The Fama French Three Factor Model

Published Date: 02 Nov 2017

After discussing the value and momentum effect mostly in terms of investment strategies, in the following the Fama-French three-factor model [1] and Carhart’s survey of mutual fund returns including momentum [2] will be discussed. Both models mainly concentrate on stocks, although Fama and French include bonds in their 1993 study. In section 4, the multi-factor model extension across asset classes by Asness, Moskowitz and Pedersen [3] will be examined.

The three- and four-factor asset pricing models shed light on value and momentum as risk factors shaping the cross section of asset returns. The factor models attribute portfolio returns to risk drivers reducing the return ascribed to alpha and manager ability, respectively. Factor-based asset allocation popular after the 2007-2009 crisis focuses on portfolio selection along risk factors rather than asset class silos and is "a natural extension of portfolio attribution analysis" [4] . Portfolio attribution analysis was made popular by Sharpe, determining an investor’s actual asset mix by "measuring exposures to variations in returns of major asset classes" [5] and to evaluate the manager’s ability.

3.1 The Fama-French Three-Factor Model

The CAPM explains the cross section of stock returns with the assets’ correlations to the market portfolio. Returns are divided into an active (alpha) and a passive (beta) factor. For US stocks from 1941-1990, Fama and French show that the CAPM leaves a considerable part of the excess return share unexplained, resulting in high alpha values. [6] 

Since value pays a positive risk premium, according to the CAPM, value firms should exhibit higher beta values than growth firms. Nevertheless, Fama and French find the opposite to be true: the betas for value portfolios’ are less than one, while the growth portfolios’ betas are above one. [7] According to the CAPM, value stocks are not riskier than growth stocks. Consistent excess returns of the value strategy are therefore regarded as anomalies.

Examining beta, in their 1992 paper Fama and French find that the traditional calculation of beta is closely tied to size, since betas are assigned to size sorted portfolios. Beta is therefore distorted, comprising elements of size as well. Therefore, Fama and French construct 100 size and historical beta sorted portfolios, producing a much wider range of beta values in size portfolios. Immediately, homogeneous returns illustrate the weak explanatory power of beta. Afterwards, the 100 portfolios are assigned post-ranking, historical beta values, which are then used for the Fama-MacBeth cross-sectional regression. Again, size-unrelated beta has little explanatory power, even if it is the only explanatory variable. The relation between average returns and beta is more or less flat. [8] 

The Fama-French three-factor model [9] is the first model including three factors explaining the cross section of returns on a portfolio basis. Both value and size (first studied by Banz in 1981 [10] ) can diminish the explanatory power of beta. Importantly, the value measure book-to-market is not a function of beta: book-to-market ranked portfolios exhibit similar betas. [11] The Fama-French three-factor model transforms size and value into risk factors or "state variables of special hedging concern to investors" [12] . Consequently – contrary to Lakonishok, Shleifer and Vishny [13] - Fama and French focus on fundamentals explaining the size and the value effect. According to the authors, value stocks’ excess return is due to higher distress risk and not due to investors’ overreaction. Underperforming firms, thus firms with high book-to-market, are more likely to be in financial distress. [14] Similarly, also the size effect is linked to fundamental risk factors like default risk. [15] 

In the following, the Fama-French three-factor model (equation 1) and its variables will be described thoroughly. The dependent variables are the average monthly returns on the NYSE. The explanatory variables are β, size (ln ME) and value (ln BE/ME). In inclusion of value and size in the multi-factor regression cause R2 to rise substantially.

The BE/ME ratio has more explanatory power than the size effect: the spread between the portfolio with the highest and the lowest BE/ME (1.53% per month) is double the spread of the size portfolios. However, size is still significant even if value is included. [16] A possible explanation is the negative correlation (-0.26) between value and size. Concluding, small sized firms tend to have high BE/ME values, poor growth prospects and high ex ante risk premiums. [17] 

BE/ME captures the effect of the earnings yield (E/P), while E/P does not capture BE/ME. Consequently, the explanatory power of E/P is due to its link to BE/ME. Additionally, forecasting returns with E/P only makes sense if the ratio is positive, which is not the case for BE/ME. This fact is reflected in the U-shaped returns of E/P-ranked portfolios, while average returns for BE/ME-ranked portfolios steadily increase. [18] 

Moreover, BE/ME also captures the effect of leverage. Leverage is measured by A/ME (market leverage) and A/BE (book leverage). While market leverage is significantly positively related, book leverage is significantly negatively related to average returns. The difference of market and book leverage correspond to the BE/ME ratio. A high A/ME implies "involuntary, market imposed leverage" [19] . Thus, high A/ME and BE/ME ratios imply distress risk premia.

High BE/ME firms exhibit superior ex post returns, accounting for low future earnings prospects and "relative distress risk" [20] priced in the market. Thus, investors approximate high BE/ME ratios for distress risk and demand high (ex ante) premia for holding high BE/ME stocks. Since not every high BE/ME firm is distressed, value leads to higher average (ex post) returns. Thus, the Fama-French three-factor model is consistent with rational asset pricing theories. However, the authors also point out irrational explanations for value like over- and underreaction. [21] 

Summing up, Fama and French find that beta cleared of the size effect does not satisfactorily explain average returns in 1963-1990. Second, BE/ME accounts for leverage effects. Third, E/P is absorbed by the size and value. [22] 

In their 1993 paper, Fama-French transform the independent variables to factor-mimicking portfolios and include two bond factors: TERM and DEF. Instead of valuation multiples, SMB (Small Minus Big) and HML (High Minus Low) are used as explanatory variables in a multifactor regression on the cross section of stock and bond returns. SMB and HML are returns on zero-investment, long-short portfolios based on ME (SMB) and the ratio of book equity (BE) to market equity (ME, HML). Equation (2) is the multi-factor regression with excess asset returns on the left hand side and alpha, the market, size and value return factors on the right hand side of the equation.

In their 1996 paper, Fama and French perform asset pricing tests with cash flow/price, sales growth, as well as long and short term past returns – confirming momentum and the long term reversal effect. [23] The three-factor model as in (2) is able to explain proxies for value such as long term past returns. The economic explanation is that long term losers are more likely to be small, distressed stocks. However, the continuation of short term past returns, i.e. momentum, is not captured by the three-factor model. Instead, it predicts the reversal of both short and long term returns. [24] The authors conclude that the assumption of rational asset pricing can be maintained, but that the three-factor model misses one risk factor. [25] 

3.2 Carhart’s Four-Factor Model

While Fama and French are reluctant to add an additional factor to their three-factor asset pricing model, Carhart publicizes a four-factor model including momentum in his 1997 paper on the persistence of mutual fund performance. [26] He finds that common risk and expense factors are driving fund return persistence. Analyzing fund managers’ abilities, Carhart attributes performance persistence to a passive momentum strategy, instead of individual manager ability. Consequently, some "mutual funds just happen by chance to hold relatively larger positions in last year’s winning stocks" [27] , so the outperformance is not persistently repeated. Funds actively pursuing a one-year momentum strategy [28] are found to underperform after expenses due to frequent portfolio rebalancing. [29] 

Carhart compares the CAPM, the Fama-French three-factor model and a four-factor model including one-year momentum (PR1YR). While the three-factor model exhibits lower pricing errors, thus higher validity, than the CAPM, it leaves short term past returns largely unexplained. Moreover, returns predicted by the three-factor model are not economically different from returns predicted by the CAPM. [30] 

The four-factor model attributes performance to RMRF (excess market return), SMB (size), HML (value) and PR1YRt (momentum) as depicted in equation (3).

The dependent variable on the left hand side of the equation is the excess portfolio return. RMRF is the Fama-French value weighted market proxy. SMB and HML are the Fama-French value weighted, zero-investment, factor-mimicking portfolios. PR1YR is calculated based on equally weighted US stock returns [31] (12-1 months), underweighting big and recently successful firms. The momentum premium is calculated as the spread of the 30% best and worst performers. [32] 

The results of the four-factor model exhibit high variance of and low correlations among RMRF, SMB, HML and PR1YR. The model improves the pricing errors of the CAPM and the Fama-French three-factor model, thus captures returns satisfactorily. [33] 

In the next section Carhart forms deciles-ranked, annually rebased portfolios of mutual funds’ lagged one-year performance. The spread of the worst and best performers is 1% per month. Since the betas of the portfolios (return-unrelated) are very similar, the CAPM cannot explain this return variation, which is why alpha (interpreted as the manager’s ability) fills the gap. Indeed, the CAPM suggests an outperformance of the best decile by 2.6% p.a. (equivalent to alpha) and an underperformance of -5.4% for the worst decile. Thus, while best and worst funds are at the same information level and bear similar market risk, they use this information differently: "worst funds appear to use this information perversely to reduce performance". [34] 

However, multi-factor models diminish alpha. Momentum alone explains half of the return spread of the top and the bottom deciles (67 basis points). Interestingly in terms of active momentum strategies, Carhart tests, albeit not significantly, show that the bottom decile funds are subject to higher expenses and higher turnover than the best performers. Nevertheless, these factors cannot explain the return difference. [35] 

The analysis of fund ranking time series shows a kind of persistency, since funds in the very top and in the bottom decile tend to be the best respectively the worst funds in the next year (Figure , Contingency table of initial and subsequent one-year performance rankings. Source: Carhart (1997), p. 71.). Superior performance reverses more quickly than underperformance. However, 80% of funds are in different deciles in subsequent time periods. [36] 

Figure , Contingency table of initial and subsequent one-year performance rankings. Source: Carhart (1997), p. 71.

The hypothesis that active momentum strategies earn abnormal returns is rejected, since momentum is "not an investable strategy at the individual security level" [37] . Carhart concludes that the top mutual funds follow passive momentum "strategies", so by not rebalancing they hold winning stocks in higher proportions and save transaction costs. [38] 

3.3 Conclusion Multi-Factor Models

Both the three-factor model proposed by Fama and French and the extended four-factor model by Carhart help to diminish alpha and thereby the unexplained performance part. Alpha, left to the manager’s investment secrets, is justifying high management fees and a sense of mystery. Concluding, from an individual investor’s perspective, these and further models increase transparency and cost efficiency of mutual funds’ performance, for example in comparison to passive investing.

Based on the Fama-French multi-factor model, various models have been constructed. For instant, similar to Zhang [39] , Chen et al. establish a multi-factor model for equities, linking investment and profitability. The authors claim that their model "reduces […] abnormal returns of […] anomalies-based trading strategies" [40] . According to Chen et al., the Fama-French three-factor model does not account for market anomalies like momentum and earnings surprises or financial distress and net stock issues. Their model includes a market, an investment and a return-on-equity factor. The model is based on the assumption that firms invest during times of high profitability. On the other hand, low investment firms had lower past returns and low current valuations. This characterizes a value stock with typically higher future ex post returns. Moreover, controlling for investment higher profitability (higher ROE) leads to higher expected returns. [41] 

However, the applicability of the referred as well as similar studies is limited due to its restriction to equity. Since value and momentum exist across markets and asset classes, the interpretation of value in terms of investment risk, for instance, is flawed. Asness, Moskowitz and Pedersen demand that value and momentum should be discussed in a global, asset class comprehensive approach. [42] Their research is the result of step-by-step extensions of the original Fama-French three-factor model.

In 1998, Fama and French examine and find evidence of a value premium in international equity markets. The authors deploy a two-factor model on developed and emerging equity markets. [43] Contrary to Fama and French (1992 & 1993), they set up, next to the market factor, just one relative distress factor (the global value spread captured by BE/ME). The authors also test for E/P, C/P and D/P. However, every ratio produces a premium that can „be described as compensation for a single common risk" [44] . They find that, contrary to the international CAPM which explains well local market returns, a two-factor model – via a time series regression – also captures the performance of local value portfolios. More specifically, global value can explain value on a country level. This is supportive for the existence of a globally inherent risk factor, since between 1975 and 1995, 75% of the global portfolio consisted of US and Japanese stocks. However, correlations for value premia are low, albeit positive (0.09 on average). Still, 75% of the global market and value factor’s variance is explained by country co-variances. [45] 

4. Value and Momentum in Every Asset Class

This section analyzes the extension of multi-factor models by Asness, Moskowitz and Pedersen [46] to other asset classes, namely international individual equities, international equity indices, government bonds, commodities and currencies.

Momentum is measured across asset classes as the "past 12-months cumulative raw return on the asset […], skipping the most recent month’s return" [47] . For value, the measurement is less straightforward. While equity value is measured via book-to-market, the book value for commodities and bonds is defined as the spot price 5 years ago. Value in currencies is calculated as the negative 5-year return adjusted for the interest earned. [48] The dataset is comprised of 4 individual country stock returns, 18 country index futures, 10 spot exchange rates [49] , 10 government bond inflation-adjusted yields and finally 27 commodity futures. [50] Market and funding liquidity is measured, for instance, by on-the-run and off-the-run spreads (market liquidity) and the TED spread, which is the daily local 3-month interbank LIBOR minus local 3 month T-Bill rate (funding risk). [51] 

Asness, Moskowitz and Pedersen detect a common, underlying risk factor, which is driving asset class returns. Assuming rational asset pricing, value and momentum risk premia therefore compensate for this underlying risk factor. Apparently, it is best made visible by comparing value or momentum throughout asset classes. [52] 

The research questions are 1) the co-movement of value and momentum across markets and time; the correlations of value and momentum and of a combination strategy over time, 2) how factor diversification can be obtained by value and momentum, 3) what economic drivers and dynamics shape value and momentum returns and their correlations and 4) the implementation of an empirical model capturing cross-sectional returns.

The analysis of value and momentum across time exhibits declining profitability of value and momentum strategies, though they are becoming "more (negatively) correlated across asset classes" [53] , which supports the view that diversification can be obtained by combining value and momentum. In equities, the average correlation between value and momentum is -0.60, for other asset classes it is -0.54. The correlations of average return series confirm this pattern, but they are even higher, due to less weight given to outliers. [54] 

The correlation structure of global value and momentum indicates the existence of an underlying risk factor. Indeed, the correlations of value and momentum strategies across asset classes are higher than the correlations of the asset classes themselves. Average individual style correlations are positive across asset classes. [55] 

The co-movement of value and momentum across asset classes is remarkable due to the differing investment backgrounds of the asset class universe, such as market and informational structures. Due to the negative correlation of value and momentum, which cannot be explained by asset class correlations, the 50/50 value/momentum combination-strategy outperforms the single strategies (Sharpe ratio: 1.43). The superior performance of the combination strategy is due to diversification. Meanwhile, the global single strategy Sharpe ratios are higher than the local single strategy Sharpe ratios, suggesting the existence of an underlying risk factor.

To detect this risk factor, Asness, Moskowitz and Pedersen perform a principle component analysis for individual stocks in the US, UK, Europe (ex. UK) and Japan and for value and momentum strategies in individual stocks globally, country equity indices, currencies, government bonds and commodities. Figure , First Principal Component of All Asset Classes. Value and Momentum Portfolios. Source: Asness/Moskowitz/Pedersen (2011), p. 35. depicts the largest eigenvector values of the co-variance matrix (returns to value and momentum strategies) for all assets. [56] While the first principle PC loads in one direction for value strategies, it loads in the other direction for momentum strategies.

Figure , First Principal Component of All Asset Classes. Value and Momentum Portfolios. Source: Asness/Moskowitz/Pedersen (2011), p. 35.

This first PC accounts for 53.6% of the co-variance matrix of global equity value and momentum portfolios and for 22.7% for all asset classes. It can be interpreted as the (either long or short) momentum-value factor. [57] While an underlying factor seems to have been detected, there is still a need for a name. Many style- and asset class-specific factors, that have been discussed (consumption, business cycle or distress risk), are not applicable, since this factor has to apply to all asset classes.

One possible risk factor is liquidity risk. The positive relation of liquidity to momentum and the negative relation to value help to explain the correlation structure of value and momentum. In comparison to market liquidity risk, global funding liquidity is more important in explaining value and momentum, albeit it seems to explain only a part of both value and momentum returns. Otherwise, the 50/50 value/momentum combination strategy could not gain excess returns, since it hedges (funding) liquidity risk. Concluding, asset returns also include a premium for another risk factor next to liquidity risk. [58] 

In the following passage, the calculations are being expounded. The authors construct value and momentum factors based on asset returns. Rather than using factor-mimicking portfolios based on spreads, the returns are weighted according to equation (4). The portfolios are then constructed as zero-cost long-short portfolios.

Each asset return in each asset class is giving a non-zero weight. Due to this and due to the positive linear relation of the weight and the signal, the factors outperform the simple factor-mimicking portfolios. [59] Additionally, 50/50 value-momentum combination portfolios and a zero-cost return spread (high-low) portfolio are tested.

In cross-sectional regressions, 48 portfolio returns are examined as dependent variables (equation 5): high (best 30%), middle, low (lowest 30%) value and momentum portfolios for 8 asset classes (individual equity portfolios for the US, UK, Europe ex. UK, Japan, equity indices, government bonds, currencies and commodities). Value or momentum in one asset class is tested against value and momentum in all other asset classes. Therefore, the dependent variable is not part of the explanatory variable VAL or MOM. An R2 of 0.55 and an average absolute alpha of 22.6 basis points show the relation between value/momentum in one asset class and the rest of the market.

Referring to Fama-French (1993) and Carhart (1997), the authors perform a second regression using constant zero-cost, signal and equal volatility weighted [60] value and momentum factors. Similar to Fama-French the factors are long-short portfolio returns, but contrary to spread portfolios every asset return is given a non-zero weight.

48 excess portfolio returns are regressed against global value and momentum. Since the regressors are the same in every regression, R2 rises to 0.707 with an average absolute alpha of 18 basis points. In addition, the authors test their model against the CAPM, the four-factor model and a six-factor model including a default (DEF) and maturity (TERM) factor. DEF and TERM were introduced by Fama and French in their 1993 paper as bond risk factors. Although stock factors only play a limited role for bond returns (except for low-grade corporate bonds), Fama and French found a link between the stock and bond market via the term factor. [61] 

The results show that the three-factor model by Asness, Moskowitz and Pedersen (AMP-model) has greater explanatory power than preceding models. The R2, for example, is substantially higher for the AMP model than for the CAPM (0.449), the four-factor (0.554) and the six-factor model (0.601). However, the Fama-French dataset is comprised of US stocks and bonds, while the AMP-factors are based on global and asset class comprehensive data. Referring to this objection, Asness, Moskowitz and Pedersen test the 4 models on the Fama-French 25 size-BE/ME and 25 size-momentum portfolios of US stocks [62] , which naturally results in much better results for the four- and six-factor Fama-French models (R2 of 0.772 and 0.766 respectively). Still, the AMP has much greater explanatory power (R2 of 0.642) than the CAPM (-0.316). The R2 of value-momentum combination portfolios decreases to 0.36, which indicates that by combining value and momentum, a substantial part of the underlying risk can be diversified away. Therefore, only a small fraction of return variance of the combination portfolio can be explained by value and momentum everywhere factors. [63] 

In the following, Asness, Moskowitz and Pedersen analyze possible macro risks as return drivers. The most prominent effect is found for the TERM and DEF factors in all asset classes. While global equity momentum is significantly negatively related to DEF, global equity value loads positively on it. Furthermore, also GDP growth and recessions exhibit a significant negative relation to momentum in all (non equity) asset classes. [64] 

Furthermore, funding and market liquidity risk approximations [65] are tested as possible risk sources. Funding liquidity risk is negatively related to value but positively to momentum, which may in part explain their negative correlation. Moreover, funding liquidity risk may be the reason or the result of rising risk aversion and rising risk premia. The combination strategy of value and momentum exhibits lower relation to funding risk. Therefore, the combination portfolio hedges funding liquidity risk. The excess return after accounting for funding liquidity indicates that liquidity risk does not capture the whole picture. On the other hand, US market liquidity risk shows a positive but weak relation to value (and even less for global liquidity risk factors). Concluding, while market liquidity seems to be insignificant, funding liquidity shocks help to explain both excess returns and the correlation structure of value and momentum.

Furthermore, more evidence on the effect of funding liquidity is detected in a Fama-MacBeth regression of several macroeconomic factors on 48 value and momentum portfolios. First, being the only explanatory variable, liquidity risk pays a significant, substantial premium of 24 basis points per month. In a second regression, only GDP growth, long term consumption growth, TERM and DEF are tested. TERM and DEF are strong with t-statistics of 2.19 and 2.18 respectively. Third, liquidity risk is added to these four risk factors, diminishing the power of TERM and DEF. Compared to the single factor regression the explanatory power of liquidity decreases (t-statistic of 2.29 vs. 3.05). Finally, market, value and momentum factors (equal volatility weighted average of all value/momentum strategies) are included. This eight-factor regression exhibits lower significance of liquidity risk, but strong and significant t-statistics on the value and the momentum effect. Therefore, value and momentum proxy for risks first attributed to global funding liquidity risk.

Finally, value and momentum patterns over time are being examined. While the correlation of value and momentum, respectively, across asset classes increases over time, the correlation of value to momentum is relatively stable. This development is, according to Asness, Moskowitz and Pedersen, due to the rise of quantitative arbitrageurs. As a result, Sharpe ratios for both strategies have been reduced. Still, the value-momentum combination’s Sharpe ratio has not changed much over time. [66] 

The analysis of the correlation structure of value and momentum reveals that both value and momentum became more correlated across asset classes over time, while the correlation of value to momentum became more negative, which is even more pronounced when Sharpe ratios and correlations before and after the 1998 LTCM crisis, as a major liquidity event, are compared. Again, after 1998 only the Sharpe ratios and correlations of value and momentum strategies changed, less so the 50/50 combination strategy.

Moreover, the effects of recessions are tested on correlations and Sharpe ratios over time: value does better during recessions than momentum with relatively stable correlations. [67] Conversely, other evidence suggests that value stocks often underperform in times of low liquidity, often coinciding with recessions. [68] The combination strategy exhibits similar Sharpe ratios for recessions and non-recessions (1.52 and 1.35) and thus consistent performance irrespective of the economy’s state. What is more, negative funding liquidity shocks see rising Sharpe ratios for value, while the contrary is true for momentum. However, this is only true after the 1998 LTCM crisis. Underlining this finding, the authors test the explanatory power of (funding) liquidity shocks on the correlations of value, momentum and value-momentum. They find a relation of liquidity to momentum, but only after 1998. Accordingly, this crisis was a major turning point for the characteristics of value and momentum in separate. During that time, value and momentum, as well as highly leveraged quantitative trading strategies gained more popularity. The liquidity drain of 1998 and the changed characteristics of value and momentum point to an underlying factor such as liquidity risk, which drives part of the returns. Also, limits-to-arbitrage will prevent the exploitation of these effects and are linked to funding liquidity risk. [69] 

Summing up, by studying average, global value and momentum returns jointly within eight asset classes, Asness, Moskowitz and Pedersen detect common factor exposure due to higher cross sectional variance in a global portfolio than in a single local portfolio. The influence of funding liquidity on value and momentum effects thus can only be observed by looking at global value and momentum factors across markets and asset classes. Moreover, for government bonds, value and momentum are first being confirmed. For currencies and commodities, a value premium was first detected. [70] 

5. Diversification Context

In the first part of the thesis, value and momentum risk premia and their possible sources were analyzed. Models assuming rational or irrational asset pricing have been distinguished. The characteristics of ex ante and ex post risk premia have been discussed, as well as the time dependency of risk premia. The migration of value and momentum – from anomalies in a CAPM setting – into risk factors in multi-factor models by Fama and French, Carhart as well as Asness, Moskowitz and Pedersen were depicted. In the following, the findings will be presented in an asset and factor allocation context.

The objective of portfolio selection is to attain diversification. Diversifying unsystematic risk reduces portfolio volatility without reducing the expected return, or increases the expected return without raising portfolio volatility. While diversification reduces downside risk in the long term, rising asset correlations can cause failure in the short term during severe crises. In the aftermath of 2007-2009, traditional diversification approaches alongside asset class silos have been questioned.

First asset allocation generations were based upon modern portfolio theory and its concept of mean-variance optimization and its distinction between systematic and idiosyncratic risk. Portfolios were built around expected returns and co-variances. [71] Selecting market risk exposure along the capital market line, the traditional 60/40 policy portfolio consists of 60% equity and 40% government bonds. By rebalancing the fixed weights growth assets are underweighted, introducing a value component. Moreover, in the traditional 60/40 policy portfolio, volatilities and correlations are assumed to be stable, neglecting changing risk premia as argued in section 2. An important risk source of the traditional 60/40 portfolio is high market directionality, which even increases during market turmoil due to higher equity volatility. Also, the bond-stock correlation is positive (0.1 in the US since 1900) during normal times and even rises during crises. [72] Apparently, this high equity risk exposure has been misjudged during the 2007-2009 crisis. [73] 

The second asset allocation generation is the extension of the local 60/40 portfolio to a global 60/40 portfolio. While market directionality is still high, global portfolios hedge against country-specific risk factors. In their 2011 study Asness, Israelov and Liew study international diversification benefits for local investors. [74] Overcoming the home bias can protect long term investors from portfolio concentration. Indeed, in the long term country specific growth is more important than short term market turmoil. [75] However, in the short term international diversification reduces volatility, but increases negative skewness. The volatility reduction incentivizes investors to increase their market exposure. Adversely, such diversified portfolios are becoming even riskier. [76] 

Over the long term international diversification benefits increase. Indeed, negative skewness is more pronounced in the short term. For the long term (longer than 3.5 years) the difference in skewness vanishes. An analysis of returns, decomposed into multiple expansion or contraction (i.e. capital gains or losses) and actual economic growth proves that multiple expansion is important in the short term, but over the long term economic growth gains in importance. In fact, multiple expansion explains 96% of quarterly returns, but over 15 years, country-specific growth explains 39% of returns. Thus, international diversification protects against portfolio concentration in countries with poor long term growth paths. Concluding, while internationally diversified portfolios exhibit short term weakness due to rising correlations, long term portfolio return is driven by long term country growth differentials. [77] 

The third generation is the endowment model, investing in typically illiquid alternative asset classes. The endowment model states that high competition and liquid markets make it harder to gain superior returns simply by fundamental research. But for alternative assets, such as real assets, venture capital and private equity, fundamental research can provide an edge. The specialty of the endowment model is the long term horizon, which allows investors to accept short term mark-to-market losses. [78] 

The fourth generation allocates alongside risk factors. Risk allocations to macro factors such as growth, inflation and liquidity can be modeled by exposure to value, momentum, carry and trend-following. This strategy is comparable to the risk parity approach, which targets "balanced contributions of various risk exposures to portfolio risk" [79] . Still, diversification takes place alongside asset class silos rather than risk factors. [80] 

Ilmanen and Kizer test diversification benefits of factor or asset class diversification. The asset class diversification portfolio exhibits a slightly higher but unimpressive Sharpe ratio of 0.48 due to the high correlations between portfolio constituents (+0.38). Contrary, the factor diversification portfolio has a Sharpe ratio of 1.44, which can be explained in regard to low correlations between the portfolio constituents (-0.02). Notably, transaction costs, which are larger for factor strategies, are not accounted for. The factor-diversified portfolio implies less equity exposure, thus lower market directionality: its correlation to US stocks is 0.64 instead of 0.87 for the asset class diversified portfolio. Consequently, the factor diversified portfolio exhibits smaller drawdowns, but disadvantageous skewness and kurtosis. Still, it has higher returns during recessions than the asset class diversified portfolio and similar returns during expansions. [81] Figure , Average (60-month rolling) Pair-Wise Correlations of the Five Constituents. Source: Ilmanen/Kizer (2012), p. 21. depicts average pair-wise correlations among the constituents of the asset class diversified and the factor diversified portfolio.

Figure , Average (60-month rolling) Pair-Wise Correlations of the Five Constituents. Source: Ilmanen/Kizer (2012), p. 21.

While the factor-diversified portfolio still exhibits common liquidity risk – the dry up of liquidity can make previously uncorrelated factors correlated –, Ilmanen and Kizer claim it to be "more manageable than concentration risk". [82] While factor diversification as an elaborate diversification approach implies high transaction costs, there is evidence that also simple international diversification provides an edge. Rising correlations across countries during economic downturns had questioned the advantage of global investing, because of "more severe tail events in global portfolios than in local portfolios" [83] .

Ang and Bekaert empirically examine this pattern and show that the reluctance of investors to invest globally is due to a home bias and not to rising volatility and correlations during economic downturns. Assuming time-varying risk premia, correlations and volatility, investors’ exposure to global markets during regime changes are relatively stable. Different regimes refer to times of high correlation and low returns (crisis) and to times of low correlation and higher mean returns (expansion). [84] 

6. Empirical Analysis

The empirical analysis is conducted in two parts. First, a scoring analysis is performed. Via zero-cost portfolios value and momentum premia are examined in three asset classes: equity indices, government bonds and currencies. The underlying assumption of the scoring approach is the persistence of value and momentum.

Second, an asset allocation model is applied. It combines forecasting asset returns via multiple regressions and mean-variance optimization. Lagged value and momentum valuation measures (rather than factor-mimicking portfolios) predict asset returns. The single expected asset returns are summarized by asset class. Then, the asset allocation is optimized based on the risk/return profiles using historic co-variances (1995/1996-2006).

The validity of this approach depends on the one hand on the predictive power of value and momentum: are value and momentum coefficients significant? It will be shown that the usage of yearly overlapping instead of monthly returns considerable improves the significance of the coefficients. While this may partly be due to the use of overlapping data, the relative strength of value in particular over longer time periods is not surprising.

On the other hand, the outcomes depend on mean-variance optimization and its restrictions. As can be seen in Figure , Asset Allocation based on Monthly Data. and Figure , Asset Allocation based on Yearly Data., mean-variance optimization can lead to extreme allocations and high portfolio turnover. The approach selected in this thesis minimizes volatility while setting the return constraint at historic mean portfolio returns.

6.1 Data

In the following, the data set used in the empirical analysis will be described. Monthly as well as yearly overlapping equity index, government bond and currency returns are analyzed. In equity, six MSCI indices (all measured in USD) are used; three developed and three developing country indices, namely MSCI North America, MSCI EAFE, MSCI Pacific [85] , MSCI Emerging Europe, MSCI Latin America and MSCI Emerging Asia. The monthly returns are derived from the total return index.

For government bonds, data is obtained from Citigroup’s currency hedged total return indices (measured in USD) for bonds with maturities from 7 to 10 years. First available data starts in 1995, which means that the first value measure is only obtainable in 2000. Twelve government bonds are included: Australia, Canada, Denmark, France, Germany, Italy, Japan, Norway, Sweden, Switzerland, UK and US. Finally, ten currencies are measured via spot rates against the USD, concentrating on industrial countries currencies: GBP, DKK, EUR, NOK, SEK, CHF, CAD, JPY, AUD and NZD.

The dataset ends in 30.11.2012. The data availability for certain asset classes is limited, therefore time series differ in their starting point, Were possible, I made use of the longer time series (for example in Table , Returns for Value and Momentum Factor-Mimicking Portfolios. and Table , Sharpe Ratios for Value and Momentum Factor-Mimicking Portfolios.) to include as much information as possible. In the scoring part, the zero-investment value and momentum factors in currencies are first available by 31.07.1990; for equities by 31.07.1995 and for government bonds by 31.07.2000. Yearly rebalancing takes place at the end of June. Only for reasons of completeness it should be noted that momentum in government bonds is available as early as 31.07.1996, which is depicted in Figure , Returns on Value and Momentum in Equity., but not included in the calculations.

Since yearly rebalancing is not necessary in the forecasting model, value and momentum valuation multiples are first at hand for currencies by 1990, for equities by 1995 and for government bonds by 2000.

6.2 Value and Momentum Measures

In the following it will be described how value and momentum are measured both in the scoring and the forecasting part of the analysis. In equities, value is measured threefold: well-known ratios such as E/P and B/P [86] are contrasted with negative 5-year returns. [87] However, it will be shown that this last measure severely underperforms E/P and B/P. Also, it negatively correlates with E/P and B/P, which is not supported in the literature. Due to the existent research on this topic [88] , the extent of the underperformance of negative 5-year returns in contrast to E/P and B/P is surprising. It is probably due to the limited time window and due to a possible distortion caused by the aggregated information level supplied by the indices.

The inverse of the first value ratio, E/P, is provided by Bloomberg and calculated on the basis of the last price divided by trailing 12 months earnings per share before extraordinary items. Notably, E/P ratios based on 12 months trailing earnings are more volatile than the well-known Shiller P/E, which accounts for 10 years of trailing earnings, thus one or two business cycles. [89] The inverse of B/P is also provided by Bloomberg and calculated with the latest available price (similar to E/P) and the most recently reported data (which may be quarterly, semi-annually or annually) on the book value per share. For government bonds and currencies the negative 5-year return is used as a value measure. In all asset classes the return is based on the average spot price of 5 years earlier plus/minus one month.

Similar to Asness, Moskowitz and Pedersen, the measure for momentum is the same in every asset class. It is calculated as the twelve months return skipping the most recent month. [90] 

6.3 Scoring Analysis

In the scoring part, it will be tested whether value and momentum exist in the selected asset classes by establishing zero-cost long-short portfolios (without leverage). The portfolios are yearly rebalanced in the end of June. Returns are measured monthly. The breakpoints are the 30th and 70th percentile of the value and momentum range.

For reasons of comparability and to enrich the equity space of the analysis, the Fama-French value (HML) and momentum (WML) factors for developed markets published on Kenneth R. French’s website [91] are included to study these factors’ interaction with the factors calculated independently. The factors are available from June 1990 to January 2013 and are in USD. The breakpoints for the Fama-French value and momentum factors are the 30th and the 70th percentile. Rebalancing takes place once a year in the end of June. The returns of HML and WML are equally weighted, hence independent of market capitalization. Fama-French measure HML via the book-to-market ratio. WML is the "stock’s cumulative return for month t-12 to month t-2" [92] . Contrarily, my momentum factor is based upon the return for month t-13 to t-2.

Table , Factor Correlations across Asset Classes.

Table , Factor Correlations across Asset Classes. summarizes the correlations of the zero-cost factor-mimicking portfolio returns. It shows very low correlations of FF value and the other value/momentum factors. The same is true for FF momentum, although both FF factors nicely interact with FX value and momentum. Expectedly, B/P and E/P are positively correlated. However, both are negatively correlated to equity value measured by negative 5-year returns. E/P is positively correlated to equity momentum, and both E/P and B/P are positively related to GB momentum. This is consistent due to the negative correlations of value and momentum factors in each asset class, specifically in equity when using negative 5-year returns. Although correlations are low in general, Table , Correlations of Value and Momentum. clarifies that there is some connection between value and momentum throughout the asset classes.

Table , Correlations of Value and Momentum.

In Table , Returns for Value and Momentum Factor-Mimicking Portfolios. the long-short portfolios’ returns are summarized. The returns achieved show substantial outperformance of some of the well-known value and momentum factors, albeit single factors substantially underperform, especially negative 5-year return. Since there are differences in the data availability, for equity and currencies there are longer time windows accessible (row 1). In the second row the time window is the same for every asset class (31.07.2000-30.11.2012).

Table , Returns for Value and Momentum Factor-Mimicking Portfolios.

Table 3 and Table 4 show that the time window makes a substantial difference. Skipping the years 1995-2000 leads to a major improvement for B/P while momentum returns decrease by a sizeable amount. This can be interpreted in terms of time-varying risk premia – therefore the years 1995-2000 saw good momentum, but little value, measured by B/P. Notably, value measured by negative 5-year return fails to perform, but doesn’t change a lot in a longer time frame. Returns for government bonds are small for value and momentum. In currencies only the value strategy works. Table 4 shows the Sharpe ratios for the zero-cost portfolios. In general, value produces considerable Sharpe ratios, especially equity (E/P) and currency value.

Table , Sharpe Ratios for Value and Momentum Factor-Mimicking Portfolios.

As is visible in Table , Returns for Value and Momentum Factor-Mimicking Portfolios. and Table 4, value generally outperforms momentum. Only in the case of government bonds, where both value and momentum are weak, momentum and value returns are comparable. When selecting an equity value ratio for the forecasting part of this study, E/P seems to be the most promising. The time series of value and momentum strategy returns are depicted in Figure , Returns on Value and Momentum in Equity., Figure , Returns on Value and Momentum in Currencies. and Figure . Returns on Value and Momentum in Government Bonds..

Figure , Returns on Value and Momentum in Equity.

Figure , Returns on Value and Momentum in Currencies.

Figure . Returns on Value and Momentum in Government Bonds.

Summarizing, value and momentum factors exist in the given dataset, except for currency momentum. The finding of value and momentum in government bonds is backed by the results of Asness, Moskowitz and Pedersen, as well as the value premium in currencies. [93] 

6.4 Forecasting Model

In this section, a forecasting model combined with an asset allocation approach will be applied. The assets and the asset classes are the same as before, therefore monthly returns for 28 assets are used (6 equity indices, 12 government bonds, and 10 currency pairs). While the time series length differs for equities, government bonds and currencies, the return data within each asset class are uniform. Based on the findings of the first part of the empirical analysis, for equity, P/E is used as an equity value measure. Value in government bonds and currencies is depicted via the negative 5-year return. Momentum in every asset class is the previous 12-months return skipping the most recent month.

The returns (the dependent variable) used in the forecasting model (until 2006) are referred to as ex post returns subsequently. Based on the results of this multiple regression, future returns are estimated, starting in 2007, and are referred to as ex ante returns. The multiple regression and the calculation of the ex ante returns will be described in the following section.

6.4.1 Multiple Regression Analysis

First, the significance of the value and momentum indicators for each asset’s monthly realized return (ex post) is tested in a multiple regression analysis. The value and momentum coefficients are estimated based on the relevant time series until 2006 [94] .

The endogenous variable is the return of asset i in month t. The exogenous variables are the respective value and momentum measures as described above, at time t-1. Contrary to Fama-French (1993) and Asness, Moskowitz and Pedersen (2011) the regressors are valuation multiples and not returns.

While equation (7) refers to the month-to-month effect, in a second step also overlapping yearly returns will be used, especially to better capture long term return reversal patterns. More specifically, the yearly return in month t of asset i will be tested on the value and momentum indicator available in month t-12. The usage of overlapping yearly returns is necessary due to the limited data range of some time series, especially government bonds.

The β coefficients for value and momentum obtained by these multiple regressions in combination with the resulting intercepts are used to obtain ex ante returns (equation 9 & 10). The calculation of value and momentum measures is unchanged.

For each asset from January 2007 to November 2012 monthly and yearly overlapping returns, respectively, exist. On the basis of these estimated returns an asset allocation model will be applied. For reasons of clarity and comprehensibility, the assets’ returns will be aggregated on an asset class level (equally weighted average returns).

For each month from January 2007 to November 2012, a mean-variance optimization will be applied, using historic co-variances (for 1995 – 2006 for the monthly return part and 1996 – 2006 for the yearly overlapping return part). This asset allocation will be back-tested with monthly ex post returns for each asset class. The asset class ex post returns are the equally weighted average returns of each asset in the respective asset class. Thus, the asset allocation model and the forecasting power of value and momentum will be tested out-of-sample for 5 years and 11 months.

First, the regressions’ results will be described. In Table 5 the results of regression (7) are depicted. Except for equities, significances are very low. The signs of the coefficients are negative for Equity North America and EAFE since P/E and not E/P is used in this regression. The coefficients’ signs for value as negative 5-year return should be positive, which is the case for currencies, albeit not significant. Most government bonds exhibit negative coefficients. It will be shown that this pattern is consistent throughout this data sample.

Table , Regression Results. Monthly Returns.

While value is weak, momentum turns out to have slightly more forecasting power on the month-to-month level. Especially government bonds show significance levels in four cases at α = 10% and in one case at α = 5%. Still, similar to value, the coefficients’ signs are negative. These results imply that the selected government bonds in the referred time window exhibit negative value and momentum: thus the returns tend to reverse in the short term with sustained long term momentum. In regard of the secular interest rate cycle and long term capital gains for government bonds, seen in 1990 to 2000, this pattern is, however, less surprising.

Summarizing the regression results of equation (7), value and momentum in general possess rather weak predictive power on the month-to-month level. For this reason, the same value and momentum measures will be used to explain overlapping yearly returns (equation 8). The results of this multiple regression are shown in Table , Regression Results. Yearly Returns.. First, the value coefficients are displayed. In general, significance levels are noticeably higher than before. All currency pairs exhibit highly significant value coefficients. While value is less pronounced for equities, the majority of the government bonds are featuring significant value effects. As before, however, the value coefficients’ signs are negative. It should be noted that the usage of overlapping data might contribute to higher significance levels for yearly returns. Nevertheless, it is intuitive that value as a long term strategy becomes stronger over longer time windows.

Table , Regression Results. Yearly Returns.<