Optimal Portfolios For Two Assets

Let’s discuss the proper way to mix two risky assets in a portfolio.  I explained mixing one risky asset with cash before. Let’s up the complexity. 

I’m going to continue exploring how to mix assets³ with the aim of maximizing your long term wealth through the geometric return. If you haven’t read the prior post on building portfolios, please start there. This post builds on top of that one, further aiming to implement the concepts of the Kelly criterion.

A well known mathematical formula exists for maximizing the long term return of two assets,¹ but the pure math doesn’t provide the context needed to truly understand the nature of the process. The pure math is just numbers with no meaning, so we’re going to ignore it for now. Let’s understand the mechanics behind the optimal portfolio construction by tackling the problem graphically.

Also at this time, I’m going to ignore cash. I’ll remove this constraints later, but for now, let’s just keep it simple.

The Assets

I’ll use the following theoretical two assets:

Asset 1:  7% Arithmetic Return, 20% Standard Deviation

Asset 2:  10% Arithmetic Return, 30% Standard Deviation

First, let’s create the mixing ranges as explained in the prior post, by subtracting the variance (standard deviation²) from the Arithmetic Return.

Graphically, here is how this looks:

The chart is similar to mixing cash and a risky asset, but this time both assets have volatility. In order to “see” the optimal portfolio, we have to perform one more transformation to the chart.  Take the “wings” on asset #1, and move them around asset #2 (You can go the other way too if you want, just remember you’d be finding the % of asset #2 if you do).

Now, compare the geometric average of asset #1, to the expanded mixing range.  Mathematically this means:

A#1 Geometric Average – Bottom of Mixing Range = % of A#1 in the portfolio.
A#1 Standard Deviation² + A#2 Standard Deviation²                                                  .

In our specific case this is:

5% – (-1%)   =  46%
30%² + 20%^{2          .}

Optimally, the portfolio should consist of 46% asset #1 and 54% asset #2. 

Plotting the geometric return of the portfolio vs. the composition of the two assets shows a peak right where expected. Furthermore, notice the combined portfolio has a higher geometric return than either asset does by itself. The combination of the two assets improves the geometric return.

Only Mix Assets If Ranges Overlap

If the two asset ranges don’t overlap before the transformation, then mixing them provides no benefit to the long term return.  As we’re about to see, asset correlation expands and contracts these ranges, potentially changing this conclusion. But after this change the concept still holds. If you want to mix the two assets, the modified ranges need to overlap.

Let’s Bring In Correlation

Correlation measures how the two assets move in relation to each other.  When no relation exists, the correlation is zero. If the correlation is 1, the two assets move in lockstep with each other. If the correlation is -1 (negatively correlated), the assets move in the opposite directions.  One will go up, and the other will go down.  Partial correlation lies in between – sometimes moving in the same way, or sometimes moving opposite.

When mixing two assets, assets correlation expands or contracts the mixing range.  A correlation of 0 leaves the range as unchanged (as above).  A positive correlation shrinks the range. Negative correlation expands the mixing range.  Let’s start by examining negative correlation (-1).

Correlation of -1.

For two fully negatively correlated assets (-1), the mixing range expands on the top and the bottom by the geometric average of the two individual mixing ranges.  In our example, the assets have mixing ranges of 9% and 4%.  So the geometric average of the two ranges are:

(9% x 4%)^1/2  = 6%

Now lets add this amount onto both ends to create the new modified mixing range.

And now, compare the geometric average of asset #1, to our modified mixing range. 

____A#1 Geo Average ___ – ___ Bottom of Mixing Range_______ = % of A#1 in the portfolio
A#1 St. Deviation² + A#2 St. Deviation² + 2 x Geo Avg (A#1,A#2)                                .                                                 

5% – (-7%)  =  48%
25%          .

Interestingly, the preferred portfolio doesn’t change much from the uncorrelated portfolio.  More on this later. 

Notice how large the mixing range is now. It should be clear that assets with very low geometric returns, even negative geometric returns will still help the overall portfolio. Negative correlation makes even “bad” assets very useful to portfolios.

Because of the negative correlation, the portfolio return chart also shows a fairly large increase in expected long term returns – nearly a 1.5% increase.

Now lets look at what happens with fully positive correlation of 1.

Correlation of 1.

When assets become correlated with each other, you reduce the mixing range by the geometric average of the two assets. 

Then compare the geometric average of asset #1 to our shrunken mixing range.

5% – (5%)  =  0%
1%         .

Unlike our uncorrelated portfolio, positive correlation makes a huge difference, quickly making asset #1 no use to the portfolio.  It makes more sense to just hold asset #2.

As correlations increase toward one, diversification becomes less and less beneficial, to the point that these two assets which have very similar geometric returns no longer work with each other in a portfolio. This aspect clearly comes through in the portfolio return chart.

In-between Correlations.

To address in-between correlations, scale the geometric average by the asset’s correlation.  As an example, for a correlation of 0.5, reduce the mixing range by half the geometric average of the asset mixing ranges. And vice versa for negative correlations. Universally this translates to:

Geometric Average of Mixing Ranges X Correlation = Change in Mixing Range

6% * 0.5 = 3%

And our calculation of the optimal portfolio in this case is:

5% – (2%)  =  43%
7%          .

Correlation’s Effect on Portfolio Construction is Not Linear.

As shown above, negative correlations don’t effect the optimal portfolio much, but positive correlations will. Plotting the optimal portfolio as a function of the correlation, you get the following chart.

You can see how negative correlations roughly leave the the optimal portfolio in the same spot.  But as the assets become more and more correlated, the optimal portfolio composition moves dramatically.

This chart doesn’t always have to point down. It can flip and bend up, depending on which asset has a higher geometric average. But it always has a similar overall shape, accelerating away as the correlation moves towards 1. If you’re dealing with a highly positive correlation, it’s important to get the prediction correct, as an error here can have large consequences.

Other Key Takeaways

A couple other key points:

• If two assets have the same geometric return, they should always be mixed 50/50. The arithmetic return does not matter. The correlation does not matter.
• Negative correlations are deeply valuable in portfolio construction, adding to the long term return. Positive correlations are harmful, limiting the benefit of diversification.

In a following post I’m going to add cash to the mix. Cash complicates the math, which is why I’ve focused here on communicating the relationship between the asset properties and optimal portfolio construction. What you need to always remember:

• Mix assets by comparing their Geometric Returns.
• The mixing range for the geometric returns is the combination of each asset’s variance, expanded or contracted based on the correlation between the two assets.
• Negative correlation is wonderful.

1- The formula is (page 9):

3- To be clear I’m only discussing assets that are reasonably symmetrical in their distribution over the time frame between re-balancing. Think stocks and bonds. Some investments, like options, are skewed and don’t necessarily follow this math.