ETF based Portfolio construction and Alternative Risk Measures
Background
Efficient Portfolio construction based on Harry Markowitz’s Modern portfolio Theory(MPT) has been around for more than half a century and it still is very much in use both by traditional wealth managers and the new crop of robo-advisors. It is based on mean-variance optimization of historical (or sometimes expected) returns for a bunch of individual stocks or asset classes. Instead of focusing too much on picking an individual stock or asset class, if we study the relative performance of multiple stocks or asset classes over time and construct a portfolio based on an individual’s risk appetite, it is possible to get better risk adjusted returns over such a diversified portfolio. The overall notion is, investors must strive to invest in a portfolio that has the least risk for a target expected return (or the greatest expected return for a target desired risk appetite) by picking a diversified portfolio along the positive curve of the efficient frontier.
Alternative Risk Measures
While variance (or its square root — Standard Deviation) as a standard risk measure has been widely used, there are other alternative risk measures like VaR and Expected Shortfall (ES) that have gained prominence, especially to communicate firm-wide risk exposure at financial firms. These measures can also be used to express risk levels for individual portfolios. Expected Shortfall is more helpful in some contexts to better understand potential risk exposure and its also mathematically referred to as a good coherent risk measure (follows the four axioms of sub additivity, monotonicity, positive homogeneity and translation invariance. For more in-depth reading, please refer to the references section below) that can be used to construct efficient market portfolios using convex optimization.
Expected Shortfall also sometimes referred to as Conditional Value at Risk (cVaR) or Average Tail Risk at a q% confidence level over a time period p is defined as the average expected loss in the 1-q% cases over a time period of p. For e.g. — For a portfolio of $1 Million at 99% confidence level and 1 month time period, if the ES is $100K, it means that the average loss over any 1 month time period is about $100K in those remaining 1% chance cases.
ES is more natural to understand in some ways, especially for people who strongly believe that variance on the positive side of returns is not bad at all!. It’s only the worst case scenarios that matter (major market downturns) and not the day-to-day market volatility.
Analysis
let’s review how an ETF based portfolio constructed using MPT looks like under standard mean-variance optimization. Let’s also analyze how alternative risk measures like ES look like for such portfolios and touch upon the question of — can we do a better job at constructing individual portfolios by using ES as the standard risk measure instead of variance?
A simple portfolio construction application is developed that uses convex optimization on mean-variance measures for a set of ETFs @http://vespanalytics.com/PortfolioOptimizer.
How to use:
- ETF’s — From the list of ETF symbols, enter at least 2 or more separated by commas
- Interval — Select the interval period over which risk, return measures are calculated. For e.g.: ES is expressed as the average loss sustained over a time period
- End Date — Pick a date up to which we want to get market returns data for the selected ETFs. Start date is selected as either 1/1/2000 or the earliest common start date for the selected ETFs, whichever is later
- Confidence Level — Expressed in percentage. Commonly used values are either 95 or 99
- Invested Dollar Amount
Note: Average Return values are decimal representation of percentages. e.g: 0.095 should be interpreted as 9.5%
Let’s go over an example portfolio construction case:
Average monthly returns for this portfolio range from roughly 0.4% to 0.6% and the mean-variance optimized portfolio mix for a target average return within this range can be obtained by selecting a value from the drop-down.
To compare how the risk measures — Standard Deviation(SD) and ES fare against each other for these mean-variance optimized portfolios, they are plotted against their corresponding target mean returns. Interestingly, in this case, while we see that SD increases with increase in target average return (which is to be expected since these portfolios are optimized for minimum variance), ES follows a somewhat opposite trend.
To be fair, this divergence in risk measures is not a common occurrence. For many portfolio combinations, both measures more or less follow similar trends. But nonetheless, for cases like these, we can demonstrate that it is worthwhile to explore alternative risk measures such as ES for portfolio construction.
References
Expected Shortfall — A natural coherent alternative at Value at Risk. https://arxiv.org/pdf/cond-mat/0105191.pdf
Modern Portfolio Theory — http://www.investopedia.com/terms/m/modernportfoliotheory.asp
