Two Essays on Asset Pricing. Ryan A. McKeon. (Under the direction of Christopher T. Stivers) Abstract

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1 Two Essays on Asset Pricing by Ryan A. McKeon (Under the direction of Christopher T. Stivers) Abstract The first essay examines whether an active fund manager can outperform the benchmark index using trading strategies based on asset pricing anomalies. I consider an active manager who faces the following constraints on his investment activity: abnormal performance and tracking error are measured relative to a benchmark index, performance is evaluated regularly, and short positions are prohibited. The manager implements size, book-to-market and momentum strategies by over-weighting small, value or winner stocks and under-weighting large, growth or loser stocks relative to the benchmark index. When the fund manager uses the Fama-French SMB, HML and UMD portfolio weights as the guide for implementing these size, book-to-market and momentum strategies the investment portfolio does not outperform the benchmark. However, alternative size, book-to-market and momentum strategies do outperform the benchmark in some circumstances. However, the performance is far less impressive than the performance of investment portfolios which include short positions. When the Russell 3000, a common real-world benchmark for fund managers, is used as the benchmark index, size and book-to-market strategies are unable to provide excess performance, while momentum strategies do. I furthermore find that performance is highly inconsistent form year-to-year, meaning that a fund manager who is evaluated on a regular basis is unlikely

2 to show consistent performance. I conclude that a typical active manager would find it difficult to beat the benchmark index by implementing strategies based on well-documented anomalies in the academic literature. In the second essay I examine the relation between profits from book-to-market strategies and momentum strategies. Specifically, I test two time-series hypotheses which are not mutually exclusive, but do have opposite predictions for subsequent momentum profits. First, if periods of large book-to-market profits are indicative of a large dispersion in expected returns, then subsequent momentum profits are likely to be relatively high. Second, if shifts in book-to-market profits (a period of growth dominating value shifting to a period of value dominating growth, or vice versa) are associated with changes in the market state, then subsequent momentum profits are likely to be relatively low. My results provide no support for the first hypothesis. However, I do find support for the second hypothesis. Specifically, momentum profits are negatively and reliably related to changes in book-to-market profits prior to implementing the momentum strategy. The results are consistent with the propositions that: (1) changes in the state of the market across the ranking and holding periods are associated with lower subsequent momentum profits, and (2) price adjustments to changes in expected returns are important for understanding momentum profits. Index words: Asset Pricing, Value, Growth, Book-to-Market, Momentum

3 Two Essays on Asset Pricing by Ryan A. McKeon B.Comm, The University of Kwa-Zulu Natal, 1998 B.Comm (Hon), The University of Kwa-Zulu Natal, 1999 A Dissertation Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Athens, Georgia 2008

4 c 2008 Ryan A. McKeon All Rights Reserved

5 Two Essays on Asset Pricing by Ryan A. McKeon Approved: Major Professor: Christopher T. Stivers Committee: Annette B. Poulsen Tyler R. Henry Jin (Ginger) Wu Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia December 2008

6 Table of Contents Page List of Figures v List of Tables vii Chapter 1 Can an Active Manager Beat The Benchmark Index Using Size-, Book-to-Market- and Momentum-based Strategies? Introduction Literature Long-Only Investment Portfolios Conclusions The Information in Book-to-Market Profits for Momentum Introduction Existing Theory Data and Methodology Results and Discussion Conclusion Appendix A: The Effect of Changes in Expected Returns 76 Bibliography iv

7 List of Figures 1.1 Time series of values for λ 0 values for the SMB*, HML* and UMD* portfolios Time series of values for λ 0 values for the S-B*, H-L* and U-D* portfolios Time series of relative performance of the SMB* strategy versus the Russell Time series of relative performance of the S-B* strategy versus the Russell Time series of relative performance of the HML* strategy versus the Russell Time series of relative performance of the H-L* strategy versus the Russell Time series of relative performance of the H-only strategy versus the Russell Time series of relative performance of the UMD* strategy versus the Russell Time series of relative performance of the U-D* strategy versus the Russell Time series of relative performance of the U-only strategy versus the Russell Value Premium: June 1927-Dec Momentum Premium: Dec 1927-May One-Year Buy-and-Hold Momentum Returns: June July v

8 vi 2.4 Two-Year Buy-and-Hold Momentum Returns: June July One-Year Buy-and-Hold Book-to-Market Returns: June July Two-Year Buy-and-Hold Book-to-Market Returns: June July Absolute HM L-changes from August July

9 List of Tables 1.1 Summary performance of the Fama-French component portfolios Fama-French Long-Short Portfolios Long-Only Anomaly-Based Portfolios Optimal Combinations of the Market and Fama-French Anomaly Portfolios Optimal Combinations of the Market and Over-Weight/Under- Weight Anomaly-Based Portfolios Performance of Pure Value-Weighted Fama-French Long-Short Portfolios Performance of Pure Value-Weighted Over-Weight/Under-Weight Anomaly-Based Portfolios Optimal Combinations of the Market and Value-Weighted Only Over-Weight/Under-Weight Anomaly-Based Portfolios Performance of Portfolios formed from the Long Side of Fama- French Anomaly Portfolios Optimal Combinations of the Market and S-, H- and U-only Portfolios Summary performance of the Russell Indices, with Information Ratios compared to the broad Russell 3000 Index Performance of Various Anomaly-Based Investment Strategies Relative to the Russell 3000 Index Momentum Strategy Simulation Summary statistics for the returns on various momentum and bookto-market portfolios (Data is monthly from August 1962 to July 2005) 79 vii

10 viii 2.3 Absolute changes in book-to-market profits and subsequent momentum profits (Data is monthly with date ranges as indicated) Absolute changes in book-to-market profits and subsequent momentum profits (Data is monthly with date ranges as indicated) Absolute changes in book-to-market profits and subsequent momentum profits (Data is monthly with date ranges as indicated) Absolute changes in book-to-market profits and subsequent momentum profits, including ex post analysis (Data is monthly with date ranges as indicated) Comparison of levels of HM L returns over the momentum holding period to longer-run historic HML levels Robustness checks for absolute changes in book-to-market profits and subsequent momentum profits (Data is monthly from August 1962 to July 2005) Contemporaneous book-to-market profits and momentum profits (Data is monthly with date ranges as indicated) Conditional contemporaneous book-to-market profits and momentum profits (Data is monthly with date ranges as indicated) Recent long-run levels of B/M profits and subsequent momentum profits (Data is monthly with date ranges as indicated) Recent long-run levels of B/M profits and subsequent momentum profits (Data is monthly with date ranges as indicated)

11 Chapter 1 Can an Active Manager Beat The Benchmark Index Using Size-, Book-to-Market- and Momentum-based Strategies? 1.1 Introduction This paper considers the performance of an active fund manager who implements investment strategies based on size, book-to-market ratio and momentum, whilst facing constraints on his investment activity. Specifically, the manager s performance is measured relative to a benchmark index, assessed on a regular, periodic basis, and the manager is prohibited from short selling. These restrictions are common real-world constraints on an active fund manager. 1 Reilly and Brown (2006) provide a straightforward definition of active equity portfolio management as an attempt by the manager to outperform, on a risk-adjusted basis, a passive benchmark portfolio (page 607). This paper considers a very specific experiment: whether an institutional fund manager who is allocated money by a plan sponsor and given a specific benchmark to beat can successfully beat that benchmark by loading on various size, book-to-market and momentum strategies inspired by the academic literature. The asset pricing literature documents that portfolios constructed by sorting on size, book-to-market ratio and recent past performance outperform the market index. These abnormal returns are termed anomalies because they cannot be explained by the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965). 2 Motivated by the data show that only 2.95% of mutual funds reported holding short positions to the SEC [Almazan, Brown, Carlson, and Chapman (2004)]. 2 The word anomaly implies market inefficiency to many. However, in this paper I do not use the word to imply market inefficiency, but rather as a convenient catch-all word for the three strategies considered. 1

12 2 failure of the CAPM to explain these results, Fama and French (1993) propose and test a three-factor model. The model includes a market factor, a size factor (called SMB for Small Minus Big) and a book-to-market factor (called HML for High Minus Low). Fama and French (1996) find that this three-factor model does not explain the momentum effect of Jegadeesh and Titman (1993), and so a momentum factor portfolio (called UMD for Up Minus Down) has since been added to the model. 3 These factor portfolios have been the source of much discussion in the asset pricing literature. One debate is whether these portfolios capture mis-pricing in the market, or systematic risk factors which drive returns. I am concerned with the practical implementation of these strategies. How much improvement over the performance of a benchmark index is possible for a realistically constrained active manager who attempts to implement these strategies? Existing literature examines the issue of how easily an investor can actually implement the profitable strategies documented in the academic literature. Houge and Loughran (2006) examine this issue generally by looking at well-known indices. Specifically, the authors compare the average returns of the S&P 500/Barra Value and S&P 500/Barra Growth indices over the January 1975 to December 2002 period and find that the difference in average returns is not statistically different to zero. The authors find a similar result for a comparison of average returns generated by the Russell 3000 Value and Russell 3000 Growth indices. The authors also analyze the performances of mutual funds from the CRSP Survivor Bias- Free US Mutual Fund Database. They find no evidence that the average returns of funds categorized as Value produce higher returns than those categorized as Growth over the 1965 to 2001 period, even when funds are split into large- and small-cap categories. Other papers examine the impact of transaction costs on profits (for example Lesmond, Schill, and Zhou (2004), Hanna and Ready (2003) and Bushee and Raedy (2005)), the impact on profits of the price impact of sizeable trades (for example Korajczyk and Sadka (2004) 3 This factor is based on the momentum factor introduced in Carhart (1997).

13 3 and Chen, Stanzl, and Watanabe (2002)) and the impact of restrictions on the weights in individual assets (see for example Bushee and Raedy (2005).) I contribute to this literature on whether abnormally profitable anomaly-based strategies are easily implemented by studying the effect on profits of specific short sale constraints, benchmarking and regular performance evaluation. I consider a specific experiment designed to shed light on how easily real-world fund managers are able to capture the superior performance of size, book-to-market and momentum strategies. I consider an institutional fund manager who is allocated funds by a plan sponsor. The plan sponsor has determined the desired asset allocation and supplies money to the fund manager based on this allocation decision. The fund manager is then charged with the task of security selection in an effort to beat an appropriate benchmark index. Busse, Goyal, and Wahal (2007) note the importance of these institutional fund managers in the market: At the end of 2004, over 50,000 plan sponsors (public and private retirement plans, endowments, foundations, and multi-employer unions) allocated over $6 trillion in assets to about 1,500 institutional asset managers (Money Market Directory (2004)). In this paper I conduct an ex post analysis to examine whether a fund manager could have beaten a broad stock market index over the last three decades by loading up on small, value or winner stocks in the investment portfolio. The fact that the analysis is ex post introduces a hindsight bias which biases the experiment in favor of finding that the manager beats the market. I furthermore ignore execution costs 4, which also provides a bias in favor of the manager. I interpret all results with these limitations of the experiment in mind. Given these features of the analysis, a finding that managers still cannot beat the benchmark would be particularly convincing. The basic procedure followed by my hypothetical fund manager is as follows: 1) the plan sponsor allocates money to the fund manager. 2) the plan sponsor specifies a benchmark index against which the fund manager will be 4 Execution costs includes both direct transaction costs related to commissions and bid-ask spread, as well as the price impact of any trading.

14 4 evaluated. 3) the manager attempts to beat the benchmark index with superior security selection ability. In each case, the security selection is based on one or more of the strategies under consideration: size, book-to-market ratio or momentum. The manager constructs an active portfolio based on one or more of these strategies, and then combines that active portfolio with the passive benchmark portfolio to arrive at the final investment portfolio 4) the manager s performance is evaluated on two levels: the performance of the investment portfolio over the entire period, and the performance of the investment portfolio in each calendar year. In all cases I use the Information Ratio (IR) to determine whether or not the fund manager beat the benchmark index 5. Although I do not attempt to exhaust all possibilities in the paper, I do consider different benchmarks, active strategies and performance evaluation measures in the analysis. The system of portfolio construction employed by the fund manager is similar to the well known Treynor-Black model frequently included in standard text books. 6 Under this model the investor is able to conduct analysis of a limited number of stocks in order to identify potentially non-negative alphas. An active portfolio is then constructed from any securities thus identified. The investor s ultimate investment portfolio is then a combination of the passive market portfolio and the actively constructed portfolio, combined in such a way as to produce a portfolio with a higher Sharpe ratio than the market portfolio alone. I initially use the CRSP value-weighted index as the benchmark index. I employ portfolio weights of the Fama-French SMB, HML and UMD portfolios as a proxy for size, value and momentum strategies. Later, I consider other methods of implementing these strategies and a real-world benchmark index in the Russell 3000 index. The fund manager implements portfolios to maximize the IR. 5 The IR, which I define specifically later, is a measure which rewards the investor for producing a return in excess of the benchmark index, whilst accounting for the active risk (the extent to which the investor s portfolio differs from the benchmark. 6 See, for example, Bodie, Kane, and Marcus (2008) pages 611 to 614.

15 5 When the fund manager uses the Fama-French portfolio weights as the active portfolio for deviating from the benchmark index, the resulting portfolios do not outperform the benchmark index. The short sale prohibition is a significant constraint using SMB, HML and UMD, making it difficult for the fund manager to deviate substantially from the benchmark index in an attempt to generate excess returns. In order to provide more freedom for the manager to deviate from the composition of the benchmark index, I construct long-short size, book-to-market and momentum portfolios in the spirit of the Fama-French portfolios, but without the equal weighting of component portfolios. These portfolios are named S-B, H-L and U-D. When the stock weights from these portfolios are used as the guide for deviating from the benchmark index the fund manager is able to construct long-only portfolios which outperform the benchmark index. Once the equal weighting of component portfolios is dropped, the short sale constraint is less prohibitive, allowing the fund manager to deviate sufficiently from the benchmark index to generate superior performance. There is the potential problem of a higher tracking error, but this does not seem to hurt the performance of these portfolios. The book-to-market and momentum strategies show marginal ability to beat the benchmark index. Finally, I consider naive long-only strategies which simply buy the component portfolio with the highest average returns. In this strategy, a manager simply buys the S portfolio (small stocks) when implementing a size strategy, buys the H portfolio (value stocks) when implementing a book-to-market strategy, or buys U (winner stocks) when implementing a momentum strategy. Here, book-to-market and momentum strategies do show evidence of being able to beat the benchmark index. As the paper considers the investing activities of real-world fund managers, I consider alternative benchmark indices to the CRSP Value-Weighted portfolio, since this CRSP portfolio is not used as a benchmark in practice. I obtain composition data for the Russell 3000 index developed by the Russell Investment Group and examine the ability of the various size, book-to-market and momentum strategies to beat this index. Over the sample period

16 6 January 1981 to December 2003 I find that book-to-market strategies were not able to beat the Russell index, while some size and momentum strategies were. A year-by-year breakdown of the performances of the anomaly-based strategies illustrates how difficult it is to consistently beat a benchmark with these strategies. For both size and book-to-market strategies it is rare to find back-to-back calendar years in which the strategies beat the benchmark. Therefore, a fund manager whose performance is being evaluated on a regular basis would have some years when he beats the benchmark, but such success would be difficult to sustain. One momentum strategy offers more consistent returns. Here, the fund manager would have at least matched if not beaten the benchmark in consecutive years quite often. The evidence suggests that short sale and tracking error constraints are impediments to achieving real-world success with size, book-to-market and momentum strategies, but they are impediments which can be overcome with particular variations of these strategies. Regular performance evaluation, however, is an impediment which all but one of the strategies considered are unable to overcome. Given the hindsight bias and the fact that I ignore transaction costs, these results reinforce the notion that a simple anomaly-based strategy is unlikely to result in consistent performance gains. However, it is also worth noting that I do not consider market timing in the analysis, which could bias the results against a finding of beating the index consistently. Avramov and Chordia (2006) show that considering business cycle indicators may improve an investors performance when undertaking real-time portfolio optimization. For example, the authors show that performance may be improved by weighting less heavily on winner stocks during recessions. The rest of this paper is organized as follows: Section II reviews the relevant literature on the size, book-to-market and momentum anomalies and their implementation in practice. Section III discusses the construction of the Fama-French size, book-to-market and momentum portfolios, and illustrates how these portfolios can be used to generate the portfolio weights for various long-only size, book-to-market and momentum trading strategies. This section documents and compares the performances of these various strategies. Section

17 7 IV offers some conclusions regarding the ability of investors to beat the market or a given benchmark index using size, book-to-market and momentum strategies. 1.2 Literature Banz (1981) and Reinganum (1981) document the size effect; small-capitalization firms on the NYSE earned higher average returns than predicted by the CAPM over the period. Barsky (1977) provides formal evidence of a value premium in the market by documenting that firms with high earnings-to-price ratios earn abnormally positive returns relative to the predictions of the CAPM. This result, and results specific to book-to-market ratios documented in Stattment (1980) and Rosenberg, Reid, and Lanstein (1985), are a precursor to the finding in Fama and French (1992) that high book-to-market stocks earn higher returns than CAPM can account for. The momentum effect is documented by Jegadeesh and Titman (1993). They find that past winners (defined as stocks with superior returns over the previous year) deliver higher average returns than past losers (stocks with inferior returns over the previous twelve months) over a holding period of three to twelve months. The three-factor model of Fama and French (1993) does not explain the short-run momentum anomaly (Fama and French (1996)). Furthermore, Schwert (2003) documents that the momentum effect has persisted since being documented in The failure of the three-factor model to capture momentum led to the creation of a momentum factor, based on Carhart (1997). The four-factor pricing model, which includes the market factor, SMB, HML and UMD, is commonly used in the asset pricing literature. Daniel (2004) documents that inclusion of the Fama-French portfolios in portfolio construction would have dramatically improved the ex-post performance of an investor over the 1968:7 to 2003:12 period. 7 Daniel documents the ex-post Sharpe ratio for optimal portfolios constructed when the available assets are: the market, SMB, HML and UMD. Daniel 7 Where maximizing the Sharpe ratio is the optimization criterion.

18 8 suggests that it is possible to almost quadruple the Sharpe ratio of the market index by including the Fama-French SMB, HML and UMD portfolios as available assets! However, this impressive performance relies on the investor being able to costlessly implement the long-short Fama-French portfolios, and invest an equivalent weight in the risk-free asset. I consider the class of investors who are restricted to long-only portfolios. Houge and Loughran (2006) investigate whether or not investors actually capture the value premium in practice. The authors examine a sample of equity indexes, mutual funds and large-cap and small-cap stocks and find no evidence that high book-to-market firms earn higher returns on average than low book-to-market firms in practice. For example, a comparison of the Russell 3000 Value Index versus the Russell 3000 Growth Index over the Feb Dec 2002 period reveals a difference in average returns of 0.11%, a statistically insignificant figure. My paper addresses a similar question to Houge and Loughran (2006), but examines specific real-world constraints which could account for the failure of documented academic results to translate into superior real-world performance. Almazan, Brown, Carlson, and Chapman (2004) analyze the extent of short sale constraints on mutual fund managers and the effect of these constraints. The authors document that 73.3% of mutual funds that filed Form N-SAR 8 in 1994 reported that their investment policies formally prohibited short selling. Although this percentage had fallen to 66.1% by 2000, restrictions on short selling by institutional investors are clearly widespread. Furthermore, even when not specifically reporting a prohibition on short selling, institutional managers remain reluctant to do so. Of those managers not specifically reporting a short sale prohibition, only 11.0% actually did short. In summary, the 1994 data indicate that only 20 of the 679 reporting funds (2.95%) actually implemented short positions! Therefore, a large fraction of fund managers consider only long positions when trying to exploit the asset pricing anomalies documented in the academic literature. 8 This is short for Form N, Semi-Annual Report, and is the report that registered investment companies must file with the SEC twice a year.

19 9 Several papers have examined the extent to which transaction costs reduce the abnormal returns documented for various trading strategies related to the anomalies. For example, Lesmond, Schill, and Zhou (2004) examine the profits generated by the momentum trading strategies and conclude that such strategies require frequent trading in relatively high cost securities. The authors conclude that real-world transaction costs would eliminate the theoretical profits documented in the literature. The authors include commissions, price impact costs, taxes, short-sale costs, opportunity costs and immediacy costs in their analysis. 9 Korajczyk and Sadka (2004) analyze the price impact of momentum trading strategies. The authors find that as much as 5 billion dollars could be invested in some momentum-based strategies before the apparent profit opportunity would disappear. This result contrasts with the findings of Chen, Stanzl, and Watanabe (2002), who find that size, book-to-market and momentum trading strategies are all severely affected by the price impact of trading. Hanna and Ready (2003) also report that controlling for transaction costs and trade delays severely reduces the returns to long-short strategies based on book-to-market ratio and momentum. Bushee and Raedy (2005) ask whether various anomaly-based trading strategies are implementable in practice. The strategies they study include: size, book-to-market, momentum, cash-flow-to-price (Fama and French (1996)), reversal(debondt and Thaler (1985)), operating accrual (Sloan (1996)) and post-earnings announcement drift (PEAD, Bernard and Thomas (1989)). The real-world constraints include restrictions on the weight of any single asset in the portfolio, short sale constraints, price impact of the trades and transaction costs. The authors find that the size and reversal strategies are not robust to their implementability constraints, while cash flow-to-price, momentum and PEAD are robust. The book-to-market and operating accrual strategies enjoy mixed success, depending on the mix of constraints employed. 9 The authors note that passive traders (using limit orders) incur opportunity costs because their trades are not always executed, while active traders (using market orders) incur sizeable immediacy costs.

20 10 Rather than stressing transaction costs, in particular the costs of establishing and maintaining short positions, this paper examines constraints which are self-imposed, or mandated by the plan sponsor. 10 The goal of this paper is similar to that of Bushee and Raedy (2005). Differences include the imposition of the tracking error constraint on the portfolio construction, and the use of the IR as the performance evaluation tool to specifically account for the level of active risk in the portfolio. 1.3 Long-Only Investment Portfolios Anomaly Portfolios Fama and French (1993) propose factor portfolios to measure the exposure of stocks to certain risk factors, namely risk related to the market factor, risk related to size and risk related to book-to-market ratio. 11 Carhart (1997) introduces an additional factor to capture risk exposure related to momentum. 12 The factor portfolios take the form of long-short portfolios which are long the stocks with characteristics suggesting high returns (eg. low size, high book-to-market ratio, etc.) and short the stocks with characteristics suggesting lower returns. Following Fama and French (1993), I create six portfolios based on size and book-tomarket ratio. Each year, stocks are sorted by size into two categories: small (market capitalization below the median of stocks on the NYSE) or big (market capitalization above the NYSE median). Stocks are also sorted by book-to-market ratio into three categories: low (book-to-market ratio in the lowest 30%), middle (book-to-market ratio in the middle 40%) or high (book-to-market ratio in the highest 30%). Six portfolios are formed based 10 See D Avolio (2002) for a thorough treatment of costs related to short sales activity. 11 Exposure to a systematic risk factor is one proposed explanation for the size, book-to-market and momentum effects. Market mis-pricing is a prominent competing theory. 12 Carhart calls this factor PR1YR, for prior one-year returns. The Fama-French UMD factor, for Up Minus Down, is similar to Carhart s PR1YR factor.

21 11 on the intersection of the size and book-to-market sorts. The following table illustrates the classification of the portfolios: (0% Size < 50%) (50% Size 100%) Low market cap( Small ) High market cap( Big ) (70% B/M 100%) High book-to-market Small Value stocks (S/H) Large Value stocks (B/H) (30% B/M < 70%) Middle book-to-market Small Middle stocks (S/M) Large Middle stocks (B/M) (0% B/M < 30%) Low book-to-market Small Growth stocks (S/L) Large Growth stocks (B/L) These six component portfolios are the ingredients for the Fama-French size and market-to-book factor portfolios. Let W SH denote the n 1 vector of portfolio weights for the Small Value component portfolio, where n is the number of stocks in the market index. 13 For each individual stock i, the weight is: w i,sh = V i V SH (1.1) where V i is the market capitalization of stock i and V SH is the total capitalization of all stocks in the Small Value component portfolio. W BH, W SM, W BM, W SL, and W BL are similar weight vectors for the other five component portfolios. If a stock is not a member of the component portfolio, then it carries a weight of zero in the vector. Following Fama and French (1993), I define W SMB as the vector of weights to be employed in determining the returns on the SMB portfolio, and calculate it as follows: W SMB = 1 3 W SH W SM W SL 1 3 W BH 1 3 W BM 1 3 W BL (1.2) The return on the SMB portfolio, R SMB, is calculated as follows: R SMB = W SMBR (1.3) where R is the n 1 vector of returns for all stocks in the market. The returns on the HML portfolio are calculated as follows: R HML = W HMLR (1.4) 13 The t subscript has been suppressed, but this vector would be different for each time at which weights are determined. t subscripts will be suppressed throughout the paper.

22 12 where W HML, the vector of portfolio weights for the HML factor portfolio, is defined: W HML = 1 2 W SH W BH 1 2 W SL 1 2 W BL (1.5) Note that stocks in the Small Middle and Large Middle component portfolios carry weights of zero in W HML. For the momentum factor, stocks are sorted by size as before. A further sort is conducted based on cumulative returns registered in the period from t 12 to t 2 before classification. Stocks are sorted by momentum into three categories: winners (prior returns in the highest 30%), flat (prior returns in the middle 40%), and losers (prior returns in the lowest 30%). The following table illustrates the classification of the portfolios: (0% Size < 50%) (50% Size 100%) Low market cap( Small ) High market cap( Big ) (70% R T 12,T 2 100%) High Return( Up ) Small Winner stocks (S/U) Large Winner stocks (B/U) (30% R T 12,T 2 < 70%) Flat Return( Flat ) Small Flat stocks (S/F) Large Flat stocks (B/F) (0% R T 12,T 2 < 30%) Low Return( Down ) Small Loser stocks (S/D) Large Loser stocks (B/D) These six component portfolios are the ingredients for the Fama-French momentum factor portfolio. Let W SU denote the n 1 vector of portfolio weights for the Small Winner component portfolio. As before, each component portfolio is value-weighted. W BU, W SF, W BF, W SD, and W BD are similar weighting vectors for the other five components. W UMD is the vector of weights employed to determine the returns on the UMD portfolio, and is calculated as follows: W UMD = 1 2 W SU W BU 1 2 W SD 1 2 W BD (1.6) Note that weights of stocks in the Small Flat and Large Flat components are zero in W UMD. The return on the UMD portfolio, R UMD, is calculated as follows: R UMD = W UMDR (1.7) Tables 1 reports the performance of the component portfolios of SMB, HML and UMD. Panel A confirms that value stocks deliver higher Sharpe ratios than growth stocks. Small

23 13 stocks outperform large stocks, but the difference is small. This reflects the fact that the size anomaly has disappeared since first being documented (see Schwert (2003)). The portfolio of small value stocks produces the most impressive performance with a monthly Sharpe ratio of The relatively high return generated by this portfolio is to be expected to some degree, since the portfolio captures both the size and book-to-market effects. The monthly Sharpe ratio for the market index is Panel B shows that previous winners, both small and large, produced Sharpe ratios in excess of the market index, whilst previous losers deliver Sharpe ratios not significantly different from zero. The performance of small losers is particularly bad, generating a negative Sharpe ratio. Clearly, prior losers do not benefit from the size effect. The portfolio of small prior winners produces the most impressive performance with a Sharpe ratio of Table 2 summarizes the performance of the Fama-French portfolios over the July December 2003 period 14. SMB delivers a Sharpe ratio below the market index, whilst HML and UMD have higher Sharpe ratios. The correlation matrix indicates low correlations between these factor portfolios and the market index, as well as low correlation amongst themselves Long-Only Anomaly-Based Investment Portfolios I examine the reasonable case of an active manager who cannot sell short due to institutional constraints. Reilly and Brown (2006) provide a straightforward definition of active equity portfolio management as an attempt by the manager to outperform, on a riskadjusted basis, a passive benchmark portfolio. The manager attempts to profit from, for example, the value anomaly by over-weighting value stocks and under-weighting growth stocks relative to their benchmark index weights. There are a variety of different ways that a manager might try to implement such a strategy. Initially, I assume that the value strategy uses the weights from the Fama-French HML 14 The choice to begin the analysis at July 1968 is a result of this being the time period analyzed in Daniel (2004), which was a starting point for this paper.

24 14 portfolio. The weights in the investor s portfolio are: W = W M + λw HML (1.8) where W, W M and W HML are n 1 vectors representing portfolio weights, benchmark index weights and Fama-French HML weights respectively. λ is a scalar which measures the degree to which the fund manager s portfolio weights deviate from the benchmark index weights. The following must hold: ı W = 1; ı W M = 1 and ı W HML = 0, where ı is the unit vector. A passive index portfolio would have λ = 0. As the manager increases λ, the portfolio weights for value stocks increase relative to the benchmark index and the portfolio weights for growth stocks decrease relative to the benchmark index. The weights on middle bookto-market stocks remain equal to their benchmark index weights. In this way, λ acts like a dial, adjusting the extent to which the portfolio is tuned away from the benchmark index and towards the value strategy. This system of portfolio construction is similar to the well known Treynor-Black model frequently included in standard text books. 15 Under this model the investor is able to conduct analysis of a limited number of stocks in order to identify potentially non-negative alphas. An active portfolio is then constructed from any securities thus identified. The investor s ultimate investment portfolio is then a combination of the passive market portfolio and the actively constructed portfolio, combined in such a way as to produce a portfolio with a higher Sharpe ratio than the market portfolio alone. In my analysis the fund manager simply uses the Fama-French factor portfolio as the active portfolio in this exercise. I am interested in investment portfolios where the anomaly strategy is implemented to the maximum extent possible. Therefore, I solve for the highest value of λ for which there are no negative weights in W. Consider the value strategy outlined in equation (1.8). Of the six component portfolios which comprise HML the small growth stocks (S/L) have the lowest market capitalization. These stocks have negative weights in W HML. Therefore, as the fund 15 See, for example, Bodie, Kane, and Marcus (2008) pages 611 to 614.

25 15 manager increases λ the weights in W for small growth stocks will decrease and eventually become negative. The value of λ which results in a zero weight for all small growth stocks (i.e., when the long-only constraint binds) is the maximum extent to which the investor can overweight value stocks and underweight growth stocks in the portfolio. Let λ HML 0 denote this value. For any small growth stock, its weight in the investment portfolio is given by: w i,sl = w i,m λ 1 2 V i V SL (1.9) I set this weight equal to zero and solve for λ HML 0. Recalling that w i,m = V i V M, it follows that λ HML 0 is: λ HML 0 = 2 VSL V M (1.10) The value of λ HML 0 varies from year to year as the relative market capitalizations of small growth stocks and the market as whole change. λ HML 0 represents the maximum extent to which the investor can overweight value stocks and underweight growth stocks when faced with a long-only constraint. Let HML* denote the value over-weight/growth under-weight portfolio where λ is set to λ HML 0. The stock weights for this portfolio are: W HML = W M + λ HML 0 W HML (1.11) In other words, HML* is the investment portfolio implemented by the fund manager when following a book-to-market strategy in an attempt to beat the benchmark index. For the small-cap over-weight/large-cap under-weight portfolio, denoted SMB*: where, W SMB = W M + λ SMB 0 W SMB (1.12) λ SMB 0 = 3 VBH V M (1.13) where V BH is the total market capitalization of the large value stocks in the SMB portfolio. For the winner overweight/loser underweight portfolio, denoted UMD*: W UMD = W M + λ UMD 0 W UMD (1.14)

26 16 where, λ UMD 0 = 2 VSD V M (1.15) where V SD is the total market capitalization of the small loser stocks in the UMD portfolio. Figure 1 plots the values for λ 0 for each factor portfolio over time. λ 0 can be thought of as the implementability of the strategy in any given year. The higher the level of λ 0, the greater the degree to which the investor can over-weight stocks with higher expected returns and under-weight stocks with lower expected returns. For SMB*, the level of λ SMB 0 is high when the market value of large growth stocks is high relative to the market as a whole. Therefore, the investor is most readily able to tilt his portfolio towards the SMB factor portfolio precisely during times when the size effect is least pronounced. Indeed, Figure 1 shows that λ SMB 0 was particularly high during the 1980 s, when small stocks did not exhibit higher average returns than large stocks (Schwert (2003)). The level of variation for λ 0 is far less pronounced for the book-to-market and momentum effects, and the magnitudes of the values are much smaller. This suggests that under the long-only constraint the fund manager will find it difficult to deviate substantially from the benchmark index Performance Analysis Table 3 summarizes the performance of the constrained SMB*, HML* and UMD* portfolios. The Sharpe ratios for HML* and UMD* are much lower than those delivered by the HML and UMD portfolios reported in Table 2:A (SMB* is slightly higher than SMB), and not significantly higher than the market index. These constrained portfolios, which contain no short positions, are not able to deliver Sharpe ratios as high as the unconstrained long-short portfolios. Furthermore, the table indicates that the correlations between these portfolios and the market, and between any two of these constrained portfolios, is extremely high.

27 17 In performance evaluation, the metric which compares the returns on a fund with returns on the benchmark index is called the Information Ratio (IR) 16. The IR is given by: IR = R P R B P T t=1 (R P,t R B,t ) 2 (1.16) T 1 where R P and R B are the average return on the investment portfolio and the benchmark portfolio respectively, R P,t is the return on the fund portfolio at time t, R B,t is the return on the benchmark portfolio at time t and T is the number of periods for which returns are recorded. As one can readily see, the IR rewards the fund manager for producing returns in excess of the benchmark portfolio, whilst penalizing the manager for straying too far from the composition of the benchmark index. The denominator can be viewed as the tracking error of the portfolio. While tracking error is most often thought of in the context of index funds, as a measure of how well the manager is indeed matching the performance of the index, it has a similarly important role in the evaluation of actively managed portfolios. The tracking error is referred to as active risk in the context of active management. The interpretation of the IR is straightforward: a positive ratio indicates that the fund manager has outperformed the benchmark index and a negative ratio indicates that the fund manager has underperformed the benchmark index. The IR s reported in Table 3 indicate that SMB*, HML* and UMD* did not outperform the benchmark index (MKT). For each of these portfolios, the IR is not significantly different from zero. This clearly illustrates that when short selling is prohibited, the fund manager is not able to construct a portfolio which beats the benchmark using the Fama-French portfolios. Daniel (2004) reports the economic magnitude of abnormal performance resulting from use of anomaly portfolios. The author considers an investor who views the market, SMB, HML and UMD as available assets for investment, and constructs a portfolio from these 16 See Grinold and Kahn (2000))

28 18 assets which maximizes the ex-post Sharpe ratio of the portfolio. The analysis considers the 1968:7 to 2003:12 time period. Daniel reports substantial increases in Sharpe ratio over the market index as the investor adds Fama-French portfolios in constructing his portfolio. Table 4 replicates the analysis in Daniel (2004) where the optimal portfolio i.e. the portfolio with the highest Sharpe ratio is found for a variety of portfolios. The analysis begins with the market index, and then subsequent results show the effect of combining the market index with one or more Fama-French portfolios in the optimum weights. Since the Fama-French portfolios are self-financing long-short portfolios, the net investment in these portfolios is zero. The analysis assumes that the investor allocates an investment weight to the risk-free rate equal to the total weight in the Fama-French portfolios. The risk-free rate is taken as the average for a one-month T-bill over this period: 0.55% per month. Under these assumptions the manager can dramatically increase the Sharpe ratio by adding long-short Fama-French portfolios to the market index in the investment portfolio. The market Sharpe ratio of can be increased to by optimally combining all three of the factor portfolios. This suggests that substantial improvement in performance over the market is possible using anomaly-based portfolios. Table 5 reports the performance of portfolios constructed in the same spirit as Daniel (2004), but with the long-only MKT, SMB*, HML* and UMD* portfolios as the available assets. The resulting optimal portfolios are compared to MKT using the IR criterion. The table shows that even optimal combinations of these constrained over-weight/under-weight portfolios cannot outperform the benchmark index. 17 The table shows that the IR s for a variety of optimal combinations of these portfolios are not significantly different to zero. The high correlations between the long-only portfolios severely limit the benefits of diversification. The p-values indicate that none of these portfolios produced Sharpe ratios which were statistically different to the market. Simply put, the optimal portfolio was not able to beat the benchmark index. 17 Changing the optimization rule to maximizing the IR instead of the Sharpe Ratio, which would in fact be the active manager s goal, does not alter the conclusions of the analysis.

29 Pure Value-Weighted Construction for the Anomaly Portfolios The HML* and UMD* portfolios considered in section C above didn t deviate significantly from benchmark weights due to short-sale constraints. This is largely due to the artificial equal weighting of component portfolios in the Fama-French portfolios. As a robustness check, Table 6 reports results for a slightly different portfolio construction exercise. Recall that the Fama-French portfolios were constructed by equally weighting the component portfolios. For example, see equation 1.5 for the calculation of weights in HML. In this section, I relax the equal weights on component portfolios used in calculating the Fama-French portfolio weights. The stock weights for the size portfolio, now called S-B, are calculated as: W S B = W S W B (1.17) where W S represents a value-weighted portfolio of small stocks (stocks below the median NYSE size; there is no sorting by book-to-market) and W B represents a value-weighted portfolio of large stocks (stocks above the median NYSE size; again there is no sorting by book-to-market). Note that here I use the relative weights of stocks within its component portfolio for the value weighting, not the original weight of the stock in the market. In other words, the weight of a stock in W S is its relative weight in the portfolio of small stocks, not its weight in the market portfolio. This is true of all analogous portfolios which follow. The stock weights for the book-to-market portfolio, now called H-L, are calculated as: W H L = W H W L (1.18) where W H represents a value-weighted portfolio of value stocks (stocks in the highest 30% book-to-market ratios; there is no sorting by size) and W L represents a value-weighted portfolio of growth stocks (stocks in the lowest 30% book-to-market ratios; there is no sorting by size). Weights on the middle 40% of book-to-market ratio are zero. The stock weights in the momentum portfolio, now called U-D, are calculated as: W U D = W U W D (1.19)

30 20 where W U represents a value-weighted portfolio of winner stocks (stocks in the highest 30% of prior returns; there is no sorting by size) and W D represents a value-weighted portfolio of loser stocks (stocks in the lowest 30% of prior returns; there is no sorting by size). Weights on the middle 40% of prior returns are zero. Table 7 reports results for the over-weight/under-weight portfolios which are constructed in a similar fashion as before, with the long-only constraint imposed. λ 0 is calculated in the same manner as before to determine the extent of over-weighting/under-weighting. λ S B 0 = V B V M, λ H L 0 = V L V M, and λ U D 0 = V D VM. Figure 2 illustrates the time series for these values, and Table 7 reports the average values for lambda, indicated by λ 0. The values of lambda are much higher than they were for SMB*, HML* and UMD*, suggesting that the manager will now have a greater ability to deviate from the benchmark index in an attempt to generate superior performance. Let S-B*, H-L* and U-D* denote the constrained size, book-to-market and momentum portfolios respectively. The weights for these over-weight/under-weight constrained portfolios are calculated as follows: W S B = W M + λ S B 0 W S B (1.20) W H L = W M + λ H L 0 W H L (1.21) W U D = W M + λ U D 0 W U D (1.22) Table 7 reports that the H-L* and U-D* portfolios did outperform the benchmark. This is in contrast to the results for portfolios constructed using SMB*, HML* and UMD*. This suggests that a short-constrained fund manager may be able to beat the benchmark index with value or momentum strategies, but not by using the HML or UMD weights when constructing the over-weight/under-weight portfolio. The reason is apparent when comparing Figure 1 to Figure 2. These graphically illustrate the values for λ 0 for various portfolios over time. It is clear that removing the equal weighting of component portfolios in SMB, HML and UMD results in far greater implementability of size, book-to-market and momentum strategies. However, the magnitude of this outperformance shown here is still far less impressive