Consider the mean-variance utility used in CAPM. The budget line when allocating a risk-free and a risky asset is the line connecting the $r_f$ and the risky asset.

Suppose that I have fixed amount of wealth and I want to allocate between two risky asset.

In the Arrow Debreu framework, the budget line when allocation between two risky asset is also the line connecting the two assets.

How about the mean-variance framework?

My guess is that, if we are allocating two risky assets, then the budget "line" is $\bf not$ the line connecting the two assets. For example, considering the allocation between two identical assets with identical mean and variance but independent correlation. Then, the allocated portfolio reduces the variance but keeps the mean the same: the "budget line" is not the line connecting two assets.

Consider the mean-variance utility used in CAPM. The budget line when allocating a risk-free and a risky asset is the line connecting the $r_f$ and the risky asset.

Suppose that I have fixed amount of wealth and I want to allocate between two risky asset.

In the Arrow Debreu framework, the budget line when allocation between two assets is also the line connecting the two assets.

How about the mean-variance framework?

My guess is that, if we are allocating two risky assets, then the budget "line" is $\bf not$ the line connecting the two assets. For example, considering the allocation between two identical assets with identical mean and variance but independent correlation. Then, the allocated portfolio reduces the variance but keeps the mean the same: the "budget line" is not the line connecting two assets.