## Abstract

In this article, the authors compare the characteristics of a low-volatility strategy and low-risk strategies based on extreme Value at Risk (eVaR) constructed from a large universe of international stocks in the period from 1995 to 2015. They find that low-eVaR strategies have better risk and performance statistics exhibiting a less-pronounced sector concentration, a common problem in low-volatility portfolios. Although they find no significant differences in the structure of the risk premium conditioned on periods of economic expansions, the authors do find some differences conditioned on times of recessions. However, their analysis cannot explain the improved downside risk profile that remains a strategy-specific characteristic, caused most likely by the more diversified sector allocation. Their findings are robust to variations in the methodology of eVaR calculation.

Low-volatility strategies have been found empirically to be conservative portfolios delivering returns comparable or higher than the market portfolio (see, for example, Blitz and Vliet [2007] and Walkshausl [2014]). There are two main methods of construction: The portfolio can be obtained by (1) minimizing portfolio variance subject to some weight constraints or (2) a two-step process in which stocks are ranked by a risk-related criterion (e.g., historical volatility or beta) and then low-volatility stocks are selected and weighted on an ad hoc basis using equal weights or cap weights.

A problem of low-volatility portfolios documented in the literature is the concentration in low-volatility stocks (see DeMiguel et al. [2009] and Clarke et al. [2011]). Such biases can lead to concentrations in certain industries that tend to be less volatile in certain time periods (see Chan et al. [1999]) and are a source of strategy-specific risk.

In this article, we explore a different methodology for defensive low-risk portfolio construction that is not based on volatility but on a tail-risk measure called extreme Value at Risk (eVaR). Our approach follows the second method; that is, stocks are ranked by their eVaR—the ones with the lowest risk are selected and equally weighted. The eVaR measure is the Value at Risk (VaR) of the return distribution of each stock computed at a low-tail probability level using extreme value theory (EVT) to get a more accurate approximation of the low quantile needed because of the non-Gaussian behavior of stock returns (see Stoyanov et al. [2011] and Schmielewski and Schwehm [2014]). Our method, therefore, belongs to the class of low-risk strategies in which risk refers to a tail-risk measure.

We run an extensive empirical study on a large universe of international stocks to compare the properties of the low-eVaR portfolio and the low-volatility portfolio based on the same ranking methodology. Our first main finding is that the low-eVaR portfolio has better risk and performance characteristics and a better downside risk profile. Our second main finding is that the sector and regional concentration of the low-eVaR portfolio is lower than that of the low-volatility portfolio. To make sure the conclusions are robust, we explore four different estimations of eVaR and find similar results. Finally, we use the Fama–French–Carhart factors to compare the risk premium structure across the two methodologies and also to look for a possible explanation of the better downside risk profile of low-eVaR portfolios. Although we find statistically significant differences by conditioning on U.S. recessions, the systematic exposures cannot explain the better downside risk profile of the low-eVaR strategy, which, as a result, turns out to be a strategy-specific characteristic.

**METHODOLOGY**

The portfolio construction method is based on the eVaR of each stock, which is a forecast of VaR computed through EVT. There are different ways in which EVT can be applied (see Embrechts et al. [2004] and McNeil et al. [2005]). To verify the robustness of the conclusions, we adopt four different methods. In this section, we first describe the methods and then outline the portfolio construction approach, and, finally, we explain the risk and performance characteristics used for strategy comparison.

**eVaR**

EVT provides a model for the extreme tail of the distribution and has been applied in finance to estimate probabilities of extreme losses or extreme loss quantiles. The appeal of EVT is that it provides a model for the extreme tail without heavy assumptions on the return distribution.

There are two ways to apply EVT in practice: (1) the *block-of-maxima method* (BM) and (2) *peaks-over-threshold method *(POT) (see McNeil et al. [2005]). BM divides the sample period into blocks of equal size, which typically match the risk horizon (e.g., one month or one week) and computes the maximum loss in each block. The asymptotic distribution of the maxima is the Generalized Extreme Value Distribution (GEV). POT sets a high threshold and considers the extreme losses beyond the threshold. The asymptotic distribution of the excess losses beyond the threshold is the Generalized Pareto Distribution (GPD).^{1}

In our empirical study, we compute eVaR in two different ways: (1) we apply both methods directly on the stock returns and (2) we combine both methods with a GARCH(1,1) filter. The GARCH (generalized autoregressive conditional heteroskadasticity) model explains the clustering of volatility of the stock returns, and EVT is applied to the residual. In the literature, both techniques have been used (see, for example, Stoyanov et al. [2017] and the references therein). In total, we have four different ways of calculating eVaR.

The method of applying GPD and GEV directly to the return distribution is straightforward; the distribution parameters are estimated using the maximum-likelihood method. The block size is set to 10 working days, and the high threshold in the POT is set to the 90% quantile of the daily loss distribution, which is a common choice (see Stoyanov et al. [2017]). eVaR is then estimated as the 99% quantile of the fitted distributions.

In the case of a GARCH(1,1) filter, we assume the return has the following structure,

1where
= σ_{tZt}. The conditional eVaR at time *t* forecasted for 10 working days ahead is given by

where *I _{t}
* denotes the information available at time

*t*and is estimated as the 99% quantile of the GPD or the GEV distribution estimated from the normalized residual

*Z*

_{t}. The parameters of the filter are estimated from daily data; the estimation of GEV and GPD follows the steps shown here.

**Portfolio Construction**

The portfolio construction method consists of the following steps. For a given eVaR calculation method, at each rebalancing that occurs every quarter, the next steps are followed:

1. The eVaR of all stocks is calculated using the most recent 750 daily returns in local currency.

2. The stocks are ranked by their eVaR in increasing order.

3. The top 200 low-risk stocks are selected and are equally weighted.

^{2}No constraints of any type are applied.

Furthermore, we apply a one-way transaction cost of 50 basis points. Also, our universe of stocks has no survivorship biases—that is, if a stock gets delisted, the portfolio loses the invested amount in that stock. We use returns in local currency in the eVaR calculation to have a ranking invariant of currency risks.

Finally, apart from eVaR we apply the same algorithm for a strategy in which stocks are ranked by their historical volatility using the same estimation time window. This allows us to compare the properties of the volatility-based strategy to the eVaR strategies on an equal basis.

**Risk and Performance Characteristics**

To compare the performance of the strategies, we use some standard performance measures and some that are more sensitive to tail losses. We look at the standard realized return, volatility, and Sharpe ratio, but we also consider the downside deviation, the 99% VaR of the realized return distribution, and the drawdowns. Finally, we calculate the pain index (PI),

3and the ulcer index (UI),

4in which (*x*)_{+} = max(*x*,0),
denotes the running maximum, and
, *t* = 1, …, *T*} denotes the number of drawdowns. For a motivation, see Schmielewski and Schwehm [2014].

**DATA**

The stock universe consists of 9,472 stocks in total from 48 developed and emerging markets. The criteria for selection is that market capitalization should be above $1 billion.^{3}
Exhibit 1 provides details on the number of stocks per sector and region globally. The time period for backtesting the performance of the strategies is from April 1995 to December 2014.

The factor returns of the global market, size, value, and momentum factors are taken from the data library of K. French that is available online.^{4} The regressions are estimated with monthly returns. To define the time periods of U.S. expansions and recessions, we use data from the National Bureau of Economic Research.^{5}

**EMPIRICAL RESULTS**

In this section, we describe the empirical results starting with the risk and performance characteristics. We find that the eVaR strategies have better tail-risk characteristics measured by a variety of different measures, better returns than the low-volatility portfolio, and better Sharpe ratios. Both low-risk strategies significantly outperform the MSCI World Index, which we use only as a broad benchmark because the constituent stocks are not the same. We also conclude that the concentration of the low-eVaR portfolios in both sectors and regions is lower than that of the low-volatility portfolio.

Second, we compare the structure of the risk premium of the low-eVaR and the low-volatility strategies using the classical Fama–French three-factor model. We find that the exposures to the traditional factors are very similar in the entire period and also if we condition on U.S. expansions. Although we find some differences in the factor structure during U.S. recessions, they do not explain the better downside risk profile of low-eVaR strategies.

**Risk and Performance**

Exhibit 2 provides a comparison of the performance of the strategies. The plot shows the evolution of the value of each portfolio with $100 invested in April 1995. The risk and performance statistics are available in Exhibit 3.

The low-eVaR strategies exhibit slightly higher realized returns and slightly better volatilities, which translate into higher Sharpe ratios than the low-volatility strategy. Their downside risk—as measured by downside deviation, eVaR, the different drawdown statistics, PI, and UI—is also better than that of the low-volatility strategy.

Across the low-eVaR strategies, it is interesting to see that the GARCH filter does not contribute any value: The risk and the performance statistics of those that utilize the filter are quite similar to those that do not. Even though econometrically it makes sense to utilize it, the lack of value added is most likely due to the big difference between the data frequency used for estimation and the rebalancing frequency.

The MSCI World Index in Exhibit 3 is used only as a broad benchmark, because its constituents are a subset of the constituents of the other strategies; clearly, the risk premium of the strategies is driven by factors beyond the market factor. It is nevertheless important to consider the improvement in the risk statistics of the strategies relative to that index: They all exhibit much lower downside risks, even though the stock universe is larger and includes riskier stocks on a stand-alone basis.

To measure the concentration of the strategies in different sectors and regions, we use the notion of effective number of stocks translated into sectors or regions. The effective number of sectors/regions is defined as

5where *k* denotes the number of sectors/regions and *w _{i}
* is the total weight allocated to the corresponding industry/region. In the case of equally weighted allocation, the

*EN*coincides with the actual number of sectors/regions. A lower

*EN*statistic indicates a higher concentration.

Exhibits 4 and 5 illustrate the allocation to the regions and sectors through time for the low-volatility strategy (left plots) and the low-eVaR strategy based on GPD with no GARCH filter (right plots). The corresponding plots of the other low-eVaR strategies are very similar. Even though one can find similar trends in the allocations, the low-eVaR strategy exhibits a visibly lower concentration. The trade-off in Exhibit 4 appears to be primarily between the allocation to North America and the emerging markets. The fact that there are similar trends is natural given that tail-risk measures are sensitive to the scale parameter (see Stoyanov et al. [2013]).

In contrast, Exhibit 5 shows more complicated dynamics. In the low-volatility regime before the financial crisis of 2008, the low-volatility strategy tended to concentrate on the financial and utilities sectors (about 70% of the total allocation). As the crisis unfolded and volatilities increased across all sectors, the sector allocations gradually become less concentrated, with the financial sector shrinking to a small fraction of its allocation in 2004. In contrast, the low-eVaR strategy appears to be much more balanced, which can be explained by the fact that the allocation is affected not only by volatility but also by the stocks’ tail risk; low volatility does not necessarily mean low tail risk.

Those observations on the relative concentration are confirmed by the numerical data in Exhibits 6 and 7, which is based on the average sector allocation from 1995 to 2015. The *EN* statistic is higher for all low-eVaR strategies; the GPD method tends to produce a relatively more diversified allocation either with or without the GARCH filter.

**Structure of the Risk Premium**

The performance results in Exhibit 3 indicate that the low-eVaR strategies tend to dominate the low-volatility strategy in two aspects: (1) the superior realized performance over the entire period and (2) better downside risk profile of the return distribution. Next, we look at the exposures in the traditional Fama–French–Carhart four-factor model (see Fama and French [1992] and Carhart [1992]) to see if these differences arise from different exposures to the systematic factors. We also look for a possible explanation for the second aspect mentioned, which in theory might be caused by a factor that is significant for the low-eVaR strategy and insignificant for the low-volatility strategy.

We run the factor model unconditionally using the data in the entire period from 1995 to 2015 and also by conditioning on U.S. expansions and recessions. The estimated exposures, confidence intervals, and goodness of fit are available in Exhibit 8, which is organized in three panels. We include the regression results of the GPD and GEV strategies, which are representative of all low-eVaR strategies.^{6}

The results in Panel A, which are based on the full sample, show no significant differences in the structure of the risk premium of the three strategies. The three exhibit a significant exposure to the market factor and a value tilt. The size and momentum exposures are insignificant. The GPD strategy appears to have a statistically significant higher exposure to the market factor at the 95% level than the other two. The source of the excess performance is either the intercept term (GEV) or a combination of the intercept and a higher exposure to the market factor (GPD).

Panel B reveals a similar picture. The case of recessions in Panel C shows a different structure in the risk premium. None of the strategies has a significant intercept. The GEV strategy has a value and a momentum tilt, while the GPD has a significant exposure only to the market index. The low-volatility strategy has a statistically significant value tilt.

Even though the differences in the exposures can explain the differences in the average returns during recessions, they cannot account for the differences in the downside of the return distribution. The GPD strategy has a significant exposure only to the market factor and yet it exhibits a difference in both the extreme and the average drawdowns. As a result, we conclude that the reason for the differences in the downside is strategy specific and is contained in the residual; that is, although in theory there may be a factor causing these differences, it is not one of the common factors suggested in the literature. Most likely, it is a consequence of the more balanced allocation across sectors, which exposes the strategy to smaller sector-specific risks.

**CONCLUSION**

In this article, we compare a low-volatility strategy and low-risk strategies based on eVaR constructed from a large universe of international stocks in the period from 1995 to 2015. We adopt four different methodologies to compute eVaR based on EVT and a GARCH(1,1) filter, which is a common model for the clustering of volatility effect. The strategies are compared in terms of the risk-adjusted performance and also in terms of the concentration of the allocations to sectors and regions. We also compare the structure of the risk premium using the common Fama–French–Carhart four-factor model.

We find that the low-eVaR strategies exhibit better risk and performance statistics and also a better downside of the return distribution measured by a variety of statistics. The common problem of high sector concentrations of low-volatility strategies is less pronounced in the low-eVaR strategies because of the additional information contained in the tail behavior. Finally, we find that the structure of the risk premium is very similar in expansions; and although we detect some differences in recessions, they cannot explain the better downside risk profile of the low-eVaR strategies. We conclude that the better downside is a strategy-specific feature most likely caused by the more diversified sector allocation.

More generally, low-eVaR strategies represent an interesting alternative to low-volatility strategies. Low-eVaR strategies are methodologically rather different and can be viewed as a way to diversify the risk of crowding into one particular low-risk methodology.

## ENDNOTES

↵

^{1}The two families of distributions are defined as where*G*_{ξ,σ,μ}(*x*) denotes the GPD, and*H*_{ξ,σ,μ}(*x*) denotes the GEV. In both cases, the parameter ξ is referred to as the tail index and depends on the degree of heavy-tailedness of the loss distribution. The parameters μ and σ are a location and scale parameter, respectively. The eVaR at a tail probability of*p*is simply the (1 −*p*) quantile of the GEV or GPD distributions—that is, it is a loss threshold violated with a probability of*p*.↵

^{2}We also implemented two modifications of this method—selecting the top 50 and the top 100 stocks—and found similar conclusions. The difference between the three modifications is the relative stability of the sector and country allocations.↵

^{3}We also applied the same strategies to the subuniverse of the U.S. stocks and found similar results.↵

^{4}The factor returns are available at mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.↵

^{5}The U.S. business economic cycles are available online at www.nber.org/cycles.html.↵

^{6}For better readability of the regression output, all returns used in the regressions are in percentages. That is, the coefficients are effectively scaled by a factor of 100.

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