# Naive Diversification under mean variance

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 get , which makes me a bit more confident..

• @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?
– T123
Mar 26 at 10:30