Two-Asset Return Constraints Can Determine Weights Directly

Summary

The document asks whether a minimum-return constraint in a two-risky-asset portfolio can determine the allocation without using the assets’ variances or covariance. With weights summing to one, the expected return is an affine function of the first asset’s weight when the two asset returns are treated as constants. Setting this expression equal to a specified target gives one weight, provided the target is attainable and the asset returns differ.

The answer explains that this direct solution follows from the return constraint itself: there is only one free allocation variable and the equality fixes it. It does not demonstrate that the allocation minimizes variance or solves a general Markowitz problem with a minimum-return inequality. If the constraint is merely a lower bound, or if the target can be exceeded by many feasible portfolios, risk and covariance can still matter to the optimization. The short exchange supplies no numerical example and leaves those broader cases unstated.

Key ideas

• With two assets and weights summing to one, only one allocation weight remains free.
• A fixed expected-return equality is affine in that weight.
• When attainable and asset returns differ, the equality determines a unique allocation.
• That calculation alone does not solve every minimum-variance problem with a return floor.
• Variance and covariance matter when the return condition leaves multiple feasible allocations.

Tags

# Markowitz Optimization with 2 assets


# Markowitz Optimization with 2 assets












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.

## Answer by Bob Jansen (score 2)

https://quant.stackexchange.com/a/75275

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.