Markowitz Bullet Train

Introduction

I have spent the last couple of weeks ironing out my approach to constructing portfolios of hedge fund programs in anticipation of publishing some model portfolios. A process like this forces me to go back to basics to make sure I have a solid grasp of where I am going. To refresh my intuition on the standard approaches to portfolio construction, and why I DO NOT think they are right, I went looking for a primer on the Markowitz Bullet. I could not find one, so I decided to write my own.

The best way to understand something is to work with as simple an example as possible. A two-asset portfolio is too simple, as the solution is easily derived. A three asset portfolio is just complicated enough to illustrate the key aspects of the problem, while still being simple enough to visualize.

Three Asset Portfolio

For the rest of this article I am going to use the following three assets:

  • Their expected returns are 1.0%, 0.75%, and 0.5% per month.
  • Their volatilities are 5%, 4.5%, and 3.75% per month, respectively.
  • Assets 1 and 2 have a correlation coefficient of -0.2.
  • Assets 1 and 3 have a correlation coefficient of +0.8.
  • Assets 2 and 3 have a correlation coefficient of 0.0.

Obviously, this is sufficient to derive the 3 x 3 covariance matrix for the assets. Notice the Sharpe Ratios (0% risk-free rate) are approximately 0.7, 0.6, and 0.5.

If we examine all possible combinations of the three assets at a resolution of 1%, there will be 5,151 different portfolios, well within the bounds of exhaustive search. The number of portfolios arises from asset 1 taking values from 0 to 100% in increments of 1%, asset 2 taking values from 0 to (100 – asset 1)%, and asset 3 representing the balance of 100%.

The following two charts show, for every combination of assets 1 and 2, what the expected portfolio return and volatility are:

Expected portfolio returns for all possible portfolios
Expected portfolio volatility for all possible portfolios

The Ideas Behind the Markowitz Bullet

We have a set of assets to which we wish to allocate capital (in the present example, three assets). We have estimates of:

  • The expected returns of each asset.
  • The covariance relationships amongst the assets.

We are subject to constraints:

  • We must allocate all our capital (the allocations sum to 100%).
  • We must allocate some minimum portion of our capital to each asset (0% in our example).
  • We cannot exceed some maximum allocation to any one asset (100% in our example).

Note: The first constraint is an equality constraint, the remaining two are inequality constraints. This is important to remember!

 

The first key intuition is that investors seek returns, but are risk averse. The risk is measured by the volatility of the portfolio of assets (I fundamentally disagree with this notion). For our example, if we plot the return vs. volatility of every possible portfolio (in 1% increments) we get a chart like this:

 

The Markowitz Bullet - Three Asset Portfolios

The next key intuition is that a rational investor presented with two portfolios offering the same expected return would choose the less volatile one. In terms of the chart above, those portfolios are the ones for which there is no other portfolio BOTH higher and further to the left. These are called “Pareto Optimal” portfolios – you can read my post “Multiple Objective Optimization” for more.

Our mission is to identify, for any given level of expected portfolio return, the portfolio with the lowest volatility. The portfolio that meets this criterion is the most “efficient” portfolio. As we explore the range of possible expected returns, we build a set of efficient portfolios. These portfolios lie on the efficient frontier.

Let’s add the efficient frontier:

 

The Efficient Frontier - Three Asset Portfolios

Unconstrained Portfolios

You will have noticed that there is a “kink” in the efficient frontier. As you might have guessed, this is due to the constraints we have imposed. It happens to be the inequality constraints that cause this effect: We would probably like to short the poor-performing assets and use the proceeds to leverage our position in the high-performing assets. We cannot do this because we have imposed a maximum allocation of 100% and a minimum of 0%.

Let’s see what happens if we relieve these constraints (our allocation must still total 100%). We want to find the portfolio that minimizes volatility for a given expected return while constraining total allocation to 100%. Time for a little math, feel free to skip it!

Stating the first sentence more formally, we want to solve the following for w:

Minimize \mathbf{w'.\Sigma .w} subject to \mathbf{w'.1}=1 (weights sum to 1) and \mathbf{w'.\mu} =\mu _{p} (expected portfolio return).

Where w is a column vector of asset weights (w’ is transposed to a row vector), 1 is a vector of 1’s, μ is a vector of individual asset expected returns.

We can solve this problem using a Lagrangian:

\mathfrak{L}(\mathbf{w}, \gamma , \lambda )=\frac{1}{2}\mathbf{w'.\Sigma .w} - \gamma.(\mathbf{w'.1} - 1) - \lambda.(\mathbf{w'.\mu} - \mu_{p})

Differentiating by w, γ, and λ gives us a set of simultaneous equations we can solve for w.

Solving the Lagrangian for a range of expected portfolio returns from 0 to 15%, we get the following unconstrained efficient frontier. We can also see much more clearly how this image earned the name “The Markowitz Bullet”:

 

 

The Markowitz Bullet - Unconstrained Asset Weights

Why can’t we use the same approach for constrained portfolios? Because you cannot use the Lagrangian solution for inequalities! We will explore how to solve this problem in a later post.

The Grand Tour

In the image below, I have added a set of landmarks to The Markowitz Bullet. These are the places we will visit on our grand tour.

All possible two asset portfolios superimposed on the  Markowitz bullet

Single Asset Portfolios

The red dots labeled “1”, “2”, and “3” represent portfolios made up of a 100% allocation to a single asset. As you would expect they are located at the expected return and volatility of each individual asset.

Notice that on the right-hand side of the bullet, there are three “tails” as the portfolios become increasingly dominated by a single asset. There will always be as many tails as there are assets. These correspond to the three corners on the first pair of charts showing returns and volatility vs. portfolio. The portfolios map out a triangle in the asset 1 vs. asset 2 space. These three points are that same triangle mapped into return vs. volatility space.

The very first point on the efficient frontier is the single asset portfolio using the asset with the highest expected return (because there cannot be another portfolio with a higher return). Remember this detail, as it is the first step in finding the efficient frontier using Markowitz’ Critical Line Algorithm.

Incidentally, the last point on the efficient frontier is the minimum variance portfolio (because there cannot be another portfolio with a lower variance).

Two Asset Portfolios

Each pair of points labeled 1 through 3 is connected by an arc. Each arc represents all different 2-asset portfolios made up of the assets connected by the arc. I have added “+” marks at each 10% increment. If we traverse the arc from 1 to 3, the first “+” indicates 90% 1, 10% 3, the second is at 80% 1, 20% 3, etc.

Notice that some arcs reach further to the left and have more curvature than others. This is due to the combined effect of correlation and variance. If two assets had a correlation of 1.0, they would be connected by a straight line. If the correlation was -1, the arc would reach all the way to zero – it would look like a wedge. The lower the correlation the more the arc is curved towards zero volatility. The volatility of each asset in the pair controls the starting points of the arc.

The adjacent chart illustrates these effects using three assets with the same returns as above, but with equal volatility (5%), and correlations of 0, 0.3, and -0.95 – you should be able to figure out which is which. Note there is a limit to how far you can push the correlations apart until you get to infeasible solutions and the correlation matrix blows up (for example, you can’t have 2 and 3 with -1, 1 and 2 with +1 and 2 and 3 with 0):

The Markowitz Bullet with extreme correlations.

Three Asset Portfolios

The grey dots show all the three asset portfolios. Notice they form a grid. Each space represents a change of +-1% in one of the assets. For example, the red arc is the portfolio made up of x% asset 1 and (100 – x)% of asset 3. The adjacent row of grey dots represent the portfolios made up of x% asset 1 and (99 – x)% of asset 3, and 1% of asset 2.

Notice how in all but one cases, replacing 1% of 3 with 1% of 2 made the portfolio better (it moved up and to the left – it had higher return with less volatility). The remaining case was where asset 3 was already 0% and could not be further reduced.

Constraints

It is really important to understand how the constraints come in to play.

  • All the grey dots honor the equality constraint: every portfolio sums to 100%.
  • One inequality constraint is in play along each of the arcs: The asset NOT in the 2-asset portfolio is at 0%.
  • At the points marked “1”, “2”, and “3”, all the inequality are in play: two of the assets are at 0% and one is at 100%

Turning Points

The final and most interesting point to note is that the efficient frontier and the arc between assets 1 and 2 are the same all the way to the kink (or “turning point”) in the efficient frontier. This is not a coincidence. The efficient frontier is bumping up against one of the inequality constraints all the way from point “1” to the turning point. That constraint is the requirement that asset 3 represent at least 0% of the portfolio. The gold colored triangles show that if we have no such constraint, we would happily short asset 3 and load up on assets 1 and 2.

At the turning point, the set of active inequality constraints is changing. In this case, we are going from the lower bound on asset 3 being active, to none of the inequality constraints being active. This feature is REALLY important for being able to map out the efficient frontier when you have multiple assets. It is the foundation of the Critical Line Algorithm.

While not strictly a turning point, the last point on the efficient frontier (the lowest and left-most point) is the minimum-variance portfolio. By definition, there cannot be a portfolio with a lower variance, so it has to be the “end of the line”.

There and Back Again

As with all good tours, we return to our starting point with a better understanding than before we left. Let’s map what we have learned back on to our charts of expected portfolio returns and expected portfolio volatility.

The following charts add a set of black points which correspond to the set of portfolios that deliver a 0.8% return (just picked as an example). That set of portfolios is then added to the volatility chart to see how volatility varies along the portfolios that deliver 0.8% return. You can see how the volatility reaches a minimum and then begins to rise again. That minimum is the efficient portfolio for a 0.8% return.

The green dots show where all the rest of the efficient portfolios lie. They are found using the same approach of finding all the portfolios that deliver x% return and locating the one that has the lowest volatility.

Notice how the turning point is really obvious. The efficient portfolio travels down the edge of the feasible portfolio space (i.e. along one of the constraints) and then abruptly turns left into the unconstrained area. I hope you can see the connection between the edges of each of these surfaces, and the 2-asset portfolio arcs in my earlier charts.

Expected returns with constant return line and efficient portfolio added
Expected volatility with constant return line and efficient portfolio added.

Conclusion

I hope you have enjoyed this rather exhaustive tour of the Markowitz Bullet and the efficient frontier.

The more assets you have the harder it is to picture what is going on – lines become surfaces, surfaces become volumes, ad infinitum. All you need to keep in mind is that the same basic principles are still operating but in more dimensions.

In the near future, I will post about the solution Markowitz came up with for finding the efficient frontier. He called it the “Critical Line Algorithm”. It involves the brilliant insight that the efficient frontier is made up of a series of edges connecting “turning points”.

Once we understand how the whole process works, we have earned the right to criticize it. At that point, I can present what I believe is the right way to build a portfolio of hedge fund strategies.

Ping me if you want the R-Scripts I used to generate the images in this post so you can play with the values yourself.

Image: Bullet Train by Kazokuda on Pixabay

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