Whereas the Cartesian coordinate system imposes a rectangular grid on the plane, the barycentric coordinate system imposes a triangular grid on the plane. We leave it as an exercise to show that the equation of a straight line is In general, the equation is a hyperplane in R n . A Portfolio is a Point. We are now in a position to view a portfolio as a point in space given in barycentric coordinates. This means that we have a proportion w 0 of our portfolio held in U.S. Dollars, a proportion w 1 of our portfolio held in asset 1, a proportion w 2 of our portfolio held in asset 2, etc.
Negative weights indicate a short position in an asset.
Note that the vertices of the tetrahedron are themselves portfolios. As expected, the volatility between any two portfolios is the distance between the two points. The portfolio's return is where z is the excess return of the n assets over the risk-free rate r 0 on U.S. Dollars, The Return Axis. The coefficients r i determine the slope and the return determines the location. All portfolios with return and all portfolios with a different return form two parallel hyperplanes.