Portfolio Optimization using Rank Correlation

Introduction

Since the introduction of modern portfolio theory (Markowitz, 1952), quantitative analysis of financial data has contributed tremendously to informed decision making in finance, such as portfolio selection, risk management, and asset pricing, to name a few. As we witness the tremendous growth in the financial markets both in terms of the quantum of data generated and innovative financial products created, effective risk management has become an even more pressing need for the financial industry.

Apart from the so-called risk-free assets, such as treasury securities, most investments offer returns with some form of risk. While individual assets may carry varying degrees of return-risk characteristics, collectively, when incorporated within a portfolio of investments, the resulting portfolio is expected to yield a certain diversification of the risks so as to offer the investor with an acceptable return-risk profile. The success of this process is largely dependent not only on the specific exposure (or allocation) in each asset, and thus, its own marginal distributions of return, but also on the correlations among returns of the underlying assets. A diversification is expected to lessen risk exposure since each asset class has a different correlation to the others; when stocks rise, for example, bonds often fall. At a time when the stock market begins to fall, real estate may begin generating above average returns. Therefore, a specific investment allocation is influenced significantly by the underlying correlations among different assets.

Given the current price of an asset, its return (over a specific period of time in the future) is the price difference (positive/negative if the future price is higher/lower than the current) divided by the current price. Since the future price is unknown, asset return is uncertain and it may be modeled as a random variable. Considering a specific historical period of (observed) asset prices, the future asset return distribution may be hypothesized, and its key statistical parameters estimated, such as the mean and variance. Suppose there is a set of assets under consideration and their asset returns are shown to be uncorrelated. It is well known that by increasing the number of such assets in a portfolio, risk in portfolio return is diversified, and in the limit when the number of such assets becomes infinite, the resulting portfolio risk asymptotically disappears. However, the existence of such a large group of perfectly uncorrelated instruments defies the common-logic in the market place. Consequently, the investor is confronted with the question of how to diversify the portfolio risk in the face of existing correlated assets. The seminal work by Harry Markowitz (1952) is an attempt in this direction, whereby an optimal set of weights is determined for each asset simultaneously to achieve a prescribed level of expected return in the future whilst associating such a portfolio with a minimum level of risk, with risk being described by portfolio variance (Elton, Gruber, Brown, & Goetzmann, 2006). In this approach, the basic rule is simple: low correlation makes for good diversification and highly correlated assets or asset classes are to be avoided. Thus, correlations play a fundamental and important role in the investment selection process.

Throughout the decades since, the above quantitative framework has inspired a large body of further work in financial data analysis and modeling, resulting in a wide range of software programs and tools that serve as decision support or help execute automated trading. There are also several subtle issues associated with estimating correlations, such as appropriate data transformation to ensure the assumed distribution, choice of statistical models, and predictions based on historical and simulated data.