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5.5 Data, style definition and test results 5.5.1 Data From Jan 1980 to Dec 2004, at the end of each June/December, all U.K. stocks are divided into 2 parts based on previous 6-month firm characteristic value X (to be consistent with previous two chapters, here X is APC, BM, DY, respectively) 26. Only stocks with positive X values and denominated by local currency (£) are included in the study. Stocks denominated by foreign currencies are excluded since their returns are also affected by foreign exchange rate fluctuations.
Following the literature, stocks that belong to the financial sectors are also excluded because their firm characteristics (e.g. APC, BM, DY) do not have the same meanings as that of non-financial stocks. To avoid the sample selection bias, all delisted stocks are retrieved and added back to the sample during the time that they are still “alive”. If a firm is delisted, the proceeds from the sale of this stock are invested equally in other firms in the style that it belongs to. After cleaning the data, at the end of each June/December, qualified stocks are ranked independently in ascending order by X and market value (MV). All sorted stocks are further allocated to 3 equal-sized MV and 3 equalsized X groups, resulting 9 (interaction) style portfolios. After styles are defined at the end of each June/December, the style category of a stock belonging to will be maintained fixed for the next 6 months, regardless whether the underlying stock’s characteristic value X is changed or not.
Chapter 3 shows that based on the role of the predicted risk premias from the
state variables zt and the pricing errors in the observed style premiums, it is suggested that the size premium and value premiums on stocks based on characteristics of APC and BM are likely related to the unpredicted component of the vector zt, while value premium based on DY seems to represent compensation for bearing business cycle risk. Such relative style returns are mainly driven by the predicted component from the state vector. In this conditional style timing policy problem that is linear on zt, style portfolios based on company characteristics of APC and BM are still included to study because zt may be significant predictor of the optimal style weights although it may fail to predict the style return moments.
Based on this procedure, monthly style return series are generated. 9 equity styles are prepared here (i.e. SV, SB, SG, MV, MB, MG, LV, LB and LG), both with value weighted and equally-weighted time-series returns from Jan 1981-Dec 2004.
Table 5-1 reports the summary statistics of the returns of simple style investing strategies during the sample period (Jan 1981 – Dec 2004).
It also reports the descriptive statistics of the 4 business cycle related variables used in this chapter. To be consistent with Chapter 3, the 4 macroeconomic variables used are default risk premium (def), dividend yield (div), the term spread (term) and short-term interest rate (yld). def is the yield spread between the lower- to higher- bond and is measured as the yield on corporate bonds less the yield on long-term U.K. government bonds. div is the dividend yield on the overall market index as proxied by the Datastream U.K. market index.
term is the difference between the 20-year gilt and 3-month Treasury bill yields and the short-term interest rate yld is proxied by the 3month Treasury bill yield. It is generally believed that these variables convey information about the macroeconomy and business cycle conditions and therefore affects the inter-temporal behaviour of equity style returns.
Table 5-1 shows that during the sample period raw monthly returns derived from simple style investing strategies are both positive and significant based on standard t test (sample size 288). Regardless which firm characteristic variables to define the value style dimension, each month on average equally-weighted value investing outperform growth investing by 1.48%, 0.94% and 0.77% based on APC, BM and DY sorting, while value-weighted style return differentials would be 1.40%, 0.77% and 0.73%, respectively. Likewise, each month on average an equally-weighted portfolio with small stocks and positive APC, BM and DY values could beat the counterpart portfolios with large stocks only by 0.78%, 0.36% and 0.50%, respectively. Such return differentials are generally significant in a t-statistics sense.
Since both value and small styles could beat their growth and large counterparts, arguably a style investing with stocks that capture the interaction of value and size effects could generate even better results.
Indeed, as Table 5-1 suggests, investing equally on the small value (SV) stocks earns average monthly returns of 2.79% if sorted by APC (2.10% and 2.00% based on BM and DY, respectively). The same strategy with large growth (LG) stocks yields monthly average returns of 0.99% by APC sorting (0.77% and 0.84% based on BM and DY, respectively).
Similar results obtained for value weighted scheme. When comparing the return differentials (spreads) between SV and LG stock groups to those with broad small-large and value-growth stocks, regardless how returns are calculated, it shows that the style return spread between SV and LG stocks is the largest, indicating that they do capture the principle investment characteristics of value and size styles and hence represent better risk-return structure. This justifies the selection of 4 styles (i.e. LV, LG, SV, SG) rather than the all 9 style portfolios in the style allocation process discussed later in this Chapter.
Table 5-1 Descriptive statistics of the performance of simple style investing strategies From Jan 1980 to Dec 2004, at the end of each June/December, all U.K. stocks (excluding financial sectors, dead/delisted stocks retrieved and dealt with properly) are sorted according to previous 6-month firm characteristic values of APC, BM and DY (only stocks with positive research values are studied). All sorted stocks are further sorted according to the market capitalisations, resulting 9 (intersection) style portfolios. Based on the sorting simple style investing returns are calculated. All returns are denominated by £, equally-weighted (EW) and value weighted (VW) schemes are reported.
5.5.2 Style definition and investor type The underlying investment opportunity set is investor specific because different investors have different preferences. Assume a hypothesised multi-style investor has access to the following equity style portfolios
in the market:
1. Small and large stocks (2 styles)
2. Value and growth stocks (2 styles)
3. Small Value (SV), Small Growth (SG), Large Value (LV) and Large Growth (LG) (4 styles)
4. Small Value (SV), Small Blend (SB), Small Growth (SG), Middle Value (MV), Middle Blend (MB), Middle Growth (MG), Large Value (LV), Large Blend (LB) and Large Growth (LG) (9 styles) The assumption of these investment instruments are reasonable in today’s financial market, in particular given the rapid development of Exchange Traded Funds (ETF) that track a specific market or market segments. For example, Vanguard follows a nine-box style box to form US stock ETF funds with holdings distributed by primary investment styles like growth, value, or blend and market segment (large-, mid-, and small-cap companies). The value-growth and small-large of (1) and (2) are typical two dimensions of equity style definition, while (3) and (4) offer more options based on the interactions of size and valuegrowth definition and hence represent specific risk-return structure.
Assume that the investors are mean-variance optimisers in traditional Markowitz paradigm with degree of risk aversion of 5. Assume that
these investors can be broadly divided into two types:
1. Sceptics – these investors disregard business cycle effect in their asset allocation process and hence implement the unconditional optimal style investing;
2. Doctrinaires – these investors trust that business cycle condition could affect their asset allocation decision, and therefore apply conditional optimal style investing incorporating the business cycle information;
The Doctrinaires can also be subdivided into those who follow the traditional two-step approach and those apply Brandt and SantaClara (2006) when timing their investings. At this stage it is assumed the Doctrinaires are Brandt and Santa-Clara (2006) followers.
Consider monthly and quarterly return frequencies 27. The optimal multi-style investing (i.e. ‘portfolio of style portfolios’) are first derived using the initial 120 (60) monthly (quarterly) returns, then using the 121 (61) observations, and so on, …, and are finally rebalanced using the T-1 observations, where T = 288 (T = 96) denoting the sample size based on monthly (quarterly) returns. The expected one-period-ahead excess investing returns are obtained from multiplying the optimal style weights of period t-1 by period t realised style excess returns28.
The time-series of this recursive scheme are recorded and analysed.
5.5.3 Test results and discussion
There are many test results based on various controlling parameters.
The motivation to use different control variables is to obtain a general insight of the findings for the research questions. The definition of equity styles is sometimes ambiguous in the literature. For example, value stocks can be defined as those with low price to cash-flow ratios, or high book-to-market ratios or stocks with high dividend yields. This Chapter use firm characteristics of APC, BM and DY to form portfolios on the value-growth dimension. Arguably, using different variables to The sample data length (288 months returns or 96 quarterly returns) does not allow the test of annual returns using 4 or 9 styles due to loss of degree of freedom.
The minimum number of observations to test the 9 styles investing under Brandt and Santa-Clara (2006) approach is 46.
28 Two excess returns are used in the study, one is based on risk-free rate and the other is based on market index (not reported here). The optimal style investing based on excess returns on market index captures the gain from beating an index with low tracking error, and is equivalent to an “active indexation” strategy with optimal weights interpreted as “active weights”.
sort stocks into value-growth styles can help generalise the findings.
Table 5-2 below lists the control variables used in the study.
Basically, the test results largely confirm the hypothesis proposed above. As an example and for concise purpose, the test results based on style portfolios sorted on stock characteristics APC only is reported below. Results can be quantitatively different with BM and DY sorted style portfolios nevertheless they all qualitatively support the same conclusion.
Table 5-3 provides estimates of single-period optimal style investing.
Panel A is for monthly return frequency and Panel B for quarterly frequency. Each panel reports the time-series average weights for different styles. R(tangent) refers the average expected monthly returns of the tangency style investing portfolios and R(predicted) refers the average monthly-equivalent one-period ahead optimal style investing returns according to the optimal style investing weights (namely style investing policy). The sample is from January 1981 to December 2004 (288 months or 96 quarters). The first 120 months (60 quarters) is used to estimate the initial optimal weights of the style investing policy and then form out-of-sample monthly (quarterly) “portfolio of style portfolios” using those weights in the next period.
Every subsequent period the style timing policy is re-estimated by enlarging the sample. The t-ratios for unconditional optimal style weights and for the business cycle variables of conditional investing are obtained based on Britten-Jones (1999) approach, and the corresponding standard errors are retrieved from these t-ratios. Note the t-ratios reported in this table are calculated from the time-series average coefficients and the time-series average standard errors. The * refers that it is significant for at least 10% level.
Table 5-3 first suggests that investors using value-weighted portfolio strategies would generally give up a large fraction of wealth to have access to large stocks. A style portfolio with fund equally distributed to all constituent stocks tends to outperform that based on market capitalisation to allocate funds (i.e. value-weighted scheme). This is because small stocks tend to outperform large stocks in the long run.
While equally-weighted investing generally yield higher volatility, their out-of-sample Sharp Ratios are also generally higher than valueweighted schemes. Consistent with the literature about the divergent returns of value-growth stocks and small-large stocks, regardless of return horizons, all types of investors are shown to significantly long value stocks and small stocks, and also tend to significantly short growth stocks or large stocks. In more detailed market segments, it can be seen that investors tend to long SV, LB, LV and short SG, MG and LG, and the long positions on SV stocks are overwhelmingly significant on both monthly and quarterly horizons.
The unconditional style investing and the conditional style investing using business cycle information are very much different. First, investors who disregard the business cycle predictability are relatively conservative with respect to their overall net equity exposures. While these Skeptics also overweight some specific styles both at long and short directions, they eventually all end up with allocating part of their wealth to cash. In sharp contrast, investors who have strong prior beliefs about the business cycle information are very aggressive in equity investing and therefore generally end up with large long exposures to equities that must be leveraged by borrowing.