Momentum Across Asset Classes

Home , Momentum (finance)

MOMENTUM ACROSS ASSET CLASSES

Loes van der Poel (207332) Bsc. Tilburg University

A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science of Finance.

Tilburg School of Economics and Management Tilburg University

Supervisor: Prof. Lieven Baele

Second Reader: Prof. Adri Verboven

November 29, 2019

Abstract

This thesis focuses on analysing the impact of momentum on a multi-asset portfolio, consisting of: equities, currencies government bonds, corporate investment bonds, corporate high-yield bonds, commodities and a real estate index. The proﬁtability of asset-class momentum strategies is analysed by using actual price data on indices of the aforementioned asset classes. The approach that was introduced by Jegadeesh and Titman is applied to assess the success of momentum strategies across asset classes. Eight strategies were investigated and the results add convincing evidence in favour of multi-asset momentum profitability. The momentum strategies generate abnormal returns, during the period 1995-2018, ranging from 6.79% to 11.45%. Strategies that employed shorter formation periods were found to be more successful during this sample period. A multi-asset momentum strategy can thus be beneficially exploited, implying that momentum is indeed present across asset classes.

Table of contents

Introduction ...... 4 1. Literature review ...... 6 1.1 Momentum within asset classes ...... 6 1.1.1 Momentum in Equity ...... 6 1.1.2 Momentum in exchange markets ...... 8 1.1.3 Momentum in commodities ...... 9 1.1.4 Momentum in bonds ...... 10 1.2 Momentum across asset classes...... 10 1.3 Explaining momentum ...... 11 2. Data and Methodology ...... 13 2.1 Data ...... 13 2.2 Methodology ...... 15 3. Empirical results ...... 19 3.1 Returns of Trading Strategies ...... 19 3.2 Performance statistics on momentum returns ...... 22 3.3 Systematic biases ...... 23 3.4 Momentum during subperiods ...... 24 3.4.1 Sample period divided ...... 24 3.4.2 Momentum during recession ...... 25 4. Robustness tests ...... 26 4.1 Momentum of volatility adjusted returns ...... 26 4.2 Transaction costs ...... 26 Conclusion ...... 28 Bibliography ...... 30 Appendix ...... 34

Introduction

On October 29, 2010, JP Morgan launched a multi-asset, long-only momentum index: the J.P. Morgan ETF Efficiente 5 Index. This index allocates capital between twelve asset classes and a US$ index based on their price momentum during the past six months. On December 28, 2016, JP Morgan launched another multi-asset, long-only momentum strategy: the JP Morgan Mozaic II index. This strategy similarly holds six out of twelve assets based on their past six month performance. The two strategies operate different risk management methodologies. However, they both rely on the hypothesis that the theoretical foundation for observed momentum effects may hold across asset classes as well. JP Morgan believes that these indices provide a diversified asset allocation that will generate stable returns. The fact that JP Morgan sees potential in multi-asset class allocation based on momentum signals raises questions on the potential for momentum investing across asset classes.

Momentum is the empirically observed proclivity of assets to maintain recent price trends in the future. The intuition behind this market anomaly is straightforward as it logically assumes that assets that have performed well in the recent past will continue to do so and vice versa. Price momentum strategies aim to exploit such a trend by buying assets that have recently appreciated and selling the ones that have recently declined. That kind of trading strategy would be profitable if and only if price momentum exists.

The occurrence of a momentum effect effectively defies one of the cornerstones in traditional finance theory, namely the efficient market hypothesis. As it suggests that historical prices can be used in order to predict future performance, this would enable investors to consistently outperform the market.

Momentum is a style factor. Style factors are traditionally defined as characteristics that persistently explain returns within asset classes, whereas so-called macro factors are supposed to capture effects across asset classes. Price continuation was first documented for equity, Levy (1967), Jensen and Benington (1970) and Jegadeesh and Titman (1993) developed various methods for selecting stocks based on their relative strength. The methodology of Jegadeesh and Titman (1993) was subsequently adopted as the leading methodology to compose momentum portfolios. Since 1993 consecutive articles, on momentum in a variety of asset classes, have been published. Apart from equity, momentum effects have been documented for commodities ((Erb & Harvey, 2006), (Miffre & Rallis, 2007) and (Fuertes,

Miffre & Rallis, 2010)), fixed income (Asness, Moskowitz & Pedersen, 2011) and currencies (Okunev & White, 2003) and (Menkhoff, Sarno, Schmeling, & Schrimpf, 2012)). The evidence of momentum in this wide variety of asset classes has motivated the multi-asset approach conducted in this thesis.

The main goal of this research is thus to analyse whether an investment strategy based on momentum across asset classes is worthwhile. Consequently, this paper will aim to prove a consistent excess return premium that can be achieved if the above-mentioned strategy is put into practice.

The remainder of this paper is organized as follows: Chapter 1 contains a literature review of momentum within each major asset class and summarizes the relevant literature on momentum across asset classes. Chapter 2 elaborates on the data collection procedure and the methodology of the empirical analysis. Next, Chapter 3 delivers the empirical results and the corresponding implications, while Chapter 4 addresses a variety of robustness checks. Finally, Chapter 5 is concentrated on concluding remarks. All tables and graphs that are referred to can be found in the appendix, starting from page 34.

1. Literature review

1.1 Momentum within asset classes

The majority of studies regarding the momentum anomaly have focused predominantly on equity momentum. Additionally the momentum effect has been proven to exist within various other asset classes as well as across asset classes. The contents of this sub-chapter are divided such that section 1.1.1 illustrates the evidence for momentum within equity, section 1.1.2 reviews the academic literature on momentum in currencies. Section 1.1.3 will expand further on momentum in commodities and finally section 1.1.4 will conclude with a review on momentum in bonds.

1.1.1 Momentum in Equity

Jegadeesh and Titman (1993) laid the groundwork for the momentum strategy as it is currently known. In their paper they found that statistically significant abnormal returns could be realized by buying US stocks that performed well over a recent period and selling US stocks that performed poorly over the same period. The results were tested for systematic risk and a lead-lag effect, Jegadeesh and Titman also checked if the profitability of their strategy was confined to any specific subsample and if the result held for subperiods. They documented an average annually compounded return of 12.01% for the long-short strategy that selects stocks based on their past six months returns and holds them for six months. Their most profitable (12-3) momentum strategy generated a monthly return of 1.49%. The study was conducted over the period 1965-1989. Moreover they found that momentum seemed to be a temporary effect, applying the strategy generated positive excess returns in the first year, but these returns were offset by negative average returns the second and third year. In fact Reversal effects were documented for longer holding periods by De Bondt and Thaler (1985, 1987) and Asness (1997). Jegadeesh and Titman concluded their findings by suggesting that the momentum-effect exists due to the fact that the market underreacts to information about the short-term prospects of firms but overreacts to information about their long-term prospects.

Following this research, further evidence for the presence of persistent return continuation for medium-term investments was provided by Rouwenhorst (1998). He used a similar approach to demonstrate momentum in European equity throughout 1980-1995. Momentum was found

6 to exist in all twelve sample countries, supporting results from previous studies. A positive relationship seemed to arise between the ranking period and the excess returns of the momentum strategy. Regardless of their ranking period, strategies documented decreasing excess returns as the length of the holding period increased. However, irrespective of their formation and holding periods, past winners consistently outperformed past losers by roughly one percent per month. Corresponding results were also found by Dijk and Huibers (2002) and Nijman, Swinkels and Verbeek (2004). Rouwenhorst observed a correlation between the European equity momentum returns and the US equity momentum returns and surmised the existence of a common intercontinental factor. Lui, Strong and Xu (1999) extended the field of research by using weekly returns instead of monthly returns, making the outcomes of their empirical research more robust. Analogously to previous discoveries they found a short-run momentum in UK stocks. They pose some alternative explanations that could possibly explain the momentum effects in stocks in addition to the behavioural theories that have formerly been presented.

Correlation between emerging stock markets and other stock markets has historically been low (Harvey, 1995). However both stock markets display similar compensation for momentum in average stock returns (Rouwenhorst, 1999). These results are later confirmed by Cakici, Fabozzi and Tan (2012), who tested for short and medium-term return continuation in four regions (Latin America, Eastern Europe, Asia and the three of them combined) and found momentum to be present in all of them with the exception of Eastern Europe. They also documented a small stock momentum premium, similar to results in preceding research.

Momentum strategies applied in the U.S. and Europe, also generated significant profits in Asia, with the exception of Korea, Indonesia and Japan (Chui et al. 2000). The magnitude and pervasiveness of these results is however not as robust it was for momentum in the U.S. and Europe. Furthermore Chui, Wei and Titman (2000) established that the momentum effect in Asia is stronger for firms with smaller market capitalizations and higher turnover. These findings are consistent with former evidence stemming from research on U.S. equity momentum.

The methodology posed by Jegadeesh and Titman (1993) can be extended to test for momentum patterns in other asset classes. An extensive body of research exist for momentum within equity, support to these momentum studies is also found in numerous other asset classes.

1.1.2 Momentum in exchange markets

A substantial amount of literature has focused on the success of implementing technical trading strategies in the foreign exchange market. ((Sweeney (1986), Levich & Thomas (1993), Kho (1996), Neely, Weller & Dittmar (1997), LeBaron (1999), and Marsh (2000)). These early studies indicate that the currency market must display some degree of market inefficiency, due to the fact that abnormal returns can be generated by employing non- fundamental analysis. Taylor and Allen (1992) provide evidence that short-term statistically significant trends and a limited degree of positive serial correlation exist in foreign exchange markets. This is similar to results by Levich (1989) and Muga and Santamaria (2007) who documented positive serial correlation and mean reversion for exchange rate fluctuations. Okunev and White (2003) have reported excess returns for long-short momentum strategies in currency markets over the period 1970-1990. They compensate for various risks and consider a number of robustness checks, leaving them to conclude that the abnormal returns must be at least partly due to the autocorrelation structure of currency returns. Bianchi et al (2004) find confirming results, however they add that the profitability of the momentum anomaly attenuates after compensating for transaction costs, to the point that it is unlikely for private investors to achieve significant profits. Furthermore they observe that currency momentum is largely skewed to shorter formation periods.

Menkhoff et al. (2012) conducted an empirical research on currency momentum strategies. Their findings substantiate prior research in this field, generating 6-10% excess returns by exploiting momentum strategies. The returns can partially be explained by transaction costs. Moreover they find a negative correlation between the holding period and the profitability of momentum strategies (with a 1month-1month strategy generating the highest profits). However strategies with short holding- and formation periods are even more sensitive to transaction costs. Additionally the best- and worst performing currencies have higher transaction costs, making a momentum strategy particularly costly. However, momentum in the foreign exchange market is still significant and profitable even when taking into account the relatively high transaction costs of this asset class. Currency momentum is volatile but allows for high excess returns and Sharpe ratios.

1.1.3 Momentum in commodities

Academics have universally recognized the value of commodities in strategic asset allocation, due to benefits including but not limited to: low correlation with other asset classes (Ankrim & Hensel, 1993; Becker & Finnerty, 1994; Anson, 1999; Jensen, Johnson & Mercer 2000, Abanomey & Mathur, 2001; Georgiev, 2001; Gorton & Rouwenhorst, 2006) and the ability to hedge against inflation (Bodie & Rosansky, 1980; Bodie 1983, Erb & Harvey, 2006; Adams, Füss & Kaiser, 2008). However, academic researchers provide documentation of short-term return continuation in the commodity market. This suggests that commodities will also add value to a tactical asset allocation

In their paper of 2006, Erb and Harvey reported positive excess returns of 7% and a Sharpe ratio of 0.45 for a simulated winner portfolio (going long the in an equally weighted portfolio of the four best performing commodity futures (performance is measured in returns over the past year)). The highest excess returns of 10.8% and a Sharpe ratio of 0.55 was reported for following a long-short strategy as defined by Jegadeesh and Titman. They also found abnormal returns for a time series momentum strategy, which goes long or short individual commodity futures as they generate respectively positive or negative returns over the past year. This trend following portfolio generated a return of 6.54% with a Sharpe ratio of 0.85. The time series momentum portfolio therefore achieved the best risk-adjusted returns (returns compensated for volatility) and both momentum portfolios achieved higher (risk- adjusted) returns than the long-only GSCI.

Miffre and Rallis (2007) further developed the understanding that pursuing active investment strategies has historically been profitable in commodity futures markets. They documented an 9.38% yearly average return for long-short momentum portfolios as opposed to a loss of 2.64% for an equally-weighted long-only portfolio of commodity futures. They added that the commodity-based long-short momentum portfolios have lower transaction costs. Transaction costs for the commodity future market range from 0.0004% to 0.033% (Locke and Venkatesh, 1997) as opposed to the assumed 0.5% transaction costs for the equity market, used by Jegadeesh and Titman (1993) or the 2.3% estimate of Lesmond, schil and Zhou (2004). Furthermore commodity futures are liquid and impose no restrictions on short-selling as is often seen in the equity market. Miffre and Rallis however point out that implementing the momentum strategy in futures markets has its own drawbacks including: initial margins, roll- over contracts and the monitoring of margin accounts.

1.1.4 Momentum in bonds

Little research has been pursued on the subject of momentum in bond markets, presumably due to difficulties working with bond data. Some preliminary work in this field was carried out by Gebhard, Hvidkjaer and Swaminathan (2005), who found no significant indication for momentum among investment-grade corporate bonds. However they did find a momentum spill over effect from equity to investment grade bonds of the same company. Khang and King (2004) also failed to register evidence for momentum in bond returns, they did however discover a reversal effect for investment grade corporate bonds.

Momentum is first shown to present itself as a profitable investment strategy for high yield bonds by Jostova, Nikolova, Philipov and Stahel (2010). They report momentum profits over the period 1991-2011 of 59 basis points per month across all bonds and 192 basis points per month for high-yield bonds. For investment-grade bonds they find no significant momentum profitability, in accordance with aforementioned studies. In 2011, Asness et al. also reported significant time-series momentum returns for global government bonds. Furthermore Luu and Yu (2012) studied momentum within government bonds and found significant abnormal returns were obtained by applying this trend-following strategy.

1.2 Momentum across asset classes

Until recently momentum was only proven to exist within several asset classes. In 2008, Blitz and van Vliet explored the innovative concept of momentum across asset classes. They applied the original methodology of Jegadeesh and Titman (1993) across twelve asset classes and found that the strategy generated statistically significant abnormal returns over the period 1986-2007. They reported annual excess returns of 8,1% for a (12-1) long only momentum strategy. Additionally they found 7.9% excess returns and a Sharpe ratio 0.61 for the long- short strategy. The results were still significant after being adjusted for transaction costs, structural biases towards certain asset classes and various factors such as the Carhart momentum factor. The results obtained in this study are in line (both in direction and magnitude) with previous results that have been obtained for momentum within specific asset classes. Blitz and van Vliet argue that a behavioural explanation for the existence of momentum at the asset class level might be that it is an inefficiency that is simply not arbitraged away yet.

Time-series momentum was already documented within asset classes in one study by Moskowitz, Ooi and Pedersen (2012) for the four major asset-classes: equity, currency, commodity and bond futures since 1970. Hurst, Ooi and Pedersen (2012) validated significant momentum premiums across asset classes over a period of 135 year, they used 67 markets all belonging to one of the four major asset classes. They also considered performance for each decade separately. “The performance has been remarkably consistent over an extensive time horizon that includes the Great Depression, multiple recessions and expansions, multiple wars, stagflation, the Global Financial Crisis and periods of rising and falling interest rates.” (Hurst et al., 2012).

Asness et al. (2013) also support the existence of momentum across asset classes in their study ‘momentum and value everywhere’. They document annual excess returns of 9.2% and a Sharpe ratio of 0.98 for the long-only momentum strategy and excess returns of 5.0% with a Sharpe ratio of 0.67 for the long-short momentum strategy..

Geczy and Samonov (2016) extended the topic of price momentum in various directions. They observed significant abnormal returns within each asset class as well as across asset classes over the long run, in accordance with previous findings. However contrary to preceding studies they find negative returns for specific subperiods. This observation is most likely due to the fact that more historical data is included in this study since Geczy and Samonov argue that momentum crashes occur more frequently before 1950. Daniel and Moskowitz (2016) have previously conducted research on the manifestation of momentum crashes, and argue that they indeed exist and are partially forecastable. Additionally Daniel and Moskowitz further investigated momentum crashes, in their studies they found exceptional performance of momentum strategies using a dynamically weighted momentum strategy. Combining a momentum strategy with dynamical weighting across asset classes, generates a Sharpe ratio of 1.19, which is remarkably high.

1.3 Explaining momentum

A number of behavioural theories has been proposed to justify momentum profitability. For instance the theory that momentum occurs as a consequence of underreaction to information, such as earnings announcement. This proposition is tested by Chan, Jegadeesh and Lakonishok (1996), who find that most recent earnings surprise also helps to partially predict future returns. Apart from the behavioural factors that have been put forward such as an

11 underreaction to earnings announcement, several studies present corroborating evidence for a risk-based explanation to momentum. The risk based explanation originates from Fama and French (1993), and is adapted by Asness, Moskowitz and Pedersen (2013) who find that momentum strategies across (as well as within) asset classes display significant comovement which indicates the presence of a common risk factor. Cooper, Mitrache and Priestley (2017) provide a model that tries to tie momentum (and value) to five macroeconomic risk factors.

2. Data and Methodology

2.1 Data

The dataset used for the main analysis in this thesis consists of a set of asset classes, proxied by a number of indices. A position into one of these indices can be taken via index futures, however there are a number of constraints on short selling and the entrance of future contracts. A more accessible approach to trade asset classes is to invest in exchange traded funds (ETF) as they can be purchased and sold short with few to none constraints. There is however a limited amount of data available for ETFs, since most ETF launches occurred in the recent past. Therefore return data of indices will be used in this thesis.

Exhibit 1 provides a summary of the asset classes that constitute the multi-asset universe for this research, the indices used to track these asset classes and the datatype obtained to generate excess returns. The total number of asset-classes taken into account amounts to seventeen. I consider five asset classes related to the equity market (US equity market, the European equity market, the Japanese equity market, the Pacific equity market excluding Japan and the emerging equity market). I also include four government bond markets (US, Europe, Japan and emerging markets), two corporate bond markets (US high yield and US investment grade), three commodities (oil, gold and industrial metals), two currencies (US and Japan) and a US real estate index. Most alternative asset classes such as cryptocurrencies and valuable inventory are not taken into consideration due to a lack of data.

The earliest date for which weekly return data is available for every asset class in the dataset is 07/12/1995. Because I implement various lookback-windows ranging from three to twelve months, the analysis effectively starts between 10/11/1995 and 07/10/1996 depending on the chosen time frame. The end date of the sample is 12/31/2018. Consequently this analysis covers roughly 24 years of data from 1995 through 2018.

For each asset class I gather the daily total return index from Thomson Reuters DataStream. The total return index tracks the price movements of the underlying asset class but also includes dividends, interest, and other cash distributions as opposed to the price index that only tracks price movements. Therefore it is a more solid measure of return that requires no additional adjustments.

The weekly Wednesday index returns are used to calculate returns in the following manner:

퐼푅푤 − 퐼푅푤−1 푟푤 = 퐼푅푤−1

Where 푟푤 is the weekly return and 퐼푅푤 is the total return index of week 푤.

Using weekly Wednesday returns eliminates the weekend effect, which is a common phenomenon for stocks. Local risk-free returns are subtracted from these returns to obtain the excess returns of the asset classes. Local risk-free returns are proxied by the 1-month LIBOR of the corresponding country, European risk-free rates before 1999 are proxied by the Germany EU-MARK 1M middle rate. LIBOR rates are obtained from the website www.iborrate.com. For emerging markets the US LIBOR will be subtracted. For the three commodities in our dataset, excess returns were downloaded directly. The excess returns will not be normalized prior to the analysis, however normalized measures of the results will be considered in order to obtain an objective comparison of the outcomes.

Exhibit 2 gives descriptive statistics of each asset class over the full sample period. All the indices except the Japanese Yen offered positive average excess returns over the entire sample period. However when taking the compounded annualized growth rate (CAGR) also known as the geometric average, we document negative returns for industrial metals as well. Pacific ex Japan equity in our selection of asset classes appears to generate the highest annualized excess average returns of 8.13%. Whereas Emerging markets government bonds document the highest average compounded returns of 7.08%. Compounded annual growth rates provide more information about the actual performance of an asset over time, in particular for volatile asset classes. It is therefore often considered a more accurate measure of return than the average annualized return. The Sharpe ratio, which is employed as risk-adjusted performance measure, is however calculated with the arithmetic mean and its standard deviation. Oil has by far the highest annualized volatility of 30.61%. The least volatile asset class Japanese government bonds registers an annualized volatility of merely 3.62%. The Sharpe ratio is calculated as it is defined by William Sharpe. The US high yield corporate government bonds delivers the best risk adjusted performance with a Sharpe ratio of 0.89.

Recession dates are obtained from the website of the National Bureau of Economic Research.

2.2 Methodology

The study focuses to investigate the impact of momentum strategies applied on the asset-class level by comparing the momentum-strategy returns to benchmark returns. I assess the performance of seventeen asset classes by means of a selected trend signal, excess return is a commonly used trend signal (see Jegadeesh and Titman (1993), Fama and French (1996) and Asness et al. (2013)). The asset classes will be ranked in descending order according to their excess returns during a specified formation period K and divided into portfolios based on this ranking. The winner portfolio consists of the three top ranked asset classes, whereas the loser portfolio consists out of the three bottom ranked asset classes. This approach is conventional for constructing a cross-sectional momentum strategy, where one applies a momentum signal that is established relative to a selection of other asset classes (as opposed to the asset class’ own performance (Time-series momentum)). The performance of each portfolio is then recorded for a given holding period of 1 month.

For comparison purposes we construct two different kinds of strategies. The first strategy is a long-only strategy consistent with the strategy proposed by Blitz and van Vliet (2014). The assets are equally weighted in each portfolio and the strategy will take only a long position in the top-ranked portfolio until the end of the holding period. This process repeats itself each month. The second strategy employs a long-short strategy that equally weights the asset classes in each portfolio. The strategy will take a long position in the top-ranked portfolio and a short position in the bottom-ranked portfolio. This position will be closed out of, at the end of the holding period.

I study formation periods of 3, 6, 9 or 12 months, because weekly data is used, these periods are defined as respectively 13, 26, 39 and 52 weeks but referred to as 3, 6, 9, 12 months. That results in four possible momentum portfolios per strategy. The selection of a one month holding period guarantees an increase in statistical significance in comparison to longer holding periods. More portfolios can be created within the same span of data. A one month holding period largely captures the effect of overlapping holding periods, which is a less straightforward approach applied by amongst others: Jegadeesh and Titman (1993) and Moskowitz and Grinblatt (1999). Strategies with shorter holding periods like the one in this research will incur greater costs due to the high turnover and the resulting transaction costs. Transaction costs and their effect on the profitability of a momentum strategy will be touched upon later in a sensitivity analysis.

Quite a few studies on equity-momentum implement a one month delay prior to the portfolio holding period, due to documented short-term reversals as a result of a lead-lag effect (Lo and Mackinlay, 1990), (Boudoukh, Richardson & Whitelaw, 1994)), lagged reactions (Jegadeesh, 1990), Lehman, 1990)) and bid-ask pressure. Academics have been indecisive over the inclusion of a lagged holding period, moreover the justification for skipping the month preceding the holding period in equity studies does not hold in more liquid markets (Asness, et al. (2013)). Therefore the one month delay between formation and holding period will not be applied in this analysis of all asset classes. From now on a momentum strategy will be denoted by its formation- and holding period. For example a long-only strategy that measures excess returns over the past 12 months is described as a 12-1 long-only strategy. The portfolios and their returns will be constructed in MatLab, the Matlab code for constructing a 3-1 strategy is provided in Exhibit 3. There is a distinction between a buy and hold approach and a rebalancing approach. Buy and hold is a passive investment strategy, where the investor actively selects asset classes but has no concern for short-term price fluctuations and therefore holds its position without adjustments. The rebalancing strategy will regulate its asset allocation in such a way that is does not deviate from the chosen equal weights. Throughout the holding period I will assume a buy and hold approach.

I will consider a number of profitability indicators to form an opinion of the performance of our momentum portfolios. In this analysis a geometric average is used to calculate average returns for the construction of our performance and risk metrics. Geometric means take into account the compounding of returns and is therefore more accurate especially for volatile numbers. The performance-risk analytics I use, include: annualized average returns, annualized compounded growth rate, annualized standard deviation, skewness, kurtosis and the adjusted Sharpe ratio. Definitions of these performance and risk metrics are provided in exhibit 4.

In order to evaluate whether momentum exists across asset classes, one wants to verify that the momentum portfolios generate abnormal returns that are statistically significant. Therefore I compare the results on risk and return with those of a benchmark portfolio. The benchmark portfolio is composed of equally weighted holding positions in all asset classes, which are rebalanced on a monthly basis. To properly assess the statistical significance of the abnormal returns two measures will be considered.

1) Welch’s t-test. This test is used to test whether two populations have equal means, we prefer Welch’s t-test over the student’s t-test because it does not assume equal 16

variances. Nonetheless the normality assumption that student’s t-test poses, is maintained, however for large sample sizes the t-statistic approximately has a normal distribution.

푋̅ − 푋̅ 푊푇 = 1 2 푠2 푠2 √ 1 + 2 푁1 푁2

The degrees of freedom associated with Welch’s t-statistic is

Where 푊푇 stands for Welch’s t-statistic and 푣 for degrees of freedom. 푋̅1 is the mean return of the momentum strategy with its corresponding standard deviation 푠1. 푋̅2 is the mean return of the benchmark portfolio with its corresponding standard deviation 푠2. 푁1 and 푁2 are the sample sizes of respectively the momentum strategy returns and the benchmark returns. 푣1 is the degrees of freedom associated with the variance estimate of the momentum strategy 푁1 −

1 and 푣2 the degrees of freedom associated with the variance estimate of the benchmark portfolio 푁1 − 2

2) Regression of the momentum returns on the benchmark returns to evaluate to what extent they are explained by the benchmark. I use a student’s t-statistic to assess the statistical significance of the alpha of this regression.

I assess the results in different sub periods and under different macro regimes. Furthermore I consider the possibility of the momentum portfolios displaying structural biases and I pose some approaches to check and adjust for these biases. For instance a high rate of return for the momentum strategy could occur if the strategy is biased towards highly volatile assets. I therefore also consider the momentum strategies as described previously where asset classes are ranked according to their volatility adjusted returns. In order to do so ex-ante volatility is measured by an exponentially weighted moving average (EWMA):

2 2 휎푡 = √휆휎푡−1 + (1 − 휆)푟푡−1

2 2 Where 휎푡−1 is the variance of the previous observation and 푟푡−1 is the squared return of the previous observation. The estimate of ex-ante volatility is initialised with the standard deviation measured over the first 52 weeks of the sample set. Therefore the analysis of this momentum approach will start at the earliest on 10/09/1996. The value of lambda is often set to 0.97, this will be the value of lambda employed in this thesis.

3. Empirical results

3.1 Returns of Trading Strategies

Exhibit 5 (panel A) documents the annualized average excess returns of all momentum trading strategies for the long-only strategy, where the rows and columns represent respectively the formation and holding periods. The annualized average excess returns of a long-only momentum strategy range from 8.46% to 11.45%. All returns observed are thus positive and their Welch’s t-statistics vary widely from 16.81 to 8.90. Therefore it can be concluded that all strategies are statistically different from the benchmark return. The probability that the momentum returns are not statistically distinguishable from the benchmark return is 0.01%. A visual representation of the cumulative returns for each strategy is added in Exhibit 6. From this graph it is immediately clear that momentum returns prospered in the period 2002-2007, recording high returns and low volatility. A positive trend can be observed for the whole sample period, but average performance might be overestimated.

Contrary to previous studies on equity momentum (for instance the leading work on equity momentum by Jegadeesh and Titman (1993)) I obtain the highest results for shorter formation periods, profits clearly decline as the formation period increases. This relationship however, only seems to emerge from 2007 onwards. The graph of cumulative returns in exhibit 6 clearly illustrates the separation of cumulative returns based on their formation periods beyond 2007. Whereas the formation period of the momentum strategy didn’t seem to have much explanatory power for returns before 2007. The inclination towards shorter formation periods might be due to the difference in observed assets or the differing sample periods or to the fact that for a multi-asset portfolio, more recent past returns have more predictive power.

The MSCI World momentum index, applies a long-only 12-6 momentum strategy on stocks. The results of the MSCI World momentum index are compared to the multi-asset 12-6 long only momentum strategy. During the analysed sample period from 07/10/1996 until 12/19/2018 the (12-1) long only multi-asset momentum strategy generates a return of 8.46% which is approximately equal to the 8.34% reported over the same period of the MSCI World Momentum Index. The multi-asset momentum strategy in this thesis and the MSCI World Momentum Index report an annual growth rate of respectively 7.54% and

6.93% and sharpe ratios of 0.62 and 0.50. It appears that the multi-asset momentum strategy delivers a slightly better risk-adjusted performance than the equity momentum strategy of MSCI. It is however noteworthy to mention some differences between the two. The MSCI World Momentum Index rebalances biannually and does not apply overlapping holding periods, which makes the strategy less sensitive to transaction costs compared to the 12-1 month multi-asset momentum strategy. Furthermore transaction costs have not yet been taken into account whereas the returns of the MSCI World Momentum Index are already compensated for transaction costs.

Exhibit 5 (Panel B) documents the annualized average excess returns of all momentum trading strategies for the long-short strategy. I observe average excess returns ranging from 6.79% to 10.66% over all strategies for the long-short momentum approach. Welch’s t-statistics vary from 3.7 to 10.2 which is evidently lower than it was for the long only strategy. However the long short momentum returns are still all statistically different from the benchmark returns. Given the Welch’s t-statistics, the null hypothesis that the momentum strategy returns do not deviate significantly from the benchmark returns, can be rejected at the 99.9% confidence level. Since the long-only strategy convincingly outperformed the benchmark, I infer from these lower t-statistics that the long-short strategy has less ability to outperform the benchmark. This conflicts with results obtained in equity momentum studies, where long-short strategies consistently outperformed the long-only strategy. This dissimilarity is most likely due to positive returns in the bottom portfolio. As the long-short strategy takes a long position in the winner portfolio and a short position in the loser portfolio, abnormal returns generated by the winner portfolio will be partially offset by the positive returns in the loser portfolio. The limited presence of negative returns in the loser portfolio is not due to a lack of negative returns in our dataset, as 45.26% of the weekly returns in our dataset are negative. Therefore it must be concluded that the momentum anomaly in the multi-asset dataset mainly manifests itself in increasing returns, whereas for negative returns, past performance does not seem to consistently predict future returns.

The difference in performance of our long-short portfolio with those of equity-momentum studies can intuitively be explained by the fact that individual stocks will report more extreme observations than asset classes, which are more diversified. Therefore the bottom portfolio of only stocks will, given the continuation of returns, often contain negative returns.

When looking at the cumulative returns of the long-short strategies in exhibit 6 it can visibly be detected that the long-short momentum strategies are more volatile than the long only strategies. Apart from this obvious difference both approaches seem to follow the same trend. Similar to the long only strategies, long-short momentum thrived between 2002 and 2007, and analogously after 2007 the strategies diverge based on their formation period.

Because I test for an amount of eight strategies, the probability of finding a difference between the momentum portfolio returns and the benchmark returns is increased just by the play of chance. When applying multiple testing, t-statistics should be corrected for this increased chance in order to prevent recording a false positive. A regular approach to address this concern is to adjust the Welch’s t-statistics by the Bonferroni correction. The Bonferroni correction compensates for the increased chance by testing each hypothesis at 훼 a significance level of where 훼 is the preferred significance level and 푚 is the number 푚 of tests conducted (in our study 32). For the preferred significance level 훼 = 0.05 I obtain a Bonferroni adjusted significance level of 훼퐵 = .00625. This is equal to obtaining t- statistics of approximately 2.807 or higher, as all the t-statistics are above this critical value all strategies remain statistically significant at the 95% confidence level.

The second way proposed to assess the statistical significance of the multi-asset momentum returns is to regress them on the benchmark returns. Results of these regressions are documented in exhibit 7. The regression exemplifies to what extent the momentum returns can be explained by the benchmark returns. the alpha of the regression is the average momentum return above or below the predicted momentum returns. The long-only momentum returns are positively correlated with the benchmark returns as the betas of their regression are all close to one. The alpha’s descend as the formation period of the strategy becomes longer, resulting in a statistically insignificant alpha for the (12-1) strategy. The three remaining alpha’s report values ranging from 7.14% to 3.82%, all significant at the 10% level. The intuition of these alpha’s is clear, assuming benchmark returns of 0%, the momentum portfolios will generate excess returns between 3.82% and 7.14% conditional on the formation period of the strategy. The long-short momentum returns are negatively correlated with the benchmark returns, this is to be expected because this strategy takes a negative position in three asset classes, as opposed to the benchmark that holds a positive position in these asset classes. The alpha’s of the long- short strategy are all significant at the 1% level and vary from 12.85% to 8.69%.

All momentum strategies exhibit positive correlations with each other, this is to be expected because they are theoretically supposed to capture the same effect. Correlations range from 0.22 to 0.83. Long only and long-short strategies logically display lower correlations with each other. Particularly, high correlations occur for strategies that have similar formation periods. Correlations decrease when the differences between formation periods becomes larger, for instance the lowest correlation is observed between the (3-1) long-short and (12-1) long only strategy. A correlation matrix is added in exhibit 8.

3.2 Performance statistics on momentum returns

Exhibit 5 provides the performance statistics of the eight strategies. The portfolio that selects asset classes based on their returns on the past 3 months and then holds them for 1 months generates the highest annualized average excess return of 11.45%. The (3-1)Long only strategy is based on excess returns the most successful. This strategy has a compounded annual growth rate of 10.68%, an annualized standard deviation of 12.72%. Combined these statistics produce a Sharpe ratio of 0.90 which measures return per unit of risk. The Sharpe ratio assumes that returns are normally distributed, which is not the case for our momentum returns. Therefore I study an adjusted Sharpe ratio, which takes into consideration skewness and kurtosis. Skewness is a measure of asymmetry of the probability distribution of the momentum returns, it measures the distinct to which returns are not distributed evenly around the mean. Negative skewness indicates that the distribution is left-tailed; there will be more (extreme) observations below the mean than above the mean. Positive skewness implies the opposite, the distribution is right-tailed, which means that more (extreme) observations are located right from the mean. Skewness is an important measure because it explores the extremes of a dataset rather than the average. Skewness of for example the (6-1) long-short strategy is -0.11 which means that the distribution is approximately symmetric, there will be a few more returns deviating from the normal distribution above the average return. There are however some strategies that report a higher skewness such as the (12-1) long only strategy with a skewness of -0.45, meaning that the distribution of returns for this strategy is moderately skewed to the left. Kurtosis measures the observations in both tails, when they exceed the tails of a normal distribution. High kurtosis makes a distribution leptokurtic and points towards a lot of extreme values. A normal distribution has kurtosis of 3. The (3-1) strategy has

22 kurtosis of 1.73 which means that the actual kurtosis is 4.73 so there is more extreme data then there would have been if the returns were distributed normally. Momentum strategies display relatively high kurtosis, with a minimum of 1.35 for the (12-1) long only strategy and a maximum of 6.44 for the (3-1) long-short strategy. The long-short strategies have higher kurtosis, which means that they have a lot of extreme values. The adjusted Sharpe ratio takes into account these two additional risk measures by introducing a penalty for excess kurtosis and negative skewness. Because the momentum returns evidently exhibit some extent of skewness and kurtosis, most of the adjusted Sharpe ratios are expected to be lower than the original Sharpe ratios. For instance the (3-1) strategy has the highest adjusted Sharpe ratio of 0.80 (as opposed to a Sharpe ratio of 0.90) which means that it generates the highest risk adjusted returns. While the long-short strategies generate high returns, their risk adjusted performance is significantly worse than the risk adjusted performance of the long only strategies.

3.3 Systematic biases

The asset classes present in the winner portfolio are taken into consideration, in order to determine if the abnormal results observed are driven by the returns of consistently outperforming asset classes. This would occur if the selection process systematically selects the same asset classes, which implies that the observed abnormal results are not due to the momentum effect but are instead caused by repeatedly holding the same outperforming asset classes. This would make our results systematically biased. I find that the frequency for which one asset class is present in the winner portfolio varies considerably for each formation period. The asset classes most selected in the winner portfolio for each formation period is oil. This asset classes reports an annualized volatility of 30%. This is the highest volatility of all asset classes considered in the sample. It is therefore not surprising that this asset class occurs often in the winner portfolios as it will probably have more extreme returns than asset classes with low volatility. Looking at the bottom portfolio, I observe that oil is often included in that portfolio as well. On average it presents itself 10.56% of the time in the loser portfolio and 12.55% of the time in the winner portfolio. US Investment grade bonds are noticeably underrepresented in the winner portfolios, they are the only asset class that is ever present less than 1% of the time. However they appear sparsely in the loser portfolios as well, on

23 average 2.44% of the time. Apart from Investment grade bonds, the occurrence of the remaining asset classes in the winner portfolio seems to be distributed quite evenly. No asset class occurs in the winner portfolio more than 13.53% of the time. The Japanese yen appears to occur in the loser portfolio the most, however it is also included in the winner portfolio on average 4.37% of the time. Exhibit 10 shows the frequency (averaged over all formation period) in each the winner and loser periods for each asset class.

To further broaden our understanding of potential systematic biases in our momentum portfolios I create static reference portfolios. These portfolios have a position in each asset class with weights equal to the exposures of the momentum portfolios to that asset class. Naturally these reference portfolios feature the same average exposures to the asset classes as the momentum portfolios. I now take the returns of the static reference portfolios as our benchmark and look if the momentum portfolio outperforms this benchmark with statistically significant results. Average excess returns of the static reference portfolios remain between 5.06% and 5.23%. I will test if the lowest of the long only momentum returns is statistically different from the return of this static reference portfolio. The (12-1) strategy generates the lowest excess return of 8.46%. I find that the return of the (12-1) strategy is statistically significant after adjusting for a structural bias towards profitable asset classes. This automatically implies that the other strategies with even higher returns are also statistically significant after adjustment. I can thus confidently conclude that our strategies do not display a large structural bias towards one asset class.

3.4 Momentum during subperiods

3.4.1 Sample period divided

Since different patterns were observed in the cumulative returns, the sample period is divided into three subperiods in order to separately assess the performance of momentum during each time frame. Visually, three different patterns, associated with the periods 1995-2001, 2002-2007 and 2008-2018, can be distinguished. From the performance statistics of these subperiods I derive that the momentum returns are not stationary over time. Exactly as the graph of the cumulative returns predicts, performance of the momentum strategies is superior between 2002 and 2007 when compared to the other two subperiods. The subperiod 1995-2001 also renders eminently positive returns, especially

24 the long-short portfolio are above average. The most recent subperiod 2008-2018 documents performance statistics that are unmistakably inferior to the performance statistics of the other two subperiods. The long-short portfolios that wielded a nine or twelve month formation period even obtained negative returns. It can thus be presumed that the average results might not be appropriate to predict future returns as they are heavily skewed by historical performance.

3.4.2 Momentum during recession

Next I look at the performance of the momentum strategies during periods of recession. I consider performance of the momentum strategy in recession periods of March 2001 until November 2001 and December 2007 until June 2009. During the recession in 2001 excess returns are noticeably lower than they were for the entire sample period. Excess annualized returns of the four long only strategies fluctuate between -16.55% and -3.81%. The long-short strategies evidently perform better during this period, reporting returns between -11.24% and 11.26%. Remarkable is the seemingly large impact of formation periods on the average excess returns. The benchmark portfolio reported a return of -8.37% in combination with a volatility of 7.91%. One can instantly see that most of the momentum strategies underperform the benchmark during this crisis. During the recession from 2007 until 2009 average excess returns of long only momentum range from -15.68% to 16.94% while long-short momentum returns varied between -12.62 and 44.38%. Volatilities of momentum returns during this period are also extremely high, reaching a maximum of 39.35% for the long short (9-1) portfolio. As seen before during the previous recession, formation periods seem to influence the excess returns of a strategy majorly. The shorter formation periods momentum portfolios generate positive returns during this period for both the long-short and long only approach. Whereas the longer formation periods generate negative returns. The benchmark return amounts to -9.27%. Depending on the formation period it is in theory possible to outperform this benchmark but the outcomes are extremely volatile.

4. Robustness tests

4.1 Momentum of volatility adjusted returns

The first robustness test takes into account the asset class returns that are adjusted for volatility. Naturally some assets are more volatile than others, subsequently returns of the equally weighted momentum portfolios could be manipulated by positions in volatile asset classes. Furthermore, volatile asset classes are most likely to appear in either the winner or the loser portfolios as they are inclined to produce the most extreme returns. In the interest of avoiding aforementioned effects volatility adjustments are applied. The volatility adjusted returns will make the asset classes more suited for comparison. Exhibit 10 exhibits the performance statistics for this approach. It can be concluded that the results remain strong when adjusting the original returns for their volatility. The majority of the momentum portfolios actually deliver slightly improved risk-adjusted returns.

4.2 Transaction costs

A practical implementation of momentum strategies necessitates the assessment of transaction costs. Momentum strategies often involve high turnover, this makes them sensitive to transaction costs. Particularly the strategy employed in this study will in all probability have a high turnover due to its short holding periods. In every month t the long only strategy holds three positions in asset classes, at the beginning of each month the strategy closes out of these positions and buys three new positions. However One asset class can occur in the winner portfolio consecutively, consequently there are not inevitably three transactions each month. A sensitivity analysis is applied to estimate the profitability after transaction costs for the momentum strategies. As a rather conservative measure of transaction costs I utilize the percentage employed by Jegadeesh and Titman is used. I also consider two percentages that are more moderate. The results of this analysis are provided in exhibit 11.

For example. the 3-month formation period strategies participates in 510 transactions throughout the sample period. This means that 510 assets are traded for the duration of this period. This comes down to 22 transactions per year on average.

The long-short strategies especially incur high transaction costs as they do not only have three long positions to maintain but additionally they also have three short positions. Transaction costs for long-short strategies can be expected to be approximately twice the transaction costs of long only strategies.

The most profitable strategy of long only (3-1) momentum is profitable for all the considered alternatives of transaction costs. However the estimate of 0.5 bps transaction costs results in negative returns for all the remaining strategies. The long only strategies are however able to produce positive returns for 0.25 and 0.1 bps transaction costs. The long-short strategy is only profitable for 0.1 bps transaction costs.

Conclusion

A broad arsenal of research has been conducted on factor investing and more specifically on momentum investing. Momentum portfolios have repeatedly been proven to outperform the market within numerous asset classes. There is however a paucity of research on the existence of momentum across these asset classes, the objective of this paper was to fill this gap. This thesis aims to assess whether the momentum anomaly is present at the asset-class level and to evaluate the impact of such a strategy on a multi-asset portfolio.

When analysing the performance of the eight momentum strategies, it can be recapitulated that all of them produced positive excess returns that were significantly higher than the excess returns on the benchmark, that equally weighted all asset classes in the sample. Strategies that employed shorter formation periods appear to obtain higher excess returns, this effect manifests itself after 2007. While long-short strategies are able to deliver returns nearly as high as the returns of long only strategies, they clearly fall behind when it comes to risk- adjusted performance (based on both Sharpe ratio and adjusted Sharpe ratio). The strategy that ranks asset over a three month periods and then holds the three best performing assets for one month generates on average superior results. It reported the highest average return (11.45%) and the highest Sharpe ratio (0.90). This strategy also delivered the best risk- adjusted performance throughout all three subperiods.

Subsequently it turns out that the momentum profits do not display a bias toward one asset class. All asset classes appear to some extent evenly in the winner and loser portfolios except for US Investment grade corporate bonds. The static reference portfolios that take the same exposure to the asset classes as the momentum strategies still underperforms the momentum strategy. Furthermore the original momentum methodology is applied once more on the volatility adjusted returns of the asset classes in the sample. This approach gives excess returns comparable to the original momentum returns. It can thus be concluded that the strategy is not biased towards extremely volatile asset classes, which would explain the abnormal returns.

Momentum investment appears to be exceptionally risky during recessions. This paper isolated the performance of the eight strategies during the recessions of 2001 and 2007-2009. Excess returns diverged immensely conditional on formation periods and whether the strategy was long only or long-short. Overall the momentum portfolios delivered a highly volatile

28 performance while often documenting negative excess returns. For the duration of recessions, the benchmark portfolio presented a better risk-adjusted return.

This empirical analysis also includes a sub period investigation of multi-asset momentum. It was found that all eight momentum strategies in the periods 1995-2001 and 2002-2007 were profitable. Intriguingly is the exceptional performance of momentum strategies during 2002- 2007. The momentum anomaly across asset classes is undeniably present from 1995-1-2007. However throughout 2008-2018 the momentum effect attenuates severely, the (9-1) and (12- 1) long-short strategies even produce negative excess returns. Even though the remaining six strategies are still profitable they are also incredibly volatile, making their risk adjusted performance less attractive.

As a robustness check, a sensitivity analysis on transaction cost was added to the analysis. Long-short strategies turned out to be more sensitive towards transaction costs due to their higher turnover. It follows that the long only strategies were evidently superior to long-short strategies under transaction costs. Conditional on the estimate chosen for transaction costs it is possible to obtain positive excess returns on a multi-asset momentum portfolio.

In conclusion it can be surmised that the momentum anomaly definitely presented itself across asset class throughout 1995-2007, the effect however diluted over the past ten years (2008- 2018). The effect is not biased towards specific asset classes but does not perform well during recessions. For practical applications it is essential to add that in case of moderate transaction costs, long only strategies will still be profitable.

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Appendix

Exhibit 1: The asset classes used for construction of momentum portfolios with their corresponding index and the datatype gathered to obtain excess returns.

Exhibit 2: Descriptive statistics on asset classes of data sample, based on annualized excess returns. The sample period runs from July 1995 through December 2018.

Exhibit 3: Matlab code for constructing a 3-1 long only and a 3-1 long short strategy

%% Setup the Import Options and import the data opts = spreadsheetImportOptions("NumVariables", 17);

% Specify sheet and range opts.Sheet = "Weekly excess returns"; opts.DataRange = "C32:S1256";

% Specify column names and types opts.VariableNames = ["Equity", "VarName4", "VarName5", "VarName6", "VarName7", "VarName8", "Commodities", "VarName10", "VarName11", "Governmentbonds", "VarName13", "VarName14", "VarName15", "Corporatebonds", "VarName17", "Currencies", "VarName19"]; opts.VariableTypes = ["double", "double", "double", "double", "double", "double", "double", "double", "double", "double", "double", "double", "double", "double", "double", "double", "double"];

% Import the data MomData51S4 = readtable("C:\Users\poell\OneDrive\Documenten\MomData (5) (1).xlsx", opts, "UseExcel", false);

%% Convert to output type MomData51S4 = table2array(MomData51S4);

%% Clear temporary variables clear opts %% Defining some variables MomDataS1=MomData51S4;

%% %Define the number of weeks and number of asset classes in our dataset [totalweeks, totalassets] = size(MomDataS1);

%Calculate the weeklly benchmarkreturns (the returns of an equally weighted portfolio of all the asset classes in our sample). MomDataS1T = transpose(MomDataS1); benchmarkret(1,:) = mean(MomDataS1T(1:totalassets,:)); benchmarkretT = transpose(benchmarkret); Benchmarkwvol=std(benchmarkret(14:1225)); Benchmarkweekly = benchmarkretT(14:1225);

%Calculate monthly benchmarkreturn, starting from the 4th month in the sample (so from the 14th week onwards). for i=1:303

start(i)=(4*i)+10; stop(i)=start(i)+4-1; end numberofmonths = length(start); for p=1:numberofmonths benchmarkretmonthly(p,1)=mean(benchmarkretT(start(p):stop(p),1 )); benchmarkvol(p,1)=std(benchmarkretT(start(p):stop(p),1)); end

Benchmark = benchmarkretmonthly;

%% clearvars -except Benchmark* MomDataS1* totalweeks totalassets

%% (3-1)MOM Portfolio construction

%First we define the formation period and the holding period, in this %case 3 months formation periods (13 weeks), 1 month holding period (4 weeks) %Because we use weekly returs we will define the periods in weeks as well PF =(13); PH = (4); T = round((1225-PF)/4); % T is the number of times we want to run our formation and holding periods, %it depends on the definition of our formation and holding periods and %their respective start and stop dates. It can be found by solving the %formula (1226-PF-PH)/4

%Next we define the start and end dates for the formation and holding %periods, we want a new formation and holding period to start every month %(every 4 weeks) for i=1:T startF(i)=(4*i)-3; stopF(i)= startF(i)+ PF-1; startH(i)=stopF(i)+1; stopH(i)=startH(i)+PH-1; end formationperiods = length(startF); holdingperiods = length(stopF);

%Next we want to calculate the average returns for each asset class during %the formation period for p=1:formationperiods Freturn(p,:)=mean(MomDataS1(startF(p):stopF(p),:)); end

%Now we sort the returns for each month form high to low

FreturnT = transpose(Freturn); [FreturnOr,idx] = sort(FreturnT, 'descend');

%Define asset classes in winner portfolio for each month ncandidates=round(0.2*totalassets); idwinner33(1:ncandidates,1:T)= idx(1:ncandidates,:); idloser33(1:ncandidates,1:T) = idx(end-ncandidates+1:end,:);

%We now want to calculate the returns of the winner and loser portfolios %over the holding period.

%First we calculate the return in the holding months for all asset classes for m=1:holdingperiods Hreturn(m,:)=mean(MomDataS1(startH(m):stopH(m),:)); end

HreturnT=transpose(Hreturn); for i=1:holdingperiods a(i)=idwinner33(1,i); b(i)=idwinner33(2,i); c(i)=idwinner33(3,i); d(i)=idloser33(1,i); e(i)=idloser33(2,i); f(i)=idloser33(3,i); end

L33returns= zeros(1,T); W33returns= zeros(1,T);

%Then we take the return over the holding month of the three best and worst %performing asset classes measured over the respective formation period T_=T*4

38 weeklywinnerret3 = zeros(1,T_); for i=1:(totalweeks-PF) AA(i) = ceil((1/4)*i); end for i=1:T_ weeklywinnerret3(i)=mean(MomDataS1T([a(AA(i)) b(AA(i)) c(AA(i))],PF+i)); weeklyloserret3(i)=mean(MomDataS1T([d(AA(i)) e(AA(i)) f(AA(i))], PF+i)); end for i=1:T_ weeklyLSret3(i)=weeklywinnerret3(i)-weeklyloserret3(i); end weeklywinnerret3=transpose(weeklywinnerret3); weeklyLSret3=transpose(weeklyLSret3); stdevLS3=std(weeklyLSret3) stdev3 = std(weeklywinnerret3); for i=1:holdingperiods W33returns(i)=mean(HreturnT([a(i) b(i) c(i)],i)); L33returns(i)=mean(HreturnT([d(i) e(i) f(i)],i)); end

%Lastly we calculate the returns of the long-shorts strategy Longshortret33 = W33returns-L33returns;

Lret33 = transpose(W33returns); LSret33 = transpose(Longshortret33); %% clearvars -except idloser* weeklyLS* idwinner* LSret* stdev* weeklywinnerret* Lret* L*returns MomDataS1* Benchmark* totalweeks totalassets

Exhibit 4: Definitions of descriptive statistics.

1 Average return (휇 ) ∑푇 푟 푅 푇 푖=1 푖 1 푇 Average compounded return ` (∏푖=1(1 + 푟푖))푇 − 1

∑푇 (푟 −푟̅)2 Standard deviation (휎 ) √ 푖=1 푖 푅 푇−1 3 푇 푇 푟푖−푟̅ Skewness (푠푘푒푤 ) ∑ ( 휎 ) 푅 (푇−1)(푇−2) 푖=1 √푁 4 2 푇(푇+1) 푇 푟푖−푟̅ 3(푇−1) Kurtosis (푘푢푟푡 ) ∑ ( 휎 ) − 푅 (푇−1)(푇−2)(푇−3) 푖=1 (푇−2)(푇−3) √푁

휇푅 Sharpe ratio (푆푅푅) 휎푅 푠푘푒푤 푘푢푟푡 Adjusted Sharpe Ratio (퐴푆푅 ) 푆푅 ∗ (1 + 푅 푆푅 − 푅 (푆푅 )2) 푅 푅 6 푅 24 푅

Exhibit 5 Performance statistics of momentum strategies, based on excess returns. Sample period 1995-2018.

Exhibit 6: Cumulative returns of the momentum portfolios over the sample period 1995-2018.

Exhibit 7: Regression of excess returns of momentum strategies on excess returns of benchmark portfolio.

Exhibit 8: The frequency of each asset class in the winner and loser portfolios.

Exhibit 9: Performance statistics of momentum portfolios during subperiods 1995-2001, 2002-2007, 2008-2018.

Exhibit 10: Momentum performance statistics using volatility adjusted returns as momentum signal, sample period 1995-2018.

Exhibit 11: Excess returns of momentum portfolios after transaction costs.