Constrained Portfolio Projection with a Tracking-Error Limit
Summary
The document formulates a portfolio-construction problem that balances closeness to a risk-budgeted target portfolio against a tracking-error limit relative to a benchmark. Some benchmark holdings are unavailable, so the candidate portfolio must assign zero weights to those assets. The objective is to minimize covariance-weighted distance from the target while enforcing the tracking-error constraint and the asset restrictions.
The author asks whether the solution can be understood as a matrix-valued blend of the target portfolio and the minimum-tracking-error portfolio available under the asset restrictions. The proposed intuition is that the target should result when the tracking-error constraint does not bind, while a binding constraint pulls the solution toward the feasible portfolio closest to the benchmark. The document gives a Lagrangian and an expression for the restricted minimum-tracking-error portfolio, but does not provide a derivation or solution. It also leaves details such as budget and other weight constraints unspecified, so the proposed blend is a question rather than a demonstrated general result.
Key ideas
- The objective minimizes covariance-weighted distance from a target portfolio.
- A tracking-error threshold limits deviation from a benchmark portfolio.
- Assets unavailable for investment are represented by zero-weight constraints.
- The author conjectures that the solution blends the target and the constrained minimum-tracking-error portfolio.
- The document poses the derivation problem without establishing the proposed blending formula.
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Full text
# Optimal portfolio as combination of target and minimum tracking error portfolios?
# Optimal portfolio as combination of target and minimum tracking error portfolios?
Dear Quant StackExchange
I seek some intuition for how my portfolio behaves given constraints.
In a universe of say 5 assets, I have a "target portfolio" with weights that are found from risk budgeting, $w_T$. But I also have a benchmark ($w_{BM}$) that I care about minimizing my tracking error volatility towards. That benchmark is invested in some assets (say asset 4 and 5) that I cannot invest in, which gives rise to a further constraint $w_i=0$ for $i=4,5$.
So I'm interested in finding a portfolio of weights ($\hat{w}$) that is as close to the target portfolio $w_T$ as possible at the same time subject to the constraint that tracking error volatility towards the benchmark is below some threshold $\tau$. The other constraint is that weights in asset 4 and 5 in $\hat{w}$ is zero.
I have set up the Lagrangian: $$L = (w-w_T)'\Sigma(w-w_T) + \lambda_1Aw + \lambda_2[(w-w_{BM})'\Sigma(w-w_{BM}) +\tau] $$ where $\Sigma$ is the covariance matrix and $A$ is a $5\times5$ matrix of zeros except for the 4th and 5th diagonal entries being 1.
My intuition says that the resulting portfolio should lie somewhere between the target portfolio, $w_T$, and the "minimum tracking error portfolio" (that I get to be): $$w_{mte} = \underset{\text{s.t. }A'w=0}{\text{arg min }(w-w_{BM})'\Sigma(w-w_{BM})} = w_{BM}-\frac{\lambda_1}{2}\Sigma^{-1}A$$
But when solving it, I'm not able to show that the optimal portfolio is something like a complex convex combination of the two (think: $\hat{w} = \Gamma w_T + (1-\Gamma)w_{mte}$ where $\Gamma = f(\Sigma^{-1}, \lambda_1, \lambda_2)$)? I guess $\Gamma$ should simplify to the identity matrix when the second constraint is not binding, i.e. $\lambda_2=0$ thereby yielding the target portfolio.
Any help is much appreciated! I'm I on the wrong track here, or is somebody able to derive it?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.