Formulating Residual Variance and Alpha Portfolio Optimization
Summary
The document proposes a quadratic program for finding an efficient frontier between expected alpha and a portfolio’s residual variance relative to a benchmark. It expresses residual variance as portfolio covariance risk less the benchmark variance scaled by the squared portfolio beta, then uses that matrix in a mean–variance objective with a risk-aversion parameter. The constraints described include fully invested weights and per-asset minimum and maximum bounds.
The author reports that ordinary mean–variance optimization works with the same constraints, while solvers fail when given the proposed residual covariance matrix because it is symmetric but not positive definite. The question raises whether negative values from this calculation indicate a conceptual or implementation error. It offers no resolution or empirical validation, so the formulation’s assumptions and the conditions needed for a valid positive semidefinite residual-risk matrix remain open issues.
Key ideas
- Residual variance is formulated as portfolio variance minus benchmark variance scaled by squared portfolio beta.
- The proposed quadratic objective trades off expected alpha against residual variance using a risk-aversion parameter.
- The optimization includes a full-investment equality and per-asset weight bounds.
- The author reports that the residual covariance matrix may fail positive definiteness checks, preventing standard quadratic-program solvers from proceeding.
- The document poses, but does not answer, whether negative computed residual variance signals a modeling error.
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Full text
# Residual Covariance Matrix, and MVO for Residual Variance and Alpha
# Residual Covariance Matrix, and MVO for Residual Variance and Alpha
My overall goal is to find an efficient frontier using QP in terms of $\alpha$ and residual variance ($\omega^2$) for a portfolio $P$ given a benchmark $B$.
We know the equation for residual variance is (excuse my bad latex skills):
$\omega_P^2 = \sigma^2_P - \beta^2\sigma^2_B$
Using an n X n covariance matrix $\Sigma$, a weight vector $\bar{w}$, a beta vector $\bar{\beta}$ made of each portfolio holding's beta compared to the benchmark, and a scalar $\sigma^2_B$ which is the calculated variance of the benchmark, we can derive a QP compatible objective function:
$\omega_P^2 = \bar{w}^T_P(\Sigma_P - \sigma^2_B\bar{\beta}\bar{\beta}^T)\bar{w}_P$
So in the Quadratic Programming Framework, given the above, a vector of alphas ($\bar{\alpha_P}$), a risk aversion parameter ($\lambda$):
$\min_\limits{\bar{w}_P} \frac{1}{2}\lambda\bar{w}^T_P(\Sigma_P - \sigma^2_B\bar{\beta}\bar{\beta}^T)\bar{w}_P - \bar{w}^T_P\bar{\alpha_P}$
subject to:
$A\bar{w}_P = b$
$G\bar{w}_P \le h$
Where $A$ is a vector of ones, and $b$ is 1 (sum of weights must equal 1) and G and h define min and max weights for each holding.
I know that my constraints are valid, as if I change this to be a typical MVO (replace $\Sigma_P - \sigma^2_B\bar{\beta}\bar{\beta}^T$ with $\Sigma_P$) the QP optimizer returns optimized portfolios with the proper bounds on weights.
The problem however, seems to be that the resulting matrix (residual covariance?) $\Sigma_P - \sigma^2_B\bar{\beta}\bar{\beta}^T$ is not positive definite (though it is symmetrical), and thus the QP solvers fail. In other words, my residual covariance matrix can yield portfolios with negative residual variance. I am still wrapping my head around what a negative residual variance would mean (and perhaps this is a sign that something is awry). Any thoughts on how to progress (or where I've gone wrong) would be greatly appreciated.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.