F
ACTOR
I
NVESTING IN THE
E
QUITY
M
ARKET
R
ESEARCH ON WHETHER THE OUTPERFORMANCE OF FACTORS IS A COMPENSATION FOR RISKNicole B. Gertsen
Master Thesis
Date: 22-07-2015
Student number: 6173071 Master: Econometrics
Specialization: Financial Econometrics University: Universiteit van Amsterdam Supervisor: Dr. M. Pleus
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A
BSTRACTResearch on investing in factors, like Size, Value and Momentum has widely increased over the past number of years. Factor portfolios show an outperformance over the market portfolio based on historical returns, but investing in factors also increases risk. Two methods are used to test the significance of the outperformance on a risk-adjusted basis. The two methods have different outcomes; the first method checks the significance of the CAPM-alpha. Most alphas are significant, implying a substantial excess return over the market. The second method checks if the outperformance is caused by an increase in risk using the Sharpe ratio test that is based on the block bootstrap suggested by Ledoit and Wolf (2008). This test checks whether the Sharpe ratio of a factor portfolio is significantly different from the market Sharpe ratio. The null hypothesis that the Sharpe ratio of one of the factor portfolios is equal to the Sharpe ratio of the market portfolio is not rejected. This implies that the Sharpe ratio of a factor portfolio is not significantly higher than the market Sharpe ratio; thus it cannot be stated that the higher return generated by investing in the factor is not just a compensation for risk.
3
C
ONTENTS1. Introduction ... 4
2. Rationale behind Factor Investing ... 7
2.1 History ... 7
2.2 ‘Behavioural finance’ point of view ... 8
3. Data and Methodology ... 9
3.1 Data ... 9
3.2 Methodology ... 9
3.2.1 Portfolio Construction ... 9
3.2.2 Long-only versus Long-short ... 11
4. Empirical Results ... 12 4.1 Approach ... 12 4.2 Results ... 14 4.3 Risk adjustments ... 16 4.3.1 Single-factor portfolios ... 19 4.3.2 Multi-factor portfolios ... 21 5. Robustness Checks ... 24 6. Conclusion ... 29 References ... 31 Appendix ... 34
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1.
I
NTRODUCTIONUp until the 2007-2009 financial
crisis, diversification was one of the most profound
investment concepts. Investing in a variety of assets and combining investment categories can diversify a portfolio. By diversifying a stock portfolio, the portfolio’s risk can be reduced without affecting expected returns. In late 2007, portfolios were more widely diversified than ever and should have been better able to absorb shocks (Koedijk et al., 2013, p.7). However, after the Lehman Brothers investment bank went bankrupt, causing almost all investment categories to decline, the concept of portfolio diversification received a lot of critique and questions began to arise whether it is really possible to effectively diversify a portfolio.Investors realized that diversification was possibly just compounding risk, rather than lowering risk. Investments should not be seen in isolation; they should be seen as a bundling of factors that have an impact on risk and return. After the Norwegian Government Pension Fund launched an investigation into its active management policy, this became a wide topic of interest. The investigators, Andrew Ang, William Goetzmann and Stephen Schaefer, were asked to analyse the fund’s performance, since after years of stable outperformance, the negative results of 2008 destroyed the cumulative active return of the previous past ten years. The study finds that a large part of the outperformance can be explained by exposure to systematic factors that were implicitly present in the portfolios and the investigators recommended making the exposure to factors a “top-down decision rather than emerging as a by-product of bottom-up active management” (Ang et al., 2009, p.20). As a result, other investors were triggered to question themselves whether they were aware of these systematic factors behind their investments and how a greater understanding of these factors could improve the investment results.
Research on factors has widely increased over the past years. The focus in literature has mostly been on market, Size, Value and Momentum (Koedijk et al., 2013 and Houweling et al., 2014). First, the market factor exists because the expected return of an asset depends on its sensitivity to non-diversifiable risk. Non-diversifiable risk is risk that is inherent to the entire market and is impossible to avoid completely. It is also known as market risk or systematic risk. Second, the Size Effect refers to the phenomenon that in the long run assets with a low market capitalization perform better than assets with a high market capitalization. Assets of
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small firms are riskier than assets of large firms because of higher systematic risk. When an event occurs that affects the entire market in a negative way, small firms are less resistant than larger, stable firms. This higher risk earns the investors a higher return premium. Third, the factor Value exists because it is expected that the price of an asset will converge to its true value (which is called mean-reversion) and therefore, undervalued assets eventually generate a higher return than the market. By focusing on these value-assets, investors can generate an excess return. Finally, the Momentum Effect is the effect that assets that have performed well in the past, have the tendency to perform well in de near future.
Koedijk et al. (2013) have investigated the average return on a portfolio when investing in all of these four factors separately and when investing in a portfolio consisting of all factors equally weighted. They show that the factors Size, Value and Momentum outperform the average market return for both US data and European data (except for Size in the European market). Houweling et al. (2014) investigate the concept of factor investing in the corporate bond market. They also show that most factor portfolios have outperformed the market portfolio. Their research is also based on US data but since factor investing is such an upcoming investing strategy, it is interesting to see how this strategy would perform in other markets, such as the Chinese market and the Malaysian market.1 Also, the Japanese market is an
interesting market since it is one of the largest markets in the world.2
This thesis focuses on two issues regarding factor investing. The first issue concerns the fact whether not only factor investing in the equity market of North America outperformed the market, but also investigates whether this is the case in the markets of Europe, Japan and Asia Pacific. The second issue stems from the fact that investing in a factor portfolio increases the return (i.e. they outperform the market), but also increases risk. This raises the question whether the higher returns of the outperforming factor portfolios are just a compensation for risk. To answer this question, the factor portfolios are also evaluated on a risk-adjusted basis. The returns are adjusted for risk in two ways; the systematic risk of a factor portfolio is corrected for by regressing its return on the market premium and returns are measured relative to total risk using the Sharpe ratio measure. The commonly used test is the test of
1China and Malaysia are the numbers one and three of the top 20 of the Bloomberg Best Emerging Markets
2014 List.
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Jobson and Korkie (1981), where the null hypothesis is that the difference between two Sharpe ratios is zero. However, as Ledoit and Wolf (2008) argue, this test is not valid here since the distributions of returns have heavier tails than the normal distribution. They suggest another inference method, namely one that is based on block bootstrap. This method is used here, because it surpasses the general used method, which tends to over-reject the null hypothesis. The thesis is setup as follows. Chapter 2 describes the rationale behind factor investing and Chapter 3 describes the data and methodology. Section 4 presents the main empirical results and analyses single-factor portfolios and multi-factor portfolios. Chapter 5 checks the robustness of the results and Chapter 6 concludes.
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2.
R
ATIONALE BEHINDF
ACTORI
NVESTINGWhen choosing factors, criteria like rationality and ‘behavioural finance’ explicability are important. Factors must have a strong theoretical basis and a clear defensible economic intuition (Koedijk et al. 2013). This chapter explains the origination of factors that are of interest in this thesis.
2.1HISTORY
In the early 1960’s the Capital Asset Pricing Model (CAPM) was introduced by Jack Treynor (1961, 1962), William F. Sharpe (1964), John Lintner (1965a,b) and Jan Mossin (1966) independently, building on the earlier work of Harry Markowitz on diversification and modern portfolio theory. This model is used to describe the returns of a portfolio using one variable, the market risk factor. It provides an appealing explanation of the relation between risk and asset returns.3 However, the model does not fully explain this relationship and therefore other
researchers investigated this model and found additional factors that can help explain the risk-asset return relationship. Subsequently, these researchers have extended the CAPM.
Banz (1981) found that small cap stocks, i.e. stocks with lower market capitalization, outperform large cap stocks. Investing in stocks of relatively small firms carries a higher risk, because these types of firms tend to have a more volatile business environment, and according to research of MSCI Research Insight, small firms are more sensitive to other underlying and observable risks such as financial distress (Chan and Chen, 1991), default risk (Vassalou and Xing, 2004) and information uncertainty (Zhang, 2006). Therefore, a higher premium is offered to investors who invest in these small cap companies. This effect is called the Size Effect and was generalized by Eugene F. Fama and Kenneth French.
In 1993 Fama and French proposed the Fama-French-3-factor-model in their paper Common
risk factors in the returns on stocks and bonds. This model is an extension of the Capital Asset
Pricing Model. Besides the Size factor, which captures the excess returns of smaller firms relative to their larger counterparts and is therefore referred to as SMB (Small Minus Big), they
3 The Capital Asset Pricing Model: Theory and Evidence (Digest Summary), Eugene F. Fama and Kenneth R.
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found that value stocks, i.e. stocks with a high book value relative to their market value (high book-to-market ratio), outperform growth stocks (stocks that have a low book-to-market ratio). Chaves et al. (2012) argue that ‘value companies’ are more risky because “value stocks co-move with some unobserved risk factor”. These types of stocks potentially increase portfolio exposure to distress, liquidity and some rare and hidden risk and therefore offer greater compensation to investors. This Value Effect captures the excess returns of undervalued companies relative to overvalued companies and is referred to as HML (High Minus Low), which stems from high book-to-market minus low book-to-market. Fama and French included these two other factors in their model: SMB and HML. Jegadeesh and Titman (1993) have shown that stocks that have done well over the past year have a tendency to continue to do well. This Momentum Effect captures the excess returns of companies that have performed well in the past relative to companies that have not performed well and is referred to as WML (Winners Minus Losers). In 1997 Carhart extended the Fama-French-3-factor model with the Momentum factor to a 4-factor-model.
2.2‘BEHAVIOURAL FINANCE’ POINT OF VIEW
In finance, the tendency to extrapolate the recent past exists, which means that investors assume that existing trends will continue. They believe that small cap firms that grow fast will continue to grow fast and, therefore, they could over-optimistically evaluate the risks that these small cap firms generate. The Value factor exists because investors are, in general, loss averse and they believe that some undervalued stock will converge to its true value. Momentum in stocks can be explained by the way in which investors process news and react to it. They can overreact, but they can also underreact. These “irrational” reactions are driven by behavioural biases such as overconfidence, conservatism bias and the aversion to realize losses.4
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3.
D
ATA ANDM
ETHODOLOGYThis chapter describes what data is used and how that data is used to construct portfolios and compute the returns of the factor portfolios. It also illustrates two different investment strategies, long-only and long-short.
3.1DATA
Monthly data from November 1990 up to and including April 2015 of the Kenneth French library website are used. The original data set also includes the months July, August and September of 1990, but for the Momentum factor there is no data available for these months. Therefore, in this thesis these months are excluded. The dataset contains monthly returns of stocks in four regions: Northern America (the United States of America and Canada), Europe (Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Italy, the Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, and the United Kingdom), Japan and Asia Pacific (Australia, Hong Kong (China), New Zealand, and Singapore (Malaysia)). The Kenneth French library provides various characteristics, including excess market returns, the returns of investing in one of the three factors SMB, HML and WML and the risk-free rate, for each month.5 The calculations, which precede the returns of investing in these factors, are based
upon multiple constructed portfolios. Koedijk et al. (2013) use the same constructed portfolios, but assume a long-only strategy and therefore use a different calculation for the returns of the factors Size, Value and Momentum.6 In the following section, the construction of the portfolios
and the computations of the returns of the factors for both strategies are explained.
3.2METHODOLOGY
3.2.1 Portfolio Construction
The market factor is the monthly return on a portfolio comprising of all stocks from a specific region, weighted to market-value, minus the risk-free rate. The risk-free rate is the American
5 They assume a long-short strategy. The difference between long-only and long-short is explained
section 3.2.2.
6They did their research for Robeco. I wrote this thesis during my internship at Alex Vermogensbank, which
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one-month Treasury bill rate for all regions. For each factor, equally weighted portfolios are constructed to keep certain effects, like the Size effect. When market weighted portfolios are used, small firms are given less weight, which removes this effect. To construct the Size and Value factor portfolios, six portfolios are constructed first. For every region, stocks are sorted into two market value groups, namely a small size group and a large size group, and into three book-to-market (B/M) equity groups, namely a high B/M group (value stocks), a neutral B/M group and a low B/M group (growth stocks). The market value of a stock is obtained by multiplying the number of shares outstanding by the current stock price. Small stocks are those in the bottom 10% of the market value and big stocks are those in the top 10% of the market value. The book value of a stock is the total value of the company's assets that shareholders would theoretically receive if a company would be liquidated. Comparing the book value to the company’s market value indicates whether a stock is undervalued or overvalued. If the ratio is above 1, then the stock is undervalued and if the ratio is below 1, the stock is overvalued. The B/M breakpoints are the 30th and 70th percentiles of B/M for the big
stocks of the region. Sorting stocks this way produces six (2x3) portfolios; two portfolios based on size times three portfolios based on B/M. On the Kenneth French Library website they are referred to as SG, SN, SV, BG, BN, BV, where S and B indicate small or big and G, N and V indicate growth, neutral and value. Table 3.1 illustrates these portfolios.
Table 3.1 2x3 portfolios formed on Size and Book-to-Market
10% smallest stocks 10% biggest stocks 70th – 100th percentile of B/M Small Value (SV) Big Value (BV)
30th – 70th percentile of B/M Small Neutral (SN) Big Neutral (BN)
0th – 30th percentile of B/M Small Growth (SG) Big Growth (BG)
To construct the Momentum factor portfolios, again six portfolios are constructed. They are based on the two Size portfolios mentioned above and on three Momentum portfolios. For the Momentum portfolios formed at the end of month t–1, the lagged momentum return is a stock's cumulative return for month t–12 to month t–2. The momentum breakpoints for a region are the 30th and 70th percentiles of the lagged momentum returns of the big stocks. The six
constructed portfolios are referred to as SH, SM, SL, BH, BM and BL, where S and B stand for Small and Big and H, M and L stand for High, Medium and Low. Table 3.2 illustrates these portfolios.
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Table 3.2 2x3 portfolios formed on Size and Momentum
10% smallest stocks 10% biggest stocks 70th – 100th percentile of Momentum Small High (SH) Big High (BH)
30th – 70th percentile of Momentum Small Medium (SM) Big Medium (BM)
0th – 30th percentile of Momentum Small Low (SL) Big Low (BL)
3.2.2 Long-only versus Long-short
CAPM is based on taking a long position in the market portfolio. Taking a long position means adding stocks to the portfolio that are expected to go up and avoid investing in stocks that have already reached their peak. When Fama and French and Carhart extended the model, they added the long-short factors SMB, HML and WML to the model. Long-short means that investors buy stocks that are expected to go up but also sell stocks that are expected to go down (‘take a short position’). In the case of investing in the market portfolio an in long-short factors, this implies that small/value/winner stocks are given more weight relatively to the market factor portfolio and big/growth/loser stocks are given that much less weight. This ensures that only the weights of the factor portfolios relatively to the weights of the market portfolio change and therefore, investing in factors is basically a zero-investment. Table 3.3 illustrates the difference between the two strategies. An illustration of the portfolio construction and of how returns of the factors based on both strategies are computed is given in the Appendix.
Table 3.3 Factor portfolios definitions, long-only strategy versus long-short strategy
Long-only strategy Long-short strategy
𝑆𝑖𝑧𝑒 = 1 3⁄ (𝑆𝑉 + 𝑆𝑁 + 𝑆𝐺) 𝑆𝑀𝐵 = 1 3⁄ (𝑆𝑉 + 𝑆𝑁 + 𝑆𝐺) − 1 3⁄ (𝐵𝑉 + 𝐵𝑁 + 𝐵𝐺)
𝑉𝑎𝑙𝑢𝑒 = 𝐵𝑉 𝐻𝑀𝐿 = 1 2⁄ (𝑆𝑉 + 𝐵𝑉) − 1 2⁄ (𝑆𝐺 + 𝐵𝐺)
𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝐵𝐻 𝑊𝑀𝐿 = 1 2⁄ (𝑆𝐻 + 𝐵𝐻) − 1 2⁄ (𝑆𝐿 + 𝐵𝐿)
In fund management, however, it is often not permitted to take a short position in stocks. The long-short strategy is a riskier strategy since not only does the manager need to select stocks that are anticipated to go up, he also has to predict which stocks are going down. This thesis focuses on the long-only strategy, since it is written in the environment of an investment bank that has no mandate to use long-short.
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4.
E
MPIRICALR
ESULTSThe first part of this chapter explains which indicators are used to analyse the performance of the factors. Next, the main results are presented. They show that most factors in the equity market outperform the market in most regions, but that it cannot be stated that these higher returns are not just a compensation for risk.
4.1APPROACH
The previously introduced portfolios, market, Size, Value, Momentum and an equally weighted portfolio, consisting of one third of each of the factors Size, Value and Momentum (EW), are analysed using a number of different performance indicators. These performance indicators include the return, the standard deviation, the skew, the kurtosis, the maximum drawdown, the Calmar ratio and the Sharpe ratio. The reported returns relate to annualized geometric averages. The standard deviation is a measure of the investment’s volatility or risk. It informs the investor how much a series deviates from its mean. The more the data are spread apart, the higher the deviation and the greater the risk.
The skew measures to what degree the distribution of the return is asymmetrical around its mean. A positive skew indicates that the right-hand tail of the distribution is longer. For financial investments this means that small losses occur frequent and extreme gains rarely occur. A negative skew indicates that the left-hand tail of the distribution is longer. For financial investments this means that there is a greater chance at extremely negative outcomes. Investors prefer a positive skew, since in this case the probability of extreme losses is smaller. The kurtosis measures to what degree the distribution of the return is more or less peaked in comparison to the normal distribution. A high kurtosis indicates that the distribution has fat tails. An investment characterized by high kurtosis means that the probability of extreme outcomes compared to a normal distribution is high. A low kurtosis indicates a relatively peaked distribution with thin tails. An investment characterized by low kurtosis means that the probability of extreme outcomes is low. Investors prefer a low kurtosis on the negative side and a high kurtosis on the positive side.
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The maximum drawdown, denoted by MD, is the maximum percentage ‘peak-to-trough’ losses suffered over the entire period. Figure 4.1 illustrates this concept. It is measured from the time a peak is reached to when a new trough is reached. Investors, who believe that observed loss patterns over longer periods of time are the best available proxy for actual exposure, interpret maximum drawdown as a measure of an investments financial risk.
Figure 4.1 Maximum Drawdown7
The Calmar ratio is the ratio of the annualized geometric average return to the maximum drawdown.8 The lower the Calmar ratio, the worse the investment performed on a
risk-adjusted basis over a specific time period and the higher the Calmar ratio, the better the investment performed.
The Sharpe ratio is the ratio of the annualized geometric average return minus the annualized average risk-free rate (the numerator) to the standard deviation of the same data series (the denominator).9 It is a measure for calculating risk-adjusted returns. The greater the value of
the Sharpe ratio, the more attractive the risk-adjusted return is. A negative Sharpe ratio indicates that a risk-free asset would perform better than the analysed risky investment.
7 The source is http://investexcel.net/maximum-drawdown-vba/
8 The drawdown time period is generally three years, but in this thesis the entire period is used. The
probability of a large drawdown over a longer period is greater than over a relatively small period. The average return is thus divided by a larger number and is therefore a conservative interpretation of this financial risk indicator.
9 The annualized risk-free rate, 𝑟
𝑓, is calculated in the same way as the annualized geometric average and is subtracted from the average return, because this way the performance associated with taking risk can be isolated; a portfolio with zero risk, such as an investment for which the expected return is the risk -free rate, has a Sharpe ratio of zero.
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4.2RESULTS
Table 4.1 shows the indicators, explained in the section above, for the five long-only portfolios for all four regions for the period November 1990 to April 2015. The first observation that strikes is that the average returns in Japan are substantially lower compared to returns in other regions. This is due to the fact that in the 1990s Japan experienced a banking crisis, causing the economy to collapse for an extended period. The results also illustrate that on average the market portfolio performed worse than all factor portfolios and the EW portfolio in all four regions, except for the Size portfolio in Europe (Figure A.1 in the Appendix illustrates this graphically). The factor portfolio Momentum performed best in all regions, except in Japan.10 The results range from 14.99% in North America, which is 37% higher than
the market, to 13.80% in Europe, which is 63% higher than the market and to 15.47% in Asia Pacific, which is 44% higher. The test statistics in table A.5 in the Appendix also illustrates that the outperformance of factor portfolios that outperformed the market is statistically significant. On the other hand, the results show that an investment in one of the factor portfolios or in the EW portfolio would have given a higher standard deviation than an investment in the market portfolio; it appears that higher returns correlate with higher risk. The Sharpe ratios, however, are higher than the Sharpe ratio of the market. This indicates that on a risk-adjusted basis, the factor outperformed the market. Section 3 of this chapter goes into more detail about this and checks whether the outperformance, on a risk-adjusted basis, of the factor portfolio against the market portfolio, is statistically significant.
For North America and Europe, the skews of all portfolios are negative, but only to a limited degree. This indicates that the left-hand tail of the distribution is slightly longer than the right-hand tail and that the chance at high negative returns is somewhat higher than would be the case if the returns were perfectly symmetrically distributed. The skews for the Japanese portfolios are somewhat higher than zero, which indicates that small losses occur quite frequently but the chance at an extreme positive outcome is also higher. The skews of the portfolios in Asia Pacific also deviate from zero, but again to a limited degree. The kurtosis estimates show that all return distributions have fatter tails than would be the case for a
10 This corresponds with results from earlier literature where authors have shown that in Japan, the
Momentum factor is hardly present. Chui, Titman and Wei (2010) state that momentum returns are stronger in cultures that value individualism and that Japan ranks low on individualism, which explains the absence of momentum returns.
Table 4.1 Summary statistics for explanatory returns: November 1990-April 2015, 294 months
This table shows performance statistics of the market and the Size, Low-Risk, Value and Momentum factors for North America, Europe, Japan and Asia Pacific over the period November 1990 - April 2015. The market factor is the return on a region's value-weight market portfolio minus the U.S. one month T-bill rate. For Size and Value, all stocks in a region are sorted into two market cap and three book-to-market equity (B/M) groups at the end of June of each year t. Big stocks are those in the top 90% of June market cap for the region, and small stocks are those in the bottom 10%. The B/M breakpoints for big and small stocks in a region are the 30th and 70th percentiles of B/M for the big stocks of the region. For Momentum, all stocks in a region are sorted into two market cap and three lagged momentum return groups at the end of each month t. Again, big stocks are those in the top 90% of market cap for the region, and small stocks are those in the bottom 10%. For portfolios formed at the end of month t–1, the lagged momentum return is a stock's cumulative return for t–12 to t–2. The momentum breakpoints for all stocks in a region are the 30th and 70th percentiles of the lagged momentum returns of the region's big stocks. The equally weighted multi-factor portfolio is an equally weighted combination of Size, Value and Momentum.
Market
North
Size
America
Value Momentum EW Market Size
Europe Value Momentum EW Average return 10.91% 14.83% 13.81% 14.99% 14.88% 8.48% 8.05% 11.60% 13.80% 11.32% Standard Deviation 14.78% 21.29% 16.92% 18.80% 17.34% 17.12% 17.45% 20.70% 17.70% 17.69% Skew -0.76 -0.27 -0.52 -0.31 -0.79 -0.65 -0.54 -0.33 -0.63 -0.64 Kurtosis 4.64 5.57 6.09 5.45 5.17 4.80 6.23 5.68 5.01 5.72 Maximum Drawdown 50.96% 59.29% 61.27% 52.65% 56.75% 58.91% 60.98% 65.36% 54.60% 60.43% Calmar Ratio 0.21 0.25 0.23 0.28 0.26 0.14 0.13 0.18 0.25 0.19 Sharpe Ratio 0.54 0.56 0.65 0.65 0.69 0.33 0.30 0.42 0.62 0.48 Market Size Japan
Value Momentum EW Market
Asia Size Pacific Value Momentum EW Average return 1.25% 2.85% 4.46% 2.72% 3.57% 10.74% 14.60% 12.77% 15.47% 14.65% Standard Deviation 19.64% 24.75% 21.67% 20.54% 21.29% 20.69% 25.66% 25.50% 22.23% 23.06% Skew 0.27 0.56 0.43 0.28 0.39 -0.42 -0.28 0.18 -1.03 -0.54 Kurtosis 3.45 4.62 4.21 3.52 3.85 5.41 5.27 7.13 6.36 5.87 Maximum Drawdown 58.05% 68.85% 63.02% 52.76% 61.22% 60.17% 68.74% 66.28% 64.84% 64.96% Calmar Ratio 0.02 0.04 0.07 0.05 0.06 0.18 0.21 0.19 0.24 0.23 Sharpe Ratio -0.08 -0.00 0.07 -0.01 0.03 0.38 0.46 0.39 0.57 0.51
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normal distribution (the kurtosis of a normal distribution is 3). This is usual for financial data, because it is much more likely that prices/returns will go to extremes, yielding fatter tails. In all regions, the maximum drawdowns are high (Figure A.2 in the Appendix illustrates this graphically). These losses were all suffered during the financial crisis of 2007. In Japan, however, the losses were suffered during the banking crisis of the 1990s.
4.3RISK ADJUSTMENTS
As was indicated by the results in the previous section, higher returns seem to correlate with higher standard deviations. The question whether these higher returns are a compensation for extra risk, arises. Therefore, two concepts that refine an investment’s return are used to measure how much risk is involved in producing the return. For the first risk-return measure the following regression is performed:
𝑅𝑖− 𝑅𝑓 = 𝛼𝑖+ 𝛽𝑖(𝑅𝑀− 𝑅𝑓)
where 𝑅𝑖 is the return on a factor portfolio, 𝑅𝑓 is the risk-free rate and 𝑅𝑀 is the return on the market portfolio. This regression can be seen as a CAPM regression, with 𝑅𝑖− 𝑅𝑓 the factor portfolio risk premium (i.e. the excess return on a factor portfolio over the risk-free rate). First of all, the alpha is a measure of performance on a risk-adjusted basis, called the active return on an investment. It is also referred to as Jensen’s alpha. In an efficient market, the expected value of the alpha is zero.11 The alpha is an indication of how an investment
performed after accounting for the risk it involved. A positive alpha means that the investment has a return in excess of the reward for the assumed risk, an alpha of zero indicates that the investment has earned a return that is adequate for the risk taken, and a negative alpha means that the investment has earned too little for its risk. Also, it is interesting to see if the alpha is significantly different from zero. Therefore, the t-statistic is also given. If the alpha is positive and significantly different from zero this means that an investment in the factor portfolio has an excess return that cannot be attributed to market returns, and thus is independent of the market returns.
11 Investopedia.com explains ‘Efficient market hypothesis’ in the following way: In an efficient market, “it is
impossible to beat the market because stock market efficiency causes existing share prices to always incorporate and reflect all relevant information”.
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Second, the beta of a portfolio is a measure of the volatility (or systematic risk) relative to the market. The measure can be seen as the tendency of the portfolios return to respond to swings in the market. A beta of 1 indicates that the portfolio’s return will move with the market, a beta of less than 1 indicates that the portfolio will be less volatile than the market, whereas a bet a that is greater than 1 means that the portfolio will be more volatile.
The other risk-return measure is based on the Sharpe ratio. As was stated in the first section of this chapter, the Sharpe ratio measures how much the return increases (decreases) relative to its risk. The higher the Sharpe ratio, the better the portfolio performed. Comparing the Sharpe ratio of one of the factor portfolios to the Sharpe ratio of the market portfolio can indicate if in fact a factor portfolio performed better than the market on a risk-adjusted basis. The status quo test is the one of Jobson and Korkie (1981), called the JK test. Based on the JK-statistic, the null hypothesis that two Sharpe ratios are equal can be tested.12 However, as Ledoit and Wolf
(2008) argue, this test is not valid when returns have heavier tails than the normal distribution or are of time series nature. The results in the previous section indicate that the distributions of the returns show excess kurtosis, which implies that the distributions have indeed heavier tails and are not normally distributed. Therefore, statistical inference about the distribution obtained from the JK test is not valid. The authors suggest an alternative inference method, namely a block bootstrap. Since the original dataset is just a sample of the total population (the total population consists of returns of all stocks over the entire history), the true distribution is unknown. By bootstrapping, inference about a population can be modelled based on a sample. The sample is resampled, so that the 'population' is in fact the sample, and the sample is in fact the resample. From the resample, inference about the sample (i.e. the ‘population’) can be obtained.
Ledoit and Wolf (2008) distinguish between i.i.d data (independent and identically distributed data) and time series data. It is well known that financial returns are generally not i.i.d. The figures in the Appendix (Table A.7) illustrate that this also applies for the returns of factor portfolios. Although the correlograms of the returns show no autocorrelation (all p-values are
12For the statistic, the asymptotic distribution is normal with mean 𝑆𝑅̂
𝑖𝑛= 𝑠𝑛𝑟̅𝑖− 𝑠𝑖𝑟̅𝑛 and variance given by 𝜃̂ =1𝑇[2𝑠𝑖2𝑠𝑛2− 2𝑠𝑖𝑠𝑛𝑠𝑖𝑛+12𝑟̅𝑖2𝑠𝑛2+12𝑟̅𝑛2𝑠𝑖2−2𝑠𝑟̅𝑖𝑟̅𝑛
𝑖𝑠𝑛(𝑠𝑖𝑛 2 + 𝑠
𝑖2𝑠𝑛2)], with 𝑟̅𝑖 (𝑟̅𝑛) the sample estimator of the mean of the factor (market) portfolio return and 𝑠 the estimated covariance matrix. The JK-statistic is given by 𝑆𝑅̂𝑖𝑛
√𝜃̂. The null hypothesis is rejected when the statistic exceeds the 95% critical value of the normal distribution, which is 1.645.
18
very high), the correlograms of the squared returns do show autocorrelation, implying there is volatility clustering.13 Therefore, the returns are not i.i.d. and the block bootstrap method for
time series data is used.
The block bootstrap for time series data consists of two procedures. In the first procedure, 𝑀 new sequences of return pairs are generated from the original return pairs (𝑟1𝑖, 𝑟1𝑛)′, … , (𝑟𝑇𝑖, 𝑟𝑇𝑛)′, with 𝑇 as the total number of months. An [𝑀𝑥𝑇]-matrix is obtained, where each sequence contains 𝑙 different blocks of return pairs, with 𝑙 =𝑇
𝑏 and 𝑏 as the block size. Each block in each sequence is obtained by randomly selecting a number between 1 and 𝑇, with replacement. This number is the index of the return pair of the original data. The authors use blocks in order to keep autocorrelations, volatility clustering and covariances intact. To illustrate this, the [𝑀𝑥𝑇]-matrix is given below:
[ [𝑟1,𝑖∗ ⋯ 𝑟𝑏,𝑖∗ 𝑟1,𝑛∗ ⋯ 𝑟 𝑏,𝑛∗ ]1,1= [ 𝑟𝑡,𝑖 ⋯ 𝑟𝑡+𝑏−1,𝑖 𝑟𝑡,𝑛 ⋯ 𝑟𝑡+𝑏−1,𝑛]
⋯
[ 𝑟𝑇−𝑏,𝑖∗ ⋯ 𝑟𝑇,𝑖∗ 𝑟𝑇−𝑏,𝑛∗ ⋯ 𝑟 𝑇,𝑛∗ ]1,𝑙= [ 𝑟𝑠,𝑖 ⋯ 𝑟𝑠+𝑏−1,𝑖 𝑟𝑠,𝑛 ⋯ 𝑟𝑠+𝑏−1,𝑛]⋮
⋮
[𝑟1,𝑖 ∗ ⋯ 𝑟 𝑏,𝑖∗ 𝑟1,𝑛∗ ⋯ 𝑟 𝑏,𝑛∗ ]𝑀,1 = [ 𝑟𝑤,𝑖 ⋯ 𝑟𝑤+𝑏−1,𝑖 𝑟𝑤,𝑛 ⋯ 𝑟𝑤+𝑏−1,𝑛]⋯
[ 𝑟𝑇−𝑏,𝑖∗ ⋯ 𝑟 𝑇,𝑖∗ 𝑟𝑇−𝑏,𝑛∗ ⋯ 𝑟𝑇,𝑛∗ ] 𝑀,𝑙 = [𝑟𝑟𝑧,𝑖 ⋯ 𝑟𝑧+𝑏−1,𝑖 𝑧,𝑛 ⋯ 𝑟𝑧+𝑏−1,𝑛]]with 𝑡, 𝑠, 𝑤 and 𝑧 random numbers from the sequence 1, … , 𝑇. It is possible that any of these random number equals 𝑇. For this special case the authors extended the sequence with 1, … , 𝑏 − 1, so that it looks like 1, … , 𝑇, 1, … , 𝑏 − 1. If one of the random number equals 𝑇, than the second return in that specific block equals the first original return.
For each of the 𝑀 sequences, the JK-statistic is calculated. This statistic is defined in the following way:
|𝛥̂∗− 𝛥̂| 𝑠(𝛥̂∗)
with 𝛥̂ = 𝑆𝑅𝑛− 𝑆𝑅𝑖, the difference in Sharpe ratios between the market portfolio and one of the factor portfolios, computed from the original data, 𝛥̂∗ the estimated difference computed from the bootstrap data and 𝑠(𝛥̂∗) the standard error for 𝛥̂∗.14 Next, the resulting 𝑀 statistics are ordered
13 Volatility clustering is the phenomenon that periods of high volatility tend to be followed by periods with
high volatility (in absolute value).
19
from small to large, which gives an empirical distribution of the statistic. Finally, the critical value is calculated and the null hypothesis of no difference in Sharpe ratios between the factor portfolio and the market portfolio can or cannot be rejected. The null hypothesis is rejected if the JK-statistic from the original data exceeds the bootstrapped critical value.15
In the second procedure, the optimal block size 𝑏 is computed. The first step in this procedure is to fit a 𝑉𝐴𝑅(1) model in conjunction with bootstrapping the residuals to the original return pairs.16 Next, 𝐵 reasonable block sizes are chosen. According to the fitted 𝑉𝐴𝑅(1) model, 𝐾 new
datasets of return pairs are generated. For each dataset and each block size, the block bootstrap described in the first procedure, is performed. From this a [𝐾𝑥𝐵]-matrix is obtained with on each entry a confidence interval 𝐶𝐼𝑘,𝑏. If 𝛥̂ lies in this interval, the entry is given value 1 and if 𝛥̂ does not lie in the interval, the entry is given value 0. For each block size 𝑏 (i.e. for each column), the number of ones is added and divided by 𝐾. This number is the percentage that indicates how often 𝛥̂ lies in the confidence interval and is defined in the following way:
1 − 𝛼̂ = #{𝛥̂ ∈ 𝐶𝐼𝑘,𝑏}/𝐾,
where 𝑏 is one of the block sizes and is fixed and 𝑘 = 1, … , 𝐾. Finally, the block size that is closest to the nominal confidence level 1 − 𝛼 (i.e. minimize |1 − 𝛼̂ − (1 − 𝛼)|) is the optimal block size.
4.3.1 Single-factor portfolios
The t-statistics in Table 4.2 illustrate that in North America and Europe, the alphas of the Value and Momentum portfolios are statistically significant. This indicates that in these regions, there is a significant excess return over the market when investing in one of these single-factor portfolios. For North America the alphas range from 0.22% to 0.32% and for Europe they range from 0.22% to 0.44%. Surprisingly, the alphas of the Size factor portfolios are never statistically significant, meaning that the hypothesis that the excess return is zero cannot be rejected. As could be expected, the alpha of the Momentum portfolio in Japan is not significant. The alpha of Value is 0.22% and is significant. Besides the alpha of the Size portfolio, the alpha
15The JK-statistic from the original data is: |𝛥̂−𝛥|
𝑠(𝛥̂). The calculation of 𝑠(𝛥̂) is also explained in the Appendix.
16The authors state that the motivation for bootstrapping the residuals is to account for some possible
20
Table 4.2 Risk-adjusted return statistics for explanatory returns: November 1990-April 2015, 294 months
This table shows the risk-adjusted return statistics of the Size, Value, Momentum and EW portfolios for North America, Europe, Japan and Asia Pacific over the period November 1990 - April 2015. The alpha and beta are the estimated CAPM-alpha and –beta. The t-stat indicates statistical significance at the 95% confidence level of a one-sided test whether the alphas of a factor portfolio are larger than 0. The JK-stat indicates statistical significance at the 95% confidence level of a one-sided test whether the Sharpe ratio of a factor portfolio is larger than the Sharpe ratio of the market portfolio. 𝐻0 rejected is No when 𝐻0 is not rejected and is Yes when 𝐻0 is rejected, based on the JK-stat*.
North
Size
America
Value Momentum EW Size
Europe Value Momentum EW alpha 0.25% 0.22% 0.32% 0.26% 0.04% 0.22% 0.44% 0.23% beta 1.19 1.03 1.05 1.09 0.89 1.13 0.94 0.99 t-stat 1.24 1.80 1.81 2.47 0.25 1.78 3.61 2.62 JK-stat 0.36 0.96 0.94 1.65 0.24 1.20 2.47 2.01 𝐻0 rejected No No No No No No Yes No Size Japan Value Momentum EW Asia Size Pacific Value Momentum EW alpha 0.22% 0.29% 0.14% 0.22% 0.33% 0.12% 0.39% 0.28% beta 1.07 1.01 0.96 1.01 1.06 1.15 0.99 1.07 t-stat 0.99 1.97 0.97 1.67 1.50 0.78 2.72 2.53 JK-stat 0.98 1.73 0.89 1.61 0.82 0.36 1.67 2.04 𝐻0 rejected No No No No No No No No
21
of the Value portfolio is also not significant in Asia Pacific. This is hardly a surprise, since the outperformance is just 2.03%, whereas the risk increases by 5%.
The betas in Table 4.2 show that in all regions, an investment in the Value portfolio is somewhat more volatile than an investment in the market portfolio. In North America and Japan, the betas of the Size portfolios are the highest, indicating that in these regions, the returns on these portfolios are more volatile than the returns on the market. In Europe and Asia Pacific the betas of the Value portfolios are the highest.
Based on the bootstrapped critical values, the null hypothesis that there is no difference between the Sharpe ratios of a factor portfolio and the market portfolio is never rejected, except in the case of the European Momentum portfolio. Not rejecting the null hypothesis in 95% of the cases does not make the null hypothesis true. So it cannot be stated that the two Sharpe ratios are equal; it can only be stated that by means of an (improved) hypothesis test, the Sharpe ratio of one of the factor portfolios is not significantly higher than the market Sharpe ratio. This means that investing in factors increases the return, but it also increases the risk substantially. The results also show that the null hypothesis, based on the general JK test, is rejected more often. According to this test, the Sharpe ratio of the EW portfolio is significantly higher than the market Sharpe ratio in all regions, except for Japan. In Europe and Asia Pacific, the Sharpe ratio of the Momentum portfolio is also significantly higher and in Japan, the Value Sharpe ratio is significantly higher. These findings indicate that the bootstrap method is more conservative than the general method; the null hypothesis is rejected less often. If the general JK test is followed, the null hypothesis tends to over-reject. Therefore assuming that outperformance of factors is not just a compensation for risk, is often not vali d. Investment banks that follow the results of the general JK-test can therefore unintentionally increase their clients’ risk. Also, the conclusions other researches made (among others Koedijk et al. (2013) and Houweling et al. (2014)) were premature, since they based their results on the general (over-rejecting) JK test.
4.3.2 Multi-factor portfolios
The surprising results of the previous section show that, from an investment bank point of view, it probably is not optimal to invest in just one of the factor portfolios. The relative risk is high (Table
22
A.5 shows high tracking errors) and the Sharpe ratios are not significantly higher than the market Sharpe ratios. To determine diversification possibilities, the pairwise correlations are given in Table A.6 in the Appendix. Panel A shows that most return correlations tend to be in the range of 80% to 95%. As Houweling et al. argue, this reflects the common exposure to the market of the long-only factor portfolios. Panel B of Table A.6 shows the correlations between the factor portfolio’s outperformances versus the market. All correlations are 65% or lower, indicating that the factor portfolios capture different effects. By combining factor portfolios in one multi-factor portfolio, an investor can benefit from diversification; the results in Table A.5 in the Appendix show that the equally weighted multi-factor portfolio has a lower tracking error than each of the single-factor portfolios. However, as the results in Table 4.2 show, it cannot be stated that the higher returns are not a compensation for risk. Thereby indicating that the portfolio is not optimally diversified. By strategically setting the weights of the factor portfolios in the multi-factor portfolio, an investor can diversify his portfolio in a more sophisticated way. For each region, based on the correlations in Panel B of Table A.6 in the Appendix and all previous results, the weights in the multi-factor portfolios are adjusted. High correlations between two factors ensure lower weights in this strategically weighted multi-factor portfolio and negative correlations ensure higher weights. Table 4.3 shows the allocation of the weights, along with the risk-adjusted return statistics.
Table 4.3 Weights and risk-adjusted return statistics for explanatory returns: November 1990-April 2015, 294 months
This table shows the adjusted weights of the multi-factor portfolio (Panel A) and risk-adjusted return statistics of SW portfolio for North America, Europe, Japan and Asia Pacific over the period November 1990 - April 2015 (Panel B). The statistics are calculated in the same way as was done in Table 4.1 and Table 4.2.
Panel A
North America Europe Japan Asia Pacific
Size 10% 20% 15% 15%
Value 45% 55% 70% 50%
Momentum 55% 25% 15% 35%
Panel B
North America Europe Japan Asia Pacific
Average return 16.20% 11.60% 4.09% 14.30%
alpha 0.32% 0.24% 0.26% 0.25%
t-stat 2.99 2.67 1.94 2.42
JK-stat 2.24 2.08 1.74 2.02
23
The results in Table 4.3 show that it is possible to construct a multi-factor portfolio with strategically chosen weights that have statistically significant alphas and Sharpe ratios. From an investors point of view this means that investing in not just one factor but in multiple factors can increase the return without increasing the risk too much. Also, for investment banks whose clients are risk-averse and who must follow a strict mandate, a strategy that involves investing in more than one factor could be an improvement.
24
5.
R
OBUSTNESSC
HECKSThis section checks whether the findings are robust to the portfolio weighting and the performance over time. First, the portfolios used to construct the factors are evaluated based on market-value weighted portfolios instead of on equally weighted portfolios.17 This means that the factor portfolios
do not benefit from the Size effect, because smaller firms are now given less weight. Second, in North America, Europe and Asia Pacific the results could be driven by the higher market volatility since 2007; Figure A.1 in the Appendix shows that after the financial crisis the market volatility increased. To check if the significance of the Sharpe ratio is nullified by this crisis, the sample period is split up in two sub periods of equal length: November 1990-January 2003 and February 2003-April 2015. In Japan, it seems like the market volatility never stabilized during the entire period. The period is split up in the same two sub periods.
As the results in Panel A of Table 4.4 indicate, the returns of almost all portfolios decreased. For the strategically weighted (SW) multi-factor portfolios, the returns dropped from 16.20% to 13.56%, from 11.60% to 9.82%, from 4.09% to 3.84% and from 14.49% to 12.43% for North America, Europe, Japan and Asia Pacific respectively. Most single-factor alphas lose their positive effect, and thereby their significance. There are only three alphas left that are significant, so the alphas do not seem robust to the portfolio weighting. For the SW portfolio the t-statistics changed from 2.99 to 1.66, from 2.67 to 1.65, from 1.94 to 2.19 and from 2.44 to 2.02. These alphas are still significant, which indicates that the alphas of the strategically weighted multi-factor portfolios are robust to portfolio weighting. However, none of the portfolios, including the strategically weighted multi-factor portfolios, pass the Sharpe ratio test. This implies, again, that investing in one of the factor portfolios increases the average return, but it also increases the risk to the extent that an investor or investment bank should question if investing in a factor is worth the extra risk. The results for the single-factor portfolios are robust to the portfolio weighting, because based on this alternative weighting the outperformance of factors is still a compensation for risk. The results for the multi-factor portfolio, however, are not robust because they now fail to pass the Sharpe ratio test.
25
Table 4.4 Robustness of risk-adjusted return statistics for explanatory returns: November 1990-April 2015, 294 months
This table shows the robustness of the risk-adjusted return statistics of the Size, Value, Momentum and SW portfolios for North America, Europe, Japan and Asia Pacific over the period November 1990 - April 2015. All statistics are calculated in the same way as was done in Table 4.1 and Table 4.2. Panel A shows the results based on market weighted portfolios over the period November 1990 - April 2015. Panel B shows the results based on the equally weighted portfolios over two sub samples: Sub period 1 contains data from November 1990-January 2003 and Sub period 2 contains data from February 2003-April 2015.
Panel A
Market Weighted
North
Size
America
Value Momentum SW Size
Europe Value Momentum SW Average return 12.06% 10.71% 13.35% 13.56% 8.08% 9.47% 11.27% 9.82% alpha 0.05% 0.00% 0.21% 0.14% 0.01% 0.05% 0.26% 0.09% t-stat 0.29 0.01 1.32 1.66 0.06 0.51 2.02 1.65 JK-stat 0.45 0.53 0.46 1.21 0.31 0.20 1.27 1.29 𝐻0 rejected No No No No No No No No Size Japan Value Momentum SW Asia Size Pacific Value Momentum SW Average return 0.52% 4.65% 2.19% 3.84% 8.05% 13.24% 14.67% 13.27% alpha 0.00% 0.30% 0.10% 0.23% -0.18% 0.17% 0.35% 0.18% t-stat 0.02 2.23 0.65 2.19 -1.16 1.02 2.32 2.02 JK-stat 0.00 1.98 0.54 2.01 1.59 0.60 1.52 1.88 𝐻0 rejected No No No No No No No No Panel B Sub period 1 North America
Size Value Momentum SW
Europe
Size Value Momentum SW
Average return 16.34% 15.04% 17.91% 18.73% 5.32% 10.40% 11.65% 9.86%
alpha 0.43% 0.39% 0.54% 0.55% -0.04% 0.34% 0.43% 0.28%
t-stat 4.24 5.51 6.35 3.06 -1.42 6.18 8.00 2.09
JK-stat 0.62 1.11 1.23 2.39 0.40 1.34 1.82 1.86
26
Table 4.4 (Continued) Robust statistics for explanatory returns: November 1990-April 2015, 294 months
Japan
Size Value Momentum SW
Asia Pacific
Size Value Momentum SW
Average return -6.61% -2.77% -3.22% -3.24% 10.98% 7.67% 12.31% 10.20% alpha 0.05% 0.25% 0.11% 0.20% 0.36% 0.06% 0.42% 0.23% t-stat -1.22 2.46 2.61 0.86 3.03 0.21 6.12 1.34 JK-stat 0.43 1.03 0.69 0.95 0.72 0.04 1.15 1.15 𝐻0 rejected No No No No No No No No Sub period 2 North America
Size Value Momentum SW
Europe
Size Value Momentum SW
Average return 13.34% 12.59% 12.14% 13.73% 10.86% 12.81% 15.98% 13.37% alpha 0.04% 0.01% 0.10% 0.08% 0.09% 0.05% 0.45% 0.16% t-stat 0.22 0.10 0.55 0.68 0.47 0.38 2.80 1.56 JK-stat 0.21 0.21 0.00 0.35 0.04 0.14 2.08 1.10 𝐻0 rejected No No No No No No No No Japan
Size Value Momentum SW
Asia Pacific
Size Value Momentum SW
Average return 13.26% 12.22% 9.02% 11.99% 18.35% 18.11% 18.72% 18.56%
alpha 0.42% 0.34% 0.10% 0.32% 0.18% 0.25% 0.31% 0.26%
t-stat 1.86 2.46 0.52 2.45 0.67 1.67 1.80 2.28
JK-stat 1.22 1.77 0.05 1.71 0.13 1.48 1.23 2.09
27
The first matter that strikes in Panel B of Table 4.4 is that the returns of the Japanese and Asian Pacific portfolios increased substantially during the second period. For the SW portfolio they increased from -3.24% to 11.99% and from 10.20% to 18.56%. The returns of all Northern American portfolios and most returns of the European portfolios, however, decreased during the second period. The results in Panel B of Table 4.4 also show that the alphas that were significant over the whole period remained significant over the first sub period. However, in North America and Europe all alphas, except the European Momentum alpha, lost their significance during the second sub period, which indicates that these regions suffered from the financial crisis. On the other hand, in Japan and Asia Pacific, the alphas that were significant during the first period remained significant during the second period, except for the Japanese Momentum alpha and the Asian Pacific Size alpha. Also, all alphas that were not significant became significant during the second period. The alphas of the Northern American and European SW portfolios decreased from 0.55% to 0.08%, from 0.28% to 0.16%, losing their significance, whereas the alphas of the Japanese and Asian Pacific SW portfolios increased from 0.20% to 0.32% and from 0.23% to 0.26, gaining significance. These results imply that the alphas are not robust to the performance across sub periods. However, again, none of the Sharpe ratios are significantly higher than the market Sharpe ratio when bootstrapped, indicating that the results for the single-factor portfolios are robust across sub periods, but the results for the multi-factor portfolios again, are not.
The results indicate that there are substantial differences in performances across the sub periods. The Chow test checks if the coefficients in the linear regressions on the two different datasets are equal. Table 4.5 shows the p-values of the Chow test. A p-value smaller than 0.05 implies that the coefficients are not equal; a better way to estimate the CAPM-alpha and beta is by performing two separate regressions instead of a single regression. As the results indicate, in all regions, except in Japan, the alphas of Value and EW were not constant over time. This implies that there is a structural break in the excess return and that the financial crisis ensured a lower excess return over the entire period. In Asia Pacific, there were breaks in all excess returns. In Japan, there were no breaks, indicating that the financial crisis was not an event that had effect on the excess return.
28
Table 4.5 P-values of Chow tests
This table shows the p-values of the Chow tests based on the regressions of one of the factor portfolios on the market portfolio. The breakpoint is set at January 2003.
Size Value Momentum EW
North America 0.0738 0.0000 0.4951 0.0001
Europe 0.0945 0.0000 0.7977 0.0002
Japan 0.3071 0.9457 0.2655 0.7790
29
6.
C
ONCLUSIONResearch on investing in factors, like Size, Value and Momentum has widely increased over the past years. Many researchers, including Koedijk et al. (2013) and Houweling et al. (2014), have shown that, based on historical data, investing in factors outperforms the market in America and Europe; the average return on a factor portfolio is higher than the average return on the market portfolio. The results in this thesis indicate that in the four regions, North America, Europe, Japan and Asia Pacific, factors also outperform the market. In North America, Europe and Asia Pacific the best performing factor is the Momentum factor, with an outperformance of 4.08%, 5.32% and 4.37% over the market. In Japan, investing in the Value factor would have given the highest outperformance, namely 3.21%. However, the standard deviation (i.e. the risk) also increases when investing in a factor. So the question whether the outperformance of factors over the market is just a compensation for increasing risk, arises.
To answer this question two concepts that refine an investment’s return are used to measure how much risk is involved in producing the return. First, a regression of a factor portfolio on the market portfolio is performed. The obtained alpha and beta indicate how much excess return the factor generated over the market and to what extent the returns are more or less volatile than the market returns. The alpha of the Size portfolio in all regions is never statistically significant, which indicates that on average, Size does not generate an excess return over the market. So even though, the risk increases there is no significant outperformance. In North America and Europe, all other alphas are significant, indicating excess returns over the market. In Japan (Asia Pacific), investing in the Value (Momentum) factor generated a significant excess return of 0.29% (0.39%).
The other concept to measure if the higher average returns are not just a compensation for extra risk is based on the Sharpe ratio. The Sharpe ratio, which is the ratio between the average return and the standard deviation (i.e. the risk), is a measure for risk-adjusted returns. The higher the Sharpe ratio, the more attractive the investment is. The results show that the Sharpe ratios of the factor portfolios are higher than the Sharpe ratios of the market portfolios. If they are significantly higher than the market Sharpe ratios, it is valid to state that the higher average returns of the factor portfolios are not just a compensation for extra risk. The general test to check their significance is the Jobson and Korkie (JK) test. However, following Ledoit
30
and Wolf (2008), this test is not valid since the returns have heavier tails than the normal distribution and are of time series nature. The authors provide an alternative method to check the significance of the Sharpe ratio: a block bootstrap. The results indicate that this alternative is more conservative than the general JK test, since it never (with the exception of Momentum in Europe) rejects the null hypothesis that the Sharpe ratio of one of the factor portfolios is equal to the Sharpe ratio of the market portfolio, whereas the general test does reject the hypothesis in some cases. When the null hypothesis is not rejected, it implies that the Sharpe ratio of a factor portfolio is not significantly higher than the market Sharpe ratio; it cannot be stated that the higher return generated by investing in the factor is not caused by an increase in risk.
The alphas are not robust to checks regarding the portfolio weighting and to the performance across sub periods. The results show that when the portfolio weighting changes from equally weighting to market-value weighted, most alphas lose their significance. Also, when the period is split up in two sub periods, alphas that are not significant in the first period, gain significance in the second period and vice versa. The results of the alphas do not seem robust. The results of the Sharpe ratio test, however, are robust to these checks. Changing the portfolio weighting and splitting the period has no effect on the results; there is no evidence that the outperformance of factors over the market is not a compensation for risk.
In conclusion of the findings in section 4.3.2 regarding the significance of the Sharpe ratio test resulting from the strategically weighted multi-factor portfolios, it would be recommendable to develop an algorithm. This algorithm could take into account the weights of the factors in the portfolio in such a way that the Sharpe ratio of this SW multi-factor portfolio is significantly higher than the market Sharpe ratio. The algorithm could be written on the basis of mean-variance optimization. However, this is beyond the scope of this particular thesis.
31
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