# Calculation of break-even correlation for diversification effect in N-assets case?

I'm thinking about a generalization of the following case: for 2 assets, there is a diversification effect as soon as i obtain a positive weight for the minimum-variance portfolio in the asset with the higher volatility.

If $$\rho_{12}$$ is the correlation coefficient and $$\sigma_1 < \sigma_2$$ then for $$\rho_{12}<\frac{\sigma_1}{\sigma_2}$$ we obtain a positive weight on $$\omega_2$$ in the minimum variance portfolio where $$\omega_2 = \frac{\sigma_1^2 - \sigma_1\sigma_2\rho_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_1\sigma_2\rho_{12}}$$ My question: regarding the correlation coefficient (-matrix), what is the general case for N-assets w.r.t $$\rho_{i,j}$$ and how to interpret this given the expression for the minimum variance portfolio weights vector as:

$$\boldsymbol{w}_{MV} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}' \boldsymbol{\Sigma}^{-1} \boldsymbol{1}}$$

where $$\boldsymbol{1}$$ is the usual unity vector and $$\boldsymbol{\Sigma}^{-1}$$ is the inverse of the covariance matrix.

EDITED: I guess similar to the 2-asset case, i have to start with the nominator $$\boldsymbol{\Sigma}^{-1} \boldsymbol{1}$$?

Thomas

• If I understood correctly you are trying to find the condition(s) for no weight negative in the minimum variance portfolio with N>2 assets? Or something else? (Not sure why you call it the "break-even correlation" though). Nov 28, 2023 at 19:40
• @nbbo2 Not really, I'm looking for a statement, for which constellations of $\rho_{I, J}$ in $\Sigma$ the volatility of the minimum variance portfolio is smaller then the volatility of the asset with the smallest volatility. I started with checking for the 2 asset case for which $\rho$ the weight for the asset with the lowest vola is not 100%.I'm looking for a generalization for the n asset case. I'm sure my original question is not clear enough so please feel free to ask. (in the train commuting home, sorry for any typos)
– T123
Nov 28, 2023 at 19:57
• Break even correlation is perhaps not the right term, I should have called it upper boundary for which the diversification effect takes place.. Hm, also not much clearer I'm afraid.
– T123
Nov 28, 2023 at 20:00

"My question: regarding the correlation coefficient (-matrix), what is the general case for N-assets w.r.t $$\rho_{i,j}$$ and how to interpret this given the expression for the minimum variance portfolio weights vector as:"

I don't have an answer for the general closed-form solution for individual weights in the MVP. However, answering the second question on interpretation of MVP weights vector,

I don't think $$\Sigma^{-1} 1$$ equals zero, actually. Because $$\Sigma^{-1} 1$$ is supposed to represent a vector of weights that are scaled by a scalar $$1'\Sigma^{-1} 1$$ that then represents the vector of minimum-variance weights. A high level mathematical proof can be understood from the perspective of the Lagrange multiplier method:

Starting with,

$$min_w \sigma^2(w) = w' \Sigma w$$

$$\sum^n_{i=1}w_i = 1$$

We can then form the Lagrangian (by adding a multiplier for the constraints):

$$L(w,\lambda) = w' \Sigma w + \lambda (\sum^n_{i=1}w_i - 1)$$

Taking the first derivative w.r.t. to the Lagrangian (to find the minima):

$$\frac{\partial L}{\partial w_i} = 2 \Sigma w + \lambda 1 = 0$$

$$\Leftrightarrow w_{min} = -\frac{\lambda}{2} \Sigma^{-1} 1 = \frac{\Sigma^{-1} 1}{1' \Sigma^{-1} 1}$$

Take note that $$-\frac{\lambda}{2}$$ is the normalization/scaling factor, which in this case is just $$1'\Sigma^{-1} 1$$. You can also think of the previous equation as the eigenvector of weights that correspond to the smallest eigenvalue that is the lowest portfolio volatility.

• Thank you for your thoughts. The derivation of the Minimum Variance Portfolio is not really my concern (know this). I discovered a similar question here: quant.stackexchange.com/questions/59694/… There, its still in the 2-asset case. I'm interested in the N-asset case. It looks like a straight-forward extention but i'm stuck today a bit..
– T123
Nov 28, 2023 at 13:31
• I edited the question, perhaps its a bit clearer now?
– T123
Nov 28, 2023 at 13:35