In this paper we construct and investigate the properties and robustness of a set of momentum factors. We also construct illustrative indexes, based on a preferred momentum definition and show that the resulting indexes exhibit a substantial exposure to momentum and relatively low levels of turnover.

We identify candidate momentum factors from a survey of the academic literature and current market practice. The candidate factors are assessed and formation and holding periods examined for the FTSE Developed universe over the period 2001 – 2014.

We note that absolute momentum may be decomposed into three component measures; namely stock specific momentum; momentum resulting from systematic risk factor exposures; and residual momentum linked to stock specific shocks.

Stock specific return (Alpha) is used to capture stock specific momentum, whilst the Residual Sharpe Ratio captures momentum linked to stock specific news. Each momentum factor shows robust performance across regions. Furthermore, each factor appears to exhibit a degree of independence.

We also consider three factors that broadly capture absolute or total momentum; the one-year cumulative return (Return); the one-year Sharpe Ratio; and the ratio of the current price to the highest price over the last 12 months (the CH12 Ratio). We note that all three measures, by construction will exhibit exposure to systematic risk factors that have performed well over the momentum formation period. Consequently, indexes premised on such measures of momentum are likely to experience significant reversals in exposure to systematic risk factors.

Measures of momentum based on raw returns have historically shown strong risk adjusted performance outcomes that are not primarily the result of country or industry effects. We prefer cumulative return measures of momentum to Sharpe Ratio measures, despite both exhibiting similar historical risk adjusted performance outcomes, since the latter has historically displayed substantial industry and country effects. The simulated performance outcomes of Return and CH12 Ratio are very similar; however the CH12 Ratio results in outcomes with a substantial bias (low beta) to systematic factors.

Our preferred measure of momentum is the Residual Sharpe Ratio, which displays relatively high risk-adjusted performance outcomes, and relatively low levels of volatility and turnover. In contrast to traditional measures of momentum, the Residual Sharpe Ratio shows limited time-varying exposure to other systematic factors.

Finally, we construct illustrative indexes based on the Residual Sharpe Ratio and show that they exhibit substantial exposure to momentum and relatively low levels of turnover.

**1. The momentum effect **

The momentum effect relies on a continuation of past patterns of stock returns. The capture of any momentum effect requires the selection of stocks based on past return or return related metrics in the expectation that such performance patterns are repeated in the future. Typically, stock performance over some past period, excluding the most recent monthly return is employed as the criterion to select stocks. The practice of ignoring the most recent performance when constructing a momentum factor is an attempt to avoid conflating momentum with short-term reversal effects. In this section we review the momentum literature; both from the perspective of assessing evidence for the existence of a momentum effect and from attempts to rationalize its existence.

**1.1 Evidence of a momentum effect **

The empirical literature examining evidence for a momentum effect is vast. Levy (1967) highlights that, stocks with higher than average past returns exhibit significant abnormal future returns. Subsequently, Grinblatt and Titman (1989), Jegadeesh and Titman (1993) and Chan, Jegadeesh and Lakonishok (1996), find that momentum is a useful indicator of future performance, that is not subsumed by market risk, size or value.

Later studies examine the effectiveness of momentum in risk adjusted performance metrics in contrast to momentum in raw returns. Biglova et al. (2004) find that risk adjusted momentum measures including the Sharpe Ratio exhibit improvements in risk adjusted performance compared to simple return based momentum factors. Bornholt and Malin (2011) show that adding volatility and other risk metrics to momentum strategies is effective. Liu et al (2010) examine measures comparing the current price to prior highs.

Momentum approaches incur high levels of turnover. Consequently, an important practical consideration is the extent of any momentum effect net of transaction costs. Li, Brooks and Miffre (2009) find that momentum effects are disproportionately concentrated amongst small and illiquid stocks. Korajczyk and Sadka (2003) show that liquidity weighted momentum strategies are superior to a capitalisation weighted approach, both in terms of abnormal returns net of transaction costs and in improving capacity.

**1.2 Explanations of the momentum effect **

A key consideration in developing explanations for the momentum effect is the level at which momentum arises; is momentum primarily an industry, country, or stock level effect? Moskowitz and Grinblatt (1999) and Sefton and Scowcroft (2005) find that momentum in the US is largely an industry effect. In contrast, Nijman et al (2004) conclude that momentum in Europe is largely a stock effect.

Daniel, Hirschleifer, and Subrahmanyam (1998) and Barberis, Shleifer and Vishny (1998) develop a behavioural finance rationale for the existence of momentum effects, founded on investor under and over-reaction to news. Jegadeesh and Titman (2000) assume that returns are driven by a one-factor model and show analytically that momentum profits have three possible origins; the degree of cross-sectional dispersion in expected returns; the ability to time momentum; and the degree of serial correlation in idiosyncratic stock returns. If momentum arises primarily from the first or second sources, then momentum profits may be interpreted as compensation for bearing systematic risk. However, if momentum arises from the third source, then momentum effects may be attributed to market inefficiency.

Momentum effects may also have more prosaic origins; the seasonality of momentum effects is well documented, exhibiting a persistent January effect that is attributed to tax-loss harvesting and window dressing behaviour by investors, see Debondt and Thaler (1985), Jegadeesh and Titman (1993), and Chu, Liu and Rathinasamy (2004).

**2. Definitions of momentum **

Momentum is typically defined as the cumulative stock return over some prior time frame ignoring the most recent period of performance. A precise definition of momentum requires design choices from several perspectives:

- Does currency play a role in momentum – should returns be calculated in local or a common currency?
- Is the distinction between price (capital) and total returns important?
- Is momentum a stock, industry or country effect?
- Do momentum effects exist for risk-adjusted metrics in addition to return measures?
- To what degree are risk adjusted measures of momentum independent of price momentum?
- What is the appropriate formation period, i.e. over what past period should past performance be calculated?
- What are the appropriate holding and rebalancing periods?
- How important are short term reversal effects? What period of recent performance should be ignored in order to avoid conflating reversal and momentum effects?

Table 1 on the next page summarises common approaches to momentum used by academics and practitioners.

**2.1 Momentum definitions in the academic literature **

In order to avoid contaminating momentum effects with currency fluctuations when examining momentum in an international context, we restrict our investigation to local currency measures of momentum. Chan, Hameed and Tong (2000) find statistically significant evidence for international momentum effects and conclude that momentum is primarily a stock phenomenon, with exchange rate dynamics playing almost no role. This suggests that the choice over local or common currency factors of momentum is unimportant. Common currency measures of momentum are used by Rouwenhorst (1998) and Nijman et al (2004), while local measures are used by Leipold and Lohre (2012), Bacmann et al (2001) and Liu et al (2010).

The majority of academic and practitioner approaches to momentum utilise total returns; the exception being MSCI who use capital returns. Furthermore, an absolute measure of return is the most common definition of momentum. Excess industry returns are examined by Moskowitz and Grinblatt (1999). Other studies consider risk-adjusted measures of momentum; for example Biglova et al (2004) and MSCI (Sharpe Ratio). Liu et al (2010) define momentum as the ratio of the current price to the past 52-week high. This approach to momentum appears to be independent of industry effects.

Guiterrez and Pitinsky (2007) and Blitz et al (2011) study residual measures of momentum for US stocks, finding risk-adjusted performance substantially in excess of that generated by absolute return measures of momentum.

The ratio of the current price to the 52-week high is examined in a US context by George and Hwang (2004). They conclude that this ratio contains incremental information to traditional momentum factors and suggest an under-reaction explanation. A number of later studies confirm the predictive power of this factor in both US and international markets; see Liu et al (2010) for a detailed literature survey.

The majority of studies use momentum factors formed (formation periods) over six, nine or 12 months and holding periods of six or 12 months. Jegadeesh and Titman (1993) examine formation and holding periods from a US perspective and conclude that nine and 12 month formation and six month holding periods exhibit the strongest momentum effects. Rouwenhorst (1998) in an international context confirms these results, highlighting nine and 12 month formation periods and a six month holding period.

Typically a one-month period between the construction of any momentum factor and its incorporation is used to mitigate reversal effects. Jegadeesh and Titman (1993) allow a one-week lag; no lag is used by Nijman et al (2004), Leippold (2012), or Biglova et al (2004).

Section 2.2 reviews the momentum definitions of other index providers. Section 3.1 proposes specific momentum definitions drawn from the academic literature and used by practitioners, which we use as a starting point for our empirical investigations.

**2.2 Index provider momentum definitions **

The MSCI Momentum indexes utilise two Sharpe Ratio measures of momentum. Six and 12 month local capital returns after excluding the most recent month and the annualised standard deviation of weekly local capital returns over a three-year period are combined. The premise of the MSCI Momentum indexes is that momentum in risk/reward measures is superior to the use of momentum in return measures. There is support for this approach in the literature – see Biglova et al. (2004) and Bornholt and Malin (2011).

The S&P 1500 Positive Momentum Tilt Index uses a more traditional approach, defining momentum as the 11 month total return to the month prior to the rebalance month.

The Russell-Axioma Momentum Indexes follow a two stage process. Initially, naïve factor indexes are constructed using the cumulative 250 trading day performance excluding the last 20 trading days. Stage two applies an optimisation approach to derive narrower indexes that track the performance of the naïve factor index whilst controlling for turnover and exposure to other risk factors.

**3. Empirical results **

**3.1 Testing momentum definitions **

From a review of the academic literature and current commercial practices, we test empirically the following measures of momentum:

- Local 12 month total returns, after excluding the most recent month (Return).
- Sharpe Ratio based on local total 12 month returns, after excluding the most recent month. Annualised volatility is calculated using daily returns over the same period (Sharpe Ratio).
- The ratio of the current local price to the highest local price over the previous 52 weeks, excluding the most recent month (CH12 Ratio).

Grundy and Martin (2001), Chordia and Shivakumar (2002) and Blitz et al (2011) note that raw return momentum strategies have time-varying exposure to systematic risk factors (e.g. market beta). A raw return measure of momentum for example, will tilt towards high-beta stocks if the market is rising over the momentum formation period and conversely towards low-beta stocks if the market is falling.

Consequently, we also examine momentum factors designed to avoid time-varying market exposure. We use the following risk model to separate systematic and non-systematic sources of return:

*R*t = α + Σk βk*F*kt+εt (1)

where *R*t is the stock local total return in period t; α is the stock specific return not explained by the risk factors ; βk is the stock exposure to risk factor k; *F*kt is the return to risk factor k in period t, and εt is the residual return. We include two risk factors – the country return and global industry return respectively. We investigate momentum in two non-systematic sources of return; stock specific return (α) and residual return (εt).

We use the Residual Sharpe Ratio measure proposed by Blitz et al (2011). The Residual Sharpe Ratio captures firm specific news that influences future returns. Gutierrez and Pirinsky (2007) and Blitz et al (2011) argue that standardising residual returns (Residual Sharpe Ratio) leads to an improved assessment of whether firm specific return shocks are news as opposed to noise. We calculate the Residual Sharpe measure of momentum in the following manners:

- Estimate equation (1) using 36 months of local total returns on rolling monthly basis for the 11 months prior to factor construction month. Each month, we calculate the average residual return for the most recent 12 months. The mean and standard deviation of the 11 month time-series of average values forms the residual momentum measure (Residual Sharpe Ratio).

The remaining source of momentum is the stock specific return. Gutierrez and Pirinsky (2007) and Blitz et al (2011) suggest that alpha should be ignored, since it captures any misspecification of the risk model. However, conditional on the given risk model, alpha may be interpreted as the stock specific return that is not the result of systematic risk factor exposures. We calculate the stock specific measure of momentum as follows:

- Estimate the annualised stock specific return from equation (1) using one-year of daily local total returns excluding the most recent month (Alpha).

The Return, Sharpe Ratio, and the CH12 Ratio measures of momentum represent total momentum measures, where momentum is predominantly a reflection of exposure to systemic risk factors, whilst Alpha and the Residual Sharpe Ratio capture momentum effects that do not originate from systematic risk exposures.

**3.1.1 Formation periods **

We begin by investigating formation and holding periods. Table 1 indicates that the most common formation periods used in the academic literature are six and 12 months.

We first perform tests on the length of the formation period for the Return, Sharpe Ratio, and CH12 Ratio measures of momentum. We rank FTSE Developed constituents on each momentum factor calculated over formation periods of varying length into quartiles. Each quartile contains equal numbers of stocks and we assess the performance of the top (high momentum) and bottom quartiles (low momentum). Stocks are weighted by free-float market capitalisation within quartiles, held for six months, and rebalanced on a March / September rebalance cycle. Chart 1 shows the spread in annualised Sharpe Ratios of the resulting momentum quartiles. All performance metrics in this paper are based on USD total returns.

Chart 1 indicates that outcomes for the Return and Sharpe Ratio momentum factors are broadly maximised and stable over formation periods of nine to 12 months. The CH12 Ratio achieves marginally superior outcomes at longer formation periods, peaking at 45 months. However, the striking feature of the CH12 Ratio results is their stability over a wide range of formation periods. A 12 month formation period for this factor results in comparable outcomes. Given the robustness of these results, we prefer to use a common formation period of 12 months for the Return, Sharpe Ratio, and CH12 Ratio measures of momentum. Our findings in this section hold, irrespective of whether an equally-weighted or a market capitalisation weighting scheme is used.

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