# Understanding the Weights of an Optimal (Mean-Variance) Portfolio

I have calculated an optimal portfolio, using a historical covariance matrix, and determined the weights of n risky assets in the optimal portfolio.

The utility function I minimize is represented by $$U(w)=w^T \mathbb{E}(R)-A\frac{1}{2} w^T \mathbb{V}(R)\, w.$$

I am wondering what makes certain assets receive high weights, and what makes certain assets receive low weights?

• How is the utility function linked to the relative wieghts of assets in your optimisation? What kind of optimisation did you perform? – Mats Lind Aug 24 '16 at 12:15

Assuming you are minimising variance/standard deviation of the portfolio, then you are trying to allocate more weights towards less risky assets. You can try this if you create the covariance matrix yourself:

> c
[,1] [,2] [,3]
[1,]  3.0  0.0 -0.1
[2,]  0.0  6.0  0.2
[3,] -0.1  0.2  1.0
> myPack::globMin(c)
Calucalated:
myPack::globMin(cov = c)

Expected return:     0.05
Standard deviation:  0.8137612
Weights:
asset 1 asset 2 asset 3
0.2430  0.0881  0.6689

So you see that if I put large variance for the asset (large diagonal element that represents the variance of the asset itself - risk) it is allocated lower weight for the global minimum variance.

Quite interesting is discussion about this in terms of eigenvectors of the covariance matrix and Random Matrix Theory application, for instance in Laloux

• Thanks. The portfolio was optimised based on the utility function (E(R)-0.5*A*variance). How does this change it? i.e. was optimized to maximize utility. – Alex Aug 24 '16 at 11:06
• I think it is the same principle, but not entirely sure now. You are looking to find highest Efficient frontier tangent to your utility function and with risk aversion (negative coefficient) it would only make the allocation even more inclined towards lower variance results - I never really studied this Markowitz utility optimisation in depth. Are you following some textbook or lectures? – Jan Sila Aug 24 '16 at 11:23

Your optimal portfolio is a compromise between high return and low variance. The simplest reason for an asset to be strongly weighted in the optimal portfolio is that this asset by itself has an above average ratio of return to variance.

Alternatively the asset correlates to other assets in a way, that the assets together have a favorable return to variance ratio. In the most extreme case that would happen, if two assets are negativly correlated but both have positive return.

Your optimal portfolio consist of assets that together maximally diversify (maybe even hedge) each other and simultaniously offer the best reward.

This is true for all utility functions that balance return and variance somehow and therefore not specific to your specific form.

Your optimal portfolio obviously lies on the efficient frontier.

• When all the returns are equal and all correlations are zero, the optimal weights are inversely proportional to the variances (so called "naive risk parity"). When some returns are better than others, the weights of the hi returns assets are raised. When correlations are non-zero the weights of pos correlated groups of assets are lowered and the neg ones are raised. – Alex C Aug 28 '16 at 11:05
• I might be stupid here, but if all correlations are zero and all returns are equal, shouldn't the optimal portfolio consist then to 100% of the asset with the lowest variance? – Ami44 Aug 28 '16 at 11:43
• It is not in your interest to put 100% in one asset because of the "free lunch" provided by diversification. With two assets of equal variance and no correlation optimal solution is 50 50. – Alex C Aug 28 '16 at 16:26
• You are right of course. I was thinking of 100% correlated assets, sorry. With zero correlation you have diversification and you should not buy only one asset. – Ami44 Aug 28 '16 at 17:40