How the portfolio weight simplex maps onto the familiar risk/return curve — and what the textbook bullet’s third dimension actually is.
Each dot below is one portfolio, colored by its weight mix. Drag the center panel to rotate the 3-D bullet; switch which coordinate plays the “third axis”; pull the flatten slider to collapse it and watch the bullet squash into the plain 2-D frontier. Toggle to 4 assets and the simplex becomes a tetrahedron, the bullet becomes a filled solid.
Elsewhere on this site I show the three-asset simplex — the triangle of all portfolios you can build from three assets — and scatter tangency points across it for various Monte-Carlo’d parameter sets. A natural question follows: how does that triangle relate to the familiar risk/return frontier plot? And if you’ve seen the classic 3-D “bullet” rendering of a three-asset frontier, you may have wondered what its third dimension actually is. The short answer is the fun one: there is no intrinsic third dimension. Here’s why.
With two assets the simplex is one-dimensional — a single slider, since w₂ = 1 − w₁. Both expected return μ(w) and variance σ²(w) are functions on that line, so the map is line → (σ, μ) plane: a 1-D object dropped into 2-D space. The result is a curve with no interior. Nothing collides; nothing fills in.
Now the simplex is the two-dimensional triangle. The actual map is triangle → (σ, μ) plane — a 2-D domain into a 2-D codomain. Because two different weight triples can share the same risk and return, the map is many-to-one, and the image is no longer a curve but a filled region. Points strictly inside that region are dominated (some other portfolio offers more return at the same risk, or less risk at the same return); only the left/upper edge is efficient.
The 3-D bullet you find in textbooks is an embedding choice, not a feature of the problem. Two of its axes are σ and μ; the third is something tacked on so the colliding portfolios separate out into a visible surface instead of piling on top of one another. That third axis is arbitrary — it can be one of the weights (w₃, w₁, …) or any portfolio statistic you like (concentration, exposure to one asset, a third moment). Whatever you pick, projecting it away returns you to exactly the same 2-D frontier, because the third axis never carried any risk-return information. It’s bookkeeping for the weight space, not a financial dimension.
Another way to see it: iso-mean lines on the triangle are straight (μ is linear in the weights), and along each one σ² is a convex quadratic with a single minimum. Collect those minima and you trace the minimum-variance frontier — a one-dimensional curve sitting inside the 2-D triangle, whose image is the hyperbola whose upper branch is the efficient frontier. The genuine frontier object is 1-D in weight space all along.
The takeaway: the dimensions that get “projected away” are coordinates of the weight space, never risk or return. The frontier itself is always a 1-D curve; everything inside it is just the shadow of a many-to-one map.