Role of median portfolios in value-premium and size-premium

Median Portfolios in VP/SP

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Role of median portfolios in value-premium and size-premium Soumya Guha Deb Professor (Finance) , Banikanta Mishra Professor (Finance) PII: DOI: Reference:

S0970-3896(16)30121-5 https://doi.org/10.1016/j.iimb.2019.07.004 IIMB 337

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IIMB Management Review

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17 November 2016 15 February 2018 15 July 2019

Please cite this article as: Soumya Guha Deb Professor (Finance) , Banikanta Mishra Professor (Finance) , Role of median portfolios in value-premium and size-premium, IIMB Management Review (2019), doi: https://doi.org/10.1016/j.iimb.2019.07.004

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Role of median portfolios in value-premium and size-premium (Short Title: Median Portfolios in VP/SP) 1) Soumya Guha Deb Professor (Finance) Associate Professor, Finance Indian Institute of Management Sambalpur, Jyoti Vihar, Sambalpur 768019, Odisha Phone: +91-993-885-3926 Fax: +91-674-230-0995 Email: [email protected]

2) Banikanta Mishra* Professor (Finance) XIMB (Xavier Institute of Management – Bhubaneswar), Xavier University, Xavier Square, Chandrasekharpur, Bhubaneswar 751013, INDIA Phone: +91-943-707-5075 Fax: +91-674-230-0995 Email: [email protected] *Corresponding author

Abstract: Analyzing Indian equity data from 1995 to 2014, we conclude that median portfolios, ignored by most researchers of value premium and size premium, are quite pivotal. Mid-cap portfolios exhibit zero VP and mid-price-to-book portfolios zero SP. A move to lower size or PB makes the premium – VP or SP, as the case may be – positive, a move toward higher size or PB makes premium negative. We are not aware of any research that finds the median portfolios to have zero risk-premium. Extended Summary: In this paper, we show that, in Indian context, there is zero VP (value premium) among mid-MC (market capitalization) portfolios and zero SP (size premium) among mid-PB (price-to-book) portfolios. That is, in the mid-MC category, the low-PB stocks earn the same return as high-PB stocks; similarly, among the mid-PB stocks, ones with low-MC earn the same return as those with high-MC. As we move away from the mid category, we observe significant VPs and SPs. More specifically, among low-MC stocks, there is positive VP (that is, low-PB stocks earn more than high-PB stocks), while, in high-MC category, there is negative VP (low-PB stocks earn less than high-PB stocks). In the same way, among low-PB stocks, there is positive SP (low-MC stocks earn more than high-MC stocks), while, in high-PB category, there is negative SP (low-MC stocks earn less than high-MC stocks). Fama and French (1993) - FF hereafter - have come up with measures of VP and SP. Value, neutral, and growth stocks denote, respectively, low-PB, mid-PB, and high-PB stocks. They measure VP by excess return on value portfolio vis-a-vis growth portfolio. Similarly, small-cap, mid-cap, and big-cap stocks denote, respectively, low-MC, mid-MC, and high-MC stocks. SP is measured by them by the excess return of small-cap portfolio over big-cap portfolio. Using Indian data from 1995 to 2014, we find that, though an overall VP or SP does not exist in India, VP exists for small-cap and big-cap portfolios and SP for value and growth portfolios. Specifically, VP is significantly positive in

small-cap portfolios, weakly evident (though often close to zero) in mid-cap portfolios, and significantly negative in big-cap portfolios. Similarly, SP is significantly positive in value portfolios, almost zero for neutral ones, and negative in growth portfolios. Thus, the mid-cap portfolio acts as a zero-premium benchmark for VP, while the neutral portfolio acts as a zeropremium benchmark for SP. On either side of the benchmark, there is a positive or negative premium. We are not aware of any work that finds the median (midcap or neutral) portfolio to be the zero-premium benchmark. We would like to iterate that our focus here is on the median portfolios and not on analyzing VP or SP per se in the Indian context. In fact, our medial portfolio analysis can be extended to other capital markets, including USA.

Keywords: Value premium; size premium; median portfolio; portfolio management; bull and bear months; LPM (lower partial moment)

JEL Codes: G11; G12; G14

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I. INTRODUCTION In this paper, we show that, in Indian context, there is zero VP (value premium) among mid-MC (market capitalization) portfolios and zero SP (size premium) among mid-PB (price-to-book) portfolios. That is, in the mid-MC category, the low-PB stocks earn the same return as high-PB stocks; similarly, among the mid-PB stocks, ones with low-MC earn the same return as those with high-MC. As we move away from the mid category, we observe significant VPs and SPs. More specifically, among low-MC stocks, there is positive VP (that is, low-PB stocks earn more than high-PB stocks), while, in high-MC category, there is negative VP (low-PB stocks earn less than high-PB stocks). In the same way, among low-PB stocks, there is positive SP (low-MC stocks earn more than high-MC stocks), while, in high-PB category, there is negative SP (lowMC stocks earn less than high-MC stocks). Fama and French (1993) - FF hereafter - have come up with measures of VP and SP. Value, neutral, and growth stocks denote, respectively, low-PB, mid-PB, and high-PB stocks. They measure VP by excess return on value portfolio vis-a-vis growth portfolio. Similarly, small-cap, mid-cap, and big-cap stocks denote, respectively, low-MC, mid-MC, and high-MC stocks. SP is measured by them by the excess return of small-cap portfolio over big-cap portfolio. Using Indian data from 1995 to 2014, we find that, though an overall VP or SP does not exist in India, VP exists for small-cap and big-cap portfolios and SP for value and growth portfolios. Specifically, VP is significantly positive in small-cap portfolios, weakly evident (though often close to zero) in mid-cap portfolios, and significantly negative in big-cap portfolios. Similarly, SP is significantly positive in value portfolios, almost zero for neutral ones, and negative in growth portfolios. Thus, the mid-cap portfolio acts as a zero-premium benchmark for VP, while the neutral portfolio acts as a zero-premium benchmark for SP. On 3

either side of the benchmark, there is a positive or negative premium. We are not aware of any work that finds the median (midcap or neutral) portfolio to be the zero-premium benchmark. We would like to iterate that our focus here is on the median portfolios and not on analyzing VP or SP per se in the Indian context. In fact, our medial portfolio analysis can be extended to other capital markets, including USA. After repeating the FF approach, we resort to three more approaches for our analysis. The first one is a modified version of the approach followed FF. They sort the firms by MC and take those above the median MC as big-cap and those below as small-cap; they do not have a mid-cap group. They also separately divide the firms into three PB groups: top 30% being growth, middle 40% neutral, and bottom 30% value. They take the intersection of these groups to form six portfolios (from small-value to big-growth). Thus, small-growth stocks are those that are both in the small-cap and growth groups; similarly, big-value stocks are those that are in both big-cap and value groups. We resort to three approaches for our analysis. The first one is a modified version of the approach followed FF. We resort to an even division for both MC and PB: top one-third, middle one-third, and bottom one-third. We take their intersections to form nine portfolios. We believe this division to be quite reasonable. In fact, FF clarify that their splits are ‘arbitrary’, though they do add that they do not expect that to influence the results. Our modification yields some new insights and turns out to be pivotal. But, we find the portfolio sizes even under this modified-FF approach to be varying quite a lot, some very small and some very large. In India, most growth stocks are big-caps and most value stocks are smallcaps. So, as reported later, it is difficult to find a critical mass of big-value or small-growth stocks. Keeping this in mind, we resort to two different ways of creating portfolios, while also

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explicitly bringing in a mid-cap group. So, we go for sequential sorting, first by MC or PB and then by the other variable. Thus, it gives rise to two separate approaches: 1) MCPB, where we sort first by MC and then each MC group by PB, and 2) PBMC, where we first sort by PB and then PB group by MC. All our divisions are tercile based; top one-third, middle one-third, and bottom one-third. So, we get nine portfolios under either of the approaches. The results under these two approaches differ marginally, but the overall direction of the findings is quite similar. Our findings under MCPB and PBMC approaches bear testimony to their usefulness of vis-à-vis FF’s approaches. The paper is organized as follows. Section II reviews the existing literature. Sections II and IV, respectively, discuss methodology and data. Section V discusses the findings, while Section VI concludes. II. LITERATURE REVIEW How VP varies across portfolios of different MC have been explored in the US market. Loughran (1997) argues that the VP in the US equity-market during 1963 and 1995 exists only for small stocks. FF also find that, during 1963-1991 in US, except for the growth portfolio, average returns tend to decrease as they move from small-cap to big-cap portfolios. Fama and French (2006) find that, though, when PB is used as the measure of value or growth, 1963-2004 VPs in USA are smaller for big-cap firms, when price-earnings ratio (PE) is used instead, 19632004 VP between big-cap and small-cap stocks virtually vanishes. Similarly, while carrying out an out-of-sample check, they find 1975-2004 VP for 14 major markets outside the United States to be similar for big-cap and small-cap stocks, whether PB or PE is used. These results suggest, they feel, that the weak inverse relation between PB and average returns observed for the topmost big-cap US stocks during 1963 to 2004 may be due to the small sub-sample of low-PB stocks among these. The CAPM (Capital Asset Pricing Model) can explain the 1926-1963 VP,

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but not that of 1963-2004. Fama and French (2006) report that their evidence on VP in US is ‘a bit more complicated’. The overall VP in US is similar before and after 1963, but they find that, during 1963-2004, value stocks have lower s than growth stocks. So, the CAPM fails to explain why value stocks get higher return than growth stocks. During 1926-1963, however, value stocks have higher s. So, the CAPM captures the VP very well during this period. They, however, clarify that it is size (or a non- risk related to size) that explains the VP, not the . Contrary to Fama and French (2006), however, Fama and French (2012) find that non-USA VPs are indeed larger for small-cap stocks. They consider the newer finding as the ‘typical’ one, since its sample size is more complete, vis-à-vis the thin sample of the earlier study. Andrade and Chacharia (2014), however, argue that two simple modifications to the Fama and French (2012) approach, who did not find significant VP in large stocks, can bring back such premium: (a) if we sort stocks on PE instead of PB and (b) if we use a global value breakpoint instead of country-specific ones. That yields global VP of 64 basis-points for the top 30% big-cap stocks vis-à-vis bottom 30% big-cap, as against 17 basis-points when we use country-specific benchmark. If we resort to PE based on earnings estimates instead of historical figures, this VP gets more pronounced. Thus, VP is not confined to only small stocks, they point out. Bauman and Miller (1997) provide an explanation based on adaptive expectation – how much forecasters rely on past performance – for the difference in the performance of value and growth stocks. Asness, Friedman, Krail, and Liew (2000) find that value portfolio performs better than growth portfolio when the spread in their valuation multiples is high and earnings growth is low. Bourguignon and de Jong (2003) carry out an experiment in six major markets, separating the structural characteristics (long term) and time effects (short term) in the PB of stocks, and find that the VP goes away as soon as the time-effect is controlled for. Brush (2007)

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argues that VP would go away if we use equally-weighted factors representing generic value and growth. How SP has varied with PB or PE ratios have, however, been hardly studied. Some of our findings share the spirit of a study by Arshanapalli, Fabozzi, and Nelson (2006), who discover, using 1962-2005 data for US, that value stocks are of low-risk, as they do quite well in distressed and recessionary markets and are unaffected by monetary policy, though they do perform poorly during periods of high credit-spreads. They do find, however, that smallcap portfolios do poorly in periods of recession, subdued capital market, and tight monetary policy, lending support to the risk-based explanation of SP. An earlier work by Jensen, Johnson, and Mercer (1998) reaches somewhat different conclusion: both SP and VP being strong under expansive monetary policy and weak or negative under restrictive policy. A similar work by Hur, Pettengill, and Singh (2014) also finds, using 1931-2006 US monthly data, that, in months other than January, there is a small-firm effect in down markets and a large-firm effect in up markets. But, some practitioners like Bogle (2001) argue that FF’s findings, like all such, are period-dependent; their study coincided with the value-investing era. In the context of Indian market, some studies establish the existence of SP and VP. Mohanty (2002) uses the 1991-2000 Indian equity data from Prowess database of CMIE (Centre for Monitoring Indian Economy) and finds the lowest MC quintile to be giving around 4% more per month than the highest quintile. Coincidentally, analyzing 1990-2003 Indian data for 482 companies of BSE-500 (equity index), Sehgal and Tripathi (2005) also find an SP of around 4% per month. Similarly, using 1990-2006 Indian equity data, Kumar and Gupta (2007) also argue that there exists VP and SP in the Indian context. Agarwalla, Jacob, and Varma (2013), who use 1993-2012 CMIE-Prowess data, however, find the SP to be negative, -0.8% per annum. So far as VP is concerned, Deb, Banerjee, and Chakrabarti (2006) and Deb (2012) find some evidence

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in support of its existence in Indian equity. Even Agarwalla, Jacob, and Varma (2013) find the annual VP in India to be a conspicuous 6%. But, using data during a period of high economic growth and stock-market development, Ebrahim, Grima, Shah, and Willams (2014) do not find VP in India, though they reconfirm it in other emerging markets and attribute the latter observation to the fact that growth firms hoard cash in anticipation of cashing in on later windows of opportunity, thus driving down their returns. Using 1995 – 2011 data for Indian equity, Dash and Mahakud (2014) show that VP can be explained by some factors but SP cannot, implying that SP has not vanished as in other developed markets. But none analyses whether VP in India is explained by CAPM and, in particular, whether it varies with firm size. Moreover, no one has studied SP in the Indian context, namely whether CAPM can explain it and especially if it varies with PB.

III. METHODOLOGY We compute return through both BAHR (buy-and-hold return) over different holding periods, from one year to five years, and AMAR (average of monthly-average-return). When we use AMAR, we take, for each year from 1995 to 2014, all stocks trading in Bombay Stock Exchange that have the required data for that year: MC on the first trading day of the year, positive PB on that date, and return for each of the twelve months. For the BAHR measure, we require return for each of the subsequent sixty months, with exceptions for the last four years. Now, to compute VP, we have to subtract the return on the growth portfolio from the return on the value portfolio. Similarly, to calculate SP, we have to find out how much more is the return on the small-cap portfolio compared to that on the big-cap portfolio. We need to define and construct these portfolios. But for that, we have to first define value stocks and growth stocks.

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When we sort the stocks in the descending order according to the PB, those in the top one-third are growth stocks, the middle one-third neutral, and the bottom one-third value. Similarly, when we sort the stocks in the descending order according to the MC, those in the top one-third are big-cap stocks, the middle one-third mid-cap, and the bottom one-third small-cap. With the help of these, we come up with nine different two-dimensional combinations, one dimension based on MC and the other on PB. We follow three different approaches. In our modified-FF approach, in which we separately sort the stocks by MC and PB, we deviate from FF’s 30%-40%-30% approach and make it 1/3-1/3-1/3.1 We also bring in a medium-PB portfolio unlike FF, who had only high-PB and low-PB portfolios. In the MCPB approach, we first sort the firms into three equal groups based on the MC dimension (small-cap, mid-cap, and big-cap) and, after that, each group into three equal sub-groups based on PB dimension (value, neutral, growth). This gives rise to nine MC-PB portfolios: SV (small-value), SN (small-neutral), SG (small-growth), MV (middlevalue), MN (middle-neutral), MG (middle-growth), and BV (big-value), BN (big-neutral), and BG (big-growth). In the PBMC approach, where we first sort by PB and then MC, our nine portfolios become VS, VM, VB, NS, NM, NB, GS, GM, and GB. SV portfolio under the MCPB approach and the VS portfolio under the PBMC approach, however, differ from the SV portfolio under the modified-FF approach. SV in modified-FF means portfolio of stocks that are both in the small-cap group (lowest one-third by MC) and value group (lowest one-third by PB). But, SV under MCPB means portfolio of value stocks (lowest one-third by PB) only among the small-cap stocks (lowest one-third by MC). Similarly, VS under PBMC means portfolio of small-cap stocks (lowest one-third by MC) only among value stocks (lowest one-third by PB). 1

In case the total number of firms leaves a remainder of two when divided by three, one firm each is added to the top and bottom group, while, if the remainder is one, one firm is added only to the middle group. For example, with 1841 (1840) firms, the three groups, from top to bottom, would be as follows: 614, 613, 614 (613, 614, 613).

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Our main focus, however, is not per se on VP or SP, but on how these vary across portfolios with different MC and PB, respectively, and especially on the median portfolios, hitherto ignored by most researchers. Specifically, we are interested in finding how SPs differ for value, neutral, and growth portfolios and VPs vary across small-cap, mid-cap, and big-cap portfolios. SP for value portfolio is calculated as SV-BV under MCPB approach and VS-VB under PBMC. Here, SV-BV means the excess return of the SV portfolio over that on BV, others being defined similarly. Thus, SPV, which equals SV – BV, can be construed to be the return on a portfolio that is long in SV and short in BV. We can similarly define VP for small-cap, midcap, and big-cap as, respectively, SV-SG, MV-MG, and BV-BG under the MCPB approach and VS-GS, VM-GM, VB-GB under the PBMC approach. Table-1 delineates all these details. Anyway, if we want to keep these portfolios unchanged over the full study-period of twenty years, we face two problems: (a) very few stocks have continuous data over these years, and (b) a stock that is a small-cap stock in the beginning may have become mid-cap or big-cap after a few years and vice versa. So, we form the portfolios annually by sorting every year, though we compute return on a monthly basis. Thus, the return on SV during each month in the first-year comes from one set of stocks, while the return on SV during each month of the secondyear comes from a possibly different set (though overlapping). The same is true of all the portfolios. The MAR (monthly-average-return) on a portfolio is measured as a weighted average of the monthly returns on the stocks in that portfolio, where the weight on each stock is equal to the MC of the stock at the beginning of the year as a proportion of the total MC of the portfolio:

MARt 

 MC

j

t

x MR jt  /

 MC , where MARt is the MAR of the portfolio in month-t, j

i

MRjt is the monthly-return on stock-j in month-t, MCj is the market-capitalization of stock-j at the beginning of the calendar-year in which month-t falls. 10

For each portfolio, we compute two different returns based on its MAR. The first one is AMAR (Average of its MAR) during N consecutive months, with N depending on the period for which we want to determine the average. The second one is the BAHR (buy-and-hold-return)

over N consecutive months, which is obtained as

, where MARt is the MAR

on that portfolio in month-t, and N is chosen to be 12, 24, 36, 48, or 60, depending on whether we want to compute the 1-year, 2-year, 3-year, 4-year, or 5-year BAHR. The above amply clarifies the differential data requirement for the AMAR and BAHR approaches. For AMAR, for each year, we select only those stocks that have return data as well as positive MC and PB for that year. For the BAHR analysis, however, we require return data to be available for the subsequent five years, except for the last four years of our study (20112014), for which we require data for four years, three years, two years, and one year, respectively. Finally, we also carry out regression analyses to check whether CAPM beta or LPM (lower partial moment) beta can explain the premiums. LPM-beta takes into account only downside risk and, thanks to the LPM-CAPM by Bawa and Lindenberg (1977), has the same sanctity as the traditional CAPM-beta, if not more, especially when stock-return distributions are non-normal or non-elliptical.

IV. DATA Our analysis spans the years 1995 to 2014. We choose 1995 as the starting point because continuous data is available for BSE (Bombay Stock Exchange) only from that year. For each year, the annual return on the market, represented by the total return (sum of capital-gains yield and dividend yield) from the first trading-date of that year to the first trading-date of the next year on the NSE (National Stock Exchange) Nifty, India’s most popular index, is given in Table 11

2. BSE is an older exchange than NSE and gives data going back a longer time, but, so far as the market-return is concerned, not only is NSE Nifty more popular among researchers, it is one for which the total return, which includes dividend yield, is available. A bull year is defined as one in which the annual market-return exceeded the risk-freerate during that year, while a bear year is where the opposite happened; ‘bull years’ refers to the set of the bull years and ‘bear years’ to the set of bear years. The table also highlights that there are two contiguous phases in our sample: the bear phase from 1995 to 2002 and the bull phase from 2003 to 2014. In addition, we also term a month as a bull month if the market-return in that month exceeded the risk-free rate in that month; the reverse makes it a bear month. ‘Bull months’ refers to the set of the bull months and ‘bear months’ to the set of bear months. In our analysis, we focus on the full period, the bull phase and the bear phase, the bull years and the bear years, and the bull months and the bear months. A phase is like a holding-period, while the years and months are not. Risk-free rate used is at the end of the month and is the cut-off yield during the most recent auction of Indian 91-day treasury-bill.2 Besides, for each stock, we collect the following data from the Prowess database of CMIE (Centre for Monitoring Indian Economy). i.

PB (Price-to-Book Ratio): This is defined as the ratio of the market price of the stock on the date the ratio is reported divided by the last reported book value of equity of the stock. Since we carry out monthly analysis, we take the PB on the first trading day of every month. Thus PB for January is the PB reported on 1st January. We use this as our growth indicator, high PB representing growth stocks, low PB value stocks, and medium PB neutral stocks. We use PB as a decimal; that is, a PB of 1:2 would be taken by us as ½ = 0.5.

2

We thank ---------- of the Clearing Corporation of India for providing us this data.

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ii.

MC (Market Capitalization): It is also reported on a daily basis and is computed as the product of the closing market price of the day and the number of outstanding shares at the end of the day. This is used as our size indicator, high MC denoting big size, medium MC mid-size, and low MC small size.

iii.

Return (Monthly): This is the total return (sum of the capital-gains-yield and dividend-yield) from the first trading date of that month to the first trading date of the next month and is adjusted for stock-splits and stock-dividends. It basically assumes that the investor buys the share at the closing price of the first trading day of the month and sells it at the closing price of the first trading day of the next month. Dividends received during the period is taken into account without any adjustment for time value of money.

As mentioned earlier, even under the modified-FF approach, not only do the portfolio sizes differ drastically across the nine portfolios and across the years, but also they can be quite small for some of the portfolios, with as low as only ten stocks. In fact, for the modified-FF approach, under both AMAR and BAHR, each of BV and SG portfolios account, across the years, on the average, for around 3% of the total number of stocks, while SV and BG around 20% each. For our MCPB and PBMC approaches, however, all portfolios in a given year have, by design, the same number of stocks; but the number varies from year to year, though much less than under the modified-FF approach. The relative stability in the portfolio size is what makes us prefer our approaches to that of FF.

V. OBSERVATIONS

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A. Portfolio risks Table-3 presents the average MC, PB, CAPM-beta, and LPM-beta for the nine portfolios. It shows that MC and PB vary directly. We find that the correlation between their log values is 0.46, which is significant at 1% level. Thus, growth firms are typically big-cap firms and value firms are typically small-cap firms. But, among the big-cap stocks, value stocks have higher risk than growth stocks for the full period and bull years, bull months, and bull phase, whether by CAPM-beta or LPM-beta.3 One of the most notable observations here is that bear-month-beta is significant for all caps – though generally higher for big-caps than that for small-caps -, while the bull-month-beta is significant only for big-caps. Thus, when market does well, it is the big-caps that gain from the upswing, mid-caps and small-caps do not. On the other hand, when the market does poorly, all caps suffer, but the big-caps suffer more than the small-caps. From this, we can take the big-caps to be more volatile than mid-caps and small-caps, but with greater potential for gaining from market upswings. Full-period LPM-betas throw out another interesting fact, namely, BG (big-cap growth stocks) and SV (small-cap value stocks) have the lowest risks. This suggests that those going for big-caps may tend to choose growth stocks and those opting for small-caps may lean toward value stocks. Therefore, value stocks should exhibit a small-cap-premium and growth stocks a big-cap-premium, which is equivalent to a negative size premium. Unfortunately, though this above is consistent with what we find, since we take firm characteristics like MC and PB only at the beginning of the year while our returns are monthly, we cannot check, as is done by Patton

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LPM-beta, to put it simply, is calculated only over the bear months, thus excluding all bull months. Just as CAPMbeta tells us how a stock performs when market goes up or down, LPM-beta tells us how a stock performs only when market does poorly (that is, it when market- return is lower than the risk-free rate).

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and Timmerman (2010), how return or risk-premium of portfolios varies with their average MC or PB. B. Portfolio returns Table-4 presents the AMAR for the full period only for the four approaches: original FF, our modified-FF, MCPB, and PBMC. There is no median portfolio under the original FF approach; anyway, strikingly, most numbers are insignificant under that approach. Under the modified-FF approach, however, median portfolios, VPM and SPN, have zero premiums. When we move to higher values of MC or PB (that is, VPB or SPG), the premiums fall; when we go to lower values (VPS or SPV), premiums rise. The changes are almost symmetric, thus making the median portfolio lie almost in the middle. We find that – but not report details here – under MCPB and PBMC approaches, there is no VP or SP. Coming to VP across different MCs, we find under both approaches that, VPM is the median portfolio having zero value, with movements towards lower (higher) MC or PB leading to positive (negative) premiums. The other observations also match with what we have found under the modified-FF approach. Researchers in investment management typically use monthly-average-return as well as buy-and-hold-return (BAHR) to measure investment performance, BAHR representing the longterm performance and monthly-average representing the short-term performance. We adopt both approaches to explore the nature of VP and SP from both short-term and long-term perspectives. Table-5 presents BAHR results for MCPB and PBMC approaches. For analyzing the premiums across different MC and PB groups, we don’t have phases under BAHR, but only different holding-periods (HPs), from one-year to five-years. We also aggregate all of them under “All Years”. Though we realize that combining the HPs do not

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make much sense, we report this, since the sample size for each HP is small and that for the combined one is relatively large. The results reported here are quite robust. The median portfolios (VPM and SPN) again have zero premiums, and when we move to lower (higher) values of market-cap or PB, the premium becomes positive (negative); many of these positive (negative) premiums are significant. The only noticeable difference between the two approaches is that, the premiums here under PBMC are quite different from those obtained using AMAR, though ones using the MCPB approach here are much closer to what we find under AMAR. Table-6 presents AMAR during bull and bear periods. Comparing big-caps and smallcaps during the bull and bear phases or years under modified-FF approach, we find that, VP for the big-caps does not vary much, while, for the small-caps, VP is strictly higher during the bear years or phase vis-à-vis bull years or phase. Comparing value and growth stocks the same way, we discover a slightly different pattern: SP among growth stocks is significantly negative and lower in bull phase or years, and SP among value stocks is significantly positive and higher in bear phase or years. These observations somewhat hold for original FF approach only for the phases. Under MCPB and PBMC, VP is also insignificant during the various sub-periods. SP is, however, positive during the bear years and is significantly higher than its negative value during the bull years. The positive value during the bear years is consistent with the higher risk, though we cannot be certain that risk would explain the full positive premium; but the significantly negative value during the bull years cannot be fully explained by risk. Value premium among big-cap stocks does not vary across bull and bear years, but, among small-cap stocks, is significantly higher during bear years or phase vis-à-vis bull years or phase. Given that value stocks among small-caps are less risky than the growth stocks in this

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group, more so during the bear years, this seems to be a puzzle. But, we believe that, under this perspective, the low risk of value stocks makes it popular among those who choose small-cap ones. If, however, we focus on the phases, value stocks among small-caps are riskier than growth stocks in this group and should command higher premium. But, SP among growth stocks is significantly negative and lower in bull phase or years as against bear phase or years, but, among value stocks, is significantly positive and higher during the bear years or phase than the bull years or phase. Since, among value stocks, small-cap ones are riskier than big-cap ones and far more so during the bear years, this is not difficult to appreciate. iii. Similarity between our approaches Since we have studied above VP and SP using modified-FF as well as our MCPB, and PBMC approaches, we naturally wonder how VPs or SPs across the approaches are related. We find that (details not reported here) SPs across the MCPB and PBMC are quite highly correlated, whether we consider AMAR or BAHR; but VPs don’t exhibit that, though they are more closely related under AMAR. Overall, though inferences based on our MCPB and PBMC approaches are not very different from what we have obtained earlier under the modified-FF approach, because the directions are often the same, we find that the magnitudes vary. On the whole, we find evidence for existence of positive VP for small-cap stocks and positive SP for value stocks, while negative VP for big-cap stocks and negative SP for growth stocks. We find these using both average of monthly average returns and buy-and-hold returns.

iv. Ability of risk to explain differential in value and size premiums

17

To check whether the risk can explain the value or size premiums obtained under the MCPB or PCMB approach, we carry out three-factor regression of VPS, VPM, VPB, SPV, SPN, or SPG. Results are reported in Table-7. If the premiums are explained by the three-factor model, the intercepts in these regressions would not have been different from zero. We, however, find that most of the intercepts obtained from the regressions over the full period or sub-periods are significant. Moreover, the intercepts mostly have the same size as the premiums under our approaches. So, VP alpha for small-cap portfolios is higher than that for the mid-caps, which, in turn, is higher than that for big-caps. Similarly, SP alpha for value stocks is higher than that for neutral stocks, which, in turn, is higher than that for the growth stocks. This persistence lends more credence to our findings. So, if we invest in a small-cap portfolio, we should go long in value stocks and short in growth stocks; on the other hand, if we choose a big-cap portfolio, we should go short in value stocks and long in growth stocks. Similarly, if we invest in a value portfolio, we should go long in small-cap stocks and short in big-cap stocks; on the other hand, if we choose a growth portfolio, we should go short in small-cap stocks and long in big-cap stocks. The table also highlights that, in the next level, the VPS-B portfolio is expected to give abnormal risk-adjusted positive return; the portfolio involves going long in VPS and going short in VPB. Going long in VPS involves choosing a small-cap portfolio with long positions in value stocks and short position in growth stocks; similarly, going short in VPB involves choosing a big-cap portfolio with short positions in value stocks and long position in growth stocks.

18

VI. CONCLUSION Following the seminal work by Fama and French (1993), many researchers have computed VP (value-premium) or SP (size-premium), respectively, as the difference between the return on portfolios of two extreme groups according to MC (market-capitalization) or PB (pricebook ratio). Using a modified version of the Fama-French (1993) approach, as well as two of our own approaches and employing Indian equity data from 1995 to 2014, we, however, conclude that the median portfolios are quite pivotal. Mid-MC portfolios exhibit zero VP and mid-PB portfolios zero SP. As we move to lower levels of MC or PB, the premium – VP or SP, as the case may be – increases, thus becoming positive, while a movement toward higher levels of MC or PB makes the premium fall, making it negative. Jurek and Viceira (2011) argue, using a dynamic portfolio model, that investors who choose only equity would lean towards value stocks in short-horizons but find growth to be a more attractive and less risky as the horizon increases. We feel that our findings are similar in the sense that equity-stories can influence the choice between growth and value stocks. It implies that, if we choose a small-cap portfolio, we should go long in value stocks and short in growth stocks, whereas, if we choose a big-cap portfolio, we should go short in value stocks and long in growth stocks. We can also combine a small-cap portfolio with a big-cap one keeping the above strategies in mind to get risk-adjusted abnormal positive return.

REFERENCES

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Agarwalla, S. K., Jacob, J., and Verma, J. R. (2013). Four factor model in Indian equities market. Indian Institute of Management Ahmedabad Working Paper. Andrade, S. K. & Chhaochharia, V. (2014). Is there a value premium among large stocks? Available at ssrn: http://ssrn.com/abstract=2437567, Accessed 31st August. Arshanapalli, B., Fabozzi, F. J., & Nelson, W. (2006). The value, size and momentum spread during distressed economic periods. Finance Research Letters, 3, 244–252. Asness, C. S., Friedman, J.A., Krail, R. J., & Liew, J. M. (2000). Style timing: Value versus growth. Journal of Portfolio Management, 26(3), 50-60. Bauman, W S. & Miller, R. E. (1997). Investor expectations and the performance of value stocks versus growth stocks. Journal of Portfolio Management, 23 (3), 57-68. Bawa, V. & Lindenberg, E. (1977). Capital market equilibrium in a mean-lower partial moment framework. Journal of Financial Economics, 5, 189–200. Bogle, J. C. (2001). The stock market universe: Stars, comets, and the sun. Remarks before the Financial Analysts of Philadelphia, February. Bourguignon, F. & Marielle, D. J. (2003). Value versus growth. Journal of Portfolio Management, 29 (4), 71-79. Brush, J. C. (2007). Value and growth: Theory and practice. Journal of Portfolio Management, 33 (3), 22-32. Dash S. R. & Mahakud, J. (2014). Do asset pricing models explain size, value, momentum and liquidity effects? The case of an emerging stock market. Journal of Emerging Markets Finance, 13, 217-251. Deb, S.G. (2012). Value versus growth: evidence from India. IUP Journal of Applied Finance, 18, 48-62. Deb, S.G., Banerjee, A., & Chakrabarti, B. B. (2006). Value premium in Indian equity market: empirical evidence. IUP Journal of Applied Finance, 12, 48-68. Ebrahim, M. S., Grima, S., Shah, M. E., & Willams, J. (2014). Rationalizing the value premium in emerging markets. Journal of International Financial Markets, Institutions and Money, 29, 51-70. Fama, E. F. & French, K.R. (1992). Cross section of expected stock returns. Journal of Finance, 47 (2), 427-465. Fama, E. F. & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33, 3-56. 20

Fama, E. F. & French, K. R. (2006). The value premium and the CAPM. Journal of Finance, 61, 2163-2185. Fama, E. F. & French, K. R. (2012). Size, value, and momentum in international stock returns. Journal of Financial Economics, 105, 457-472. Kumar, R. & Gupta, C. P. (2007). A re‐examination of factors of asset pricing in Indian stock market. In R. Nandagopal & V. Srividya (Eds.), Emerging Financial Markets (pp. 105-122). Excel Books. Hur, J., Pettengil, G., & Singh, V. (2014). Market states and the risk-based explanation of the size premium. Journal of Empirical Finance, 28 139-150. Jensen, G. R., Johnson, R. R. & Mercer, J. M. (1998). The inconsistency of small-firm and value stock premiums. Journal of Portfolio Management, 24 (2), 27-36. Jurek, J. W. & Viceira, L. M. (2011). Optimal value and growth tilts in long-horizon portfolios. Review of Finance, 15, 29-43. Lougran, T. (1997). Book-to-market across firm size, exchange, and seasonality. Journal of Financial and Quantitative Analysis, 32, 249-268. Mohanty, P. (2002). Evidence of size effect on stock returns in India. Vikalpa 27, 27-37. Patton, A. J. & Timmerman, A. (2010). Monotonicity in asset returns: New tests with applications to the term structure, the CAPM, and portfolio sorts. Journal of Financial Economics, 98, 605–625 Sehgal, S. & Tripathy, V. (2005). Size effect in Indian stock market: Some empirical evidence. Journal of Business Perspective, 9, 27-42.

21

Table 1. Premium definitions

Premium Premium Definitions VP SP

MCPB Approach

PBMC Approach

((SV+MV+BV) - (SG+MG+BG))/3 ((SV+SN+SG) - (BV+BN+BG))/3

((VS+VM+VB) – (GS+GM+GB))/3 ((VS+NS+GS) – (VB+NB+GB))/3

VPS (VP: small-cap portfolio) VPM (VP: mid-cap portfolio) VPB (VP: big-cap portfolio) VPM-B VPS-M VPS-B

SV – SG MV – MG BV – BG VPM - VPB VPS – VPM VPS - VPB

VS - GS VM - GM VB - GB VPM - VPB VPS – VPM VPS - VPB

SPV (SP: value portfolio) SPN (SP: neutral portfolio) SPG (SP: growth portfolio) SPN-G SPV-N SPV-G

SV – BV SN – BN SG – BG SPN – SPG SPV – SPN SPV - SPG

VS - VB NS - NB GS - GB SPN – SPG SPV – SPN SPV - SPG

22

Table 2 Returns for years and phases Year

Annual Return

1995

-19%

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

16% -3% 5% 60% -16% -21% 0% 79% 20% 56% 46% 28% -46% 75% 13% -9% 18% 5% 44%

Phase: Years: Return

Bear Phase: 1995-2002: 0%

Bull Phase: 2003-2014: 998%

23

Table 3 Summary statistics This table gives summary statistics for CAPM Beta, LPM Beta, Size (MC), and Price-Book ratio (1995-2014) for the nine basic portfolios in the MCPB approach. In this table, we report the summary statistics of the nine base portfolios in the MCPB approach. The monthly excess returns (that is return in excess of the risk-free rate) of the base portfolios are generated over 240 months from January 1995 to December 2014 and regressed over the monthly excess return of the market to get the beta estimates. For computing LPM-beta, we drop the months in which the market-return exceeds the risk-free rate. NSE NIFTY Total Return Index is used a proxy for the market. The average size and PB of each portfolio is obtained by taking the simple average of the 20 beginning-of-year values of MC and PB of individual stocks in that portfolio. Similar tables are also generated for the PBMC and modified-FF approaches; they are not reported here, as the results are not very different and would add little marginal information. Average monthly betas of the portfolios across different periods Period/RiskMeasure

No of Obs

Full period beta

240

Bear phase beta

96

Bull phase beta

144

Bear years beta

108

Bull years beta Bear months beta

132

0.042* 0.047* * 0.079* *

131

0.064*

Bull months beta Fullperiod LPMbeta Bear years LPM-beta Bull years LPMbeta

109

SV 0.059* * 0.090* *

SN 0.048* *

SG 0.044* *

MV 0.045* *

MN 0.053* *

MG 0.054* *

BV 0.057 **

0.046* 0.050* *

0.033 0.052* *

0.041* 0.057* *

0.052* 0.053* *

0.023 0.070* *

0.063*

0.046* 0.042* * 0.031* * 0.056* * 0.083* *

0.078*

0.030* 0.070* * 0.078* *

0.032* 0.072* * 0.102* *

0.000

-0.019

-0.012

-0.016

0.002

240

0.103

0.110

0.115

0.110

108

0.084

0.081

0.088

132

0.192

0.221

0.177

0.026* 0.072* *

0.046 0.057 **

BN 0.064 ** 0.062 * 0.056 **

BG 0.049 ** 0.053 * 0.042 **

0.014

0.021 0.093 ** 0.094 * 0.059 *

0.025 0.098 ** 0.089 * 0.082 *

0.020 0.080 ** 0.082 * 0.047 *

0.116

0.121

0.103

0.111

0.086

0.074

0.094

0.100

0.077

0.087

0.061

0.191

0.160

0.145

0.145

0.134

0.151

Average of market capitalization (Rs million)

24

Mean Minimum

Maximum

S V

S N

S G

59 9

78 5

93 9

2 1, 24 8

3 1, 76 1

2 2, 23 1

M V 4, 86 1 19 2 11 ,6 46

M N 5, 61 8 19 5 14 ,1 78

M G 6, 17 8 19 2 16 ,0 25

M V 0. 54 0. 04

M N 0. 99 0. 39

1. 31

2. 23

M G 3. 46 0. 89 17 0. 08

BV 12 2,4 87 1,7 46 39 3,1 48

BN

BG

314,6 10

480,0 40

1,753

1,741

2,905 ,872

4,380 ,108

BN

BG

2.02

7.51

0.98

2.53

3.65

120.2 3

Average of price-to-book ratio (in decimal)

Minimum

S V 0. 31 0. 00

S N 0. 59 0. 23

Maximum

1. 09

1. 68

Mean

S G 1. 87 0. 54 12 9. 31

BV 0.9 7 0.0 7 1.6 9

25

Table 4 AMAR (Average of Monthly Average Returns) for full period under different approaches This table gives the AMAR (Average of Monthly Average Returns) for the full period under different approaches. So, n = 240. FF

VPS VPM VPB SPV SPN SPG

0.09% -0.17% 0.14% -0.01% -0.11%

Modified FF

0.25%** 0.06% -0.13%* 0.12%* -0.08% -0.26%**

MCPB

PBMC

0.20%** 0.03% -0.08%* 0.08%* 0.00% -0.19%**

0.22%** 0.03% -0.13%* 0.19%* -0.07% -0.16%**

*** indicates significance at 1%, ** at 5%, and * at 10%

26

Table 5 Mean values of BAHR (buy-and-hold-return) for different holding periods: MCPB & PBMC only The ‘n’ values on top of each column indicate the number of observations used to calculate the mean. The means are calculated across the number of portfolio-formation years. Thus, for the one-year BAHR, we have 20 portfolio formation years: one for each year. For the two-year BAHR, it is 19, one less, and so on. Although the BAHR values are over a particular holding period, all the values indicated in the table are on annualized basis. For all years, the mean of all such values across all holding periods from one year through 5 years is indicated. 1 year Premium n=20 Panel A : MCPB Approach VPS 0.53% VPM 0.45% VPB -1.11% SPV -0.78% SPN -0.62% SPG -2.43%** Panel B : PBMC Approach VPS 2.22%* VPM -0.27% VPB -1.80%* SPV 2.27%* SPN -1.07% SPG -1.75%**

2 years

3 years

4 years

5 years

All Years

n=19

n=18

n=17

n=16

n=90

0.38%* 0.22% -1.08%* -0.10% 0.05% -1.56%*

0.74%* 0.38% -0.66% -0.15% 0.76% -1.55%*

0.86%* 0.36% -0.34% 0.37%* 1.24% -0.83%

0.97%** 0.41% -0.09%* 0.59%* 1.11% -0.48%

0.68%** 0.36%* -0.69%** -0.05% 0.46% -1.42%***

2.49%* -0.41% -1.75%** 2.68%* -0.64% -1.57%**

2.51%* -0.18% -1.35%** 2.47%* -0.05% -1.40%**

2.41%* -0.01% -1.03%* 2.50%* 0.35% -0.95%*

2.17%* 0.19% -0.86%* 2.20%* 0.58% -0.83%*

2.36%*** -0.15% -1.39%*** 2.42%*** -0.21% -1.33%***

*** indicates significance at 1%, ** at 5%, and * at 10%

27

Table 6 AMAR (Average of Monthly Average Returns) for bull & bear periods for various approaches This table gives AMAR (Average of Monthly Average Returns) for the bull and bear periods (both phases and years) under different approaches. So, n varies, but total n = 240.

Bull Phase n=144

Bear Phase

Bull Years

Bear Years

n=96

n=132

n=108

0.35%** -0.15% 0.49%* 0.09% -0.02%

0.01% -0.14% 0.01% -0.12% -0.14%

0.18% -0.19% 0.29% 0.05% -0.08%

0.10% 0.04% -0.11% -0.10%* -0.33%** -0.32%***

0.47%*** 0.08% -0.17% 0.46%*** 0.29%* -0.18%

0.21% -0.01% -0.11% -0.03% -0.26%** -0.35%**

0.30%** 0.15% -0.16% 0.31%** 0.14% -0.15%

Panel C: MCPB Approach VPS 0.03% VPM 0.03% VPB 0.07% SPV -0.32%* SPN -0.31%** SPG -0.28%***

0.44%*** 0.03% -0.30% 0.68%*** 0.47%* -0.06%

0.10% -0.06% 0.05% -0.17% -0.24%** -0.22%**

0.31%** 0.14% -0.23% 0.39%** 0.29% -0.16%

Panel A: FF (original) approach VPS -0.09% VPM VPB -0.17% SPV -0.09% SPN -0.13%* SPG -0.17%* Panel-B: modified-FF approach VPS VPM VPB SPV SPN SPG

Panel D: PBMC Approach VPS -0.06% VPM -0.12% VPB -0.10% SPV -0.22%* SPN -0.34%** SPG -0.27%***

0.64%*** 0.24% -0.16% 0.80%*** 0.33%* -0.01%

0.10% -0.07% -0.12% 0.03% -0.27%** -0.18%**

0.38%** 0.15% -0.14% 0.38%** 0.17% -0.14%

*** indicates significance at 1%, ** at 5%, and * at 10% 28

Table 7: Three-factor alphas under various approaches This table presents results of 3 factor models applied to the premiums to test if traditional 3 fcator model explains the premiums under the MCPB, PBMC and FF approach using AMAR. The three-factor regression is as follows: Premt = α + β1 (RMt - RFt) +β2 (RMt - RFt)+ β3 (RMt - RFt) + εt, where Premt is the size or value premium ( excess over Rf) for month t, RFt is the risk free rate and RMt is the market return. The portfolios are explained in the text.

Full period VPS 0.002** VPM 0.001 VPB -0.001 SPV 0.001 SPN -0.001 SPG -0.003** VPSB 0.004**

Bull phase 0.001* 0.001 -0.001 -0.001 -0.004** -0.004**

Bear Phase 0.005** 0.000 -0.001 0.004** 0.004 -0.002

0.002

0.006**

Full period VPS 0.002** VPM 0.000 VPB -0.001 SPV 0.000 SPN 0.000 SPG -0.002** VPSB 0.003**

Bull phase 0.000 0.001 0.001 -0.003** -0.003** -0.003**

Bear Phase 0.003* 0.000 -0.003 0.006** 0.004 0.000

0.000

0.006**

Full period VPS 0.002 VPM 0.000 VPB -0.001 SPV 0.002 SPN -0.001 SPG -0.001* VPSB 0.003**

Bull phase 0.003 0.000 -0.001 0.003* 0.000 -0.001

Bear Phase 0.000 0.000 -0.001 0.000 -0.002 -0.001

0.004**

0.001

Modified FF Bull Years Bear Years 0.001 0.004 0.001 0.001 -0.001 0.000 -0.001 0.002 -0.003** 0.002 -0.004** -0.002 0.002 0.003 MCPB Bull Years Bear Years 0.001 0.002 0.000 0.001 0.001 0.000 -0.002 0.002 -0.003* 0.001 -0.002** -0.001 0.001 0.003 PBMC Bull Years Bear Years 0.002 0.001 0.000 0.000 -0.001 -0.001 0.002 0.001 -0.001 -0.001 -0.002 -0.001 0.003

0.002

Bull months 0.004* 0.003** -0.001 0.000 -0.002 -0.005**

Bear months 0.005** 0.003** 0.006** 0.001 0.002 0.002

0.006**

-0.001

Bull months 0.003 0.002 -0.002 0.000 -0.003 -0.004

Bear months 0.003* 0.003** 0.004** 0.003 0.005** 0.004*

0.005*

-0.001

Bull months -0.003 -0.002 -0.003 -0.004 -0.004 -0.005**

Bear months 0.000 -0.002 -0.004** -0.001 -0.002 -0.005**

0.001

0.004

*** indicates significance at 1%, ** at 5%, and * at 10%

29