Portfolio Management – Why Correlation Matters
Authored By: Ranjan Bhaduri & Gunter Meissner | Publish Date: June 1, 2018
Authored By: Ranjan Bhaduri & Gunter Meissner | Publish Date: June 1, 2018
What is correlation?
While this question seems simple enough, it actually isn’t. In today’s financial environment, correlation is defined differently in different contexts:
So when we discuss ‘correlation’, the first thing we have to do is agree on which correlation concept we are talking about. As mentioned, in this paper, we will address the Pearson correlation model.
A Short History
As with many groundbreaking discoveries, there a bit of a controversy who the creator of concept of correlation is. Foundations on the behaviour of error terms were laid in 1846 by the French mathematician Auguste Bravais, who essentially derived what is today termed the ‘regression line.’ However Walker (1929) describes Bravais nicely as “a kind of Columbus, discovering correlation without fully realizing that he had done so.” Further significant theoretical and empirical work on correlation was done be Sir Walter Galton in 1886 who created a simple linear regression and interestingly also discovered the statistical property of ‘Regression to Mediocrity,’ which today we call ‘Mean-Reversion’.
A student of Walter Galton, Karl Pearson, whose work on relativity, antimatter, and the fourth dimension also inspired Albert Einstein, expanded the theory of correlation significantly. Starting in 1900, Pearson defined the correlation coefficient as a product moment coefficient, introduced the method of moments and principal component analysis, and founded the concept of statistical hypothesis testing, applying P-Values and Chi-squared distances.
Portfolio Management And Correlation
There are two critical equations in Markovitz’ 1952 Nobel Prize rewarded MPT (Modern Portfolio Theory). For a two-asset portfolio with assets X and Y, we have Equation 1:
µXY = wX µX + wY µY (1)
µXY : return of the portfolio1
wX : weight to asset X
wY : weight to asset Y
µX : return of asset X
µY : return of asset Y
Note that the weights add up to one: wX + wY = 1
The second critical equation is the portfolio standard deviation ‒ σXY or Equation 2
σXY : standard deviation of the portfolio
σ2X : variance of the returns of asset X
σ2Y : variance of the returns of asset Y
ρXY : correlation coefficient between the returns of the assets X and Y;
Importantly, the term σX σY ρXY in Equation 2 is called covariance of ‘X’ and ‘Y.’ We will minimize the covariance later to increase the Sharpe ratio.
It is interesting to note that the portfolio standard deviation of returns, also termed portfolio volatility, is interpreted in finance as risk: The higher the volatility, the higher the risk. Is volatility a good measure of financial risk? Well, most investors do not want their returns to fluctuate much, so there is some rationale. However, high volatility also means high potential profits. Therefore, minimizing volatility will also minimize potential profits. Consequently, it is better to apply more sophisticated risk concepts such as VaR (Value at Risk), Conditional VaR (CVaR), ES (Expected Shortfall), the Omega function, the Sortino ratio, or EVT (Extreme Value Theory) to measure financial risk.
Two Free Lunches
We all know that diversification is the key to investing. However, what exactly is diversification? We can define diversification as reducing the volatility of a portfolio by increasing the number of asset n in the portfolio and/or minimizing the covariance between the ‘n’ assets in the portfolio
Both actions and comprise free lunches, since they can be achieved with little or no cost.
With the first, there are different ways to show that increasing the number of assets in a portfolio reduces the volatility of a portfolio. In a two-asset portfolio, the increase in assets can be simulated with the weighting factors wX and wY. wX=wY=50% would mean a high diversification, whereas wX=1 (and consequently wy=0), or wy=1 (and consequently wX=0) would mean no diversification. If the assets X and Y have identical volatility, (i.e. σX = σY,) the equation results in the lowest portfolio volatility σXY for wX=wY=50%. This is true for any value of the correlation coefficient ρXY (See Figure 1).
However, an equal wX, wY weighting does not always minimize the portfolio volatility. The optimal weighting ratio depends on the individual volatilities σX and σY. The higher the volatility of an asset i, σi, the lower is weighting wi that derives the lowest portfolio volatility value, and vice versa (See Figure 2).
For an ‘n>2’ asset portfolio, it can be shown that the portfolio volatility decreases with increasing number of assets n. Let’s assume we have ‘i = 1, n assets’ in the portfolio and all assets have equal weights (1/n,) identical volatilities, and the assets are uncorrelated. Let’s also assume the mean return of an ‘asset i, xi’ is zero. Then the volatility of ‘asset I’ is . The portfolio volatility for equal weights ‘1/n’ is . Since we assume equal volatilities for each ‘asset i,’ we can write . Hence the volatility of the portfolio σP decreases with increasing number of ‘asset n’ as seen in Figure 3.
The second free lunch within the MPT (Modern Portfolio Theory) framework is reducing the portfolio volatility σXY, by reducing the covariance between the asset returns in the portfolio. For a two-asset portfolio this can be seen immediately from Equation 2. It shows the positive relationship between σXY and . The higher , the higher σXY, and the lower, the lower σXY. More mathematically, partially differentiating the square of Equation 2, the variance with respect to the covariance , we get Equation 3:
It confirms that there is a positive relationship between σ2XY and .
It also tells us that we have find the optimal weights wX and wY, which minimize σ2XY for market given values of σX, σY and ρXY.
Implementing the free lunch 1 is easy to achieve: Just put as many assets in the portfolio as possible. Implementing free lunch number 2 is a little trickier. We first have to find assets, which besides having a high return, have a low correlation between the returns. This may be the case for stocks versus gold, since gold is often bought as a safe haven asset; stocks versus bonds (especially in the later stages of an expansion when interest rates and stock prices rise); possibly stocks versus cryptocurrencies or weather derivatives; and so on.
From Equation 2 or 3 we realize that the lower the covariance between the returns of the assets , the lower the portfolio volatility σXY. So if we want to minimize the portfolio volatility σXY, the lower , the better. However, a very low means that the portfolio return will suffer since on average one asset will increase and the other decrease.2
Therefore, we should not only minimize portfolio volatility, but optimize the portfolio return – portfolio volatility ratio. Including a risk-free interest rate (r), this brings us to the most widely applied risk-adjusted performance measure the Sharpe ratio:
μP : Portfolio return
r : risk-free return (e.g. the return of a Treasury bill)
σP : Volatility of the portfolio return
Equation 4 is the ex-post Sharpe ratio, while the ex-ante Sharpe ratio uses the expected return of the portfolio E(μP).
Is the Sharpe ratio a good performance measure? Well, like any financial concept, it has pros and cons. The pros are that it is simple and can be easily understood. It informs the investor how well an excess return compensates for the risk taken. And, if derived in the same way, different Sharpe ratios tell the investor which portfolios have higher or lower risk-return ratios.
On other hand, it applies volatility as a risk-measure. This penalizes upside potential. In addition, volatility (the standard deviation of returns) assumes that the returns are normally distributed. However, in reality, returns often have fat tails, (higher kurtosis than the normal distribution) and can be asymmetric, (have a skewness ≠ 0 which is not captured).
As well, over the years, some hedge funds have learned to ‘game’ the Sharpe ratio. They purchase assets with a short term high Sharpe ratio, but a potentially high future price risk. An example is going long catastrophe bonds, which typically have a high return and low volatility. However, when the underlying catastrophe occurs, the price of the catastrophe bond declines sharply and volatility spikes.
Other drawbacks of it are that liquidity risk is not incorporated in the Sharpe ratio and can, as seen in the LTMC blow-up in 1997 and the2007-2009 financial crisis, have severe consequences. Plus, some hedge funds use a long time period to calculate the volatility, which typically results in a low volatility number. Hedge funds may also eliminate extreme returns or smooth returns, which again leads to a lower volatility.
Due to these drawbacks, many extensions and alternatives to the Sharpe ratio have been suggested such as the Omega function, a ranking function, which includes all moments; the Sortino ratio, which incorporate only downside volatility; the Treynor ratio, which only applies systematic risk; or the bias ratio, which measure how far the return of a portfolio differs form an unbiased distribution, just to name a few.
Despite these drawbacks, let’s derive a high Sharpe ratio by minimizing correlation. We will first do this for a two-asset portfolio. We have two equations to optimize: Equation 1 which we want to maximize and Equation 2 which we want to minimize.
Our approach will be to first empirically screen for asset combinations with a low correlation. We will then find the weights wX and wY, which minimize Equation 2 given a desired threshold level for the portfolio return of Equation 1. To start, we screen for assets with a low correlation. Assuming we found AAPL and WMT, we download the end-of-day prices of these and then calculate the returns, the standard deviation of the returns (the volatility of prices), and the correlation coefficient ρXY and the covariance of the returns. Then we find the optimal AAPL – WMT allocation by finding the weights wX and wY which minimize the covariance and the first two terms in Equation 2, given certain constraints for Equation 1. We typically use maximum likelihood methods to find wX and wY. Finally, we compare the Sharpe ratio with other Sharpe ratios.
Applying the first five steps, for a desired return threshold of five per cent, using GRG (Generalized Reduced Gradient) optimization, we derive a Sharpe ratio of 2.83 for data from 12/20/2016 to 12/20/2017. Typically, Sharpe ratios of >1 are considered ‘reasonable;’ Sharpe ratios of >2 are considered ‘good;’ and Sharpe ratios of >3 are considered ‘excellent’ (see www.dersoft.com/Sharpeoptimization.xlsm for details).
The generalization for a portfolio with n>2 assets is mathematically and computationally straight forward. The return is now and the portfolio standard deviation is , where βh is the horizontal β vector of asset weights (summing up to 1); βv is the vertical β vector of the asset weights;3 and C is the covariance matrix of the returns of the assets.
Applying these approach for the 10-asset portfolio of AAPL, NVDA, BA, JNJ, WMT, AA, NKE, 10y T-bond, gold, and bitcoin, we get a Sharpe ratio of 4.5 for data from 12/20/2016 to 12/20/2017 (see www.dersoft.com/Sharpeoptimization.xlsm for details).
These exercises show diversification consists of two free lunches. The first lunch is receiving a lot of attention since it is easy to achieve: increasing the number of assets in a portfolio decreases the portfolio volatility. The second requires some math and computing: It is minimizing the covariance between the assets in the portfolio by finding the optimal weights which minimize the weighted covariance and other terms of the portfolio volatility equation. In order to increase a certain performance measure such as the Sharpe ratio, minimizing the portfolio volatility given desired return thresholds can be applied. Typically, maximum likelihood procedures are used to find the optimal asset allocation (the optimal asset weights).
Ranjan Bhaduri is founder and president of Bodhi Research Group.
Gunter Meissner is president of Derivatives Software and adjunct professor of MathFinance at Columbia University and NYU.
For the appendix and references to this article, contact the authors.