This post illustrates a couple of things that I learned this year with an application in finance. I learned about the simplex when I was researching amino acids. I learned some nitty-gritty about portfolio theory. These combined with my pre-existing knowledge about game theory and mixed strategy solutions.

Specifically, I learned a way of visualizing all possible probabilities of portfolio outcomes. This post narrowly focuses on 3 so that I can draw a picture. But the idea generalizes to many probabilities.  

Say that I can choose to hold some combination of 3 assets (A, B, & C), each with unique returns of 0%, 20%, and 10%. Obviously, I can maximize my portfolio return by investing all of my value in asset B. But, of course, we rarely know our returns ex ante. So, we take a shot and create the portfolio reflected in the below table. Our ex post performance turns out to be a return of 15%.

That’s great! We feel good and successful. We clearly know what we’re doing and we’re ripe to take on the world of global finance. Hopefully, you suspect that something is amiss. It can’t be this straightforward. And it isn’t. At the very least, we need to know not just what our return was, but also what it could have been. Famously, a monkey throwing darts can choose stocks well. So, how did our portfolio perform relative to the luck of a random draw? Let’s ignore volatility or assume that it’s uncorrelated and equal among the assets.  

Visualizing Success with Two Assets

Say that we had only invested in assets A and B. We can visualize the weights and returns easily. The more weight we place on asset A, the closer our return would have been to zero. The more weight that we place on asset B, the closer our return would have been to 20%.

If we had invested 75% of our value in asset B and 25% in A, then we would have achieved the same return of 15%. In this two-asset case, it is clear to see that a return of 15% is better than the return earned by 75% of the possible portfolios. After all, possible weights are measures on the x-axis line, and the leftward 75% of that line would have earned lower returns.  Another way of saying the same thing is: “Choosing randomly, there was only a 25% that we could have earned a return greater than 15%.”  

Visualizing Success with Three Assets

What about three assets? The portfolio weights of three assets require something more than just a line, but it’s still doable. See the image below-left. The x-axis measures the weight on asset B and the y-axis measures the weight on asset C. Where’s the weight on asset A? It’s whatever’s leftover. The three portfolios that are composed of only a single asset lie at the vertices.  The black line below-left plots all of the asset weight combinations that would provide a return of 15%. The shaded region is the set of weights that yield a return that’s greater than 15%.

Now that we can see the set of portfolios that would have beaten our allocation’s return, we can talk about how likely our ‘success’ of 15% return was, given that there were 3 assets to choose. The entire large triangle between the vertices has an area of 0.5 (B*H*0.5=1*1*0.5). That’s the size of the universe of possible weights. The area of the shaded region has an area of 0.0625 (B*H*0.5=0.25*0.5*0.5). The shaded area composes 12.5% of the entire universe (0.0625/0.5). If we chose our portfolio allocation randomly, then there is only a 12.5% chance of beating the return of 15%. Our asset allocation performed better than 87.5% of all possible allocations.

Clearly, we can generalize this method upward and onward to many assets and for many goals, such as standard deviation, Sharpe ratio, etc. It’s not just that our portfolio was able to beat some specific benchmark or ‘the market’. Rather, we have an idea of what was possible, given the assets that we could choose from.

Basically, a bunch of what I learned this year can be boiled down to a triangle… Maybe two triangles.