Expected tail loss

Smart beta, factor beta, alternative beta... it goes by many guises, but it generally refers to risk premia investing that is more targeted in nature than broad market beta. Its origins lie in early quantitative investment approaches that seek to add value using factor based models for identifying what are now familiar return drivers, e.g. value, momentum, size and quality to name but a few.

Although factor-based models have grown in popularity over the last ten years, the wealth eroding losses experienced in the Financial Crisis of 2008 have driven allocations to mini- mum volatility strategies. Minimum volatility strategies seek to decrease expected losses during market downturns by overweighting those stocks that have historically contributed the most to a portfolio with minimal volatility either through low stock price volatility or favorable correlations. This approach lowers downside risk, but consequently also potentially limits upside. We question whether a mini- mum volatility strategy is the best means of securing this objective of reducing downside risk. This note takes a closer look at minimum volatility and considers an alternative strategy that we find more appealing: expected tail loss.

Minimum volatility – the basics

A minimum volatility approach defines risk, as the name implies, in terms of volatility. For a portfolio of stocks, this means that the portfolio’s volatility is determined by the underlying volatili- ties and correlations of the stocks held within that portfolio.

Unfortunately, volatility only adequately describes the risk an investor faces under fairly restrictive assumptions. In the real world of markets, the assumptions are nearly always violated:

• Correlation only captures linear relationships, but the co-movements of stocks are often non-linear, especially in times of crisis;
• Volatility is only an adequate descriptor when returns are normally distributed;
• Empirical evidence1 demonstrates that returns are more extreme than what can be expected under the assumption of log normality, as this paper will highlight.

Finally, the nature of the volatility statistic makes it at best a partial solution to the investor’s problem of minimizing drawdowns. Volatility is a symmetric measure. A process that minimizes volatility not only reduces downside potential but also reduces upside potential.

Expected tail loss as an alternative to minimum volatility

In order to address investors’ concern over loss mitigation, we propose constructing a portfolio around a risk measure that better captures the nonlinearities and non-normality of the distribution of stock returns. The measure we use is Expected Tail Loss (ETL). ETL allows us to estimate the features of the distribution that cannot be adequately captured using volatility. The richer measure of risk that ETL affords, allows us to better target portfolio drawdown risk. Additionally, ETL offers the benefit of not explicitly compressing the exposure to the upside of the distribution. Our simulation results show the ETL approach achieving superior risk-adjusted returns and lower drawdowns than those achieved using Minimum Volatility.

A closer look at minimum volatility

Minimum volatility portfolio construction has its origins in Modern Portfolio Theory (MPT) which holds under a restrictive assumption set, that investors’ preferences can be described using two statistical variables: expected return (mean) and risk (variance). It’s important to note that the minimum volatility portfolio is not simply a collection of stocks with the lowest individual volatilities; rather, it is the outcome of a mean-vari- ance optimization that exploits the opportunities for cross stock diversification, where the required inputs are individual stock volatility and cross stock correlation estimates. However, the conditions where the mean-variance framework adequate- ly captures the full dimensionality of risk facing the investor are rarely met in practice.

Let’s investigate this proposition further by looking at the co-movement of stocks. In a mean-variance world this co-movement is sufficiently measured by cross stock correla- tion. Correlation is, of course, a linear measure of association. Observation, however, suggests the co-movement of stocks can be non-linear / unstable at times. Consider that the diversification effect is dependent on robust correlation estimates, which are typically estimated using historical data.

Then consider the case where most of the estimation window corresponds to a relatively calm state of the prevailing market. This leads to issues during market turmoil when correlations are known to increase drastically; just when the diversification effect is of most utility. Under this minimum volatility method- ology, the correlation estimates do not accurately reflect the relationship between assets in this high risk environment, and are therefore ill-equipped to provide diversification during extreme conditions.

Further, the mean-variance framework upon which minimum volatility is based, assumes that stock returns are normally distributed. If this assumption were to hold then a plot of the histogram of stock returns would be bell shaped and symmet- ric with thin tails. But, in fact, this is not what we observe. As an illustrative example, Xiong (2010)² points out that observa- tions drawn for a normal distribution with the same standard deviation for the S&P 500 (measured from January 1926 – April 2009) predict a return of -15.5% to occur just over once in 83 years. In point of fact, the S&P 500 suffered a monthly loss of greater magnitude in at least 10 instances during the same measurement interval.

Finally, and perhaps most importantly, volatility has some properties that may make it a less desirable measure of risk from an investor’s perspective. Recall, volatility is a statistic that describes how observations are dispersed around the average value. Its symmetrical nature means that values below and above the average both contribute equally. Minimizing the volatility of a portfolio limits negative portfolio returns as desired; however, minimizing volatility will also decrease positive returns. A preferable low risk strategy would lower the magnitude of losses but preserve the ability of the portfolio to recover quickly. After a 20% portfolio drawdown, a 25% gain is required to climb out of the trough as a result of the lower asset base. The portfolio constructed to minimize volatility limits downside exposure but it also restricts upside potential.

A closer look at Expected Tail Loss

We propose a risk measure for use in portfolio construction that does not face the issues and limitations of minimizing volatility described above. The ideal measure would neither require symmetry nor assume normality and should be robust during market downswings in an attempt to prevent draw- down.

Expected Tail Loss (ETL), which is an extension of the com- monly used Value at Risk (VaR) statistic, fits these require- ments. Recall, VaR is a threshold statistic defined as the minimum amount of portfolio loss at a specified probability and horizon. For example, hypothetical portfolio might have a 5% VaR value of USD1 million. This means that 5% of the time the portfolio will lose USD1 million or more in a specified time horizon, say over a month. VaR does not, however, tell the investor how much on average they can expect to lose when losses exceed USD1 million. This is the information that the ETL statistic can relate.

² Xiong, J. (2010) “Nailing Downside Risk” Journal of Risk Management.

ETL is a more coherent risk measure that based on VaR which explains why some call it conditional VaR. ETL is defined as the expected amount of portfolio loss at a specified probability level. Instead of using VaR and knowing that the portfolio will lose at least USD1 million 5% of the time, using the ETL measure will tell us that we expect to lose on average USD1.2 million 5% of the time. Exhibit 1 shows this graphically using a histogram of hypothetical portfolio returns. The VaR value, represented by the dotted line, is the return value when 5% of the distribution is to the left. The ETL value is the expectation or average of the distribution to the left of the VaR. One note about semantics, in the above example we used the cash value of the loss, but this can also be expressed in return space, which we did for exhibit 1 and continue to do for the rest of the note.

ETL is calculated by averaging the losses that are beyond a certain threshold of a portfolio return distribution. There are many ways to create the distribution, but the simplest is to use the empirical portfolio returns. The minimum ETL portfolio optimization finds the combination of portfolio weights that result in the lowest ETL value (by summing the weighted individual asset return distributions). Minimum ETL portfolio construction uses past returns to capture co-movement of assets and does not suffer from the issues associated with a correlation estimate based upon a return distribution assump- tion. Upside reduction due to symmetry also does not exist for ETL because the statistic focusses purely on the loss side of the distribution. Additionally, assumptions of normality are not needed with ETL; the optimization process uses every observa- tion of the asset return data within the specified historical window to model the left side fatter tail in the distribution.

Most importantly the concept of ETL is in line with the investor’s true objective, reducing portfolio loss and not just portfolio uncertainty.

In simulations it yields more attractive risk-adjusted returns and lower drawdowns than its minimum volatility counterpart