Markowitz efficiency and size effect: evidence from the UK stock market

Abstract

Academics and practitioners have frequently debated the relationship between market capitalization and expected return. We apply the Markowitz efficient frontier approach to develop a portfolio performance measure that compares the return of a portfolio to its optimal return, using data from the UK stock market over the period 1985–2012. Our results show that there is a negative relationship between portfolio size and portfolio return during the period under study. When comparing actual portfolio return with achievable return for the same level of risk, we find that as the portfolio size expands, underperformance of the portfolio increases, i.e. the larger the portfolio size, the greater the underperformance. This indicates that Markowitz efficiency is difficult to achieve, particularly in large portfolios. Changing model parameters leads to alternative efficient frontiers that impact upon the measurement of performance. However, the use of alternative efficient frontiers does not affect our result of the size effect on the relative performance of portfolios. Our study shows that the size effect is present over the full period. Our findings also suggest that the excess returns found in small portfolios are likely to be associated with higher levels of diversifiable risk in comparison with larger portfolios. Furthermore, in contrast to other studies, we find no evidence to support the size reversal effect in the data.

Appendix

We address the issue concerning the impact on our results if our sample had not required survival for the period. Since this paper collected stocks of the FTSE All-Share Index at June 2012 starting from January 1985, the portfolio returns in Panels A and B in Table 2 are based on existing stocks that continue to survive from the testing periods, January 1985–June 2012 and January 2000–June 2012. To investigate whether the results differ if the sample had not required survival for the full period starting from 1985, we include four other subperiods for reporting the size effect: January 1990–June 2012, January 1995–June 2012, January 2000–June 2012, and January 2005–June 2012. In doing so, we can increase our sample size to further examine the size effect over different periods. Results in Table 9 show that the smallest portfolio persistently outperforms the largest portfolio. For example, over the period between 1995 and 2012 the monthly difference in SML (small minus large portfolio) is 0.54 %, resulting in a gain of 6.5 % per annum. Table 9 also exhibits that the monthly standard deviation of each size portfolio is relatively stable between 4.87 % (portfolio 5 over the period 1995–2012 in Panel A) and 6.89 % (portfolio 7 over the period 2005–2012 in Panel B).

Table 9 Size effect for all surviving stocks in each time period

In Table 10 we include 612 stocks on the FTSE All-Share Index by forming 5 and 10 portfolios to calculate their portfolio returns respectively. We split the sample into quintiles to form 5 portfolios and also divide the sample into deciles to form 10 portfolios. Since the maximum number of stocks in our sample is 612 stocks in 2012, many of these stocks will not be present at the start of the period under review. The formation of size portfolios can only be based on the ascending order of market capitalization of stocks at June 2012 rather than at the beginning of 1985. Although we are aware that if investors perceive the size effect they should form their size portfolios based on the forward-looking portfolio formation (i.e. at the beginning of the time when they purchase stocks), the backward-looking portfolio formation helps to explain the possibility of survivorship bias. The results shown in Table 10 indicate the final portfolio position of the available stocks in 2012. The return bias is not comparable with Table 2 in which portfolios are formed at the beginning of observational period.

Table 10 Formation of portfolios based on final year 2012 market capitalizations

Table 10 shows that the smallest size portfolio tends to have a lower return than larger size portfolios. Although at first sight these results appear to contradict our findings of the size effect, after further examination these figures clearly show that the size effect is present in the data. Since we rank portfolios based on the end of the observational period at June 2012, several small stocks when purchased at the beginning of the observational research period yielded higher returns and increased their market capitalizations, gradually expanding in size. Accordingly, the remaining stocks in the smallest size portfolio are the least successful companies, confirming the size effect in the UK stock market.

Table 11 reports the increase in market capitalization for each size portfolio from the start of each respective period of portfolio formation to June 2012 showing only companies that have survived for the entire periods of observation. The average size of the smallest portfolio in Panel A is 1,027 million pounds and the market capitalization growth rate of the smallest portfolio is 20 % per year. In contrast, all other portfolios have market capitalization growth rates between 8 and 11 % in Panel A. When investigating the short observational period 2000–2012, the smallest portfolio also exhibits the highest growth rate among all size portfolios. Although the smallest size portfolio has a high growth rate, the growth rate only counts for those stocks which continue to exist in the sample period. This growth rate for the smallest portfolio may be optimistic as the failing firms removed from the index are more likely to be in the smallest group and have significantly lower returns than surviving firms. To summarize the results in the Appendix, consideration of additional subperiods and regrouping the number of stocks in portfolios strengthens the results.

Table 11 The impact of the growth of market capitalization on formation of portfolios

To explore the survivorship bias, Kothari et al. (1995) collect two sets of stock return data from COMPUSTAT data and CRSP (Centre for Research in Securities Prices) data and they assume that the CRSP sample does not suffer from a survivorship bias problem. Kothari et al. (1995) find that the returns for small firms are 9–10 % points higher on COMPUSTAT than returns for CRSP—COMPUSTAT small firms. Since our sample to estimate portfolio returns is based on available stocks at the end of 2012, the survivorship bias in our sample can be regarded as the difference between portfolio return on the existing stocks over the entire observational period and portfolio return on the existing stocks over the different subperiods. Thus, it can be argued that portfolio return on available stocks for the full period is likely to be higher than the portfolio return on available stocks for a subperiod.

$$ {\text{SB}}_{\text{j}}^{ 1 9 8 5} { = }\overline{\text{r}}_{{{\text{j}},1985}} - \overline{\text{r}}_{{{\text{j}},2005}} $$

$$ {\text{SB}}_{\text{j}}^{2000} { = }\overline{\text{r}}_{{{\text{j}},2000}} - \overline{\text{r}}_{{{\text{j}},2005}} $$

where ${\text{SB}}_{\text{j}}^{ 1 9 8 5}$ indicates survivorship bias of portfolio j whose portfolio return is based on available stocks since 1985. ${\bar{\text{r}}}_{{{\text{j}},1985}}$ (${\bar{\text{r}}}_{{{\text{j}},2000}}$) is portfolio j return for available stocks since 1985 (2000) as shown in Table 2. The estimation of the survivorship bias based on the smallest size portfolio can be derived between Tables 2 and 9 (${\bar{\text{r}}}_{{{\text{j}},2005}}$ is portfolio j return for available stocks since 2005 as shown in Table 9). The monthly portfolio bias for ${\text{SB}}_{\text{j}}^{ 1 9 8 5}$ on the smallest portfolio is 0.31 %, resulting in a bias of 3.7 % per annum. Further, the monthly portfolio bias for ${\text{SB}}_{\text{j}}^{ 2 0 0 0}$ on the smallest portfolio is 0.38 %, resulting in a bias of 4.5 % per annum. The average survivorship bias of the smallest size portfolio is 4.1 % which is much lower than the 9–10 % points in Kothari et al. (1995). The resulting difference of our estimation is that we only consider the available stocks at the start of each period under investigation.