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I'm trying to understand where the efficient frontier comes from.

What I understand about the efficient frontier

I understand the efficient frontier is essentially a subset of the boundary of the investment opportunity set of attainable portfolios. Intuitively, it represents, for each point along the $\sigma_{p}^2$ axis, the highest return possible at that point, which carves out the subset of points seen in this diagram below.

Figure 1

enter image description here

MPT Optimization as a minimization problem

As I understand it, Modern Portfolio Theory has the following set up. Assume a portfolio is created through some combination of $m$ assets. The portfolio has a corresponding weight vector $\mathbf{w}\in [0,1]^m$ and we have a return vector $\mathbf{r} \in \mathbb R^m$. We also define $\boldsymbol{\mu}=E[\mathbf{r}], \boldsymbol{\Sigma} = \text{Cov}(\mathbf{r})$. ​ Then we have $\mu_p = \mathbf{w}^T \boldsymbol{\mu}$ and $\sigma_{p}^2 = \mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w}$.

Then we find the optimal $\mathbf{w}^*$ via

Optimization Problem 1

\begin{align*} \mathbf{w}^*&=\underset{\mathbf{w}}{\text{argmin }} \mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w} \\ &\text{s.t.}\\ \mathbf{w}^T \boldsymbol{\mu} &= \mu^*\\ \sum_{i=1}^m w_{i} & = 1\\ \end{align*}

Source: https://en.wikipedia.org/wiki/Modern_portfolio_theory

MPT optimization as a maximization problem

According to this question,

Efficient frontier doesn't look good

The efficient frontier stems from the above optimization problem. But I see a problem with that.

Consider optimization problem 1.

Figure 2

enter image description here

If I slide along the $\mu_p$ axis for different expected returns, I will be able to now find the minimum variance for each one. But this would trace out that entire boundary, not just the efficient frontier. Therefore, it would seem to me this optimization problem would be better.

Optimization Problem 2

\begin{align*} \mathbf{w}^*&=\underset{\mathbf{w}}{\text{argmax }} \mathbf{w}^T \boldsymbol{\mu} \\ &\text{s.t.}\\ \mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w} &= \sigma^{2*}_p\\ \sum_{i=1}^m w_{i} & = 1\\ \end{align*}

I feel like, although they are similar problems, the implications are different. For Optimization Problem 2, I imagine sliding along the $\sigma_p^2$ axis. For each one, I pick the max $r_p$. Intuitively, this would trace out the curve seen in the efficient frontier diagram above. This would be in contrast to Optimization Problem 1 which traced out the entire boundary, not just the efficient frontier.

My question

Are both formulation of the optimization problem for finding $\mathbf{w}^*$ equivalent?

If so, can you explain what conditions are necessary for Optimization Problem 1 to trace out only the efficient frontier and not just the entire boundary of the investment opportunity set of attainable portfolios?

If not, can you explain why the prior stack exchange question says Optimization Problem 1 describes the efficient frontier?

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    $\begingroup$ I‘d say that you need to set the target return at some level above the expected return on the minimum variance portfolio. As our investor is already a rational investor, this is no additional assumption IMO. $\endgroup$ Dec 1 '21 at 1:00
  • $\begingroup$ The point you make is a valid one. But either method can be used, with care. In method 1 you have to take care not to use a guess an initial $r_p$ that gives you a point on the inefficient side of the curve. In method 2 you have to take care not to start with a $\sigma_p$ too low to be attainable, i.e. there is no portfolio with such a small variance. $\endgroup$
    – noob2
    Dec 1 '21 at 13:02
  • $\begingroup$ @noob2 but there are more than one point along $r_p$ that may have a given level of $\sigma^2_p$. By contrast, for any given $\sigma^2_p$, the max $r_p$ would be unique, at least in standard plots I've seen for the efficient frontier. Is that your understanding as well? $\endgroup$ Dec 2 '21 at 4:47
  • $\begingroup$ @Kermittfrog How would you know that minimum level without looking at the graph? With the maximization problem, you don't need to know that it would appear, so there's no need to sort of "by hand" or artificially add the minimum return allowed. $\endgroup$ Dec 2 '21 at 15:08
  • $\begingroup$ @noob2's comment has it all. Both methods require initial knowledge of the MVP. With maximization, you trace out sigma and accept the first result that is attainable; with minimization, you trace out mu (conventionally starting from 0) and stop at the first result for which sigma is non-decreasing. $\endgroup$ Dec 3 '21 at 8:05

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