Geometric Frontier

I fear I have gone past explaining a concept and need to take step back. Recently I used this chart which has the standard deviation on the x-axis (as opposed to the % of the asset in the portfolio). I want to explain where this chart comes from.

To re-cap, the example uses two assets with the following qualities:

Asset #1 forecast:
6% Return
20% Standard Deviation

Asset #2 forecast:
10% Return
30% Standard Deviation

The assets are uncorrelated.

Portfolio Properties

The chart comes from calculating the portfolio’s returns and standard deviation for every possible portfolio.

Determining the properties of a portfolio from the properties of it’s components is fairly straight forward when there are only a few assets involved. We’ll start with the arithmetic return.

Arithmetic Return

In our instance with two assets, the arithmetic return (expected return) is:

Portfolio Arithmetic Return = Weight₁ x Return₁ + Weight₂ x Return₂

The weight is the percent of the asset in the portfolio. So a portfolio of 40% asset #1 and 60% asset #2 would have a portfolio return of:

8.4% = 40% x 6% + 60% x 10%

Standard Deviation

Standard deviation is a bit more complicated. It’s formula is:

Portfolio St. Dev. = Sqrt[ (Weight₁ x St.Dev.₁)² + (Weight₂ x St.Dev.)² + 2 x Correlation x St.Dev.₁ x St.Dev.₂ x Weight₁ x Weight₂]

And in this sample portfolio case you would find:

19.7% = Sqrt[ (40% x +20%)² + (60% x 30%)² + 2 x 0 x 20% x 30% x 40% x 60%]

Efficient Frontier

These formulas come from standard portfolio theory and are pretty boring by themselves. But when you apply them to all possible portfolios — from 0% to 100% — it gets interesting. Doing so creates a curve showing a relationship between return and risk (standard deviation).

This shape is sometimes called a “Markowitz bullet” after Harry Markowitz who developed much of this math. The top part of this curve is called the efficient frontier because each portfolio on this curve has the highest return possible at that level of risk. Therefore these portfolios are “efficient”.

Usually the Markowitz bullet and the efficient frontier are developed from many assets. But we’re going to keep this simplified as there is nothing fundamentally different when using two or three assets vs twenty.

Geometric Frontier

Now as I’ve said many times, the arithmetic return isn’t the important return. The geometric return is what matters. So to turn the Markowitz bullet into something more useful we have to convert arithmetic returns into geometric returns using the formula:

Arithmetic Return– Standard Deviation² / 2 = Geometric Return

When you take the Markowitz bullet and convert it into a geometric return, you get the charts I showed in the last post.

Taking the highest return at each level of volatility, you find the geometric efficient frontier. The geometric efficient frontier runs from the point of maximum geometric return to the point of minimum volatility (this point is usually called the Minimum Variance Portfolio).

Notice how many of the portfolios that were deemed efficient under the prior efficient frontier are now no longer considered appropriate places to invest with the geometric efficient frontier.

Adding Cash

Now we add cash to the chart. We’ll set cash’s return at 2% for the example (the equation for adding cash in the footnotes¹).

The blue dots are all possible portfolio combinations of Asset #1, Asset #2, and cash, spaced out at 4% increments. They cover the “space” over which the potential portfolios provide returns. The blue line is the same geometric Markowitz bullet discussed above: the portfolios made up of only Assets #1 and Asset #2.

The orange dashed line is the geometric efficient frontier. Every point on this line provides the best possible geometric return for that specific standard deviation. There are three key points on the geometric frontier:

1. The first is on the far left edge, and is at 100% cash. It’s a bit strange to call it this, but when cash is included in the portfolio the left side of the frontier is the minimum variance point.
2. The second, and most important point, is the maximum geometric return at the right edge of the frontier (and the top of the entire curve). The blog posts outlining portfolio construction (here, here, and here) find this point without getting into the entire geometric frontier.
3. The third point is traditionally called the tangency portfolio.² On the geometric efficient frontier, this is the point where cash falls out of the portfolio³. I consider it an “elbow point”, because the geometric efficient frontier “bends” here towards a flatter path through the risk/return space.

Along The Curve

I wanted to introduce everyone to these portfolio curves, where they came from, and how to create them for yourself. Understanding them becomes very important when thinking about errors. Strangely, the geometric frontier is not discussed much on the web, especially in easily accessible terms.

I’ve talked now about the logic of aiming “left of the peak“, potentially to even provide a higher return. One way to do this is to target a portfolio along the geometric frontier left of the maximum return.

In Part 4, we’ll go deeper into investing along this curve.

1-Portfolio with Cash

Portfolio Arithmetic Return = Weight₁ x Return₁ + Weight₂ x Return₂ + Weight_Cash x Return_Cash

The Standard Deviation is the same equation, since cash has no volatility, but if curious, the equation for three assets its:

2-Because on the arithmetic efficient frontier, this point is “tangent” to the capital allocation line.

3-Important caveat being not allowing leverage or borrowing.