Literature Review

The concept of drawdown is far from new. In the 1980s, Garcia and Gould (1987) note that in their experience, many investors put more emphasis on maximum drawdown as a risk metric than on volatility. Many studies have since contributed to a better understanding of maximum drawdown. For example, Douady et al. (2000) and Magdon-Ismail et al. (2004) derive the distribution and expected value of the maximum drawdown for a (drifted) Brownian motion;

Camara Leal and de Melo Mendes (2005) compare the maximum drawdown of several index time series to the maximum drawdown of a fitted Pareto model; Casati and Tabachnik (2013) compare empirical distributions of maximum drawdown to distributions simulated with skewness and excess kurtosis; Cheridito et al. (2012) derive the distribution of maximum drawdown for stopped processes of class sigma; Kim (2018) compare portfolio sorts based on maximum drawdown, value-at-risk, and volatility; and van Hemert et al. (2020) compute comparative statics to address

3.2 Literature Review

how changes to return, volatility, length of time horizon, and autocorrelation affect the maximum drawdown.

Although the maximum drawdown is arguably the most prominent drawdown measure, numerous other drawdown measures have been introduced. Most notably, Chekhlov et al. (2005) propose the average drawdown as part of a class of drawdown measures called conditional drawdown (CDD), which also includes the maximum drawdown.¹ In contrast to the maximum drawdown, which captures only the largest drawdown, the average drawdown is computed as the average of all drawdowns. Similarly, Martin and McCann (1989) mention a measure where all drawdowns are first squared and then averaged. Some drawdown measures, like the average continuous drawdown, are used predominantly in the denominator of drawdown-based performance ratios, cf. Schuhmacher and Eling (2011). Bradford and Siliski (2016) propose the active drawdown measure which is computed relative to a benchmark index. In addition to measures based on the intensity of drawdowns, measures related to the duration of drawdowns have also been discussed, for example, in Mahmoud (2017). Goldberg and Mahmoud (2017) introduce the conditional expected drawdown (CED), which is useful as an ex-ante risk concept but not applicable to ex-post evaluation because it requires the distribution of maximum drawdowns and cannot be computed for a single sample path. A modification of CED with demeaned returns is proposed in Molyboga and L’Ahelec (2016). The end-of-period drawdown measure, which emphasizes aspects of regret at the end of the evaluation period, is introduced in Möller (2018). Korn et al. (2019) propose two new measures, the trend weighted drawdown and the linearly weighted drawdown, as part of a comprehensive weighted drawdown framework, which includes many previous drawdown measures.

Apart from the development of drawdown measures, several other strands of drawdown literature have developed. For example, Grossman and Zhou (1993) incorporate a drawdown constraint into a continuous-time investment problem in which a specified drawdown must not be exceeded at any time. This application has attracted significant attention in the literature, including by Cvitanic and Karatzas (1995), Alexander and Baptista (2006), Elie and Touzi (2008), Sekine (2013), Yao et al. (2013), Cherny and Obłój (2013), Rieder and Wittlinger (2014), Angoshtari et al. (2016), Kardaras et al. (2017), and Roche (2019), whereas the drawdown constraint has been generalized, other constraints have been added, and results have been extended to

1In general, the CDD class includes drawdown measures where the drawdown with respect to the running maximum is continuously assessed, and the worst (1−α)·100% of these drawdowns are averaged forαbetween zero and one.

different underlying processes and portfolios. A different strand of largely mathematical literature has assessed stochastic properties of the drawdown process, see Hadjiliadis and Vecer (2006), Mijatović and Pistorius (2012), Landriault et al. (2017b), and Bai and Liu (2019). Another area of ongoing debate is whether different drawdown-based performance ratios lead to different rankings of investments, see Eling and Schuhmacher (2007), Eling (2008), Caporin and Lisi (2011), Haas Ornelas et al. (2012), Auer and Schuhmacher (2013), Auer (2015), and Korn et al.

(2019).

Additionally, other ideas regarding drawdown have been pursued. For example, Vecer (2006) and Vecer (2007) investigate the relation between drawdown and option pricing, Heidorn et al.

(2009) use the maximum drawdown to analyze risk properties of funds of hedge funds, Gilli and Schumann (2009) employ drawdown measures among other alternative risk measures in portfolio optimization, and Pospisil and Vecer (2010) define and analyze drawdown Greeks. Moreover, Zabarankin et al. (2014) develop drawdown-β and drawdown-α with respect to a drawdown CAPM, Palmowski and Tumilewicz (2017) price drawdown-type insurance contracts, and Challet (2017) use the drawdown duration to construct an estimator for the Sharpe ratio. Although drawdown measures are included in many surveys of risk or performance measures (e.g., Bacon (2008), Caporin and Lisi (2011), and Caporin et al. (2014)), they remain not as well studied as

other risk measures, such as value-at-risk or expected shortfall.

Let us turn to the literature on persistence. Because of its fairly broad literal meaning, i.e. that a phenomenon continues to exist for a prolonged period, questions of persistence have been addressed in vastly different fields of finance and economics. These include the persistence of inflation (Pivetta and Reis, 2007), the persistence of firm capital structure (Lemmon et al., 2008), the persistence of bank profits (Goddard et al., 2011), the persistence of executive compensation (Cheng et al., 2015), and the persistence of earnings, cash flows, and accruals (Hui et al., 2016).

However, the arguably most prominent strand of the persistence literature is concerned with volatility persistence, also known as ‘volatility clustering’.² Its fundamental observation dates back at least to Mandelbrot (1963) who notes that large price movements typically follow previous large movements of either sign, and small changes typically follow previous small changes of either sign. More rigorously, Ding et al. (1993) observe that autocorrelations of absolute and squared returns – or, more generally,|r_t|^d – are positive for various exponentsdand even for long lags.

The study of volatility clustering has benefited immensely from the development of (G)ARCH

2We use the terms volatility persistence, volatility clustering, and persistence in standard deviation interchangeably.

3.2 Literature Review

models by Engle (1982) and Bollerslev (1986), which explicitly model persistence in volatility.

Utilizing these models, volatility persistence has been documented with high-frequency as well as low-frequency data ranging from intraday to monthly returns (Chan et al., 1991; Jacobsen and Dannenburg, 2003).

While the results on volatility persistence are fairly unanimous, return persistence – or, more generally, performance persistence – is surrounded by much more ambiguity. Several studies in the early 1990s support claims of persistence. For example, Jegadeesh and Titman (1993) demonstrate persistence in single stocks by showing that strategies that buy past winners and sell past losers generate significant positive returns. Assessing mutual funds,³ Grinblatt and Titman (1992), Hendricks et al. (1993), Brown and Goetzmann (1995), and Elton et al. (1996) find evidence for performance persistence in terms of returns and alpha against different portfolio benchmarks. To measure persistence, they consider regression results, portfolio sorts, contingency tables, and rank correlations, respectively. In a seminal paper, Carhart (1997) finds that persistence diminishes if momentum is taken into account. First, he replicates that portfolios sorted on past return differ substantially in return and CAPM alpha in the following year. Then, he demonstrates that these differences disappear when alpha is computed with respect to a four-factor model, which includes a factor-mimicking portfolio for one-year return momentum. Incorporating this finding by measuring performance as alpha with respect to Carhart’s four-factor model, the subsequent literature contains mixed results. Bollen and Busse (2005) find short-term persistence in daily returns that disappears for longer horizons. Cohen et al. (2005) report persistence in momentum-adjusted returns and report even stronger persistence in a new performance measure that includes holdings information relative to other funds. Kosowski et al. (2006) find some evidence of persistence using a bootstrap approach, whereas Fama and French (2010) use a slightly different bootstrap procedure and find hardly any evidence of persistence. Huij and Verbeek (2007) find some persistence, especially for small-cap or growth funds, when they improve the sorting into portfolios with an empirical Bayes approach. Barras et al. (2010) argue that previous approaches do not distinguish between superior performance because of skill or because of luck; they try to build portfolios with funds that truly have skill but do not find substantial

3The majority of the performance persistence literature studies mutual funds. Analyzing hedge funds, Ammann et al. (2013) report significant performance persistence, and Eling (2009) find that persistence critically depends on the type of hedge fund. Analyzing portfolios managed by institutional investment management firms, Busse et al. (2010) find that modest persistence is present in three-factor alphas but disappears after controlling for momentum.

outperformance. Berk and van Binsbergen (2015) find persistence in mutual fund performance when combining the four-factor alpha with assets under management to obtain a ‘value-added’

performance measure. El Ghoul and Karoui (2017) compare funds with high and low corporate social responsibility (CSR) scores and observe that high-CSR funds exhibit stronger performance persistence than low-CSR funds. Harvey and Liu (2018) use a panel regression framework to estimate fund alphas and find some evidence of persistence. Overall, performance persistence is a topic of ongoing debate, and at least some persistence can be observed.

The persistence of higher-order moments has been investigated as well. Ex-post stock returns exhibit positive skewness, which has been found to persist. For example, Singleton and Wingender (1986) observe the skewness of monthly stock returns to be weakly persistent over consecutive five-year periods by computing rank correlations and transition frequencies. Using the same data, Muralidhar (1993) conducts a bootstrap test that considers the sampling distribution of the sample skewness and concludes that skewness is strongly persistent. Defusco et al. (1996) and Nath (1996) support these results by extending the data coverage and adding a new bootstrap test based on the sampling distribution of the difference in skewness. While the skewness persistence of individual stocks does not automatically carry over to portfolios, Sun and Yan (2003) find that mean-variance-skewness efficient portfolios exhibit skewness persistence. Jondeau and Rockinger (2003) document persistence for the conditional skewness and kurtosis, which they compute with parameter estimates of a GARCH model with generalized t-distributed residuals. For time series of stock indices and exchange rates, they find that both conditional skewness and kurtosis are persistent, but skewness persistence exceeds kurtosis persistence. Ergün (2011) analyzes the same question with robust skewness and kurtosis measures and obtains mixed results.

We pick up both strands of persistence and drawdown literature and examine the persistence of drawdown measures. From the persistence angle, expanding the set of analyzed quantities to the increasingly important class of drawdown measures appears worthwhile. From the drawdown angle, addressing persistence is crucial to substantiate the use of drawdown measures in investment practice.