# Ruling out corner solution in portfolio maximization problem

I am new to the econometric world. I have a portfolio maximization problem $$\max \sum_{i}^ n a_{i} x_{i} \quad \text{s.t.} \quad \sum_{i}^n a_{i}=1, a_{i} \geq 0.$$ I solved the problem but I had a corner solution which I don't wish to determine. The whole idea is about the portfolio diversification. What conditions may be added to my objective function to have an interior solution not a corner? I tried tow utility functions(CRRA and exponential utility) in case this may help to make any difference to my final solution but nothing change.

• Are you using the Harry Markowitz portfolio optimization? The general idea in modern portfolio theory is that all the investors are risk averse, they assume that the returns of the securities follow the normal distribution and hence they need to know the mean and the variance of this distribution, from all the investments they will choose those with the highest expected return given the variance or the lowest variance given the expected returns of the investment. Commented Feb 3, 2023 at 15:08
• In this case, assuming that $r_i$ is the return of the investment $i$, such that $r_i\sim N(\bar{r}_i,\sigma_i^ 2)$, for each investment $i$ (the investment is the stock return) is the outcome of the following maximization problem $$r_i^{*}=argmax\{\mathbb{E}[r_i]-\delta\mathbb{V}ar(r_i)\}$$ where $\delta$ is the constant risk aversion coefficient. The problem is generalized for a porfolio s.t. $p_i=\sum_{i=1}^{k}\alpha_ir_i$ where $\alpha_i$ is the weight of the stock (or investment) with the return $r_i$, s.t. $\sum_{i+1}^k a_i=1$. Commented Feb 3, 2023 at 15:16
• So, try to think the above problem to solve for the optimal weights $\alpha^*$ that is a vector of dimension $k$ finite. Commented Feb 3, 2023 at 15:16
• What are you choosing here: $a_i$ or $x_i$?
– Amit
Commented Feb 3, 2023 at 15:21
• The classical problem solves for the optimal portfolio or equivalently the optimal weights. Take a look here statistik.econ.kit.edu/download/doc_secure1/8_StochModels.pdf and here sites.math.washington.edu/~burke/crs/408/fin-proj/mark1.pdf it may be of some help! Commented Feb 3, 2023 at 15:31

If $$x_i$$ is the return* on an asset, then your formulation seems to maximize a specific realized return which can only be done after the fact. No wonder you get a corner solution; you would put all the weight on the realized return that happened to be the highest of all.