Guide To Factor-Based Investing
These notes are primarily based on the ideas and learnings from "Your Complete Guide To Factor-Based Investing, The Way Smart Money Invests Today", 1st Edition, by Andrew Berkin and Larry Swedroe in 2016. The description of the book mentions: "There are hundreds of exhibits in the investment factor zoo, so it is difficult to see which ones are actually worth time and money. This brings a thorough yet still jargon-free and accessible guide to applying the most valuable quantitative and evidence-based approaches to outperforming the market: factor investing - a journey through the land of academic research and an extensive review of its quest to uncover the secret of successful investing". The authors describe unique criteria and characteristics in the definition of a factor as persistent, pervasive, robust, investible and intuitive. Overall, the premise of the book is structured to remain objective and provide reputable evidence for all the claims put forward with citations for further reference.
To note, the MSCI Index returns used are gross returns, since the data for gross returns is more extensive with a longer history - as an indirect addition, the use of gross returns presents a higher and more conservative hurdle when considering whether a fund successfully captured a factor (net returns include a reduction for the impact of international taxes on dividends and interest). Also, for context, a t-statistic is a measure of statistical significance, where a value greater than about 2.0 is an indication of something meaningful above random noise - generally, depending on the sample size, a t-statistic of 1.960 is preferable for a confidence of 5% in statistical significance (although a t-statistic of 1.645 may be acceptable for a confidence of 10% in statistical significance). For distinction, in most cases, the average annual (not annualized or compounded) return is presented - in other words, this is the arithmetic average (rather than the geometric average). Finally, the use of "domestic" refers to the United States, while the use of "international" refers to other markets, including developed and emerging markets but excluding the domestic market.
Foreword
At its most basic level, factor-based investing is simply about defining and then systematically following a set of rules which produce diversified portfolios. When distinguishing these rules, it is vital to avoid data mining which leads to false factors which are only the product of randomness, selection bias, and accident. Because of this, it is vital for these rules to be based on real factors and proven with evidence to have persistence (historically deliver reasonably reliable returns), pervasiveness (deliver, on average, these returns in a variety of locales and asset classes if such tests apply), robustness (it should not be dependent on one very specific formulation but fails to work if other related versions are tested), intuitiveness (make sense to or is the evidence just going by historical performance), and investability (it is possible to actually implement the factor). In addition, it is necessary to consider whether the current modern environment is the same as the past environments in which the evidence for the factor is based (primarily concerning if the factor happens to be based on behaviour), whether the theme of the factor actually overlaps with other factors which have been proposed, and whether the discovery of a new factor will subsequently affect the performance of that factor.
Introduction
Factors can be seen as the characteristics of equities and other assets which explain performance and provide premiums due to a relative risk, such that these characteristics are cross-sectionally common across a broad set of assets. In other words, factors are a quantitative way of expressing a qualitative theme (without the qualitative theme, there is no reason to believe the factor is real), where this qualitative theme is the result of investors making decisions based on their risk preferences (although there are also other arguments focussing on behavioural aspects). In a sense, the factors are an indirect indication of this risk and underlying state variable sensitivities which cannot actually be known but must be compensated - implying that investors can access higher expected returns if they are willing and able to infer and bear the state variable sensitivities which other investors are avoiding. For example, the success of Warren Buffett in outperforming the market is not necessarily the result of skill, but it can simply be attributed to the intuitive identification of and exposure to factors (in no way does it detract from his success, but it highlights his natural ingenuity).
The aim of diversification is to build a portfolio of assets which have a low or no correlation in order to increase the certainty of an outcome. The most common form of investments have traditionally been public equities and bonds, because public equities and bonds have shown to have a fairly low or no correlation. Recently, alternative investments have become more accessible, such as private equity and hedge funds. However, the correlations of private equity and hedge funds with public equities has been found to be 0.71 and 0.79 respectively, where most of the returns of these types of alternative investments are explained by the returns of public equities. In almost all cases, alpha, which can be seen as true outperformance defined as excess returns above the appropriate risk-adjusted benchmark, has proven to be very elusive for these fund managers trying to outperform the market. Other alternative investments, such as real estate investment trusts and infrastructure, also have a relatively high correlation with public equities. In the past, commodities have shown promise of having almost no correlation to public equities (although there are no expected returns above inflation from commodities). This limits the possibilities for diversification when considering asset classes.
However, rather than viewing a portfolio as a collection of asset classes, one can view it instead as a collection of diversifying factors. In this way, it is possible to find diversification within asset classes, rather than only considering the asset classes as groups. It has been found that diversification through factors within an asset class can actually be more effective at reducing portfolio volatility and market directionality than diversification across asset classes as groups. This highlights a distinct objective of factors, where it is not only about increasing return, but it is also about decreasing the level of uncertainty for the amount of return realized.
In order for a factor to be real, it is necessary for the underlying fundamental theory of the factor to consistently provide explanatory power to portfolio returns while delivering a premium and it must be persistent (hold across long periods of time and different economic regimes), pervasive (holds across countries, regions, sectors, and asset classes), robust (holds for a variety of definitions of similar concepts or metrics), investable (holds even after considering obstacles from actual implementation), and intuitive (logical risk-based reasons or behavioural-based reasons for its premium). From over 600 factors which have been published, there are only a handful of factors which acceptably meet these criteria (others either have not passed the criteria or overlap with these remaining factors with less significant results (variations on a common theme)). Thus, these factors provide a means to explain the vast majority of differences in returns between diversified portfolios.
Capital Asset Pricing Model
The journey of attempting to quantify differences in returns begins, by building on Modern Portfolio Theory from Harry Markowitz in the 1950s, with the Capital Asset Pricing Model (CAPM) from John Lintner, William Sharpe, and Jack Treynor in the early 1960s. The CAPM was the first formal asset pricing model and provided the first precise definition of risk and how it drives expected returns. In addition, the formalization of a model allowed for an evaluation of whether a fund manager who appeared to have outperformed the market actually generated alpha or if this outperformance could simply be explained by exposure to factors. The CAPM considers risk and return through the description of a single factor, where the risk and return of a portfolio are determined only by exposure to this factor. This factor is market beta and is the measure of the sensitivity of the equity risk of a portfolio relative to the risk of the overall market. This risk is systematic, non-diversifiable, and inherent to the market, because it is not possible to diversify away no matter how many equities are in a portfolio - in other words, the risk is always proportional to the covariance with the broad market.
Market Beta
By definition, market beta expresses the degree or sensitivity to which an asset tends to move with the broad market. It is expressed mathematically as the correlation between the return of an asset and the return of the market multiplied by the ratio of the volatility of the asset to the volatility of the market (as measured by the standard deviation of returns). Alternatively, this can be described as the ratio of the covariance between the return of the asset and market to the variance between the volatility of the asset and market. Thus, a market portfolio weighted by market capitalization and consisting of all the equities in the market is a representation (or, at least, an approximation) of the broad market and must have a market beta of 1.0. Under the assumptions of the CAPM, this portfolio is the mean-variance optimal portfolio - in other words, the assets contained in the market are optimally priced to reflect their expected means and covariances, such that their weights in the market portfolio are their mean-variance optimal weights.
Simply, a market beta of more than 1.0 implies that the portfolio has taken more risk than the broad market, while a market beta of less than 1.0 implies that the portfolio has taken less risk than the broad market. For example, if a portfolio has a beta of 1.5, it is expected for the portfolio to go up 15% when the market goes up 10% and go down 15% when the market goes down 10%. Alternatively, if a portfolio has a beta of 0.7, it is expected for the portfolio to go up 7% when the market goes up 10% and go down 7% when the market goes down 10%. (In reality, it is also necessary to consider the risk-free rate, since market beta is a premium above the risk-free rate for taking additional risk, but these examples are simplified without evaluating the risk-free rate).
The risk-free rate is the rate of return on a completely riskless asset. As alluded to, market beta is a premium above the risk-free rate for additional risk. Conventionally, the risk-free rate is taken to be the return of the 1-month U.S. Treasury bill (essentially, a riskless asset under almost all historic circumstances). So, this premium can be measured as the difference in the average annual (not annualized or compounded) return between the return of the broad market and the return of the risk-free rate - in other words, this is the return which would be realized by a long/short portfolio. From 1927 to 2015, when subtracting the annual average return of the 1-month U.S. Treasury bill from the annual average return of equities in the domestic market, the premium from market beta has been 8.3% as compensation.
Considering the persistence of market beta, it has been found that, from 1927 to 2015 and in the domestic market, the premium has been positive in 66% of calendar years and, over longer periods, the odds of outperformance by equities in the broad market of the risk-free rate become even greater. While the premium from market beta has shown high persistence and this persistence has increased with longer periods, it is important to note that there is always still the chance of an adverse outcome with a negative premium no matter how long the period. This has to be the expected result, because, if this was not the case, there would be no risk associated with investing in equities as long as the investor had the ability to wait until the premium was guaranteed to be positive. It is partly the risk of underperformance which explains why the premium exists - if there was no indefinite risk, investors would bid up the prices of equities until the expected return from equities was equal to the risk-free rate, in which case there would then no longer be a premium to be earned. Thus, the implications of risk create the expectation for higher returns, although these higher returns can never be guaranteed to be realized, but the likeliness for them to be realized increases over longer periods.
This concept must be true for all factors and this risk of underperformance is why it is critical that investors do not assume more risk than they have the ability, willingness, or need to take. This is also why discipline is so crucial to successful investing, where an investor must have the ability to be patient and ignore long periods of underperformance - as said by Warren Buffett, "the most important quality for an investor is the temperament, not intellect, to control the urges which get other people into trouble in investing". As this demonstrates, even periods of 10 years are not even close to being long enough to draw substantive conclusions given the inherent volatility of the assets involved.
| Factor | Rolling Periods From 1927 To 2015 | ||||
|---|---|---|---|---|---|
| 1 Year | 3 Year | 5 Year | 10 Year | 20 Year | |
| Market Beta | 66% | 76% | 82% | 90% | 96% |
For complement, the Sharpe ratio can be defined as a measure of the return of a portfolio earned above the risk-free rate relative to the amount of risk taken - essentially, this provides a measure of risk-adjusted returns. In this case, risk is measured by the standard deviation of the return of the portfolio (although it should be noted that this is not necessary accepted as the most reliable measure of risk). It is expressed mathematically as the ratio of the difference between the return of the portfolio and risk-free rate to the standard deviation of the return of the portfolio. For example, if the return earned on an asset was 10% with a standard deviation of 20% and the return of the 1-month U.S. Treasury bill was 4%, then the Sharpe ratio would be equal to 0.3. Over the period from 1927 to 2015, the Sharpe ratio of market beta was 0.40 for equities in the domestic market.
Considering the pervasiveness of market beta, it has been found that the premium from market beta has been positive in virtually every country and region around the world since 1900. From the 2016 Credit Suisse Global Investment Returns Yearbook and measured with the return from the 1-month U.S. Treasury bill, it was reported that, from 1900 to 2015, the premium from market beta was positive in all of the 21 developed countries considered and ranged from 3.1% in Belgium to 5.5% in the United States to 6.3% in South Africa - globally, the premium was 4.2% (premium for the world ex-U.S. was 3.5% and Europe was 3.4%). Considering the last 50 years from 1966 to 2015, the premium from market beta ranged from 1.4% in Austria to 4.4% in the United States to 6.6% in Sweden - globally, the premium was 4.1% (premium from the world ex-U.S. was 4.5% and Europe was 5.4%). Thus, the premium from market beta (above the 1-month U.S. Treasury bill) has been highly pervasive throughout modern history across different countries.
| Country | 1900-2015 | 1966-2015 |
|---|---|---|
| Australia | 6.0% | 3.5% |
| Austria | 5.5% | 1.4% |
| Belgium | 3.1% | 3.4% |
| Canada | 4.1% | 2.3% |
| Denmark | 3.4% | 4.8% |
| Finland | 5.9% | 6.1% |
| France | 6.2% | 4.9% |
| Germany | 6.1% | 3.9% |
| Ireland | 3.7% | 4.8% |
| Italy | 5.8% | 1.5% |
| Japan | 6.2% | 4.0% |
| Netherlands | 4.4% | 5.2% |
| New Zealand | 4.4% | 3.2% |
| Norway | 3.1% | 4.2% |
| Portugal | 4.7% | 3.9% |
| South Africa | 6.3% | 5.9% |
| Spain | 3.3% | 3.7% |
| Sweden | 3.9% | 6.6% |
| Switzerland | 3.7% | 5.2% |
| United Kingdom | 4.3% | 4.6% |
| United States | 5.5% | 4.4% |
| World | 4.2% | 4.1% |
| World ex-U.S. | 3.5% | 4.5% |
| Europe | 3.4% | 5.1% |
Considering the investability of market beta, it is possible to easily construct a market-like portfolio with low turnover. This is usually achieved by holding a diversified number of holdings to effectively approximate the market and then weighting these holdings based on their relative market capitalizations. As a result, the weightings within the portfolio will naturally adjust as the prices of the holdings change and, in turn, the market capitalizations of the holdings change. So, there would be no need for trading to rebalance the holdings under most circumstances. In addition, even when necessary, this portfolio would encounter very minimal trading costs, as the costs of trading in the form of bid-ask spreads and commissions from brokers have dramatically decreased in recent years. Moreover, funds which implement this portfolio have had their expense ratios driven lower due to competition between investment firms offering readily available mutual funds and exchange-traded funds. Thus, it is simple and inexpensive to invest in a portfolio which reliably targets market beta.
Considering the intuitiveness of market beta, there are clear risk-based reasons for an investor to expect an excess return through a premium from market beta. Since it is required that return is coupled with risk, it is necessary for a significant explanation of risk to justify a high level of return (conversely, an explanation for a significant return can also necessitate a high level of risk). The premium from market beta is large and a possible explanation for this significance is that the risk of owning equities is highly correlated with the risks of economic cycles, where, in recessions, investors who earn a salary or own a business are exposed to the risk of both bear markets and job retrenchments or reduced sales. So, given the large body of evidence demonstrating that individual investors are, on average, highly risk averse, they must, on average, demand a sizable premium as compensation for accepting this risk of exposure to market beta (especially since the risks of the loss of income from employment or poor performance from their business are uninsurable).
Furthermore, another explanation for the premium from market beta is that a large percentage, if not the vast majority, of equities are owned by individuals with high net worths. These individuals decide the prices which they are willing to accept. However, the marginal utility (increase in satisfaction from consuming an extra unit) of wealth decreases as net worth increases, so, in order to incentivize these individuals who do not have a prominent need to take additional risk, only a sizable premium would be satisfactory compensation for them to be induced to accept the additional risk (and, since the return is large, the risk must be just as significant).
A third explanation can draw from both of the previous explanation with a focus related to the investment life cycle and borrowing constraints. Given their long investment horizons, young investors would generally prefer to invest in assets which would offer them a high expectation of return, but this would actually decrease the expectation of the premium from market beta due to their willingness to take risk. However, these young investors have a limited ability to invest, because they typically have lower levels of income earlier in their career, inopportune priorities for consumption goals (such as buying a house), and constraints on their ability to borrow through secured loans. This combination leads to an inability to take risk and prevents most young investors from building significant investments in equities. Conversely, elderly investors have shorter investment horizons and generally have less willingness and need to take risk. Since they are also in the withdrawal stage, they tend to reduce their exposure to equities as a percentage of their total assets. Overall, the result is that the risk of market beta is concentrated in middle-aged investors, who can be generally regarded as saving consumers. This group is likely to be saving for retirement, but they also have many short-term concerns, such as savings for the college education of their children. Thus, they are likely to have a higher net worth but still likely to be more risk averse than younger investors, so, as mentioned in the previous explanations, they must demand a sizable premium to be compensated for accepting the risks from market beta.
A final and quantitative explanation is that the volatility of equities is much greater than it is for a riskless asset, like the 1-month U.S. Treasury bill. For example, the annual standard deviation of the domestic market has been about 20% compared to about 3% for the 1-month U.S. Treasury bill. The risk involved with exposure to equities is also seen in periods when returns were the most negative with large drawdowns. For example, the worst 1-year return for the domestic market was -43.5% in 1931, but the 1-month U.S. Treasury bill has never experienced a loss in a calendar year. Moreover, the worst maximum drawdown for the domestic market occurred from 1929-09 to 1932-06 and lost more than 83%, while the 1-month U.S. Treasury bill returned 6% over this period - leading to underperformance of almost 90% due to the risk of market beta. For an investor to accept this risk as a possibility, they would need to obtain justification through expectations for a sizable premium.
Just to note, this risk, and any other risk, is theoretically reflected in the prices of equities through the discount rate. The price of equities is equal to the present value of expected future cash flows which is equal to the ratio of the sum of expected future cash flows and discount rate of those cash flows relative to the time horizons of those cash flows. Thus, it is the investors in aggregate who decide this discount rate through prices based on the risk they have the ability, willingness, or need to accept (in more detail, the discount rate can actually be decomposed into the real risk-free rate, expected inflation, measurement uncertainty premium, and premium from the risk of factors). It should also be noted that this assumes that the accuracy in the prediction of expected future cash flows is reliable (or, at least, the uncertainty in this prediction is insignificant relative to the magnitude of the discount rate) and, if the market is efficient, this assumption is expected to be valid through the proposal that all public and known information is already accurately reflected in the price of equities (although constantly and rapidly changing as new information, which by its nature must be random, becomes available) - in other words, there is very minimal risk in the reliability of the pricing of equities.
Therefore, the fundamental framework for market beta is robust and it reliably fulfils the qualifications as a factor. However, as a model, the CAPM was flawed and could only explain about 67% of the differences in returns of diversified portfolios. For example, comparing two portfolios with a difference in returns of 3%, the differences in exposure to market beta would be able to explain roughly 2% of the overall difference in returns. Thus, the remaining portion of the differences in returns must either be explained by luck (as a probabilistic outcome), skill (at either picking individual securities or timing the market), or other unidentified factors.
Fama-French Three-Factor Model
Over time, consistent anomalies were discovered which violated the CAPM and could not be classified as luck or skill. After various papers from Fischer Black, Rolf Banz, Sanjoy Basu, Barr Rosenberg, Kenneth Reid, and Ronald Lanstein, it was proposed by Eugene Fama and Kenneth French in 1992 that, in addition to market beta, exposure to additional factors of size and value further explained the differences in returns of diversified portfolios. This led to the Fama-French Three-Factor Model (FF3), which greatly improved upon the explanatory power of the CAPM and accounted for more than 90% of the differences in returns of diversified portfolios. For example, comparing two portfolios with a difference in returns of 3%, the differences in exposure to market beta, size, and value would be able to explain roughly 2.7% of the overall difference in returns. Again, the remaining portion of the differences in returns must either be explained by luck, skill, or other unidentified factors.
Size Factor
By definition, the size factor expresses the degree or sensitivity to which an asset tends to move with the difference between equities with small market capitalizations compared to equities with large market capitalizations. Based on the CRSP 1-10 Index, equities with large market capitalizations are defined to be within the 1st to 5th deciles of market capitalization (larger half), while equities with small market capitalizations are defined to be within the 6th to 10th deciles of market capitalization (smaller half). The expected component of return of an asset due to its exposure to the size factor is equal to the product of the sensitivity of the asset to the size factor and the difference between the annual average return of equities with small market capitalizations and annual average return of equities with large market capitalizations (referred to as Small Minus Big (SMB)).
So, it can be said that the return from equities with small market capitalizations offers a premium above the return from equities with large market capitalizations - in other words, this is the return which would be realized by a long/short portfolio (it should be noted that the use of a long/short portfolio eliminates effects from other possible factors, where it is assumed that any effects affect both the long portion and short portion equivalently). From 1927 to 2015 and in the domestic market, when subtracting the annual average return of equities with large market capitalizations from the annual average return of equities with small market capitalizations, the premium from the size factor has been 3.3% as compensation.
Considering the persistence of the size factor, it has been found that, from 1927 to 2015 and in the domestic market, the premium has been positive in 59% of calendar years and, over longer periods, the odds of outperformance by equities with small market capitalizations of equities with large market capitalizations become even greater. As with other factors, there is always still the chance of an adverse outcome with a negative premium no matter how long the period. Over this same period, the Sharpe ratio of the size factor was 0.24 for equities in the domestic market. However, alternate definitions of the size factor can have notably higher premiums (dividing equities with more resolution than halves) and combining the size factor with other factors can also be very effective.
| Factor | Rolling Periods From 1927 To 2015 | ||||
|---|---|---|---|---|---|
| 1 Year | 3 Year | 5 Year | 10 Year | 20 Year | |
| Size | 59% | 66% | 70% | 77% | 86% |
Considering the pervasiveness of the size factor, this can be evaluated by comparing the return of the MSCI EAFE Index (return from equities with large and middle market capitalizations from developed markets (Europe, Australasia, and Far East) excluding the United States and Canada) against the return of the DFA International Small-Cap Index (return from equities with small market capitalizations from developed markets excluding the United States). From 1970 to 2015, the MSCI EAFE Index returned 9.5%, while the DFA International Small-Cap Index returned 14.5%. Additionally, it has been found that, from 1982 to 2014, there is evidence of a premium from equities with small market capitalizations in each of the 14 out of 15 European markets considered, as well as in the overall consideration of these 15 European markets. Finally, for emerging markets, the Fama-French Emerging Markets Index returned 10.4%, while the Fama-French Emerging Markets Small-Cap Index return 11.7% from 1989 to 2015. Thus, the premium from the size factor has been reasonably pervasive throughout modern history across different countries.
Considering the investability of the size factor, it is evident that equities with small market capitalizations are less liquid and potentially more expensive to trade as a consequence. However, by examining the results of funds, it is possible to evaluate whether these funds have been able to successfully capture the size factor. For example, the Bridgeway Ultra Small-Cap Market Fund (BRSIX) is passively managed, invests in equities with micro market capitalizations (where trading costs are potentially a significant hurdle), has an inception date on 1997-07-31, and realized a return of 10.3% from 1997-08 to 2015-12, which successfully matched the CRSP 10 Index and outperformed the CRSP 9-10 by 0.8%. In addition, the DFA U.S. Micro-Cap Fund (DFSCX) has an expense ratio of 0.52% and realized a return of 11.8% from 1982-01 to 2015-12, which actually outperformed the Fama-French U.S. Small-Cap Index by 0.2% and outperformed the Russell 2000 Index by 1.7%, while the DFA Small-Cap Fund (DFSTX) has an expense ratio of 0.37% and realized a return of 10.4% from 1992-04 to 2015-12, which underperformed the Fama-French U.S. Small-Cap Index by 0.05% and outperformed the Russell 2000 Index by 1.4%.
A similar approach can also be applied for funds in international and emerging markets. For example, the DFA International Small-Cap Fund (DFISX) has an expense ratio of 0.54% and realized a return of 9.1% from 1999-01 to 2015-12 (actually realized a return of 6.7% from 1996-10 to 2015-12, but a comparable index is not available from this date), which outperformed the MSCI EAFE Small-Cap Index by 1.1%. In addition, the DFA Emerging Markets Small-Cap Fund (DEMSX) has an expense ratio of 0.72% and realized a return of 10.8% from 1998-04 to 2015-12, which outperformed the Fama-French Emerging Markets Small-Cap Index by 1.6% and MSCI Emerging Markets Small-Cap Index by 4.1%. Thus, although the fees are fairly high, it is possible to invest in a portfolio which reliably targets the size factor. (It should be noted that deviations from the returns of indices may be a result of the definitions of market capitalization used in the indices, deeper exposure to the size factor, noise due to shorter periods, or targeting of additional factors in the funds).
Considering the intuitiveness of the size factor, it is possible to formulate risk-based reasons for an investor to expect an excess return through a premium from the size factor. Relative to equities with large market capitalizations, equities with small market capitalizations can typically be characterized by greater leverage, smaller capital base, restrictive access to capital, greater vulnerability to variations in credit conditions, reduced ability to adapt to economic adversity, less depth of management, higher volatility of earnings, lower levels of profitability, greater uncertainty of cash flow, and less liquidity (more expensive to trade). However, it has been found that, when the size factor is isolated, there is only a significant premium in periods of expansionary monetary policy, but the size factor is not statistically significant in periods of restrictive monetary policy. Furthermore, it has been identified that equities with small market capitalizations grow faster than equities with large market capitalizations when economic indicators are positive (after economic indicators have been negative), but equities with small market capitalizations do poorly compared to equities with large market capitalizations when economic indicators are negative (during recessions when the marginal utility of consumption is highest) - in other words, equities with small market capitalizations are more procyclical than the broad market. Thus, it is logical that the size factor would vary across economic cycles and can be seen as compensation for accepting the risk from economic cycles.
An expected and quantitative explanation is that the volatility of equities with small market capitalizations is greater than it is for equities with large market capitalizations. For example, in the United States, the annual standard deviation of equities with small market capitalizations has been about 30% compared to about 20% for equities with large market capitalizations. The risk involved with exposure to equities with small market capitalizations is also seen in periods when returns were the most negative with large drawdowns. For example, during the Great Depression in 1931, the CRSP 1-5 Index lost -43.3% while the CRSP 6-10 Index lost -50.2%; during the recession and bear market in 1973 through 1974, equities with large market capitalizations lost -39.2% while equities with small market capitalizations lost -53.1%; and during the Financial Crisis in 2008, equities with large market capitalizations lost -36.5% while equities with small market capitalizations lost -38.7%. For an investor to accept this risk as a possibility, they would need to obtain justification through expectations for a reasonable premium.
However, it should be highlighted that there is abnormally poor performance of equities with small market capitalizations which can be classified as growth equities (particularly equities which have poor profitability and make relatively large investments). Over the period from 1927 to 2015 where equities with large market capitalizations returned 9.8% and equities with small market capitalizations returned 11.8%, it is found that growth equities with small market capitalizations returned only 8.7% while exhibiting higher volatility with a standard deviation of 32% in the domestic market. Ideas from behavioural finance provide possible explanations and reason that this may exist due to investors having a preference or taste for lottery tickets. Essentially, it has been found that investors have a preference for assets which exhibit positive skewness, in which returns more than the mean are fewer but farther from it than returns less than the mean - such investments provide the small chance of a huge payoff and investors find this small possibility attractive (similar observations can be made for initial public offerings, private equity, and distressed companies). The result is that positively skewed assets tend to be overpriced and, subsequently, earn negative average excess returns due to the risk which is not being compensated. However, in theory for ideal markets, it would be expected for these anomalies to be arbitraged away by investors who do not have this same preference and are not willing to hold these assets (or willing to short these assets). Unfortunately, in real markets, there are limits to arbitrage (regulations against short positions, interest costs of borrowing, limited supply of shares for shorting, re-calling of shorts before strategies are compensated, triggering of liquidation in the short-term, and potentially unlimited losses from shorting) and anomalies can persist in balance with the limits (it should be noted that this does not necessarily contradict the Efficient Market Hypothesis, but merely adapts the model for real markets to need to include additional information which would reduce to zero in an ideal market).
It should also be noted that there may be reasons to question the robustness of the size factor, such as the decline in the premium from the size factor since its initial publications and after accounting for greater exposure to market beta. However, when filtering out equities with small market capitalizations which can be classified as growth equities, its merits are seen, especially in combination with other factors, like the value factor and momentum factor. For example, when controlling for the quality factor, it has been found that equities with a very poor quality are typically very small, have low average returns, and tend to be highly illiquid, such that the returns of these equities chiefly explain the sporadic performance of the size factor - in other words, controlling for the quality factor, equities with small market capitalizations and high quality outperform equities with large market capitalizations and high quality, while equities with small market capitalizations and low quality outperform equities with large market capitalizations and low quality, but the standard effect from the size factor using a blend of equities with low quality and high quality suffers from a size-quality composition effect. Thus, the premium from the size factor would be notably higher except that these equities tend to be of lower quality, but the premium which emerges from the size factor after controlling for the quality factor is stable through time, robust to its specification, more consistent across seasons and sectors, not concentrated in equities with micro market capitalizations, not captured by an illiquidity premium, and results in a monotonic relationship between size deciles and excess returns (excess returns steadily increase moving from equities with the largest market capitalizations (actually negative for the largest equities) to equities with the smallest market capitalizations).
Value Factor
As mentioned, the value factor was proposed in the FF3. By definition, the value factor expresses the degree or sensitivity to which an asset tends to move with the difference between equities with cheap valuations (value) compared to equities with expensive valuations (growth). The valuations of an asset can be measured through a fundamental ratio, such as the book-to-market (book-to-price) ratio which compares the book value to the market value, where a cheap valuation will have a high book-to-market ratio and expensive valuation will have a low book-to-market ratio. Conventionally, using the fundamental ratio, equities with expensive valuations are defined to be within the upper 30% of equities with the lowest book-to-market ratios, while equities with cheap valuations are defined to be within the lower 30% of equities with the highest book-to-market ratios (middle portion is considered to be core, blend, or neutral). The expected component of return of an asset due to its exposure to the value factor is equal to the product of the sensitivity of the asset to the value factor and the difference between the annual average return of equities with cheap valuations and annual average return of equities with expensive valuations (referred to as High Minus Low (HML)).
So, it can be said that the return from equities with cheap valuations offers a premium above the return from equities with expensive valuations - in other words, this is the return which would be realized by a long/short portfolio (it should be noted that the use of a long/short portfolio eliminates effects from other possible factors, where it is assumed that any effects affect both the long portion and short portion equivalently). From 1927 to 2015 and in the domestic market, when subtracting the annual average return of equities with expensive valuations from the annual average return of equities with cheap valuations, the premium from the value factor has been 4.8% as compensation.
It should also be noted that the value factor is robust to various definitions with a fundamental ratio. Other metrics which can be used to distinguish cheap valuations and expensive valuations include earnings-to-price (high for value equities versus low for growth equities), cash flow-to-price (high for value equities and low for growth equities), or sales-to-price (high for value equities and low for growth equities). For example, from 1952 to 2015 and in the domestic market, the premium from the value factor was 4.1% with a t-statistic of 2.4 as measured by book-to-market ratios, 4.7% with a t-statistic of 2.9 as measured by cash flow-to-price ratios, and 6.3% with a t-statistic of 3.4 as measured by earnings-to-price ratios. Likewise, from 1980-01 to 2014-06 and in developed markets, the premium from the value factor was 6.1% as measured by book-to-market ratios, 8.0% as measured by cash flow-to-price ratios, and 7.3% as measured by earnings-to-price ratios. Therefore, it is unlikely for these results for the value factor to be the result of data mining.
Considering the persistence of the value factor, it has been found that, from 1927 to 2015 and in the domestic market, the premium has been positive in 63% of calendar years and, over longer periods, the odds of outperformance by equities with cheap valuations of equities with expensive valuations become even greater. As with other factors, there is always still the chance of an adverse outcome with a negative premium no matter how long the period. Over this same period, the Sharpe ratio of the value factor was 0.34 for equities in the domestic market.
| Factor | Rolling Periods From 1927 To 2015 | ||||
|---|---|---|---|---|---|
| 1 Year | 3 Year | 5 Year | 10 Year | 20 Year | |
| Value | 63% | 72% | 78% | 86% | 94% |
Considering the pervasiveness of the value factor, this can be evaluated by comparing the return of the Fama-French International Growth Index (return from equities with expensive valuations from developed markets excluding the United States) against the return of the Fama-French Value Index (return from equities with cheap valuations from developed markets excluding the United States). From 1975 to 2015, the Fama-French International Growth Index returned 8.6%, while the Fama-French International Value Index returned 13.8%. Additionally, it has been found that, from 1982 to 2014, there is evidence of a premium from equities with cheap valuations in each of the 15 out of 15 European markets considered, as well as in the overall consideration of these 15 European markets - it ranged from 1.5% in Ireland to 7.3% in Sweden to 4.9% on average (during this period, the premium from the value factor was 4.5% in the United States and 6.0% in developed markets excluding the United States with the strongest performance in Japan). Finally, a similar comparison can be made with the return of the Fama-French Emerging Markets Growth Index against the return of the Fama-French Value Index. From 1989 to 2015, the Fama-French Emerging Markets Growth Index returned 9.3%, while the Fama-French Emerging Markets Value Index returned 13.0%.
Considering the investability of the value factor, it is evident that funds focussing on equities with cheap valuations have existed for extended periods of time. By examining the results of funds, it is possible to evaluate whether these funds have been able to successfully capture the value factor. For example, the DFA U.S. Large-Cap Value Fund (DFLVX) has an expense ratio of 0.27% and realized a return of 9.8% from 1993-03 to 2015-12, which actually outperformed the MSCI U.S. Prime Market Value Index by 0.5% and outperformed the Russell 1000 Value Index by 0.4%. In addition, the DFA U.S. Small-Cap Value Fund (DFSVX) has an expense ratio of 0.52% and realized a return of 11.6% from 1993-04 to 2015-12, which outperformed the MSCI U.S. Small-Cap Value Index by 1.0% and outperformed the Russell 2000 Value Index by 9.7%.
A similar approach can also be applied for funds in international and emerging markets. For example, the DFA International Value Fund (DFVIX) has an expense ratio of 0.25% and realized a return of 5.9% from 1994-06 to 2015-12, which outperformed the MSCI EAFE Value Index by 0.8%, while the DFA International Small-Cap Value Fund (DISVX) has an expense ratio of 0.69% and realized a return of 7.4% from 1995-01 to 2015-12, which matched the MSCI EAFE Small-Cap Value Index. In addition, the DFA Emerging Markets Value Fund (DFEVX) has an expense ratio of 0.56% and realized a return of 9.8% from 1997-01 to 2015-12, which outperformed the MSCI Emerging Markets Value Index by 4.1%. Thus, although the fees are fairly high, it is possible to invest in a portfolio which reliably targets the value factor. (It should be noted that deviations from the returns of indices may be a result of the definitions of value and growth used in the indices, deeper exposure to the value factor, noise due to shorter periods, or targeting of additional factors in the funds).
Considering the intuitiveness of the value factor, there is a debate whether there are risk-based reasons or behavioural-based reasons for an investor to expect an excess return through a premium from the value factor. As expected, the risk-based reasons argue that the premium exists due to additional risk which is being compensated. The behavioural-based reasons argue that the premium exists due to an anomaly and as a result of persistent pricing errors in contradiction to the Efficient Market Hypothesis. These behavioural arguments are based on emotional reactions from investors, such as investors naively extrapolating past growth and overpricing growth equities which leads to expensive valuations, while overreacting to adverse news and underpricing value equities which leads to cheap valuations. However, it can be argued that there are influences from both risk-based reasons and behavioural-based reasons, although it should be noted that, if the primary reasons are based on behaviour, it would be expected for these reasons to diminish or be arbitraged away by investors who do not have this same preference - in other words, there should be no expectation for compensation in the form of a premium.
With regard to the risk-based reasons, equities with cheap valuations are often associated with distress which can be measured in the form of a reduction in dividends, high ratio of debt-to-equity, and high standard deviation of earnings - it should be emphasized that these equities have cheap valuations for a reason due to their distress. It has been found that, when these characteristics of distress were present, there were associations with portfolios of valuations with high book-to-market ratios and subsequent returns were higher than the returns from equities with low book-to-market ratios. This shows that value equities have a reason for being priced cheaply, which is due to additional risk from distress and substantial earnings risk. Further investigation has also shown an asymmetric risk of value equities, where, during economic cycles, value equities are much riskier than growth equities in recessions but only moderately less risky than growth equities under normal circumstances. In this way, it has been identified that equities with cheap valuations deliver higher performance than equities with expensive valuations when economic indicators are positive (after economic indicators have been negative), but equities with cheap valuations have poorer performance than equities with expensive valuations when economic indicators are negative (during recessions when the marginal utility of consumption is highest) - in other words, value equities are more procyclical than the broad market. When this is combined with a high aversion to risk, especially when this risk can be expected during low economic cycles (with the risks of the loss of income from employment or poor performance from their business), the result is a sizable and persistent premium from the value factor. Thus, it is evident that the value factor would vary across economic cycles and can be seen as compensation for accepting the risk from economic cycles.
Likewise, a positive relationship has been found between leverage and the returns of equities with cheap valuations compared to equities with expensive valuations. This is expected as a high level of leverage directly increases risk, specifically the risk of defaults and bankruptcy, where the book-to-market ratio (and other fundamental ratios of valuation) adds incremental explanatory power about the riskiness of the assets associated with the corresponding equities. Along with this high level of leverage of equities with cheap valuations, the high level of uncertainty of cash flow when under distress contributes to the risk of these equities to be able to pay back their debt. However, as the result, it is only rational to expect a premium from the value factor, as would be demanded as compensation by rational investors due to the increased exposure to additional risks - in other words, in order for investors to accept the additional risks of equities with cheap valuations instead of equities with expensive valuations, they would require a higher expected return.
In more detail, it is possible to examine the relationship between the value factor and macroeconomic variables, such as industrial production, money supply, and interest rates. As alluded to, in times of economic expansion when industrial production rises, value equities become less risky relative to growth equities and, consequently, the prices of value equities increase more than the prices of growth equities as discount rates are decreased. The result is that the spread between the equities with high book-to-market ratios and equities with low book-to-market ratios narrows, which corresponds with decreased expectations for the premium from the value factor in the future. In times of economic retraction, value equities become riskier relative to growth equities and, consequently, the prices of value equities decrease faster than the prices of growth equities as discount rates are increased. The result is that the spread between the equities with high book-to-market ratios and equities with low book-to-market ratios widens, which corresponds with increased expectations for the premium from the value factor in the future. A similar negative relationship exists between the premium from the value factor and money supply, where, following an increase in the money supply, the prices of all equities increase, but the prices of value equities tend to increase more than the prices of growth equities, such that the premium from the value factor declines. Conversely, following a decrease in the money supply, the prices of all equities decrease, but the prices of value equities tend to decrease more than the prices of growth equities, such that the premium from the value factor improves. There is also a positive relationship between the premium from the value factor and interest rates, where, as long-term interest rates rise, the prices of all equities decrease, as equities become less attractive relative to bonds, with the prices of value equities decreasing faster than the prices of growth equities, which leads to an increasing premium from the value factor. Conversely, when long-term interest rates fall, the prices of value equities increase faster than the prices of growth equities, which leads to a decreasing premium from the value factor. In each case, the premium from the value factor is realized due to the transition between the macroeconomic variables (however, it is not necessarily realized during the transition between the macroeconomic variables, as the expectation and likelihood of these macroeconomic variables in the future are already included in current prices) and such that the favourable effects are more significant than the adverse effects (otherwise there would be no premium from the value factor in the long-term over multiple economic cycles). All of this is somewhat expected, because, if value equities offered equivalent performance to growth equities during most periods and better performance in some periods, then there would be no risk involved in holding these value equities, as it would be guaranteed for them to match or exceed the performance of growth equities - in other words, it must be expected for value equities to have significant periods of underperformance relative to growth equities.
An expected and quantitative explanation is that the volatility of equities with cheap valuations is greater than it is for equities with expensive valuations. For example, from 1927 to 2015, the Fama-French Large-Cap Index (including value, neutral, and growth) had an annual standard deviation of 19.7% compared to the Fama-French Large-Cap Value Index (excluding utilities) which had an annual standard deviation of 26.8%, while the Fama-French Large-Cap Growth Index (excluding utilities) had an annual standard deviation of 21.5%. Likewise, from 1927 to 2015, the Fama-French Small-Cap Index (including value, neutral, and growth) had an annual standard deviation of 30.4% compared to the Fama-French Small-Cap Value Index (excluding utilities) which had an annual standard deviation of 33.4%, while the Fama-French Small-Cap Growth Index (excluding utilities) had an annual standard deviation of 33.4%. For an investor to accept this risk as a possibility, they would need to obtain justification through expectations for a reasonable premium.
Examining the ideas from the behavioural-based reasons, a popular explanation is that investors are systematically too optimistic in their expectations for the performance of growth equities and too pessimistic in their expectations for the performance of value equities. This may be due to anchoring, confirmation bias, or investors confusing familiarity with safety, where the publicity of growth equities tend to make them common holdings in portfolios, even though they are not necessarily better investments - ultimately, this leads to expensive valuations from a higher price being needed to own these overweighted equities (in this way, growth equities may be referred to as glamour equities). This mispricing hypothesis has been investigated by identifying potential ex-ante biases, comparing expectations implied by pricing multiples against the strength of fundamental ratios, and examining profitability, change in financial leverage and liquidity, and change in operational efficiency. It was found that among equities where the expectations implied by their current value or growth classification were consistent with the strength of their fundamentals the value or growth effect in realized returns is statistically and economically indistinguishable from zero; returns to traditional value or growth strategies are concentrated among those equities where the expectations implied by their current value or growth classification are incongruent ex-ante with the strength of their fundamentals; and returns to this incongruent value or growth strategy are robust and significantly larger than the average return generated by a traditional value or growth strategy. In short, the premium from the value factor was concentrated among the subset of equities where expectations implied by valuations were not aligned with the strength of the fundamental ratios of the equities - in other words, there is mispricing of these equities.
Another explanation comes from the reaction to new information about equities being realized, where investors are likely to underreact to information about growth equities which contradicts their beliefs about growth prospects or reflects the effects of mean reversion in performance. Conversely, since value equities are inherently more distressed and underweighted or neglected, investors tend to react too pessimistically to information about value equities with attitudes changing and reflections in fundamental ratios occurring too slowly. This is related to anchoring as a form of cognitive bias in which investors place an inordinate amount of importance on certain characteristics or attributes and inappropriately weight the influence of subsequent information to support their initial assessment (favour data which agrees and dismiss data which disagrees with their initial reference point). This is particularly relevant when associated with fundamental ratios, where it has been found that investors fail to adjust their future expectations of these fundamental ratios sufficiently according to mean reversion - in other words, investors expect equities with expensive valuations to be able to continue without negative adjustment, even though the evidence suggests that future performance is likely to be disappointing as these equities revert to the mean. This aligns with the conclusions that equities with low book-to-market ratios (or high price-to-earnings ratios) and weak fundamentals are persistently overvalued and equities with high book-to-market ratios (or low price-to-earnings ratios) and strong fundamentals are persistently undervalued.
A third explanation centred around behaviour is loss aversion, where investors have the tendency to be more sensitive (by placing greater utility) on losses than on gains. For example, the pain realized from a loss of $100 is much greater than the pleasure received from a gain of $100. This leads to investors requiring more than even odds to accept an even bet - in other words, the average individual would not be willing to bet on the outcome of a coin flip unless they received odds greater than 2:1 to make the bet. Typically, the greater the amount at stake, the greater the odds need to be for an investor to accept the bet - alternatively, the less the amount at stake, the less the odds need to be for an investor to accept the bet (counter-intuitive examples are lottery tickets, where small amounts are at stake and the odds are extremely poor but the return is incredibly high - in a way, this actually leads to a preference for these bets, since the level to accept the bet is so low). The degree to which an investor experiences loss aversion is also related to whether the investor has recently experienced gains or losses, where a loss which follows after prior gains is less painful due to cushioning from those earlier gains (in a way, this is due to the investor viewing the loss as coming from the gains and not the original amount at stake). Conversely, a loss which follows after prior losses is more painful due to the setback from those earlier losses. With regard to equities, growth equities tend to have good recent performance, as evidenced by their current high prices, such that investors become less concerned about any subsequent losses - as a result, investors are willing to hold growth equities even though the future expected returns are now lower. On the other hand, value equities tend to have poor recent performance, as evidenced by their current distress, such that investors become very concerned about any further losses - as a result, investors are not willing to hold value equities even though the future expected returns are now higher.
Carhart Four-Factor Model
In 1997, after research from Narasimhan Jegadeesh and Sheridan Titman, Mark Carhart proposed the momentum factor as an addition to the FF3 to increase the explanatory power and accounted for more than 95% of the differences in returns of diversified portfolios. The original definition of momentum is cross-sectional momentum and it measures relative performance by comparing the return of an asset relative to the returns of other assets within the same asset class (even if all the assets happened to have risen in value and have positive recent returns, a cross-sectional momentum strategy would still short the assets with the lowest recent returns and hold the assets with the highest recent returns).
There is an alternative definition of momentum known as time-series momentum or trend-following and it measures absolute performance by following the trend of an asset with respect to its own historic performance and other asset classes - in other words, it would only be long assets which have been rising in value and short assets which have been declining in value (if all the assets happened to have risen in value and have positive recent returns, a time-series momentum strategy would short none of the assets). This alternate definition was only recently proposed by Tobias Moskowitz, Yao Hua Ooi, and Lasse Heje Pedersen in 2012 (although trend-following has previously been an informal investing style used in trading and technical analysis). In either case, there is a premium from the momentum factor.
Cross-Sectional Momentum Factor
By definition, the momentum factor expresses the degree or sensitivity to which an asset tends to move with the difference between equities with positive recent performance compared to equities with negative recent performance. This is due to the tendency for assets which have performed well in the recent past to continue to perform well in the short-term future and for assets which have performed poorly in the recent past to continue to perform poorly in the short-term future. The momentum of an asset is defined as the last 12 months of returns excluding the most recent month (as the most recent month tends to show a reversal, which some have attributed to trading effects). Based on this definition, equities with high momentum are defined to be within the upper 30% of equities with the highest recent returns, while equities with low momentum are defined to be within the lower 30% of equities with the lowest recent returns (middle portion is considered to be core, blend, or neutral). The expected component of return of an asset due to its exposure to the momentum factor is equal to the product of the sensitivity of the asset to the momentum factor and the difference between the annual average return of equities with high momentum and annual average return of equities with low momentum (referred to as Up Minus Down (UMD)).
So, it can be said that the return from equities with high recent returns offers a premium above the return from equities with low recent returns - in other words, this is the return which would be realized by a long/short portfolio (it should be noted that the use of a long/short portfolio eliminates effects from other possible factors, where it is assumed that any effects affect both the long portion and short portion equivalently). From 1927 to 2015 and in the domestic market, when subtracting the annual average return of equities with low recent returns from the annual average return of equities with high recent returns, the premium from the momentum factor has been 9.6% as compensation.
It should be noted that the momentum factor is robust to various definitions of metrics. For example, using the last 6 months or 9 months of returns excluding the most recent month has also produced a similar premium from the momentum factor. Additionally, there are other measurements of momentum in the form of fundamental momentum using changes in fundamental ratios as the primary metric rather than price, such as earnings momentum, changes in profit margins, and changes in sentiment. Therefore, it is unlikely for these results for the momentum factor to be the result of data mining. However, it should be noted that looking at longer periods tends to show a similar partial reversal as is seen when looking at very short periods.
Considering the persistence of the momentum factor, it has been found that, from 1927 to 2015 and in the domestic market, the premium has been positive in 73% of calendar years and, over longer periods, the odds of outperformance by equities with high recent returns of equities with low recent returns become even greater. As with other factors, there is always still the chance of an adverse outcome with a negative premium no matter how long the period (just because a premium from the momentum factor was always realized over periods of 20 years historically does not mean that it is guaranteed to always be realized over periods of 20 years in the future). Over this same period, the Sharpe ratio of the momentum factor was 0.61 for equities in the domestic market.
| Factor | Rolling Periods From 1927 To 2015 | ||||
|---|---|---|---|---|---|
| 1 Year | 3 Year | 5 Year | 10 Year | 20 Year | |
| Momentum | 73% | 86% | 91% | 97% | 100% |
Considering the pervasiveness of the momentum factor, the momentum factor has been found to be present in over 40 countries, including the United States, United Kingdom, Europe, and Japan, and more than 12 asset classes, including equities, bonds, currencies, and commodities. The premium from the momentum factor was also examined in 23 markets grouped into North America, Japan, Asia Pacific, and Europe, where a high premium was found ranging from 0.64% per month in North America with a t-statistic of 1.9 to 0.92% per month in Europe with a t-statistic of 3.4, except in Japan which did not realize a statistically significant premium, such that the premium for the world was 0.62% per month with a t-statistic of 2.3. Interestingly, this premium is larger for equities with small market capitalizations, at 0.82% per month with a t-statistic of 3.1, compared to equities with large market capitalizations, at 0.41% per month with a t-statistic of 1.4. The most widespread study of the momentum factor considered 47 country equity indices, 43 government bond indices, 48 currencies, 76 commodities, 301 global sectors, and 34,795 equities in the domestic market from 1800 to 2014. The results found that the premium from the momentum factor was consistently significant within each asset class and across all of the asset classes, in which the premium ranged from 0.57% per month with a t-statistic of 6.8 for country equity indices, 0.13% per month with a t-statistic of 2.3 for government bond indices, 0.51% per month with a t-statistic of 9.6 for currencies, 0.36% per month with a t-statistic of 6.6 for global sectors, and 0.51% per month with a t-statistic of 6.0 for equities in the domestic market.
Considering the investability of the momentum factor, there are concerns for the practicality and applicability of the momentum factor after costs due to its prerequisite for frequent trading and high turnover, especially after tax considerations. For example, both the AQR Large-Cap Momentum Style Fund (AMOMX) and AQR Small-Cap Momentum Style Fund (ASMOX) have annual turnovers around 80%. However, in the current modern environment, transaction costs for trading have decreased and assets have become more liquid, so it is possible to implement robust and scalable strategies which target the momentum factor. An indirect, although less impactful, method of incorporating the momentum factor into a portfolio could be to delay the selling of equities with high recent performance, delay the buying of equities with low recent returns, advance the selling of equities with low recent returns, and advance the buying of equities with high recent returns (occasionally, this may be at the expense of some staleness in the signals on which the strategies are based). Even in a long-only portfolio, this can be useful, as it has been found that being underweight an asset relative to the market is economically similar to being short the asset (albeit with the constraint that the largest underweight can only be as large as the weight of the asset in the market - in a sense, including short positions in a portfolio is simply underweighting an asset by more than this constraint).
Considering the intuitiveness of the momentum factor, most of the evidence focuses on behavioural-based reasons for an investor to expect an excess return through a premium from the momentum factor. These reasons are focused on either the underreaction or overreaction by investors as new information becomes available. For example, underreaction may result on aggregate due to new information, such as corporate earnings or dividend announcements, travelling slower than expected to be reflected into prices, while overreaction may result from investors who chase returns and create a feedback mechanism which drives prices further away from fundamentals. It has been proposed through limited attention bias that these reactions are evident due to investors reacting differently when faced with a series of small and gradual changes or continuous information (typically leads to underreaction) compared to large and dramatic changes or discrete information (typically leads to overreaction) - even when the sum of the changes has the same cumulative impact. In either case, it would be expected for prices to eventually be corrected in the long-term, but the period of mispricing in the short-term is sufficient for the momentum factor to be realized, where there is a monotonic increase in the premium from the momentum factor for equities with discrete information compared to equities with continuous information.
Another explanation centred around behaviour is the disposition effect, where investors tend to sell winning investments too quickly to realize gains but hold on to losing investments too far in the hope of breaking even. In other words, the disposition effect creates an artificial obstacle, where, when favourable news is announced, the price of an asset does not immediately rise to the expected value reflected by its fundamentals due to premature selling (with a possible lack of buying), and, when adverse news is announced, the price of an asset does not immediately fall to the expected value reflected by its fundamentals due to reluctant selling (with a possible excess of buying). Ultimately, this leads to distortions in prices due to mispricing preventing the reflection of accurate prices and, due to the limits to arbitrage, the prices only converge or reverse slowly to the accurate prices.
There is tentative evidence in the form of risk-based reasons for an investor to expect an excess return through a premium from the momentum factor. However, it is somewhat counter-intuitive, as the risk of an asset would have to increase after positive returns or decrease after negative returns have been realized. An accepted explanation is that equities with high recent returns face greater risk compared to their previous prices due to their growth prospects having now been identified as more risky (face higher cost of capital and more uncertain cash flows), while equities with lower recent returns face less risk compared to their previous prices due to their growth prospects having now been identified as less risky (face a lower cost of capital and less uncertain cash flows). In other words, after prices have begun to rise due to favourable news, there is more risk that the actual growth is uncertain and does not materialize to justify the higher prices with greater possibilities for disappointing cash flow (which would then, conversely, justify even higher prices in the present due to the higher risk), while, after prices have begun to fall due to adverse news, there is less risk that the actual growth is uncertain and does not materialize to justify the lower prices with greater possibilities for sufficient cash flow (which would then, conversely, justify even lower prices in the present due to the lower risk). This feedback mechanism could then continue in a series but reduce in impact with each iteration until successively diminished over time. Nonetheless, unfortunately, these arguments are comparatively deficient and may lack support.
From a different perspective, a quantitative explanation for such a high premium from the momentum factor is that the momentum factor has a very high excess kurtosis of 13% and pronounced left-skew of -2.5%. In other words, the returns smaller than the mean are fewer but farther from it compared to the returns larger than the mean which are numerous but close to it. As a consequence, the momentum factor tends to experience very large drawdowns during periods with high volatility (although this tends to be focussed on the short-side and is less of a concern for long-only portfolios). For an investor to accept this risk as a possibility, they would need to obtain justification through expectations for a reasonable premium.
Time-Series Momentum Factor
As mentioned, time-series momentum, often referred to as trend-following, measures absolute performance by following the trend of an asset with respect to its own historic performance (referred to as Winner Minus Loser (WML)). From 1927 to 2014 and in the domestic market, by going long equities with positive returns in the last 12 months excluding the most recent month and short equities with negative returns during the last 12 months excluding the most recent month, a value-weighted strategy realized an average monthly return of 0.55% with a t-statistic of 5.28 (in more detail, there was an average monthly return of 0.69% with a t-statistic of 2.41 from 1927 to 1948, 0.47% with a t-statistic of 3.60 from 1949 to 1970, 0.62% with a t-statistic of 3.84 from 1971 to 1992, and 0.42% with a t-statistic of 1.91 from 1993 to 2014). Furthermore, from 1975 to 2014, it was found that time-series momentum produced positive risk-adjusted returns in each of the 13 out of 13 international markets considered (highest in Denmark with an average monthly return of 1.15% with a t-statistic of 5.06). Additionally, even with 16 different combinations of formation and holding periods, time-series momentum has been found to be profitable across asset classes, including equities, bonds, currencies, and commodities. Interestingly, with regard to equities, time-series momentum fully subsumes cross-sectional momentum, but cross-sectional momentum cannot capture the results of time-series momentum - for example, time-series momentum tends to avoid seasonal effects and crashes during reversals of the market which usually influence cross-sectional momentum.
As with cross-sectional momentum, the intuitiveness of the time-series momentum is primarily related to behavioural-based reasons through underreaction with agreement with the previous explanations involving a series of small and gradual changes or continuous information and the disposition effect creating an artificial obstacle to changes in the available information. This is in alignment with evidence, where the highest premium from the momentum factor is realized in equities with small market capitalizations at 0.78% per month with a t-statistic of 5.52, since new information tends to diffuse more slowly compared to equities with large market capitalizations, which generate a premium from the momentum factor of 0.47% per month with a t-statistic of 4.33. There are also extra behavioural-based reasons with possible links, such as anchoring, herding, confirmation bias, overconfidence, and trading activity of participants in the market who are not seeking a profit and have other intentions, like central banks, government interventions, and corporate hedging programs. However, due to the recent proposition of time-series momentum, there are no funds with extended and proven historic performance, but funds are available for investment, such the AQR Managed Futures Strategy Fund (AQMIX), although there may be noteworthy concern for the future persistence of time-series momentum as more funds try to formally capture its premium. Thus, the momentum factor considered from the perspective of time-series momentum has been persistent, pervasive, robust, (possibly) investable, and intuitive. Curiously, a combination of cross-sectional momentum and time-series momentum has shown promising results.
Profitability And Quality
Although intuitively alluded to by Benjamin Graham, David Dodd, and other investors, in 2013, Robert Novy-Marx proposed the profitability factor as an addition to the market beta, size, value, and momentum factors to increase the explanatory power and account for more than 95% of the differences in returns of diversified portfolios. In a broader context and with research from Cliff Asness, Andrea Frazzini, and Lasse Pedersen, the profitability factor can be expanded into a quality factor, where the quality of an asset is measured with a broader definition with the inclusion of profitability, growth, and management.
Profitability Factor
By definition, the profitability factor expresses the degree or sensitivity to which an asset tends to move with the difference between equities with high quality compared to equities with low quality. The quality of an asset was originally measured through the gross profits with the conventional definition as the difference between sales and the cost of the goods or services sold. Based on this definition, equities with high profitability are defined to be within the upper 30% of equities with the highest quality, while equities with low profitability are defined to be within the lower 30% of equities with the lowest quality (middle portion is considered to be core, blend, or neutral). The expected component of return of an asset due to its exposure to the profitability factor is equal to the product of the sensitivity of the asset to the profitability factor and the difference between the annual average return of equities with high profitability and annual average return of equities with low profitability (referred to as Robust Minus Weak (RMW)).
So, it can be said that the return from equities with high quality or gross profits offers a premium above the return from equities with low quality or gross profits - in other words, this is the return which would be realized by a long/short portfolio (it should be noted that the use of a long/short portfolio eliminates effects from other possible factors, where it is assumed that any effects affect both the long portion and short portion equivalently). From 1927 to 2015 and in the domestic market, when subtracting the annual average return of equities with high gross profits from the annual average return of equities with low gross profits, the premium from the profitability factor has been 3.1% as compensation (during the original period considered from 1962 to 2010 and in the domestic market, the profitability factor realized a premium of 0.31% per month with a t-statistic of 2.49).
This is somewhat surprising as profitable equities have generated significantly higher returns than unprofitable equities despite often having significantly more expensive valuations. In addition, profitable equities tend to be growth equities, although they can be value equities, and, consequently, this originally resulted in an abnormal return or alpha of 0.52% per month when a long/short portfolio based on quality was analysed with the FF3. Furthermore, the more profitable growth equities tend to have a larger market capitalization than the less profitable growth equities, but the more profitable value equities tend to have a smaller market capitalization than the less profitable value equities, so, in combination, the value and profitability factors can complement each other and provide hedging across economic cycles with reduced overall volatility. It should also be noted that the profitability factor is robust to various definitions of quality, including considerations for return on equity, earnings, cash flow, accruals, payouts, operating efficiency, or gross margins - in some instances, it has been found that cash-based operating profitability is actually a more reliable metric than the conventional gross profitability or net income-based profitability. Interestingly, alternatively measuring profitability as the ratio of gross profits to assets has roughly realized the same explanatory power as the book-to-market ratio used for the value factor.
Briefly, as mentioned, the value and profitability factors can complement each other in combination. For example, from 1962 to 2010 and in the domestic market, the value factor would have realized a premium of 0.31% per month with a standard deviation of 2.94%, while the profitability factor would have realized a premium of 0.41% per month with a standard deviation of 3.27%. However, in combination, the value and profitability factors would have realized a premium of 0.71% per month with a standard deviation of 2.89%, Sharpe ratio of 0.85, and t-statistic of 5.87. This is primarily due to the advantage realized from the value and profitability factors having a favourable correlation of -0.57 over the period - as a result, the combination of value and profitability factors never experienced a five-year period with negative returns. (Unfortunately, it is not recommended to simply focus only on equities with small market capitalizations, cheap valuations, and high quality, as the premium from this exposure may not necessarily be greater than the exposure to other combinations of factors - in other words, as an example, the greater exposure to the profitability factor from certain equities with large market capitalizations and expensive valuations may be high enough to offset the negative exposure to the size factor and value factor (for instances, \(\beta_{sz}\) of -0.1, \(\beta_{val}\) of -0.1, and \(\beta_{prf}\) of 0.5 may be just a favourable as \(\beta_{sz}\) of 0.2, \(\beta_{val}\) of 0.2, and \(\beta_{prf}\) of 0.2 in a long-only portfolio - also applicable to other combinations of factors)).
Considering the persistence of the profitability factor, it has been found that, from 1927 to 2015 and in the domestic market, the premium has been positive in 63% of calendar years and, over longer periods, the odds of outperformance by equities with high quality of equities with low quality become even greater. As with other factors, there is always still the chance of an adverse outcome with a negative premium no matter how long the period. Over this same period, the Sharpe ratio of the profitability factor was 0.33 for equities in the domestic market.
| Factor | Rolling Periods From 1927 To 2015 | ||||
|---|---|---|---|---|---|
| 1 Year | 3 Year | 5 Year | 10 Year | 20 Year | |
| Profitability | 63% | 72% | 77% | 85% | 93% |
Considering the pervasiveness of the profitability factor, this has been evaluated by considering international markets from 1990-06 to 2009-10 with results indicating the presence of the profitability factor in developed markets, including Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Hong Kong, Italy, Japan, Netherlands, New Zealand, Norway, Singapore, Spain, Sweden, Switzerland, and United Kingdom. Specifically, from 1982 to 2014, it was also found that the premium from the profitability factor has been 3.6% on average across 15 European markets, where it was only negative in Belgium and Finland (during this period, the premium from the profitability factor was 4.4% in the United States). Finally, a similar comparison can be made in emerging markets, where it has been found that, from 1998-01 to 2013-09, the premium from the profitability factor has been 6.7% (although the length of the period of consideration is partially insufficient).
Considering the investability of the profitability factor, there are no funds with extended and proven historic performance due to the recent proposition of the profitability. However, a strategy attempting to capture the profitability factor will have a low turnover and minimal trading costs. Therefore, it would be expected for realistic implementation to be possible. As mentioned, the profitability factor would be a suitable complement to a strategy developed around capturing other factors, specifically the value factor.
Considering the intuitiveness of the profitability factor, there is a conflict of characteristics between the expectations put forward from the value factor with regard to risk-based reasons. This is because, in direct contrast to the value factor, more profitability equities on an absolute basis tend to be growth equities (although not all growth equities are profitable), are often less prone to distress, and have lower leverage compared to unprofitable equities. However, since these are growth equities with profitability, there are expectations for more of their cash flow to be in the distant future and, due to the distance of these cash flows, there is uncertainty around the actual prospects of these cash flows being realized. Moreover, the opportunity of cash flow in the distant future is likely to attract increased competition which would threaten profit margins and further contribute to this uncertainty around the future cash flow. It is also possible that the opportunities undertaken by equities with high quality have the potential to be more profitable but also more insecure than opportunities undertaken by equities with low quality, which adds to the uncertainty around cash flow which will actually be realized from these opportunities in the future. Thus, this uncertainty can be seen as a risk and investors would demand a premium in order to hold these equities.
Counter-intuitively, it has been found that equities with high quality have improved returns relative to equities with low quality when economic indicators are negative compared to the return from these equities with high quality relative to equities with low quality when economic indicators are positive. In other words, equities with high quality are less susceptible to economic adversity than equities with low quality (difference in performance is due to equities with low quality experiencing more adverse effects than equities with high quality (which may still experience some adverse effects but to a lesser degree)). This is unexpected, as the marginal utility of consumption of investors increases during recessions, but the premium from the profitability factor is actually increased. Unfortunately, this makes it difficult to reconcile the profitability factor with risk-based reasons for its existence on an absolute basis.
A possible behavioural-based reason is that investors expect the performance of equities with high quality to revert to the mean faster than this performance actually does, while investors are also willing to bet on the revival to a mean of equities with low quality despite low gross profits. This mispricing is supported by findings which show that there have been differences between earnings forecasted by investors and actual earnings realized across portfolios sorted based on profitability, where it was found that there is a monotonically decreasing relationship across deciles of profitability from low quality to high quality. In other words, investors may be too pessimistic about the future performance of equities with high quality and too optimistic about the future performance of equities with low quality, which results in the expectation error from the difference between forecasted and actual earnings being less for equities with high quality (set targets too low) and greater for equities with low quality (set targets too high).
These outcomes are strange, as they indicate that investors are underweighting or, in some cases, slightly negatively weighting metrics of profitability, even though these metrics of profitability are actually reliable predictors of positive returns for equities with high quality and negative returns for equities with low quality. However, due to the limits to arbitrage, the prices only converge or reverse slowly to the accurate prices as results are realized. It also should be emphasized that this behavioural-based reasons is subtly different from the behavioural-based reasons mentioned with the value factor around growth equities, where investors naively become overly optimistic about the equities which are popular due to recent news (whether this news is encouraging or discouraging) - to emphasize, not all growth equities are profitable.
To consider the extent to which behavioural-based reasons explain the profitability factor, equities can be divided based on the difficulty to arbitrage away anomalies and level of information uncertainty. In either case, the premium from the profitability factor should be higher for equities which are more difficult to arbitrage away anomalies (greater limits to arbitrage) and higher for equities which have greater information uncertainty (more significant role from psychological biases). Using a set of standard proxies for these cases, it has been confirmed that the profitability factor is substantially stronger among equities which are more difficult to arbitrage away anomalies and equities which have greater information uncertainty - confirmed to such an extent that the profitability factor is only marginally significant or actually insignificant among equities which are more easy to arbitrage away anomalies and equities which have less information uncertainty. Essentially, the premium from the profitability factor is increased by about 1% per month among equities with smaller market capitalization, higher return volatility, higher cash flow volatility, less analyst coverage, larger analyst forecast dispersion, fewer institutional holdings, higher idiosyncratic return volatility, lower trading volume, higher bid-ask spreads, lower credit rating, higher illiquidity, and lower age. Lastly, it was found that the profitability factor is not driven by ex-post overreaction but by ex-ante underreaction to current profitability news, such that equities with high metrics of profitability are relatively underpriced and equities with low metrics of profitability are overpriced. Thus, it can be concluded that there is evidence for the profitability factor persisting due to limits to arbitrage and psychological biases which prevent the mispricing of these equities from being corrected.
Quality Factor
As mentioned, the profitability factor can be expanded into a quality factor, where the quality of an asset is measured with a broader definition. Typically, equities with high quality have low earnings volatility, high margins and profitability, high asset turnover, efficient use of capital, low financial leverage, minimal debt, low operating leverage, low fixed costs, strong balance sheet, low macroeconomic risk, and low idiosyncratic risk. The expected component of return of an asset due to its exposure to the quality factor is equal to the product of the sensitivity of the asset to the quality factor and the difference between the annual average return of equities with high quality and annual average return of equities with low quality (referred to as Quality Minus Junk (QMJ)). From 1927 to 2015 and in the domestic market, when subtracting the annual average return of equities with high quality from the annual average return of equities with low quality, the premium from the quality factor has been 3.8% as compensation. Over this same period, the Sharpe ratio of the quality factor was 0.38 for equities in the domestic market. Interestingly, exposure to the quality factor in combination with the value factor (in addition to benefiting from cheap leverage provided by insurance operations) is responsible for much of the success of Warren Buffett in outperforming the market and makes his alpha statistically insignificant.
| Factor | Rolling Periods From 1927 To 2015 | ||||
|---|---|---|---|---|---|
| 1 Year | 3 Year | 5 Year | 10 Year | 20 Year | |
| Quality | 65% | 75% | 80% | 89% | 96% |
Simplified Explanations
Since the publication of "Your Complete Guide To Factor-Based Investing, The Way Smart Money Invests Today", 1st Edition, the Fama-French Five-Factor Model (FF5) was published in 2014. This asset pricing model included factors for market beta, size, value, profitability, and investment (excludes momentum). This increased the explanatory power and accounted for more than 95% of the differences in returns of diversified portfolios. It should be noted that there are other asset pricing models, such as those from AQR Capital Management, Alpha Architect, and Q-Factor Model, but most of the factors have overlapping characteristics.
For context, the investment factor was a result of contributions from Sheridan Titman, John Wei, Feixue Xie, Gil Aharoni, Bruce Grundy, and Qi Zeng. The investment factor can be viewed as the degree or sensitivity to which an asset tends to move with the difference between equities with conservative asset growth in investment policies compared to equities with aggressive asset growth in investment policies. The asset growth can be measured through the change in book value of assets due to internal investments and returns, which suggests that equities directing profits towards major growth projects (aggressive) are likely to underperform equities directing profits towards other cash flow streams (conservative). The expected component of return of an asset due to its exposure to the investment factor is equal to the product of the sensitivity of the asset to the investment factor and the difference between the annual average return of equities with conservative asset growth and annual average return of equities with aggressive asset growth (referred to as Conservative Minus Aggressive (CMA)).
A simplified risk-based explanation can be adapted for each factor at the expense of accuracy in the formal description. This can also be linked to the pricing of equities based on the present value of expected future cash flows or, alternatively, based on the book value of equities and present value of expected residual income (valuations based on its ability to generate cash flow after all expenses have been paid and return requirements have been satisfied). To emphasize, there is a reason for a higher discount rate in the present value resulting from risk, such that only an investor willing to bare this risk (while other investors hedge against this risk) can be compensated by the higher discount rate.
- For market beta, if equities and the risk-free rate had the same expected return, there would be no reason to hold equities due to their higher volatility and, so, in order to be compensated for this volatility, a premium must be expected from market beta. This volatility is directly associated with uncertainty and the accompanying risk. This is also clearly observed with the risk-free rate being reflected in the price as a component of the discount rate.
- For the size factor, this is not directly observed in the pricing of equities. As mentioned, it could be expected for an equity with a small market capitalization to have a higher expected return than an equity with a large market capitalization due to generally have higher costs of capital and more uncertain cash flows (along with other operational obstacles), such that a premium must be expected from the size factor to compensate for this risk. However, due to the evidence, this may be due to there typically being more equities with small market capitalizations relative to equities with large market capitalizations and this allows for greater diversification in the characteristics of these equities - in other words and with regard to the value factor, there is a premium for equities with small market capitalizations and cheap valuations over equities with large market capitalizations and cheap valuations. With equivalent characteristics, there is evidence that the size factor has no further impact (as is expected from the models used for the pricing of equities).
- For the value factor, if an equity with a high book-to-market ratio and equity with a low book-to-market ratio had the same market price, it would be expected for the equity with a high book-to-market ratio to have a higher expected return than the equity with a low book-to-market ratio, as a high book-to-market ratio (low price-to-book) has a greater discount rate being applied than a low book-to-market (high price-to-book) ratio, such that a premium must be expected from the value factor to compensate for this risk. This is observed in the expected change in book value, where, if other variables are held constant constant besides the market price and discount rate, an equity with a lower market price and higher book-to-market (lower price-to-book) ratio at the same expected cash flow must have an associated risk premium in the discount rate being reflected in the market price relative to an equity with a higher market price and lower book-to-market (higher price-to-book) ratio at the same expected cash flow (otherwise there would be no reason for them to have the same market price).
- For the profitability factor, if an equity with a high quality and equity with a low quality had the same market price, it would be expected for the equity with a high quality to have a higher expected return than the equity with a low quality, as excess gross profits with higher cash flows would imply a riskier outlook discounting these gross profits compared to reduced gross profits with lower cash flows, such that a premium must be expected from the profitability factor to compensate for this risk. This is observed in the expected future earnings, where, if other variables are held constant besides expected future earnings and discount rate, an equity with higher expected future earnings at the same market price must have an associated risk premium in the discount rate being reflected in the market price relative to an equity with lower expected future earnings at the same market price (otherwise there would be no reason for them to have the same market price). It should be noted that this is on a relative basis, as opposed to the absolute basis mentioned previously.
- For the investment factor, if an equity with a conservative asset growth and equity with an aggressive asset growth had the same market price, it would be expected for the equity with a conservative asset growth to have a higher expected return than the equity with an aggressive asset growth, as a conservative asset growth (focussed on more certain income) with a lower expected change in book value would imply a riskier outlook discounting this asset growth compared to an aggressive asset growth (focussed on more uncertain growth) with a higher expected change in book value, such that a premium must be expected from the investment factor to compensate for this risk. This is observed in the expected cash flow (from investments), where, if other variables are held constant besides expected change in book value and discount rate, an equity with a lower expected change in book value (from the asset growth of investments) at the same market price must have an associated risk premium in the discount rate being reflected in the market price relative to an equity with a higher expected change in book value (from the asset growth of investments) at the same market price (otherwise there would be no reason for them to have the same market price).
Unfortunately, it is quite difficult to provide a simplified risk-based explanation for the momentum factor, whether on a cross-sectional or time-series basis, but the behavioural-based explanations are directly associated with underreaction and overreaction. However, it is still questionable whether this factor can be captured after transaction costs for trading and if it is even rational to try to capture this factor when considering taxes. Personally, if it is even targeted, the momentum factor can only be effective through incorporating it into trading which would have already been relevant - delaying the selling of equities with high recent performance, delaying the buying of equities with low recent returns, advancing the selling of equities with low recent returns, and advancing the buying of equities with high recent returns. Moreover, there is recent research indicating that the momentum factor may be the result of factor rotation.
| Factor | Rolling Periods From 1927 To 2015 | Sharpe Ratio | Ann. Premium | ||||
|---|---|---|---|---|---|---|---|
| 1 Year | 3 Year | 5 Year | 10 Year | 20 Year | |||
| Market | 66% | 76% | 82% | 90% | 96% | 0.40 | 8.3% |
| Size | 59% | 66% | 70% | 77% | 86% | 0.24 | 3.3% |
| Value | 63% | 72% | 78% | 86% | 94% | 0.34 | 4.8% |
| Momentum | 73% | 86% | 91% | 97% | 100% | 0.61 | 9.6% |
| Profitability | 63% | 72% | 77% | 85% | 93% | 0.33 | 3.1% |
| Quality | 65% | 75% | 80% | 89% | 96% | 0.38 | 3.8% |
| Diversified P | 83% | 95% | 98% | 100% | 100% | 0.96 | 5.3% |
| Diversified Q | 87% | 97% | 99% | 100% | 100% | 1.12 | 5.6% |
| Diversified P has 20% allocated to factors with profitability. Diversified Q has 20% allocated to factors with quality. | |||||||
The annual correlations between the associated factors also reveal a diversification benefit in a portfolio which attempts to capture the factors. This is due to the low, no, or even negative correlation between most of the factors. To diversify a long-only portfolio towards factors, an investor must tilt their portfolio and increase the relative percentage targeting the equities associated with the factors - in other words, their portfolio must hold more than the broad market of equities with small market capitalizations, cheap valuations, high recent returns, and high quality, while holding less than the broad market of equities with large market capitalization, expensive valuations, low recent returns, and low quality. For an example of this diversification, it is possible to consider an even allocation between each factor against an allocation to only one individual factor. The result shows a significantly increased return relative to the risk, as evident by the significantly higher Sharpe ratio, and improved odds of outperformance over rolling periods of any length. It should also be noted that it is almost always more optimal to combine strategies in a single multi-style fund to target multiple factors, rather than using separate strategies in separate funds to individually target factors.
| Market Beta | Size | Value | Momentum | Profitability | Quality | |
| Market Beta | 1.00 | 0.28 | -0.27 | -0.17 | -0.27 | -0.52 |
| Size | 0.29 | 1.00 | 0.01 | -0.12 | -0.22 | -0.53 |
| Value | -0.27 | 0.01 | 1.00 | -0.20 | 0.09 | 0.04 |
| Momentum | -0.17 | -0.12 | -0.20 | 1.00 | 0.08 | 0.30 |
| Profitability | -0.27 | -0.22 | 0.09 | 0.08 | 1.00 | 0.74 |
| Quality | -0.52 | -0.53 | 0.04 | 0.30 | 0.74 | 1.00 |
| Factor | Mean Return | Standard Deviation | Sharpe Ratio |
|---|---|---|---|
| Market Beta | 8.3% | 20.6% | 0.40 |
| Size | 3.3% | 13.9% | 0.24 |
| Value | 4.8% | 14.1% | 0.34 |
| Momentum | 9.6% | 14.1% | 0.34 |
| Profitability | 3.1% | 9.3% | 0.33 |
| Quality | 3.8% | 10.0% | 0.38 |
| Diversified P | 5.3% | 5.5% | 0.96 |
| Diversified Q | 5.6% | 5.6% | 1.12 |
| Diversified P has 20% allocated to factors with profitability. Diversified Q has 20% allocated to factors with quality. | |||
As a final note, it should again be emphasized that, with all factors, there is always still the chance of an adverse outcome with a negative premium no matter how long the period, otherwise there would be no risk associated with investing in factors as long as the investor had the ability to wait until the premium was guaranteed to be positive. Similarly, tracking error regret, due the deviation of a portfolio targetting factors from the broad market, causes investors to make the mistake of confusing ex-ante strategy with ex-post outcomes. Thus, before even considering a portfolio targetting factors, an investor must have conviction and trust the evidence behind the decisions in their plans, even though they can make the best decision possible at the time for the most reliable outcome but still not be guaranteed for this outcome to actually be the most favourable outcome (it just has the highest chance of being the most favourable outcome).