# 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? :-/

• Can you change means instead of variances and covariances? Commented Sep 20, 2023 at 13:59
• @phdstudent Unfortunately not, but i can manipulate the Merton parameters if i can't find a solution for the covariance matrix for the moment, but the ultimate target is a modification of the covariance matrix. In addition, a shift in the mean vector would shift the efficient frontier up or down, correct?
– T123
Commented Sep 20, 2023 at 14:06