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2.4 Risk-Managed Momentum

2.4.2 Risk-managed Performance

crash indicator, IC,t1, we also take into account systematic sources of risk, leading to enhanced momentum scaling. More precisely, when the indicator predicts a crash (IC,t1= 1), exposure to the momentum strategy is reversed (we are long in losers and short sell previous winners). Thus, CI not only mitigates momentum crashes but also benefits from them.

Fig. 2.5:Weights in Momentum

−2 0 2 4

1930 1940 1950 1960 1970 1980 1990 2000 2010 2020

Weight

CI DYN

Panel A: Weights in Momentum, CI and DYN (1928:09−2020:05)

−2 0 2 4

1930 1940 1950 1960 1970 1980 1990 2000 2010 2020

Weight

CVOL

Panel B: Weights in Momentum, CVOL (1928:09−2020:05)

Fig. 2.5 plots scaling weights of risk management strategies over the full sample period from September 1928 to May 2020. Panel A displays the weights of the crash indicator (CI) and dynamic scaling strategy (DYN), while Panel B plots weights with respect to the constant volatility strategy (CVOL). Following Daniel and Moskowitz (2016) and to make results comparable, all risk management strategies are scaled to have an annualized in-sample volatility of 19%.

volatility resulting from a 12%-target is roughly 16.5%, and thus close to the in-sample volatility imposed here. While weights reported in Barroso and Santa-Clara (2015) range from 0.13 to 2.00, respective values in our analysis are 0.15 and 2.41. The standard deviation of weights is equal to 0.45 and thus falls below DYN (0.80) and CI (0.92).

Fig. 2.6 presents cumulative returns of all risk management strategies over the full sample period from September 1928 to May 2020 (Panel

Fig. 2.6:Risk-Managed Performance: Cumulative Returns

100 101 102 103 104 105 106 107 108

1930 1940 1950 1960 1970 1980 1990 2000 2010 2020

Cumulative Return

CI CVOL DYN MOM

Panel A: Risk−managed Momentum (1928:09−2020:05)

0.5 1.0 1.5 2.0

1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940

Cumulative Return

CI CVOL DYN MOM

Panel B: Risk−managed Momentum (1930:01−1939:12)

1.0 1.5 2.0

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

Cumulative Return

CI CVOL DYN MOM

Panel C: Risk−managed Momentum (2000:01−2009:12)

Fig. 2.6 presents cumulative returns of original momentum (MOM), the constant volatil-ity strategy (CVOL), dynamic scaling (DYN), and the crash indicator strategy (CI). Panel A shows cumulative returns over the full sample period from September 1928 to May 2020, while Panels B (1930s) and C (2000s) display cumulative returns in crash periods.

Following Daniel and Moskowitz (2016) and to make results comparable, all risk man-agement strategies and original momentum are scaled to have an annualized in-sample volatility of 19%. Cumulative returns of Panels B and C are calculated applying full-sample returns, i.e. returns are not re-scaled within decades. The y-axis of Panel A is logarithmized to improve visibility.

Panel A shows that CI clearly outperforms original momentum, CVOL, and DYN. While a $1 investment in the CI strategy in September 1928

Table 2.5:Risk-Managed Performance: Descriptive Statistics

Statistic Full Period

MOMraw CVOL DYN CI

Mean 13.81% 17.85% 19.62% 21.33%

Median 17.64% 18.80% 10.73% 12.46%

Minimum 77.02% 28.26% 24.62% 25.39%

Maximum 26.16% 24.99% 42.18% 44.18%

Volatility 27.32% 19.00% 19.00% 19.00%

Sharpe Ratio 0.51 0.94 1.03 1.12

Skew 2.27 0.32 0.85 0.75

Kurtosis 16.58 2.02 6.47 6.07

Table 2.5 presents descriptive statistics for original momentum (MOMraw), the constant volatility strategy (CVOL), dynamic scaling (DYN), and the crash indicator strategy (CI).

Calculations cover the full sample period from September 1928 to May 2020. Following Daniel and Moskowitz (2016) and to make results comparable, all risk management strategies are scaled to an annualized in-sample volatility of 19%. Both, mean and median returns are annualized, while minimum and maximum returns are stated as monthly returns. Furthermore, we annualize volatilities and Sharpe ratios.

would have led to almost 55 million dollars, the same investment in DYN would have generated roughly 11.8 million dollars. This is equal to an outperformance of 367%. With respect to CVOL, results are even clearer as cumulative returns of CI are almost 24 times higher. Original momentum is staggeringly outperformed by roughly 55 million dollars.

As shown in Table 2.5, all risk management strategies successfully re-duce extreme losses. While original momentum exhibits monthly crashes of up to 77%, risk-managed crashes reduce to about 28.3% (CVOL), 24.6%

(DYN), and 25.4% (CI), respectively. Moreover, annualized mean returns of CI (21.33%) exceed those of original momentum (13.81%) by almost

55%. CVOL and DYN are outperformed by 19.5% and 9%, respectively.26 At a first glance, the increase in monthly returns may appear small, yet the actual improvement is strong. First, a large chunk of these benefits can be reaped precisely when marginal utility is highest, namely during crash periods (Panels B and C of Fig. 2.6). Without rescaling to 19%, average monthly returns of the CI strategy in the 1930s (2000s) amount to roughly 0.35% (0.53%), whereas returns of DYN and CVOL decrease to 0.26%

(0.48%) and 0.28% (0.43%), respectively. By rescaling strategies to 19%, results become even clearer. Average CI returns are 0.78% (1.02%), while CVOL and DYN earn only 0.25% (0.52%) and 0.30% (0.96%), respectively.

Second, Table 2.6 presentst-tests for differences in means. CI returns are significantly different from original momentum, both over the full sample and in respective states of the crash indicator. Most importantly, although not being significant over the full sample (when considering CVOL and DYN), CI returns are significantly higher in crash periods.

Thus, the crash indicator strategy particularly mitigates crash risk, yet keeping momentum’s upside potential alive.

Third, in Section 2.5 we perform robustness checks that confirm a superior performance of CI in sub-samples.27 Moreover, they reveal a significantly worse performance of the dynamic strategy when applied to

26We also construct a strategy that adjusts the Daniel and Moskowitz (2016) approach for the crash indicator (instead of a bear market indicator). Risk adjusted returns largely exceed original momentum but trail DYN.

27In a first draft, only pre-Covid data until October 2019 was available. While April 2020 has been one of the 15 worst monthly momentum returns (28%), the momentum strategy merely lost 2.8% throughout February 2020 to May 2020. Thus, the Covid induced market crash did not result in a momentum crash, which is why CI correctly did not indicate a crash. Hence, CI only lost 0.3%, whereas DYN performed worse than original momentum (3.1%). We consider this as out-of-sample evidence.

Table 2.6:T-Tests for Differences in Average Returns

Sample p-value

CV OL DY N MOMraw

Full Sample 0.2150 0.5423 0.0000***

IC= 0 0.5130 0.7141 0.0004***

IC= 1 0.0051*** 0.0390** 0.0146**

Table 2.6 presents p-values of t-tests for differences in means of the crash indicator strategy (CI) and the other risk management strategies. Positive numbers imply higher average returns of CI.IC,t1= 0 andIC,t1= 1 indicate the current state of the crash indicator proposed in section 2.3.2. Full Sample tests cover data from September 1928 to May 2020. Stars indicate significance at the 10% (*), 5% (**) and 1% (***) level.

relatively new markets.

Fourth, although CI requires less data than DYN, even non-crash re-turns (IC,t1 = 0) are higher (1.85% vs. 1.76% per month). Our results thus suggest that DYN’s outperformance with respect to CVOL is exclu-sively driven by applying variance scaling instead of volatility scaling. In fact, variance scaling in the spirit of Barroso and Santa-Clara (2015), i.e.

without having to estimateµ, earns higher full period returns than DYN (1.68% vs. 1.63%).

Finally, by ex-postscaling risk-adjusted returns to 19%, Daniel and Moskowitz (2016) circumvent a highly relevant problem of implementing their strategy: there is no hint on how investors could intuitively choose the scaling parameter λ ex-ante.28 To investigate the sensitivity with respect to the choice ofλ, we recalculate risk-managed returns for sev-eral parameter choices. As the full sampleλto achieve 19% annualized volatility is roughly 0.49, we choose values fromλ= 0.1 toλ= 1.0 to

cal-28See Equation (2.5).

Fig. 2.7:Sensitivity Analysis of DYN (1928:09 - 2020:05)

● ●

ex−post λ

● ●

ex−post λ

−100%

−80%

−60%

−40%

−20%

0.0 0.2 0.4 0.6 0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

λ

Maximum Monthly Loss Annualized Volatility

Annualized Volatility Maximum Monthly Loss

Fig. 2.7 presents maximum monthly losses and annualized in-sample volatilities of the dynamic scaling strategy (DYN) with respect to the scaling parameterλ, presented in Equation (2.5). Returns are calculated over the full sample period from September 1928 to May 2020.Ex-postλ(= 0.491) denotes the full sampleλthat is chosen to achieve an in-sample annualized volatility of 19%.

culate both maximum monthly losses and annualized volatilities. Fig. 2.7 presents results. Whileλ >0.49 slightly reduce losses and annualized volatilities, λ <0.49 result in sharply increased draw-downs, accompa-nied by large volatilities. Most importantly, withλ <0.12 investors would have lost their full investment, rendering risk management ineffective.29 Nevertheless, it is questionable whether very smallλ’s are implementable at all (λ= 0.12 would produce a maximum weight of 17.3).30 Rational µσ investors might note that Sharpe ratios are not affected byλ’s greater than 0.12. However, Shleifer and Vishny (1997) show that professional

29This is particularly relevant for the 1930s momentum crash when no empirical data was available.

30We thank an anonymous referee for this suggestion.

arbitrage is usually conducted by a small number of arbitrageurs, using the capital of less sophisticated, potentially irrational, investors who are unaware of fair values and act rather myopic. Thus, although the choice ofλ >0.12 does not influence Sharpe ratios directly, it may significantly affect monthly draw-downs, potentially leading to fund withdrawals and ultimately an indirect effect on Sharpe ratios.

Apart from that, CI and DYN offer the highest maximum returns (44.2% and 42.2%), but also exhibit the lowest medians (12.5% and 10.7%) and a high kurtosis (6.1 and 6.5), suggesting that average returns are driven by a small number of particularly high returns. However, the kur-tosis is still clearly reduced and both strategies show a positive skewness of 0.75 and 0.85, respectively. In contrast, CVOL displays a kurtosis of roughly 2.0 and returns are still negatively skewed (−0.32). Hence, all strategies effectively reduce tail risk. Finally, as volatilities coincide by construction, Sharpe ratios reflect average returns: CI displays the highest Sharpe ratio (1.12), followed by DYN (1.03) and CVOL (0.94).31

Despite offering the largest Sharpe ratio, the high variability of CI’s weights makes it reasonable to ask whether our results hold after in-cluding transaction costs. Following Barroso and Santa-Clara (2015) and Hanauer and Windmueller (2021), we therefore first calculate each strat-egy’s turnover.32 Table 2.7 presents results. For the original momentum strategy, we obtain an average monthly turnover of roughly 81.1%, which

31Note that the discussion of average returns extends to Sharpe ratios. By applying an ex-post crash indicator, average returns increase to 22.8% and the Sharpe ratio improves to roughly 1.20.

32The calculation of the monthly turnover is outlined in Appendix 2.A.4.

is similar to the results of Barroso and Santa-Clara (2015) who report a monthly turnover of 74%. However, note that their sample is limited to the period from March 1951 to December 2010.33 Average monthly turnovers of risk-managed strategies are 81.5% (CVOL), 78.3% (DYN), and 84.4% (CI), respectively. While we expected CI to display the highest turnover, results for the dynamic strategy are surprisingly low. Although the average absolute change of wtDY N1 (0.154) more than doubles the change ofwtCV OL1 (0.073), the portfolio turnover is 3.2 percentage points lower and even falls belowMOMraw. We explain this finding by DYN’s mean and median weight in the momentum strategy (0.94/0.76), which is lower than for both CI (0.97/0.76) and CVOL (1.00/0.96). In high turnover periods (when more stocks enter or leave one of the portfolios), smaller weights reduce monthly turnover, while in periods of low turnover, the larger variation in momentum weights becomes more relevant. In these periods, DYN indeed displays a higher portfolio turnover. However, we find the former effect to be stronger.34 By construction, these findings

33Hanauer and Windmueller (2021) find a clearly lower turnover of roughly 54%. This finding is caused by constructing the momentum strategy with HML-style portfolios based on 70%/30% percentile breakpoints and double-sorts instead of momentum deciles. Moreover, they report a significantly larger increase of turnover when risk-managed strategies are considered. We explain this difference by higher scaling weights (their risk-managed strategies are scaled to have the same annualized volatility as original momentum). Daniel and Moskowitz (2016) do not report turnovers.

34We examine DYN’s turnover relative to the turnover of CVOL (∆=T ODY NT OCV OL).

First, we calculate DYN’s average turnover with respect to the monthly change in weights. If the difference is above the median (0.038), we findto be 0.08 (i.e.T ODY N

exceedsT OCV OLby eight percentage points), while otherwiseis clearly negative (0.14). Although the variation of weights indeed influences monthly turnover, the positive effect of a lower mean exposure is prevailing. Moreover, we calculatewith respect to the turnover of original momentum. IfT OMOM is below (above) its median (0.796),amounts to roughly4.0 (2.5) percentage points. Thus, in high turnover periods the impact of a smaller median weight increases.

Table 2.7:Turnover and Break-even Round Trip Costs

MOMraw CVOL DYN CI

Turnover 81.11% 81.47% 78.27% 84.44%

Round-trip costs (5% level) 10.14% 17.14% 20.10% 20.65%

Round-trip costs (1% level) 7.97% 15.64% 18.54% 19.20%

Table 2.7 presents the average monthly portfolio turnover of original momentum (MOMraw), the constant volatility strategy (CVOL), dynamic scaling (DYN), and the crash indicator strategy (CI). Furthermore, we report annualized break-even round trip transaction costs, i.e. round trip costs that would render profits of each strategy insignif-icant at the 5% and 1% level, respectively. Both, turnovers and break-even round trip costs are calculated according to Appendix 2.A.4. We cover the full sample period from September 1928 to May 2020.

also come into effect when considering the CI strategy. However, in this case, turnover increases by incorporating the crash indicator.

To prove that the CI strategy remains superior after including trans-action costs, we follow Barroso and Santa-Clara (2015), Hanauer and Windmueller (2021), and Grundy and Martin (2001) and calculate the round trip transaction costs that would render profits of each risk man-agement strategy insignificant at the 5% and 1% level, respectively.35 As depicted by Table 2.7, CI’s annualized break-even transaction costs at the 1% level are 19.20% and exceed both DYN (18.54%) and CVOL (15.64%). Results at the 5% level are very similar. We thus conclude that the crash indicator strategy still outperforms CVOL and DYN after including transaction costs.

35The calculation of break-even transaction costs is outlined in Appendix 2.A.4.