PCA Factor Hedging and Residual Variance Minimization

Worked Solution

How to Think About It: PCA decomposes the covariance of asset returns into systematic factors (the top eigenvectors) and idiosyncratic residuals. When you build a portfolio, your P&L has two pieces: the part driven by common factors (market, sector rotations, etc.) and the part that is stock-specific. A stat-arb or market-neutral desk wants to hedge out the systematic piece and harvest the residual -- that is exactly what this problem sets up. The constraint "zero exposure to the first $m$ PCs" is how you say "I don't want any market or sector risk" in math. Minimizing residual variance on top of that gives you the tightest idiosyncratic portfolio possible.

Part (i): Factor and Residual Returns

The factor returns are the projection of asset returns onto the principal component directions:

$f_t = V^\top r_t \in \mathbb{R}^k$

Each component $f_{t,j} = v_j^\top r_t$ is the return of the $j$-th factor portfolio. The reconstructed systematic return is:

$V f_t = V V^\top r_t$

Geometrically, $V V^\top$ is the orthogonal projection matrix onto the column space of $V$ (the subspace spanned by the top $k$ eigenvectors). So $V f_t$ is the projection of $r_t$ into the factor subspace. The residual is the orthogonal complement:

$u_t = r_t - V V^\top r_t = (I - V V^\top) r_t$

This lives in the $(d - k)$-dimensional subspace orthogonal to the factors.

Part (ii): Portfolio Factor Exposure and Residual Variance

The portfolio return is $w^\top r_t$. Split it:

$w^\top r_t = w^\top V f_t + w^\top u_t$

The factor exposure vector is:

$\beta = V^\top w \in \mathbb{R}^k$

The $j$-th entry $\beta_j = v_j^\top w$ is the portfolio's loading on the $j$-th PC. This is the number a risk manager looks at -- it tells you how much your portfolio moves when factor $j$ moves by one unit.

The residual return of the portfolio is $w^\top u_t$. Its variance is:

$\sigma_u^2(w) = w^\top \Sigma_u \, w$

where $\Sigma_u = \text{Cov}(u_t)$ is the residual covariance matrix. Since $u_t = (I - V V^\top) r_t$, we have:

$\Sigma_u = (I - V V^\top) \Sigma (I - V V^\top)$

where $\Sigma$ is the full covariance matrix. In practice, $\Sigma_u$ is often approximated as diagonal (each stock's idiosyncratic variance is independent), which makes the optimization much cheaper.

Part (iii): The Optimization Problem

Let $V_m = [v_1, \ldots, v_m]$ be the $d \times m$ matrix of the first $m$ factor loadings. The problem is:

$\min_{w \in \mathbb{R}^d} \; w^\top \Sigma_u \, w$

subject to:

$\mathbf{1}^\top w = 1 \quad \text{(unit leverage)}$

$V_m^\top w = 0 \quad \text{(zero exposure to first } m \text{ PCs)}$

The zero-exposure constraint $V_m^\top w = 0$ is a system of $m$ linear equations. Together with the leverage constraint, we have $m + 1$ linear equality constraints.

Convexity: Yes, this problem is convex. The argument is straightforward:

• The objective $w^\top \Sigma_u \, w$ is a quadratic form in $w$. Since $\Sigma_u$ is a covariance matrix, it is positive semidefinite, so the objective is convex (in fact, if $\Sigma_u$ has full rank restricted to the feasible subspace, it is strictly convex).
• All constraints are affine (linear equalities), and affine sets are convex.
• Minimizing a convex function over a convex set is a convex optimization problem.

This is a quadratic program (QP) with linear equality constraints. It has a closed-form solution via the KKT conditions (Lagrange multipliers). Introducing multipliers $\lambda$ for the leverage constraint and $\mu \in \mathbb{R}^m$ for the factor-neutrality constraints:

$w^{*} = \Sigma_u^{-1} \left( \lambda \, \mathbf{1} + V_m \mu \right)$

where $\lambda$ and $\mu$ are determined by substituting back into the constraints. When $\Sigma_u$ is diagonal (independent idiosyncratic variances), the solution simplifies significantly and can be computed in $O(d)$ time.

Long-short version: If the leverage constraint is $\|w\|_1 = 1$ instead of $\mathbf{1}^\top w = 1$, the $\ell_1$-norm constraint is still convex (it is the sublevel set of a convex function), but it is no longer a smooth linear constraint. The problem becomes a quadratically constrained convex program, still efficiently solvable but without a simple closed-form solution.

Answer: The factor exposure vector is $\beta = V^\top w$, residual variance is $w^\top \Sigma_u \, w$, and the minimum-residual-variance portfolio subject to unit leverage and zero exposure to the first $m$ PCs is a convex QP with linear equality constraints. Convexity follows from a PSD quadratic objective and affine constraints. The closed-form solution is $w^{*} = \Sigma_u^{-1}(\lambda \mathbf{1} + V_m \mu)$ with multipliers determined by the constraints.