Formulating Mean-Variance Optimization for Index Tracking and Beta Tilts

Summary

The document presents a proposed mean-variance portfolio optimization using residual covariance from a diagonal model. Its constraints set a target expected return, require the portfolio weights to sum to one, impose a beta-related condition, and specify a desired portfolio beta through an auxiliary benchmark weight. The author asks whether setting that target beta to one would replicate a market index and whether a higher target could create a beta tilt.

No answer, derivation, or backtest is included, so the formulation is not validated in the document. In particular, it does not define a tracking-error objective based directly on benchmark-relative returns; it minimizes residual variance under the stated constraints. The relationship between the auxiliary weight, the beta constraint, and the desired exposure would need to be checked carefully before interpreting the solution as index replication or a controlled beta tilt.

Key ideas

• The proposed objective minimizes residual variance using a diagonal-model covariance matrix.
• Constraints specify a target return, a fully invested portfolio, and beta-related exposures.
• The author asks whether a target beta of one can replicate an index and whether a higher target can create a tilt.
• The formulation is not answered or tested in the document.
• Index tracking would require checking whether the objective and constraints directly control benchmark-relative risk.

Tags

# Index Tracking Problem


# Index Tracking Problem












I have set up a mean variance optimization problem,

$$min:{W}^{\prime}{\Sigma_{\varepsilon}{W}}$$ $$s.t:{W}^{\prime}{\alpha}=R_B\;,\;\;W^{\prime}l={1},\;\;W'\beta=0,\;\;W'Z=\beta_p$$

where, $W$ is an ($n+1$)vector and the last value ($W_{n+1}=\sum_{i=1}^{n}w_i\beta_i$ ),

$\;\Sigma_{\varepsilon}$ is the covariance matrix of residuals as proposed by the diagonal model,

$\alpha$ is an $(n+1)$ vector of estimated alpha values expect for the last value which is the benchmark return $R_B$,

$l$ is an ($n+1$) vector of ones expect for the last value which is $0$,

$\beta$ is an ($n+1$) vector of ones expect for the last value which is $-1$

$W'Z=\beta_p$ is a condition that the beta of my portfolio is a value that I desire, so $Z$ is a vector of $0$s expect for the last value which is 1.

I am trying to form a portfolio that replicates the market index, as you can see I have not set up this problem in the tracking error framework ( I just started reading it). I have kept the condition $W'\beta=0$ because this is the way the diagonal model was initially set up by the book I have been reading.

Can this model work to replicate a market index if I set up my exposure $\beta_p=1$? could it help me tilt my exposure say if I wanted a portfolio beta of 1.5 ?. Thanks for your help !!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.