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Suppose there are only two risky assets and we want to optimize our portfolio. Constraints are that we have a minimum return $\overline{r}$ and we can only invest $w_1 + w_2 = 1$.

Is it possible that in this setting the constraint $w_1 \times r_1 + (1-w_1) \times r_2 = \overline{r}$ always solves the problem or am I doing something wrong here?

I tried to set it up with Lagrangian: The constraint with $\lambda$ always provides me directly with the solution.

But how is that? I mean it seems strange that the solution is completely independent of the variance and covariance.

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  • $\begingroup$ No. It is not possible, unless r1 and r2 are perfectly correlated. You have to show the minimization problem that you want to solve and the Lagrangian. $\endgroup$
    – Andre
    Apr 26, 2023 at 4:14

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With $r_1$ and $r_2$ constants and your constraint the return equation reduces to an affine function (a line in the plane) which indeed has only one solution for a given level.

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    $\begingroup$ I disagree. I think his problem is he is going from a statement about a minimum return, which I would represent using an inequality constraint, to something with an equality constraint. The equality constraint is not the same as a minimum return for the portfolio, it is that the portfolio has that specific return. $\endgroup$
    – John
    Apr 19, 2023 at 17:17
  • $\begingroup$ That could very well be. Maybe something got lost in the translation to math. $\endgroup$
    – Bob Jansen
    Apr 19, 2023 at 20:00
  • $\begingroup$ "Suppose there are only two risky assets and we want to optimize our portfolio", what is the criterion that is being optimized. With only the constraint it is not clear... $\endgroup$
    – nbbo2
    Apr 20, 2023 at 16:21
  • $\begingroup$ r1 and r2 are not constant. $\endgroup$
    – Andre
    Apr 26, 2023 at 4:12

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