<p>If <span class="math-container">$x_i$</span> is the return* on an asset, then your formulation seems to maximize a specific <em>realized</em> 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.</p>
<p>Normally, however, you would choose portfolio weights aiming to</p>
<ul>
<li>maximize <em>expected</em> future return subject to some constraint regarding risk, e.g. fixed variance or value at risk (VaR) or expected shortfall (ES); or</li>
<li>minimize risk (e.g. variance, VaR or ES) subject to a constraint on <em>expected</em> future return.</li>
</ul>
<p>Such an approach is less likely to produce a corner solution.</p>
<p>*Or price.</p>

CC BY-SA 4.0
2023-02-06T08:15:24.703Z
As always with downvotes, I would appreciate a constructive comment so that I can improve upon what seems to be lacking. (Sadly, my intuition tells me this one has nothing to do with the quality of my answer; I hope I am mistaken.)
Richard Hardy
CC BY-SA 4.0
2023-02-03T21:03:45.320Z
2023-02-03T21:33:47.400Z
2023-02-03T21:33:47.400Z
Richard Hardy
<p>I am new to the econometric world.
I have a portfolio maximization problem
<span class="math-container">$$
\max \sum_{i}^ n a_{i} x_{i} \quad \text{s.t.} \quad \sum_{i}^n a_{i}=1, a_{i} \geq 0.
$$</span>
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.</p>
<p>Any advice will be appreciated.</p>

CC BY-SA 4.0
2023-02-03T15:08:23.513Z
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.
Oliver Queen
CC BY-SA 4.0
2023-02-03T15:16:22.913Z
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. $\sum_{i+1}^k a_i=1$.
Oliver Queen
CC BY-SA 4.0
2023-02-03T15:16:44.003Z
So, try to think the above problem to solve for the optimal weights $\alpha^*$ that is a vector of dimension $k$ finite.
Oliver Queen
CC BY-SA 4.0
2023-02-03T15:21:05.043Z
What are you choosing here: $a_i$ or $x_i$?
Amit
CC BY-SA 4.0
2023-02-03T15:31:00.830Z
The classical problem solves for the optimal portfolio or equivalently the optimal weights. Take a look here https://statistik.econ.kit.edu/download/doc_secure1/8_StochModels.pdf and here https://sites.math.washington.edu/~burke/crs/408/fin-proj/mark1.pdf it may be of some help!
Oliver Queen
CC BY-SA 4.0
2023-02-03T15:32:10.263Z
I have a question though. How are you supposed to solve for the optimal weights without knowing modern portfolio theory principles?
Oliver Queen
CC BY-SA 4.0
2023-02-03T17:30:37.660Z
@A.F.R.S2022 if my answer is of any help I would appreciate it if you cast a vote :)
Oliver Queen
<utility><finance><optimization>
Ruling out corner solution in portfolio maximization problem
A.F.R.S2022
# Ruling out corner solution in portfolio maximization problem

(Asked by: username_0 on 2023-02-03)

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.

Any advice will be appreciated.

**Tags:** utility, finance, optimization

> **Comments:**
> **username_1**: 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.  
> **username_1**: 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. $\sum_{i+1}^k a_i=1$.  
> **username_1**: So, try to think the above problem to solve for the optimal weights $\alpha^*$ that is a vector of dimension $k$ finite.  
> **username_2**: What are you choosing here: $a_i$ or $x_i$?  
> **username_1**: The classical problem solves for the optimal portfolio or equivalently the optimal weights. Take a look here https://statistik.econ.kit.edu/download/doc_secure1/8_StochModels.pdf and here https://sites.math.washington.edu/~burke/crs/408/fin-proj/mark1.pdf it may be of some help!  
> **username_1**: I have a question though. How are you supposed to solve for the optimal weights without knowing modern portfolio theory principles?  
> **username_1**: @username_0 if my answer is of any help I would appreciate it if you cast a vote :)  

---

## 1 answers

### Answer by username_3 on 2023-02-03. Score: 1 | Accepted answer

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.

Normally, however, you would choose portfolio weights aiming to

* maximize *expected* future return subject to some constraint regarding risk, e.g. fixed variance or value at risk (VaR) or expected shortfall (ES); or
* minimize risk (e.g. variance, VaR or ES) subject to a constraint on *expected* future return.

Such an approach is less likely to produce a corner solution.

\*Or price.

> **Comments:**
> **username_3**: As always with downvotes, I would appreciate a constructive comment so that I can improve upon what seems to be lacking. (Sadly, my intuition tells me this one has nothing to do with the quality of my answer; I hope I am mistaken.)  

---

