Quarter Past Investing: The Timing Ability of Value Stock Investors
University of Groningen Faculty of Business and Economics
Master of Finance
Supervisor: Dr. Jakob Bosma By: Alexander Obenaus
S3478645
I Content
I. Acknowledgment ... II II. List of Tables ... III III. List of Abbreviations ... IV IV. Abstract ... V
1. Introduction... 1
2. Literature Review ... 3
3. Methodology and Hypothesis ... 10
3.1 Stock selection ... 10
3.2 Performance Gap and Controlling for the Hindsight Effect ... 11
3.3 Research Hypothesis ... 13
4. Data ... 14
5. Results ... 18
5.1 Performance of value stock investors ... 18
5.2 The impact of value stocks on the performance gap and hindsight effect ... 20
II I. Acknowledgment
For valuable advice throughout my Master thesis, I would like to express my gratitude to my supervisor Dr. Jakob Bosma. Through discussions and his feedback, I received helpful guidance on how to conduct my research.
Additionally, I would like to thank my family and friends for their support throughout the entire year of my Master. In stressful times, their words gave me the motivation to continue putting in my all efforts and energy.
Thank you. Sincerely,
III II. List of Tables
Table 1 Descriptive statistics of the sample Table 2 Timing ability of value stock investors
Table 3 The impact of value stocks on the performance gap and hindsight effect Table 4 Method for determining the hindsight effect
Table 5 Descriptive statistics of growth stocks
IV III. List of Abbreviations
AMEX American Stock Exchange
BH Buy-and-hold
B/M Book-to-market
DW Dollar-weighted
GM Geometric
IRR Internal rate of return NYSE New York Stock Exchange OLS Ordinary least squares PTBV Price-to-book value
V IV. Abstract
1 1. Introduction
Value stocks, which outperform growth stocks by 7.68% per year, are companies with low earnings and exposed to distress (Fama and French, 1998) but there is no evidence whether investors purchase these stocks at the right time. Previous research demonstrates that the return of securities is different from the return earned by the skill of investors. A stock’s return is measured as the buy-and-hold (BH) return and implies that a stock is purchased and held over the entire investment period. In contrast, investors can buy and sell a stock during their investment horizon. By timing their investments well, investors can increase their returns, which are computed as the dollar-weighted (DW) return. This measure weighs returns stronger/less if the amount of invested capital is high/low, respectively. The difference between the BH return and the DW return is the performance gap and illustrates investors’ timing ability (Dichev, 2007). However, according to Hayley (2014), the measure of DW returns is prone to bias and hence estimated investor returns can be misleading. The author provides evidence that capital distributions can relatively change the weights of previous returns retrospectively. Therefore, it is required to account for this potential bias, the hindsight effect, when analysing the return of investors (Hayley, 2014). Hayley (2014) introduces a method that divides the performance gap into two parts: the hindsight effect and the actual timing effect of investors. Besides Muñoz and Vicente (2018), who examine the timing ability of mutual fund investors, previous research has not considered the hindsight effect. Furthermore, investors’ timing ability has not been analysed for individual stocks previously. However, since some investors follow a value strategy (Lakonishok, Shleifer, and Vishny, 1994) it is important and interesting for them to know how they time their investments. This becomes even more relevant when considering that the average market capitalization of value stocks increased tremendously between 1990 and 2017 (see table 1). This illustrates that a lot of capital entered the value stock market. Hence, even slightly lower DW returns (compared to BH returns) will be hurtful to investors. Due to the findings of previous research, that investors chase high returns and exhibit bad timing abilities, it can be argued that value stock investors time their investments badly. Additionally, the question arises to what degree the DW return of value stock investors is biased by the hindsight effect.
2 value stocks seem more attractive to them but due to bad timing, investors would reduce their returns. In contrast, it can be argued that value stock investors exhibit good timing abilities as value stocks are currently undervalued by the market (Lakonishok, Shleifer, and Vishny, 1994). Considering the findings of Hayley (2014) and Muñoz and Vicente (2018), that the hindsight effect is responsible for the major part of the performance gap, the aspect of value stock investors’ timing is further allured. Friesen and Sapp (2007) argue that the performance gap is negatively correlated with Fama and French’s high-minus-low (HML) factor, which implies that value stock investors time their investments well. However, Friesen and Sapp (2007) did not consider the hindsight effect. Additionally, Muñoz and Vicente (2018) identify that mid value-style funds are worse at timing than mid growth-style funds, whereas small value-style funds exhibit better timing than small growth-style funds. Since the authors have no explanation for these findings, the timing ability of value stock investors remains unanswered.
This study examines the timing ability of value stock investors for listed NYSE and NASDAQ stocks between 1990 and 2017. US stocks are analysed as Thomson Reuters Datastream (TDS) provides coverage for in the meanwhile dead US stocks and all active stocks. The time horizon of 28 years is selected as it represents the investment activity of an average investor (Keswani and Stolin, 2008). In particular, this study tries to answer the question if value stock investors exhibit bad timing skills. Additionally, it is desired to answer the question if the behaviour of value stocks is related to bad timing and biases the return of investors downwards. Answering these questions is done by selecting stocks according to their book-to-market (B/M) ratio. Fama and French (1993) construct their HML factor based on this ratio, where high B/M ratio stocks represent value stocks. As the B/M ratio of stocks can change over time, this study selects every December the current value stocks and examines for every stock the BH return, the DW return, the hindsight effect and the actual timing effect for the following year. Furthermore, in cross-sectional regressions, it is analysed if the bad timing is associated with stocks being value stocks.
3
1The definition “performance gap” was introduced by Friesen and Sapp (2007) and not used by Dichev
(2007). However, Dichev (2007) also calculates and reports the difference between the BH and DW return. regressed on the Fama and French three-factor model and the performance gap. In order to identify the effect of value stocks on the performance gap, an interaction variable of the HML factor with it is included. The positive interaction between the two variables implies that value stocks are not connected with bad timing. Thereafter the hindsight effect replaces the performance gap and the regression is applied again. The result of a negative interaction between the HML factor and the hindsight effect confirms that the returns of value stock investors are biased downwards. As a result, this study identifies that the return earned by the skill of value stock investors is better than reported and indicates the impact of the hindsight effect. Nevertheless, due to bad timing skills, value stock investors reduce their returns on average by -0.07% annually. Since the average market capitalization of value stocks increased by over ten times during the past 28 years (see table 1), a -0.07% lower return can be costly for investors. Therefore, value stock investors’ timing abilities are improvable.
2. Literature Review
Dichev (2007) addresses the difference between an investment’s (e.g. stocks) and an investor’s return. A stock’s return can be computed by its price in period 1 and its price in period 2. This return is calculated as the geometric average and implies a BH return. Such a return illustrates that the stock was purchased and held over the entire investment period. However, since investors buy and sell stocks during the investment period, the timing and amount of the additional capital flows affect the returns (Dichev, 2007). By investing at the right time, investors can increase their returns (Friesen and Sapp, 2007). DW returns capture the impact of these additional capital flows as periods with large invested capital are weighted more. Therefore, DW returns reflect the return that is earned due to the investor’s skill more precisely and hence a distinction is required between BH returns and DW returns (Dichev, 2007). Friesen and Sapp (2007) define the difference between the BH and DW return as the performance gap, which illustrates the timing ability of investors.
4 return as they illustrate how much capital was invested during which period (Dichev, 2007). In particular, Dichev (2007) determines the correlation between the distributions of a given year with the average returns during the past and future three years. With respect to the NYSE/AMEX stocks, he identifies that the distributions of a given year are significantly negatively correlated (-0.26%, p-value 0.027) with past returns and highly significantly positively correlated (0.51%, p-value < 0.001) with future returns. The NASDAQ stocks exhibit distributions that are highly significantly negatively (-0.57%, p-value 0.002) correlated with past returns and moderately positively (0.28%, p-value 0.159) correlated with future returns. This illustrates that when distributions are negative, which represent additional cash injections by investors, past returns were positive. Alternatively, when distributions are positive, which expresses capital distributions to investors, past returns were negative. In contrast, cash distributions are followed by positive future returns and cash injections are followed by negative future returns (Dichev, 2007). Therefore, Dichev (2007) concludes that investors time their investments badly as they invest after positive returns and before negative returns. On the other side, firms exhibit good timing as they distribute capital after negative returns and before positive returns (Dichev, 2007). According to Dichev (2007), these results are important for the field of finance as stock returns, which previous research articles considered the same as investors’ returns, are a key metric. He argues that the performance gap upwardly biases the equity risk premium of investors and the cost of equity of firms should be reduced by it. However, Hayley (2014) provides evidence that reducing the equity risk premiums is not justified by the investors’ bad timing ability. Nevertheless, the underperformance of investors’ returns implies that passive strategies, which avoid bad timing, might be superior (Dichev, 2007).
5
2 Muñoz and Vicente (2018) define the performance gap as the difference between the DW return and BH
return. Therefore, according to their definition, a negative performance gap implies bad timing. Thus, they report their gaps negatively. For the readers’ convenience, Muñoz and Vicente’s (2018) performance gaps are reported positively in this section.
experience worse timing than funds with an income objective The six objective categories, all having significant monthly performance gaps, are aggressive growth (0.25%), small-cap growth (0.16%), mid-cap growth (0.13%), growth (0.14%), growth and income (0.06%) and income-growth (0.03%) (Friesen and Sapp, 2007). Furthermore, Friesen and Sapp (2007) investigate the relationship between a fund’s quality and an investor’s timing ability. They identify that the best-performing funds possess the worst timing ability, by applying Fama and French’s three-factor model and Carhart’s four-factor model and by computing the alpha for each fund. For both models, the funds are ranked into deciles according to their derived alphas. All deciles of the three-factor model exhibit a positive and significant performance gap (implying bad timing), where the highest alpha deciles exhibit the highest performance gap. The monthly average alpha for funds with a positive alpha is 0.273% and the monthly average performance gap is 0.252%. Hence, investors in the best-performing funds almost entirely offset the excess returns due to bad timing. Adding the momentum factor does not change the results, as it also illustrates that the best performing funds exhibit the worst timing ability. The monthly average alpha of positive alpha funds is 0.233% with a performance gap of 0.182%, indicating that the major part of the gains is offset by bad timing (Friesen and Sapp, 2007).
6 funds, respectively. However, small-cap growth funds exhibit worse timing than small-cap value funds with annual performance gaps of 2.36% and 2.26%, respectively (Muñoz and Vicente, 2018). All derived performance gaps by Muñoz and Vicente (2018) are highly statistically significant. This illustrates that most value funds exhibit larger performance gaps than growth funds, thus indicating that value fund investors have worse timing abilities than growth fund investors. Since the results of Friesen and Sapp (2007) and Muñoz and Vicente (2018) on value funds contradict each other, research on stock level data seems suggestive. Conducting research on value-stock investors’ timing ability on stock-level data seems further interesting as Friesen and Sapp (2007) state that the best performing funds have the worst timing skills. Since Fama and French (1998) explore that value stocks outperform growth stocks annually by 7.68% it could be argued that value stock investors exhibit worse timing skills than growth stock investors. Value stocks are companies with low earnings, while growth stocks exhibit high earnings (Fama and French, 1998). Fama and French (1998) state that the value premium arises due to the exposure of distress, whereas Lakonishok, Shleifer, and Vishny (1994) argue that value stocks are undervalued and exhibit higher returns when their share price corrects.
Nevertheless, to further assess what creates the performance gap, Friesen and Sapp (2007) apply simulated data on possible ways of investor behaviour. They identify that investors withdraw capital after bad returns and inject capital randomly to better-performing funds. This illustrates that the return-chasing behaviour of investors increases the performance gap (Friesen and Sapp, 2007). These results are in line with Dichev and Yu (2011), who also examine the cause of the performance gap. By analysing the correlation between a given year’s distributions with past and future returns, it is assessed whether returns are chased or predicted (Dichev and Yu, 2011). This is the same method, which Dichev (2007) applies to examine investor’s timing ability. Dichev and Yu (2011) explore that investors make investments after previous high returns but not before future high returns. Hence, the return-chasing behaviour accounts for the major part in the performance gap (Dichev and Yu, 2011). Considering that value stocks outperform growth stocks with the findings of Friesen and Sapp (2007) and Dichev and Yu (2011) that investors chase high returns and exhibit bad timing abilities, it could be inferred that value stock investors exhibit bad timing skills. Furthermore, due to the findings of Fama and French (1998), it could be argued that Friesen and Sapp’s (2007) results contradict each other. On the one hand, they argue that the HML factor is negatively correlated with the performance gap, indicating that value stocks exhibit better timing ability than growth stocks. On the other hand, investor’s return-chasing behaviour indicates that they invest in value stocks at the wrong time. Hence, identifying the timing ability of value stock investors seems interesting for the field of finance.
7 returns are lowered by 7.7% annually because of bad timing. For 2,111 funds-of-funds during the same period, the performance gap is 6.9% per year. The results for both fund types are highly statistically significant with p-values of 0.001 and 0.003 for hedge funds and funds-of-funds, respectively (Dichev and Yu, 2011). Hence, this illustrates that Dichev and Yu’s (2011) results of investors having bad timing ability are in line with Dichev’s (2007) and Friesen and Sapp’s (2007) results. In addition, Clare and Motson (2010) examine investors’ timing ability based on UK mutual funds between 1992 and 2009. They also distinguish between private and institutional investors in order to analyse who exhibits worse timing ability. The overall reported performance gap for UK investors is 0.82%, with private and institutional investors having a performance gap of 1.17% and 0.27%, respectively. Thus, private investors are worse at timing their investments than institutional investors (Clare and Motson, 2010). Overall, Clare and Motson’s (2010) results are in the line with the results of the previous authors, that investors exhibit bad timing ability.
As previously outlined, Dichev (2007) and Friesen and Sapp (2007) apply different methods to derive their results. Dichev (2007) determines the BH returns at the aggregate level of national stock markets. He argues that aggregate distributions are highly general and thus are appropriate to compare DW returns between international markets (Dichev, 2007). Moreover, this approach does consider the amount and time period of invested capital and does not equally weight each asset (Dichev and Yu, 2011). The impact of equally weighting assets is illustrated by Dichev and Yu (2011), who report for 7,505 hedge funds during the same period a BH return of 6.4% and 13.8% when equally- and value-weighted, respectively. They argue that the low equally-weighted BH return is the result of giving relatively more weight to small and underperforming funds. The approach of aggregated data is also applied by Keswani and Stolin (2008) and Clare and Motson (2010). In contrast, Friesen and Sapp (2007) state that determining aggregate BH returns might bias the conclusion of investors timing ability. They argue that aggregate distributions suffer from the fact that positive and negative distributions might cancel each other out. Hence, potentially important information is dismissed, eventually leading to the conclusion that there is no performance gap. Additionally, applying aggregate distributions does not allow examining the fund selection ability of investors (Friesen and Sapp, 2007). Muñoz and Vicente (2018) also apply this methodology.
8 London Share Price Database, Keswani and Stolin (2008) explore for the UK an annual performance gap of -1.3%, thus indicating good timing of investors. Since the two databases’ returns are reasonably correlated (0.972) and the UK’s BH returns are close, the differences in the performance gap arise due to the coverage problem of TDS (Keswani and Stolin, 2008). Additionally, Keswani and Stolin (2008) argue that the 77-year period of Dichev (2007) does not reflect an investor’s investment horizon. Therefore, the authors divide the 77 year-period of the NYSE/AMEX stocks into three sub-periods of equal length (25 years and 8 months), which represent more the investment horizon of an investor who saves for his pension. Only the first sub-period exhibits bad timing with a performance gap of 1.8% annually, while during the second and third-period investors timed their investments well with performance gaps of -0.2% and -0.3%, respectively. Hence, the average performance gap between 1926 and 2002 of 0.4% is smaller than the performance gap of 1.3% as reported by Dichev (2007). The NASDAQ stocks’ performance gap is almost halved if extending the time horizon by four years until 2006. From 1973 until 2002 the performance gap is 5.3%, whereas it decreases to 2.9% when applying the period between 1973 and 2006 (Keswani and Stolin, 2008). Thus, Keswani and Stolin (2008) conclude that the analysed time horizon affects the magnitude of the DW return.
9 Cash leaving the market reduces the weights of future positive returns and illustrates bad timing ability. However, cash withdrawals increase the weights of poor previous returns retrospectively and illustrate the hindsight effect (Hayley, 2014). These two effects can be examined separately by evaluating each distribution’s impact on the DW return. By assuming that all returns subsequent of each distribution are equal to the buy-and-hold return, all distributions are neither well nor badly timed. Hence, the hindsight effect is responsible for all impacts on the DW return (Hayley, 2014).
10 expense ratios, which are 1.09% and 2.70%, respectively. After applying Hayley’s (2014) method and considering the hindsight effect the annual timing effect for low and high expense funds are 0.73% and 0.78%, respectively (Muñoz and Vicente, 2018). This illustrates that investors in high expense funds still exhibit worse timing than in low expense funds but reduce their return by 0.05% annually and not 1.61%. Furthermore, when controlling for the hindsight effect the timing effect of 0.6% annually is identical for cap value and large-cap growth funds. Mid-large-cap value funds still exhibit worse timing than mid-large-cap growth funds, with the annual timing effect being 1.13% and 0.61%, respectively. After controlling for the hindsight effect, small-cap growth funds still have a greater annual timing effect (0.98%) than small-cap value funds (0.84%) but these effects are smaller than the previously reported performance gaps. However, in contrast as stated above, large-cap value fund investors do not exhibit worse timing ability than investors in large-cap growth funds. The derived timing effects and hindsight effects of Muñoz and Vicente (2018) are all highly statistically significant. In conclusion, the authors suggest that considering the hindsight effect is necessary in order to make correct inferences about investors’ timing ability.
The results of Muñoz and Vicente (2018) illustrate that mid-value funds have worse timing than mid-growth funds, whereas growth funds have a greater timing effect than small-value funds and large-cap small-value and growth funds exhibit the same timing effect. Hence, no inference can be made, which security-style exhibits worse timing. Additionally, the inconsistency of Friesen and Sapp’s (2007) results on the HML factor might be biased by the hindsight effect. Thus, the timing ability of value stock investors remains open to debate and therefore needs to be examined at the stock level.
3. Methodology and Hypothesis
3.1 Stock selection
11 examines the period between 1990 and 2017, 28 portfolios are created every December between 1989 and 2016. For comparison and robustness checks, the same is done for growth stocks in each year. Following Fama and French (1998), the growth stocks consists of the bottom 30% B/M stocks but firms with negative B/M ratios are excluded (Fama and French, 1993).
3.2 Performance Gap and Controlling for the Hindsight Effect
In order to compute the performance gap it is required to derive the BH return and DW return. BH returns express the return of an investment that is held over the entire investment horizon. Until the investment is sold, BH returns imply that no other capital flows have occurred. Hence, BH returns reflect the return of stocks and not of investors (Dichev, 2007). The BH return is computed for every stock individually as the geometric average return and can be computed with the following equation:
(1) 𝑟𝑖𝐵𝐻 = [∏(1 + 𝑟 𝑖,𝑡) 𝑇 𝑡=1 ] 1 𝑇 − 1 ,
where 𝑟𝑖,𝑡 is the monthly return of stock i in month t (Muñoz and Vicente, 2018). The individual monthly stock returns are calculated by:
(2) 𝑟𝑖,𝑡 = ( 𝑝𝑖,𝑡
𝑝𝑖,𝑡−1) − 1 ,
where 𝑝𝑖,𝑡 and 𝑝𝑖,𝑡−1 express the price of stock i in period t and period t-1, respectively. DW returns illustrate the return earned by the skills of investors more precisely. This approach attributes more/less weight on returns in times in which investors have invested more/less capital, respectively. The DW return of each stock can be calculated as the internal rate of return (IRR) and illustrates the returns as an investment sequence (Dichev, 2007). Additionally, the DW return reflects the performance of an average investor (Friesen and Sapp, 2007) and can be computed with the following equation:
(3) 𝐶𝐹𝑖,0+ 𝐶𝐹𝑖,1 (1 + 𝐼𝑅𝑅𝑖)1+ 𝐶𝐹𝑖,2 (1 + 𝐼𝑅𝑅𝑖)2+ ⋯ + 𝐶𝐹𝑖,𝑡 (1 + 𝐼𝑅𝑅𝑖)𝑡 = 0 ,
where 𝐶𝐹𝑖,𝑡 illustrates the distribution of stock i in period t and 𝐶𝐹𝑖,0 implies stock i’s initial market capitalization. The distributions entering and leaving the market are the most important variables when deriving DW returns (Dichev, 2007). Following Dichev (2007), Keswani and Stolin (2008) and Hayley (2014), the distributions are calculated by:
12 where 𝑑𝑖,𝑡 is the monthly distribution of stock i in period t, 𝐾𝑡 is the monthly market capitalization of stock i in period t and 𝑟𝑖,𝑡 is the monthly stock return in period t that is computed by equation (2). Each distribution enters equation (3) with its inferred sign. Negative distributions imply an additional investment of investors, thus capital entering the market, whereas positive distributions express capital allocations from companies to investors, which illustrates capital leaving the market. The initial market capitalization of each stock is entered with a negative sign, while the stock’s ending market capitalization is entered with a positive sign (Dichev, 2007).
The performance gap is derived in order to measure investors’ timing ability (Friesen and Sapp, 2007). Following Muñoz and Vicente (2018), the performance gap for each stock is defined as:
(5) 𝑃𝐺𝑖 = 𝑟𝑖𝐷𝑊− 𝑟𝑖𝐵𝐻 ,
since a negative value for bad timing ability is more intuitive. In contrast, a positive performance gap illustrates good timing ability (Muñoz and Vicente, 2018). 𝑟𝑖𝐷𝑊 is computed as the IRR of a stock according to equation (3) and 𝑟𝑖𝐵𝐻 refers to the result of equation (1).
By applying Hayley’s (2014) approach the performance gap can be divided into two components, namely the hindsight effect and the actual timing effect. This approach starts by assuming that the stock’s average monthly BH return over a given year is the return in each month. Thus, the actual returns are replaced for the average monthly BH return. Additionally, it is assumed that every distribution is zero. Hence, the BH return and the DW return are identical initially. Thereafter, the approach requires two steps (see table 4 in the appendix for a detailed illustration). Firstly, the actual historical return is inserted for the average monthly BH return in the first month. Then the DW return is recalculated for the entire given year and the difference to the initial DW return (which was equal to the BH return) is recorded. Secondly, the historical distribution is inserted for the first month. Then the DW return is recalculated and the difference from the derived DW return in step one is recorded. This procedure is done for 2T steps, where T implies the distributions/returns, until the actual data replaces the two assumptions (Hayley, 2014). The hindsight effect is the total sum of the changes in the DW return that are based on the inserted distributions, which is done in the second step of Hayley’s (2014) approach. The total sum of the changes in the DW return due to the first step illustrates the relation between returns and previous distributions, implying the actual timing ability. This method is applied for each stock in every year.
In order to examine if value stocks are responsible for the performance gap, Fama and French’s three-factor model, including an interaction variable and the performance gap is applied:
(6) 𝑟𝑖,𝑡𝐵𝐻 = 𝛼
13 where 𝑟𝑖,𝑡𝐵𝐻is the average BH return of all stocks in year t in excess of the average 1-year US treasury yield. For equation (6), i can be v, g or v-g and represent value stocks, growth stocks or the difference between value and growth stocks, respectively. Hence, 𝑟𝑣,𝑡𝐵𝐻expresses the BH excess return of value stocks in period t. The average BH return and not the average DW return is applied as the dependent variable because the annually constructed factors of Fama and French are held throughout the year. 𝑅𝑀𝑅𝐹𝑡 is the excess return of the market in year t. 𝑆𝑀𝐵𝑡 captures the returns of small and big stocks in period t, whereas 𝐻𝑀𝐿𝑡 comprises the returns of stocks with high and low B/M ratios. The interaction variable 𝐻𝑀𝐿𝑡∗ 𝑃𝐺𝑣,𝑡 is the product of the HML factor and the performance gap of value stocks in period t. This variable is constructed in order to identify if and how value stocks interact with the performance gap. The fifth variable of the equation is 𝑃𝐺𝑖,𝑡 and denotes the performance gap of either value stocks, growth stocks or the difference between them in period t. The main purpose is to identify how value stocks interact with the performance gap but as a robustness check, the same is done for growth stocks. To further analyse how the behaviour of a stock interacts with the performance gap, the difference between value and growth stocks (no excess return) is also applied as a dependent variable. Thus, 𝑃𝐺𝑣−𝑔,𝑡 illustrates the difference of the performance gap between value and growth stocks in period t.
After computing the hindsight effect by Hayley’s (2014) method, the regression: (7) 𝑟𝑖,𝑡𝐵𝐻 = 𝛼
𝑖+ 𝛽1,𝑖𝑅𝑀𝑅𝐹𝑡+ 𝛽2,𝑖𝑆𝑀𝐵𝑡+ 𝛽3,𝑖𝐻𝑀𝐿𝑡+ 𝛽4,𝑖𝐻𝑀𝐿𝑡∗ 𝐻𝐸𝑖,𝑡+ 𝛽5,𝑖𝐻𝐸𝑖,𝑡 ,
is applied. Similar to equation (6), i can take the letter v, g, or v-g and illustrate value stocks, growth stocks or the difference between value and growth stocks, respectively. However, the hindsight effect of either value stocks, growth stocks or the difference between them, replaces the performance gap. This is done in order to examine if and how value stocks are prone to the hindsight effect. Additionally, as robustness checks the same is done for growth stocks and the difference between value and growth stocks. The factors of Fama and French’s (1993) model remain the same as in equation (6).
3.3 Research Hypothesis
14 according to the return-chasing behaviour and bad timing ability of investors, value stock investors should exhibit a worse performance gap than growth stock investors. This is indicated by the results of Muñoz and Vicente (2018), who control for the hindsight effect and report that most value-style funds have greater performance gaps than growth-style funds. Nevertheless, value and growth stocks offer incentives to investors. On the one hand, because value stocks earn higher returns on average they could attract the interest of investors more than growth stocks. On the other hand, growth stocks are currently better performing stocks with high earnings (Fama and French, 1998), thus also giving incentives to investors. Therefore, due to the investigation conducted so far, the following two research hypotheses arise:
Hypothesis 1: Value stock investors exhibit on average bad timing and have a more negative performance gap than growth stock investors.
Hypothesis 2: The actual timing of value stock investors is better as reported because the hindsight effect biases the performance gap.
Firstly, these hypotheses will be examined by computing the performance gap, hindsight effect and timing effect of value and growth stocks. Secondly, it will be analysed whether or not the performance gap and the hindsight effect depend on the stocks being value stocks. For the first hypothesis, this will be done with equation (6), with the BH excess return of value stocks (𝑟𝑣𝐵𝐻) being the dependent variable. In order to be true, the coefficient associated with interaction variable HML*PG needs to be negative, as a positive HML factor represents value stocks and a negative performance gap signifies bad timing. The second hypothesis will be tested with equation (7), also with 𝑟𝑣𝐵𝐻as the dependent variable. Since, a negative hindsight effect illustrates that the DW return is biased downwards, the interaction variable HML*HE is expected to be negative. For robustness checks, the same is done for growth stocks and with the difference between value and growth stocks.
4. Data
15 created. Secondly, with a time series request for each of these individual created constituent lists the necessary data for dead US stocks was collected. For every active and dead stock monthly data on the (adjusted) price, market capitalization and PTBV in the period between December 1989 and December 2017 was derived. The adjusted price is applied as it accounts for stock splits and dividends, thus allowing for price comparisons over time (Ince and Porter, 2006). The B/M ratio was computed as the reciprocal of the PTBV. A limitation of the applied data could be that especially during the early years of the sample period the PTBV is not reported for a number of stocks. Hence, not all stocks that were active during a year were analysed. Following Dichev (2007), Friesen and Sapp (2007), Dichev and Yu (2011) and Muñoz and Vicente (2018), monthly data is used. The disadvantage of annual data is that cash in- and outflows within a year cannot be measured precisely. Daily data is not advantageous as the feasibility of the IRR calculations would be problematic (Dichev, 2007).
16 percentile, respectively (except in 2002). This indicates that large companies drive the average market capitalization and average distribution. Hence, this supports applying data at company-level and not aggregated market-level. Thus, large distributions of large companies do not offset several distributions of small companies. Table 5 in the appendix presents the descriptive statistics of growth stocks that were analysed for robustness purposes.
Table 1 – Descriptive statistics of listed NYSE and NASDAQ value stocks between 1990 and 2017. Panel A illustrates the B/M ratios by which the stocks are selected every December in year t-1. For the B/M ratio the mean, median, maximum, minimum and standard deviation is presented. The stocks were selected if their B/M ratio was in the top 30% of a given year.
Panel B presents the monthly market capitalizations and distributions in $ million. N represents the number of analysed value stocks in a given year. For the market capitalization and distribution the mean, median, 25th percentile, 75th percentile and standard deviation are reported. A stock’s distribution is derived by 𝑑𝑖,𝑡= 𝐾𝑖,𝑡−1∗ (1 + 𝑟𝑖,𝑡) − 𝐾𝑖,𝑡, where 𝐾𝑖,𝑡 is the market capitalization of stock i in period t and 𝑟𝑖,𝑡 is the return of stock i in period t.
Panel A: Book-to-market ratios for the value stocks.
Year Mean Med. Max. Min. Std.dev.
Table 1 (continued) 17 Panel B: Average market capitalization and distribution of value stocks (in $ million).
Market Capitalization Distribution
Year N Mean Median 25th Per. 75th Per. St.Dev. Mean Median 25th Per. 75th Per. St.Dev.
18
3
T-statistics are computed under the assumption that the mean of the measure follows a normal distribution given the sample size. The t-statistics are computed as follows: The measure’s standard deviation is divided by the square root of the number of observation in order to receive the mean of the standard error. Thereafter the measure’s mean is divided by the mean of the standard error. As a result, the statistic is received. The t-test is done with the “tdist” function in Microsoft Excel 2016.
5. Results
5.1 Performance of value stock investors
19
Table 2 – Timing ability of investors.
Panel A reports the equally-weighted average BH return, DW return, performance gap, hindsight effect and timing effect for all analysed value stocks in the period of 1990-2017. BH returns of stocks are calculated as the geometric average of monthly returns. DW returns are computed as the IRR, with the stock’s market capitalization in December of year t-1 entering the calculation negatively and the ending market capitalization in December of year t is added to the last distribution of December. The stock’s distributions between January and December enter the IRR calculation with their inferred signs. The performance gap is the difference between DW returns and BH returns. By applying Hayley’s (2014) method, the hindsight effect and timing effect are computed. The sum of these two numbers adds up to the performance gap.
Panel B reports the average BH return, DW return, performance gap, hindsight effect and timing effect of growth stock investors. All numbers in table 2 are presented as percentage per year and were calculated on a monthly base but for reporting purposes, they are presented annually. Hence, some figures might not add up exactly. ***, ** and * illustrate significance at a 1%, 5% and 10% significance level, respectively.
Panel A: Performance of value stocks between 1990 and 2017.
Year BH return DW return PG HE TE 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 -39.88 32.91 29.66 21.15 -0.35 27.03 19.51 30.77 -9.83 10.63 0.56 29.23 0.53 63.10 22.69 4.99 17.05 -10.81 -64.52 48.50 20.65 -14.92 18.79 34.24 0.60 -11.74 16.29 7.00 -39.98 32.83 29.72 20.39 -0.43 26.84 19.60 30.53 -10.22 10.45 0.36 28.95 4.34 62.73 22.31 4.93 16.49 -11.19 -65.14 48.26 19.51 -14.98 18.23 34.03 0.22 -13.06 16.25 6.69 -0.09 -0.08 0.06 -0.76 -0.08 -0.18 0.09 -0.24 -0.39 -0.18 -0.20 -0.28 3.81 -0.36 -0.38 -0.06 -0.56 -0.38 -0.63 -0.24 -1.14 -0.06 -0.57 -0.21 -0.38 -1.32 -0.04 -0.31 0.02 -0.01 -0.03 -0.61 -0.06 -0.04 0.06 -0.06 -0.36 -0.06 -0.08 -0.33 3.24 -0.09 -0.36 0.02 -0.38 -0.21 -0.54 0.14 -1.51 -0.09 -0.34 -0.29 -0.29 -0.97 -0.05 -0.02 -0.11 -0.07 0.09 -0.15 -0.02 -0.14 0.04 -0.18 -0.03 -0.12 -0.13 0.05 0.57 -0.27 -0.02 -0.08 -0.18 -0.17 -0.09 -0.38 0.37 0.03 -0.22 0.08 -0.09 -0.35 0.01 -0.29 Mean 10.85 10.67 -0.18 -0.12 -0.07 St.Dev. 0.000 0.007 0.002 t-Statistic -7.18*** -5.27*** -11.03***
Panel B: Average performance of growth stock investors between 1990 and 2017.
BH return DW return PG HE TE
Mean -1.19 -1.38 -0.18 -0.04 -0.15
St.Dev. 0.013 0.009 0.005
20 resulting in a timing effect of 0.37%. On average, value stocks had superior returns during the first months of 2010. Additionally, investors made large capital injections, which reduced the weights of those superior returns relatively. Hence, the performance gap in 2010 is biased downwards.
For robustness purposes, the same procedure is done for growth stocks. Following Fama and French (1998), the stocks whose B/M ratio is within the bottom 30% at December in t-1 are considered a growth stock. Stocks with negative B/M ratios are excluded (Fama and French, 1993). Panel B of table 2 presents the average BH return, DW return, performance gap, hindsight gap and timing effect of growth stocks during the period 1990-2017 (see table 6 in the appendix for an annual overview of growth stock investors’ performance). The average BH return of growth stocks is -1.19% per year. Hence, value stocks outperform on average growth stocks during 1990-2017, which is in line with the findings of Fama and French (1998). Due to the average DW return being -1.38%, growth stock investors also exhibit a performance gap of -0.18% per year. This gap is highly significant with a p-value of 0.000. Thus, value stock and growth stock investors suffer from a performance gap of the same magnitude. However, after controlling for the hindsight effect, which is -0.04% on average per year, the timing effect of growth stock investors is -0.15% per year. The timing effect of growth stock investors is highly significant with a p-value of 0.000, while the hindsight effect is statistically insignificant with a p-value of 0.204. Nevertheless, the results of table 2 clarify that the timing ability of value stock investors is actually better than the timing ability of growth stock investors. As a result, it can be said that hypothesis 1 is partly true. Value stock investors’ performance gap is on average negative but not by a larger amount than the performance gap of growth stock investors. According to the results of table 2, hypothesis 2 is true. The hindsight effect biases on average the performance gap downwards. Hence, value stock investors have a better timing ability than the performance gap expresses. These results are in line with Hayley (2014) and Muñoz and Vicente (2018) and illustrate that ignoring the potential hindsight effect in DW returns can lead to erroneous inferences. Furthermore, since value stocks exhibit on average better performances and timing abilities than growth stocks, these results are contrary to the findings of Friesen and Sapp (2007), who argue that the best performing funds have the worst timing abilities. These opposite findings may arise as Friesen and Sapp (2007) did not consider the hindsight effect. 5.2 The impact of value stocks on the performance gap and hindsight effect
21
Table 3 – The impact of value stocks on the performance gap and hindsight effect.
All six regressions are based on the Fama and French (1993) three-factor model. Regressions I and II include the performance gap and its interaction with the HML factor for value and growth stocks, respectively. The hindsight effect and its interaction with the HML factor are included in regressions IV and V for value and growth stocks, respectively. Regressions III and VI present value and growth stocks in a joint model and include the interaction of the HML factor with the performance gap and hindsight effect, respectively. All regressions were estimated by the ordinary least squares (OLS) method. For each variable, the coefficient is presented with the t-statistic in parentheses. The t-statistics are adjusted for White-Hinkley heteroskedasticity consistent standard errors and covariance. ***, ** and * illustrate significance at a 1%, 5% and 10% significance level, respectively. All regressions were computed with EViews 10.
Variable I II III IV V VI Intercept -4.189 (-2.22**) -15.625 (-7.21***) 10.376 (3.88***) -3.986 (-2.17**) -15.583 (-6.90***) 10.329 (3.93***) RMRF 1.060 (8.36***) 1.172 (8.17***) -0.15 (-1.10) 1.037 (8.78***) 1.160 (8.12***) -0.146 (-1.02) SMB 0.868 (7.61***) 0.490 (3.38***) 0.448 (3.08***) 0.860 (7.11***) 0.505 (3.40***) 0.488 (2.87***) HML 0.600 (3.47***) -0.049 (-0.46) 0.644 (8.57***) 0.548 (4.44***) -0.062 (-0.50) 0.637 (6.08***) HML*PG 0.170 (0.40) 0.168 (1.31) 0.444 (3.00***) PG 5.326 (2.81***) -0.732 (-0.44) -2.868 (-0.66) HML*HE -0.093 (-0.18) 0.266 (1.12) 0.673 (2.67**) HE 6.850 (2.52***) -1.883 (-1.07) -2.905 (-0.49) Dep. Variable 𝑟𝑣𝐵𝐻 𝑟𝑔𝐵𝐻 𝑟𝑣−𝑔𝐵𝐻 𝑟𝑣𝐵𝐻 𝑟𝑔𝐵𝐻 𝑟𝑣−𝑔𝐵𝐻 R2 0.9454 0.9207 0.7607 0.9455 0.9182 0.7456 Observations 28 28 28 28 28 28
22 interaction variable can be interpreted that if the HML factor increases (implying returns of value stocks), the performance gap increases too. Considering that an increasing performance gap implies good timing of investors, value stock investors do not exhibit bad timing. Hence, hypothesis 1 is not true. However, the interaction variable is statistically insignificant with a p-value of 0.693. Nevertheless, this finding supports Friesen and Sapp’s (2007) result of the HML factor being negatively correlated with the performance gap (representing good timing abilities of value stocks). The R2 of regression I signifies that 94.54% of the variation of the value stocks’ return can be explained by the model. For a robustness check, regression II in table 3 illustrates equation (6) for growth stocks, with 𝑟𝑔𝐵𝐻 indicating the excess return as the dependent variable. The HML factor is negatively correlated (-0.049) with the dependent variable, thus representing growth stocks. However, with a p-value of 0.653 it is not statistically significant. Similar to regression I, the interaction variable HML*PG is positive (0.168) and insignificant (p-value 0.205), while the performance gap is negative (-0.732) but also insignificant (p-value 0.664). Therefore, these variables suggest that growth stock investors exhibit bad timing abilities. To further assess the interaction between value stocks and the performance gap, regression III is applied. This regression is also based on equation (6) and the dependent variable,𝑟𝑣−𝑔𝐵𝐻 is the difference between the BH return of value and growth stocks. This implies a long position in value stocks and a short position in growth stocks. The performance gap in regression III is also the difference between the gap of value and growth stocks. The highly statistically significant and positive intercept variable of 10.376 can be explained by the average value stock BH return being positive and the average growth stock BH return being negative during this study’s sample period. Furthermore, the statistically insignificant market risk premium of -0.15 illustrates that the dependent variable, which is a difference, cannot be explained by market returns. Since the HML factor with 0.644 is positive and highly statistically significant, it can be said that value stocks contribute positively to the difference of the dependent variable. This seems reasonable as the dependent variable expresses a long value stock minus short growth stock position Additionally, the interaction variable of the HML factor with the performance gap is positively correlated (0.444) with the dependent variable and with a p-value of 0.007 highly statistically significant. This illustrates that the dependent variable behaves more as a value stock as the performance gap increases and indicates that value stocks are not connected to bad timing. The negative performance gap of -2.868 indicates that the difference between value and growth stocks’ BH returns increases as the performance gap gets negative.
23 significant. Similar to the results of regression I, this implies that the selected stocks correspond to small and value stocks. Additionally, the hindsight is positively correlated (6.850%) with the BH excess return. Furthermore, the hindsight effect is statistically significant with a p-value of 0.019. Hence, this expresses that if investors’ DW return is biased upwards/downwards by 1% the BH excess return increases/decreases by 6.85%, respectively. The interaction variable HML*HE has a coefficient of -0.093 and thus is negatively correlated with the dependent variable. The negative interaction between the two variables implies that as the HML factor increases, the hindsight effect decreases. This illustrates that the return of value stock investor’s is biased downwards and supports that hypothesis 2 is correct. Furthermore, this finding supports the results of panel A in table 2, which illustrates that the performance gap of value stock investors is biased downwards. However, similar to regression I, the interaction variable HML*HE has a p-value of 0.860 and hence is statistically insignificant. As an additional robustness check, regression V presents the interaction of growth stocks with the hindsight effect. The HML*HE variable has a coefficient of 0.266 (p-value 0.277) and signifies a positive interaction. Since a negative HML factor indicates growth stocks, a negative hindsight effect illustrates that the reported returns of growth stock investors’ returns are lower than they actually are. Hence, the HML*HE variable of regression IV also supports the results panel B in table 2, that growth stock investors’ DW returns are biased downwards. Regression VI measures the behaviour of stocks in regards to the hindsight effect. Hence, 𝑟𝑣−𝑔𝐵𝐻is the dependent variable and the hindsight effect is also the difference between value and growth stocks. Similar, to regression III, the highly statistically significant and positive intercept (10.329) of regression VI can be explained by the long-short strategy. In addition, the statistically insignificant market risk premium states that it cannot explain the difference between BH returns of value and growth stocks. Furthermore, the positive HML factor of 0.637, which is also highly statistically significant (p-value 0.000), indicates that the dependent variable increases by 1% if the return of value stocks exceed the return of growth stocks by 0.637%. The interaction variable HML*HE is statistically significant (p-value of 0.014) and positively (0.673) correlated with the dependent variable. Hence, as the hindsight effect increases the dependent variable behaves more as a value stock and the difference between value and growth stock’s BH return increases.
In conclusion, the research suggests that value stock investors’ returns are not as bad as reported because their returns are biased downwards. Furthermore, bad timing does not arise due to the behaviour of value stocks.
6. Conclusion
24 longer being value stocks, the performance of value stocks is analysed from January to December of a given year. Furthermore, to avoid that big value companies absorb the capital distributions of small value companies, the data is applied at stock-level. After computing the BH return and DW return for value stocks during 1990-2017 the average performance gap is -0.18%. However, this measure is biased downwards by -0.12%, resulting in an actual timing effect of -0.07%. As a robustness check, this is carried out for growth stocks during the same period. The average performance gap of growth stock investors is -0.18%, implying that value and growth stock investors have the identical bad timing skills. After controlling for the hindsight effect, which is -0.04%, the actual timing effect of growth stock investors is -0.15%. Hence, considering the hindsight effect signifies that it can change the inference about the timing ability of investors. Furthermore, all of these measures are highly statistically significant (except the hindsight effect of growth stocks), which suggests that the findings of Hayley (2014) and Muñoz and Vicente (2018) on the hindsight effect are supported by this study. The results of this study also indicate that value stocks themselves are not responsible for a negative performance gap. This is supported by the positive interaction of the HML factor and the performance gap of value stocks. Additionally, value stocks interact negatively with the hindsight effect. This illustrates that the return of value stocks is biased downwards and supports this study’s results. As a result, it can be said that the timing ability of value stock investors is not as bad as reported. Moreover, the hindsight effect should be considered when evaluating investors’ timing performance. Since, value stocks outperform growth stocks and value stocks investors have better timing skills than growth stock investors, it seems advantageous to invest in value stocks. Nevertheless, value stock investors reduce their returns on average by -0.07% as they invest at the wrong time and hence there is potential for improving their timing skills.
A possible limitation of this study could be attributable to the data being dominated by small stocks. Due to equally-weighting the stocks and the high coefficients of the SMB factors, it can be questioned if the results of this study are also representative for big value stocks. Given the fact, that the SMB factor in all six applied regressions is positive and highly statistically significant, it seems that this limitation holds.
VI V. Appendix
Method to compute the hindsight effect
Table 4 illustrates Hayley’s (2014) method for the value stock “Calithera Biosciences”, whose BH return in 2017 was 8.18%. This stock is selected, as it is a recent example and illustrates the possible magnitude of the hindsight effect. Since, the DW return of “Calithera Biosciences”-investors is 2.73%, the performance gap is -5.45%. However, the actual timing effect is -1.19%, whereas most of the performance gap is due to the hindsight effect of -4.26%.
The unnamed rows in the first column of table 4 exhibit the monthly returns. Rows with an “r” express the IRR calculation after inserting the corresponding monthly return and computing the new ending distribution in column M12. The distributions in r-rows are always the same as of the previous d-row (e.g. row 4r has the same distributions as row 3d), because it is desired to identify how the previous distributions were timed. Rows with a “d” express the IRR calculation after inserting the corresponding monthly distribution. Since the actual returns are already inserted in d-rows, this illustrates the assumption of Hayley (2014) that investors can decide on the amount of distributions after they know the securities’ return. Also due to the known returns, d-rows have the same ending distribution (in M12) as the previous r-row. Only in the last step, the actual ending distribution replaces the assumed ending distribution.
The first row of table 4 illustrates the first assumption of this method that all returns equal the BH return. Row “0” implies the second assumption of this method, that all distributions are equal to zero. Hence, all grey fields in table 4 imply Hayley’s (2014) two assumptions. As stated by Hayley (2014), due to the two assumptions, the initial IRR (or DW return) is with 8.18% identical to the stock’s BH return. Then, the actual monthly return for month 1 (in January the return was 115%) replaces the assumed BH return and the IRR is computed. Next, the actual monthly distribution replaces the assumed distribution of zero and the IRR is calculated again. Thereafter, this procedure is done for all twelve months. The sum of the changes on the IRR due to the actual monthly returns is the timing effect. The hindsight effect is the sum of the changes on the IRR due to distributions.
Table 4 - Method to identify the hindsight effect
This table illustrates the hindsight effect for Calithera Biosciences during the year 2017. M0 illustrates the company’s market capitalization in December 2016 and enters the IRR calculation negatively. In the first column, “r” illustrates the IRR for the given month after the actual corresponding return is inserted. The “d” for the given month expresses that the IRR calculation includes the actual corresponding distribution. TE is the timing effect and is the difference between the IRR after inserting the historical return (e.g. row 2r) and the previously computed IRR (e.g. row 1d). HE expresses the hindsight effect and is the difference between the IRR after inserting the historical distribution (e.g. row 1d) and the previous IRR (e.g. row 1r). The final distribution in M12, which illustrates December 2017, is based on the calculation: 𝑀0 ∗([∏12𝑡=1(1 + 𝑟𝑡)]
1
12− 1), where 𝑟
𝑡 expresses the return in month t in the unnamed rows.
IX Descriptive statistics for growth stocks
Panle A of table 5 illustrates the B/M ratios of the analysed growth stocks. Similar to value stocks, the B/M ratio of growth stocks increases during crises years (e.g. 1990 and 2008). Panel B presents the average market capitalization and distributions of the growth stocks. The average market capitalization for growth stocks increased during the sample period by six times, from around $ 2 billion in 1990 to over $12 billion in 2017.
Table 5 – Descriptive statistics of listed NYSE and NASDAQ growth stocks between 1990 and 2017. Panel A illustrates the B/M ratios by which the stocks are selected every December in year t-1. For the B/M ratio the mean, median, maximum, minimum and standard deviation is presented. The stocks were selected if their B/M ratio was in the bottom 30% of a given year.
Panel B presents the monthly market capitalizations and distributions in $ million. N represents the number of analysed value stocks in a given year. For the market capitalization and distribution the mean, median, 25th percentile, 75th percentile and standard deviation are reported. A stock’s distribution is derived by 𝑑𝑖,𝑡= 𝐾𝑖,𝑡−1∗ (1 + 𝑟𝑖,𝑡) − 𝐾𝑖,𝑡 , where 𝐾𝑖,𝑡 is the market capitalization of stock i in period t and 𝑟𝑖,𝑡 is the return of stock i in period t.
Panel B: Book-to-market ratios for the annual value stocks.
Year Mean Med. Max. Min. Std.dev.
Table 5 (continued) X Panel B: Average market capitalization and distribution of value stocks (in $ million).
Market Capitalization Distribution
Year N Mean Median 25th Per. 75th Per. St.Dev. Mean Median 25th Per. 75th Per. St.Dev.
XI Results for growth stocks
All stocks with a positive B/M ratio in the bottom 30% were considered a growth stock. For each stock in every year the BH return, DW return, performance gap, hindsight effect and timing effect were computed. The equally-weighted averages of each year are presented in table 6. During crises years, growth stocks experienced strong negative returns (e.g. 2000-2002 and 2008), leading to an overall negative BH return.
Table 6 – Detailed overview of growth stock investors’ timing ability.
All numbers are presented in percent per year. The measures report the equally-weighted average BH return, DW return, performance gap, hindsight effect and timing effect for all growth stocks in the period of 1990-2017. Following Fama and French (1993), companies with negative B/M ratios were excluded from the sample. BH returns of stocks are calculated as the geometric average of monthly returns. DW returns are computed as the IRR, with the stock’s market capitalization in December of year t-1 entering the calculation negatively and the ending market capitalization in December of year t is added to the last distribution of December. The stock’s distributions between January and December enter the IRR calculation with their inferred signs. The performance gap is the difference between DW returns and BH returns. By applying Hayley’s (2014) method, the hindsight effect and timing effect are computed. The sum of these two numbers adds up to the performance gap. Since, all numbers were calculated on a monthly base but presented annually for reporting purposes, some figures might not add up exactly.
XII VI. References
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