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I'm looking for a way to introduce naive diversification bias in a mean variance framework and had the idea to model it as some sort of "aversion to extreme portfolio weights" of the following form:

$$ (1-\delta)\omega^{'} \Sigma\omega+\delta\omega^{'}\omega \rightarrow Min!\\ s.t.1^{'}\omega=1 $$

Here, $\omega$ is the weightsd vector, $\Sigma$ the covariance matrix and $\delta$ some preference parameter. By setting up the Lagrangean and solving for my lagrange multiplyer and then with this multiplier for the optimal weights vector i get something that looks like:

$$ \omega^*=\frac{1^{'}((1-\delta)\Sigma+\delta)^{-1}}{1^{'}(1-\delta)\Sigma+\delta)^{-1}1} $$

however, this doesn't look quite right as i expected for $\delta=0$ to obtain the min.variance portfolio and for $\delta=1$ i hoped to get a weights vector containing $\frac{1}{n}$ where $n$ is the number of assets/ length of my vector $1$. I'm sure i made a mistake somewhere along the way, would someone mind checking where i made a mistake and point me to the right way?

Thanks a lot for your help Thomas

EDIT: As mentioned in the comments here are some details how i got to this result: The Lagrangean of the above problem is

$ L(\omega,\lambda)=(1-\delta)\omega^{'} \Sigma\omega+\delta\omega^{'}\omega-\lambda(1^{'}\omega-1)$

The FOC for $\omega$ is:

$2(1-\delta)\Sigma\omega+2\delta\omega-1^{'}\omega-1)=0$ (the FOC for $\lambda$ is the usual constraint.)

from which i get $\omega=\frac{\lambda}{2}((1-\delta)\Sigma+\delta)^{-1}1$. Using this to obtain $\lambda^{*}$ i get $\lambda^{*}=\frac{1}{1^{'}((1-\delta)\Sigma+\delta)^{-1}1*2}$.

Inserting this for $\lambda$ in the first FOC i get the expression for the weights-vector as stated above. Setting $\delta=0$ i get the usual expression for the variance minimum portfolio, setting $\delta=1$ (i.e. naive diversification is my only target), then i get: $\omega^{*}=\frac{1}{1^{'}1}$. I think the expression ${1^{'}1}$ gives me the length of the asset vector of $n$ as I'm summing over all 1's.

Can someone confirm that this is correct? Furthermore, I'm not sure how to interpret the expression $((1-\delta)\Sigma+\delta)^{-1}$ as it contains a scalar $\delta$ and a matrix $\Sigma$ times a scalar? Is there something missing?

Running this in my little portfolio-toy and removing some specific constraints i getenter image description here, which makes me a bit more confident..

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  • $\begingroup$ @nbbo2 Thank you for giving it some thought. I suspect that this might not be the reason, I guess it's rather my poor understanding of algebra. As soon as I have access to my pc I will update this with some details, maybe then it's easier to spot my errors. The idea with $\omega^{'} \omega$ is rather simple as I'm trying to minimize vola plus the Herfindahl index. Perhaps this is not the right way to do it? $\endgroup$
    – T123
    Mar 26, 2023 at 10:30

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