# Budget line for mean variance utility

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.

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.

• What should the axes be? Commented Apr 18, 2022 at 7:55
• @MichaelGreinecker Maybe take the most convenient axes? If the axes are mean and variance, then the budget line will be actually a curve, which is not very convenient. Commented Apr 18, 2022 at 17:07
• The most convenient axes will make this as simple as the problem of dividing one's money between burgers and bus tickets. There's no real difference. Commented Apr 18, 2022 at 17:40

As Michael Greineker points out, axes matter. A scooter is shorter and lighter then a truck, while a Rolls-Royce falls inbetween in both dimensions. Possibly there is a scooter and a truck for which $$70\% \cdot \text{length}_{\text{scooter}} + 30\% \cdot \text{length}_{\text{truck}} = \text{length}_{\text{Rolls-Royce}}$$ and $$70\% \cdot \text{weight}_{\text{scooter}} + 30\% \cdot \text{weight}_{\text{truck}} = \text{weight}_{\text{Rolls-Royce}},$$ thus in this coordinate system the Rolls-Royce will be on a line between the scooter and the truck. Yet this is not a budge line: for approx. \\$50,000 I could buy a truck and a scooter, but it would not be enough for a Rolls-Royce.