a linear problem...

\$ w_{a} = B^{-1} \cdot w_{f} \$

Unfortunately, it is a linear problem with multiple solutions in most cases...

*1. Case "No restrictions on portfolio leverage, no restrictions on asset weights"*A solution can be found with a pseudo-inverse matrix operation, e.g. Moore-Penrose matrix inversion. With \$ B^+ \$ being the pseudo-inverse of B...

\$ w_{a} = B^+ \cdot w_{f} \$

Additional solutions can be found with...

\$ w_{a} = B^+ \cdot w_{f} + ( I - B^+ \cdot B ) \cdot x \$

x is a vector with arbitrary real numbers.

**"**

*2. Case "Restrictions on portfolio leverage, no restrictions on asset weights*Portfolio leverage is a linear equality constraint. Example: fully invested portfolio.

\$ w_{a} = B^+ \cdot w_{f} \\ \mathbf{1} \cdot w_{f} = 1 \$

Equality constraints can be handled by including "dummy variables" in \$ w_{a} \$ and $ B^+ \$, no problem.

**3. Case "Restrictions on portfolio leverage, restrictions on asset weights"**Restrictions on asset weights are inequality constraints (example: long-only portfolios). Inequalities can be handled with so-called slack variables. An alternative might be using a constrained generalized inverse, e.g. the Bott-Duffin inverse.

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For real-world investment purposes, the big issue is the stability of the factor betas over time, market regimes and asset universes.

Meucci(2007): "Risk contributions from generic user-defined factors" proposes a solution if the factor model is not complete: the asset weights are determined such that the residual risk is minimized. In this approach, the asset weights are determined as regression coefficients with a simple formula. Unfortunately, the approach is not able to handle restrictions.

Roncalli/Weisang(2012): "Risk Parity Portfolios with Risk Factors" discuss the issue in the context of risk budgeting with risk factor exposures. They interpret $ ( I - B^+ \cdot B ) \cdot x $ as an idiosyncratic risk component due to the fact that the pseudo-inverse is itself a least squares solution to a system of linear equations.

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