This blog post will likely be slightly more mathematical than my past posts, though I am hoping to keep it high level enough that it is still just as accessible.  I will be using matrix notation in the mathematics, but will step through each part of the calculation to explain it in plain English -- so please stick with me!

In this post, I want to provide a intuitive framework for understanding how unconstrained mean-variance optimization finds the optimal solution for the maximum Sharpe ratio portfolio.  By creating an intuition, we can understand why unconstrained mean-variance optimization can be dangerous and unstable due to sampling noise in estimating expected excess returns, variances, and correlations.

First, we will begin with our definition of the Sharpe ratio: the expected excess returns per unit risk:

sharpe

Here,

mu

is a vector of expected returns for our assets,

variance_covariance_matrix

is our variance-covariance matrix, and

risk_free_rate is our risk-free rate.  For simplicity, we will define the expected return in excess of the risk-free rate as the term:

excess_return

Assuming that the excess return of the minimum-risk, fully invested portfolio is positive, then the weights that solve for the maximum Sharpe ratio portfolio have the closed-form solution:

optimal_weights

This solution does not provide much intuition; the solution for our weights is a set of simultaneous equations built around the relationships of expected returns, variances, and correlations.  Fantastically unclear!

To gain some understanding of how this relationship plays out, let's consider a fairly trivial case.  Let's assume that we know for certain that the returns of our assets are entirely independent of each other (which implies that their correlations are zero).  This means that our variance-covariance matrix is now simply a matrix with variance terms on the diagonal and zeros on the off-diagonals.  This makes the computation of the inverse variance-covariance

inverse_variance_covariance_matrix

a trivial procedure: diagonal terms need to only be replaced by their reciprocals.  The term:

variance_adjusted_excess_returns

then becomes a vector of excess returns divided by variances.  The term:

sum_variance_adjusted_excess_returns

is simply the sum of those values.  Therefore, our optimal weights for independent assets are proportional to their relative expected excess returns per unit risk, where "risk" is measured as return variance.

Another way to think about this solution is that in the case of independent assets, mean-variance optimization will first leverage every asset so that its variance is equal to the maximum variance, and then set weights proportionally to the leveraged expected returns.  So if we consider the case where we have two assets, SPY and AGG,

SPY

AGG

whose returns are independent, the weights for the maximum Sharpe ratio portfolio of these two assets would be found by:

  1. Coming up with our scaling factor: the multiple that will scale AGG's variance to be in line with SPY's.  The solution is: spy_agg_scaling_factor
  2. Scaling AGG's expected excess return by this factor: gamma_agg
  3. Weighing SPY and AGG in proportion to their expected excess return terms

Our solution is:

spy_agg_optimal_weights

Now we have developed a fairly good intuition for how unconstrained MVO works for independent assets; how can we translate this to the non-trivial case where we have assets with a non-zero correlation structure?

Fortunately, statistics provides us with a tool for translating a non-zero correlation structure into a zero correlation structure via Principal Component Analysis (PCA).  PCA will take an NxN variance-covariance matrix of our assets and create a set of N linearly-uncorrelated principal portfolios (made up of our assets; sometimes referred to in literature as an eigen-portfolio) and their corresponding variances.  

The expected excess return for each principal portfolio is simply computed by the weighted average of expected excess returns.  So if our first principal portfolio has weights:

weights

then its expected excess return is defined by:

gamma_principal_portfolio

With the expected excess return, variance, and the statistical guarantee that the returns from these principal portfolios are linearly-uncorrelated, we can easily apply our above base-case intuition: leverage the principal portfolios until they have equal variance and then the optimal weights are the relative proportion of their leveraged expected excess returns.

So what, then, is the intuition behind the principal portfolios?  This one is a bit more difficult to answer.  Traditionally, the intuition behind each principal portfolio is that it represents some sort of independent risk factor.  What those risk factors are depends on the assets in the portfolio we are exploring.  When the assets are all U.S. domestic equities, the principal portfolio with the largest variance is typically a systematic risk factor; the rest of the portfolios typically correspond highly to sector or industry factors.  When the portfolio is made of global assets, we tend to find equity risk factors, interest rate risk factors, commodity factors -- and frequently geographic risk factors.

Now that we have a fairly "simple" framework for understanding how unconstrained MVO creates a solution for correlated assets, we can discuss the risks.  The biggest risk comes from the fact that in reality, we never know the true variance-covariance structure, but rather must construct a sample estimate, E, from historical returns.  Random Matrix Theory (RMT), a branch of probability that deals with matrix-shaped random variables, tells us that when we perform a principal component analysis on E, the composition of the principal portfolios with the lowest variance are likely dominated by sampling error.  In other words: they're noise.

Remember that unconstrained MVO will leverage the low variance portfolios until they have equal variance as the highest variance portfolio.  If a low-variance principal portfolio has a high relative expected excess return, by applying leverage it can balloon to dominate the portfolio.  This is why unconstrained MVO is often considered unstable: from one sampling period to the next, measurement noise can end up dominating portfolio composition.

Corey is co-founder and Chief Investment Officer of Newfound Research, a quantitative asset manager offering a suite of separately managed accounts and mutual funds. At Newfound, Corey is responsible for portfolio management, investment research, strategy development, and communication of the firm's views to clients.

Prior to offering asset management services, Newfound licensed research from the quantitative investment models developed by Corey. At peak, this research helped steer the tactical allocation decisions for upwards of $10bn.

Corey is a frequent speaker on industry panels and contributes to ETF.com, ETF Trends, and Forbes.com’s Great Speculations blog. He was named a 2014 ETF All Star by ETF.com.

Corey holds a Master of Science in Computational Finance from Carnegie Mellon University and a Bachelor of Science in Computer Science, cum laude, from Cornell University.

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