How Covariance Changes Shift a Mean-Variance Efficient Frontier
Summary
The document asks how to modify a portfolio covariance matrix so that the resulting mean-variance efficient frontier moves right by a fixed amount in volatility while retaining its original shape. The author uses an analytical frontier derived from Merton’s 1972 parameters and compares the original frontier with a shifted target. Changing the volatility vector alone alters both the curve’s curvature and position, so it does not produce the desired match.
As a possible approach, the author reports that minimizing the sum of squared differences through adjustments to the Merton parameters comes close to the target, then asks whether a better method exists. The document provides no covariance matrix, asset inputs, equations for the target shift, or numerical results, so it does not establish whether an exact covariance transformation exists. It is best read as an open portfolio optimization question about the geometry of efficient frontiers, rather than as a worked procedure or validated solution.
Key ideas
- The author seeks a covariance matrix change that shifts an efficient frontier right by a fixed volatility amount.
- The frontier is derived analytically using Merton 1972 parameters.
- Changing asset volatilities alone alters the frontier’s curvature as well as its position.
- Minimizing squared differences through parameter adjustments is suggested as an approximate fitting method.
- The document does not provide enough inputs or results to verify an exact solution.
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# How to change the covariance matrix for a parallel-shift of the efficient frontier? # How to change the covariance matrix for a parallel-shift of the efficient frontier? I'm trying to obtain a parallel shift in my efficient frontier based on the Merton 1972-parameters. As i think a picture tells you more than 1000 words here is what i tried: The setting of my problem is the following: Correlation Matrix: Std.and Mean vector: To derive an analytical solution for the efficient frontier (as i don't want to numerically search every time i try something) i obtained the following frontier: I try to shift this efficient frontier to the right (e.g. increasing the volatility of all combinations of assets). Here is an illustration. The two green frontiers are what i get if i use the original market setting and if i shift the volatility of the efficient frontier by a fixed shift (here by 5% to illustrate my problem). The blue line is the analytical frontier that i hope to get after changing the covariance matrix (after recalculating my Merton 1972-Parameters given the changed covariance matrix) such that it matches with the shifted efficient frontier (the right one in green). I tried to shift the volatility-vector but this changed the curvature and the position of the blue efficient frontier but doesn't match the right green one. My question is therfore: How can i manipulate the covariance matrix such that my blue efficient frontier matches the right green frontier? Please let me know if something is not clear, i will provide details then. As always i appreciate your help and suggestions. Thomas EDIT: I played around a bit and here is one thought: Minimizing the sum of squared differences by manipulating my Merton parameters gets me somehow close. Any better ideas? :-/
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.