Testing the international diversification benefits of using small-cap
stocks in stock portfolios
Abstract:
This thesis examines whether the use of small stocks in international diversification can reduce portfolio riskiness and improve portfolio performance compared to international diversification using large stocks over the period 2002-2016. I find that stock return correlations are significantly lower for small stocks compared to large stocks during the entire sample period. Moreover, the value-at-risk for stock portfolios using small stocks is significantly lower while the Sharpe ratios of portfolios using small stocks are significantly higher than for portfolios using only large stocks, suggesting that investors can benefit from the use of small stocks to diversify.
MSc Finance Asset Management
Master Thesis by Enzio Fruijtier
Supervisor: Florencio Lopez de Silanes Molina
Statement of Originality
This document is written by Student Enzio Fruijtier who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Table of contents
1. Introduction
4
2. Literature Review
5
3. Methodology
8
3.1 Hypotheses
8
3.2 Data
9
3.3 Dynamic Conditional Correlation
12
3.4 Value-at-Risk
13
4. Results
14
4.1 Time-varying Correlations
14
4.2 Value-at-Risk
18
4.2.1 Historical VaR
19
4.2.2 Modified VaR and Sharpe ratios
20
5. Robustness
23
6. Conclusion
28
1. Introduction
Investors are interested in diversifying their portfolio, in order to reduce the idiosyncratic risk component of the portfolio. To diversify, investors usually diversify across different industries or across different countries. However, the world has become more integrated over the past decades and international equity returns have become more correlated (Longin and Solnik, 1995). This development has reduced the diversification benefits that international diversification provide to the point that some people doubt that international diversification is still useful.
Most investors that invest abroad invest exclusively in the larger, better known firms (Kang and Stulz, 1997). These firms are generally multinationals that are very affected by global economic
developments. An investor internationally diversifying using only large firms will therefore still be exposed to major global developments due to the high correlations between these large firms. Small firms however, often operate very locally and are therefore less exposed to global developments. Using firms with low market capitalizations in an internationally diversified portfolio may therefore provide useful diversification benefits that portfolios containing only firms with large market
capitalizations cannot provide, because the stock returns of these firms are less correlated with those of other firms. For this reason, Eun, Huang and Lai(2009) proposed the use of small firms for
international diversification and found that the stocks of these firms are less correlated with those of other firms and that internationally diversified portfolios that make use of small stocks have higher Sharpe Ratios.
Only limited research has been done on the use of small firms for international portfolio
diversification so far. This paper contributes to existing research by adding to Eun, Huang and Lai’s research on the diversification benefits of small firms by taking a sample period that is more recent and therefore more relevant to current investors than that used in their paper. Eun et al. used constant correlations for their research and evaluated performance using the Sharpe Ratio.
Literature however shows that correlations vary over time (Longin and Solnik (1995), Longin and Solnik (2001) and Chua, Kritzman and Page (2009)) and the Sharpe Ratio is flawed in that it uses standard deviation to measure risk, while investors generally care more about downside risk than upside risk when they invest. To deal with this, this paper contributes by evaluating time-varying correlations and comparing tail risk of portfolios containing only large firms with portfolios
containing both large and small firms by comparing the Value-at-Risk. The ultimate goal of this paper is to find if the use of firms with a small market capitalization for international diversification can provide useful diversification benefits beyond those of internationally diversified portfolios containing only large firms
I find that small stocks are significantly less correlated with other stocks during the entire sample period than large stocks which leads to the finding that including small stocks for international diversification significantly reduces tail risk for investors and that the portfolios including small stocks also have much higher Sharpe Ratios.
The paper is organized as follows: Section 2 reviews the background and existing literature. Section 3 describes the methodology used. Section 4 presents the empirical results. Section 5 tackles some robustness issues. Section 6 contains the conclusion and discussion.
2. Literature review
Levy and Sarnat (1970) were the first to write about the benefits of internationally diversifying an equity portfolio, finding that the low correlation between stock returns across countries can help reduce stock portfolio volatility. The rationale behind this finding is that firms operating in different countries will not be affected by the same shocks. The stock price of firms operating in different countries will therefore be less correlated than those of firms operating in the same country, leading to lower portfolio volatility. When this paper was written, international diversification significantly reduced portfolio riskiness. Over the past few decades however, the world has become increasingly
more integrated, due to globalization and technological improvements like electronic trading. Indeed, Longin and Solnik (1995) found that correlations in international equity returns have become significantly higher in the period 1960-1990. Longin and Solnik(2001) later find that correlations do not increase during periods of high volatility but rather that correlations increase in bear markets while they do not increase in bull markets. When international equity return correlations increase, the risk reduction of diversifying your portfolio internationally decreases. This increased integration has caused some people to doubt that international diversification is still useful.
Kang and Stulz (1997) find that when investors invest abroad, most of them invest almost exclusively in the larger firms. Large-cap firms are often multinationals and are consequently more exposed to developments in global factors. This is why Eun, Huang and Lai (2009) propose the use of small-cap firms in portfolios for international diversification. Small-cap firms usually operate very locally making them less affected by shocks elsewhere in the world. Due to small firms operating more locally, they could be interesting to use for international diversification, as the stock returns of these small firms can potentially be less correlated with other stocks from different countries. Indeed, Eun et al. find that the stock returns of small-cap stocks are significantly less correlated with stock returns of foreign firms than large-cap stocks, suggesting that these small stocks may be useful to use for international diversification. Furthermore, they find that the inclusion of small cap stocks in internationally diversified portfolios can be used to increase the Sharpe ratios of the portfolios, suggesting that the use of small stocks can lead to a better risk-reward payoff. Because small-cap stocks are generally not as liquid as large-cap stocks, transaction costs for small-cap stocks are higher. However, Eun et al. (2009) find that small-cap stocks provide significant international diversification benefits even after controlling for these higher transaction costs.
However, their paper has some limitations that my paper adresses. In their paper, they only used constant correlations. This is unrealistic considering that Longin and Solnik (1995) found that correlations were increasing over time and that correlations increase in bear markets (Longin and
Solnik, 2001). Furthermore, Chua, Kritzman and Page (2009) find that the correlation of US and foreign stocks when the returns of both are one standard deviation below the mean is 76% higher than normal and that the correlation is 17% lower than normal when both market’s returns are a standard deviation above the mean. Assuming correlations to be constant can lead to misleading results if correlations vary over time. Correlations increasing during an economic downturn are especially bad for diversification as it means that diversification benefits are lower than usual at the moment they are needed the most. You and Daigler (2010) also find that using only constant correlations to measure diversification benefits is flawed. To deal with this problem, they propose the use of the DCC-GARCH (Dynamic Conditional Correlation - Generalized Autoregressive
Conditional Heteroscedasticity) model by Engle (2002) as an effective way to calculate the
conditional correlations of international equity returns. According to Engle, the DCC-GARCH is often the most accurate way to measure correlations when compared to MV-GARCH and several other measures. Like You and Daigler, I deal with correlations changing over time by calculating the time-varying stock return correlations using the DCC-GARCH method to find out if correlations are still lower for small stocks than for large stocks when the time-varying nature of correlations is taken into account.
Another limitation of Eun et al.’s paper is that the sample they use spans from the beginning of 1980 to the end of 1999. It seems likely that the world has become even more integrated since this period, causing international equity returns to have become even more correlated. Therefore, it should be tested if their results still hold when a more recent period is considered. The sample I use spans from the beginning of 2002 to the end of 2016, which should be more relevant to current investors.
The final limitation of the paper of Eun et al. is that they use the Sharpe ratio to measure the
portfolio performance. The Sharpe ratio uses the standard deviation of stock returns as a measure of risk. However, when people want to reduce the riskiness of their portfolio, they are mostly interested in reducing downside risk while the standard deviation describes the volatility in both directions.
Evaluation of the risk of using small stocks in international diversification therefore is better achieved by considering the tail risk of investments. A popular measure for assessing tail risk is the Value-at-Risk (VaR). This measures the maximum loss that will occur in a set time period at a specific confidence interval. According to prospect theory, investors are loss averse and the utility loss of a decreasing value of investments outweighs the utility gain if the investments increase by the same amount (Kahneman and Tversky, 1979). A loss reduces utility by more than an equally sized gain does. The tail risk is therefore very important in the eyes of investors. For portfolios containing small-cap stocks to be perceived as less risky to investors, they should have a lower VaR than portfolios only containing large-cap stocks. To contribute to the existing literature I will therefore also evaluate the tail risk of the portfolios by calculating the VaR. For comparability with Eun et al. and
completeness, I however also report the Sharpe ratios but consider them to be less important than the VaR.
One downside to using Small-Cap stocks for international diversification is that transaction costs are higher for Small Cap stocks (Chiyachantana, Jain, Jian, & Wood, 2004). This is caused by the trading volume for these stocks being lower, making Small Cap stocks less liquid than stocks with a higher market capitalization. Chiyachantana et al. find that the additional trading costs for small-cap stocks are roughly 1,10-1,28% based on an annual turnover of 100%. Taking this into account, Eun et al. still find significant diversification benefits when using small stocks, provided annual turnover does not surpass roughly 150%.
3. Methodology
3.1 Hypotheses
Based on the literature review, I derive a number of hypotheses to test. The first hypothesis concerns the correlation of small stocks returns compared to large cap stock returns. In order for small stocks to indeed be useful for international diversification, small stocks have to be less correlated with other
stocks than large stocks are. It is especially important for small stocks to not be more correlated with other stocks than large stocks when stock returns are poor, as investors most require the benefits of diversification during bad times. If the correlations are not significantly lower for small stocks,
investors will not gain additional diversification benefits when they use small stocks in their portfolio. Therefore, the first hypothesis is as follows: Time-varying international equity correlations are lower for small-cap stocks compared to large-cap stocks.
If this hypothesis is found to be true, I can test whether these lower correlations actually reduces the risk of the portfolio. Evaluating the risk is not simply done by calculating the standard deviation: the standard deviation takes both negative and positive deviations into account. When investors choose to diversify, they care most about reducing the downside risk of portfolios. To measure how using small stocks for international diversification affects risk, I test whether the following hypothesis holds: Internationally diversified portfolios containing small-cap stocks have a lower Value at Risk than internationally diversified portfolios that only contain large-cap stocks.
Finally, while the Sharpe ratio is not a perfect measure due to both upwards and downwards deviations being counted equally, it does still have some value in measuring performance. While I therefore consider the importance of the performance based on the Sharpe ratio to be considered secondary to the performance based on the VaR measure, if the use of small stocks for diversification leads to higher Sharpe ratios, it will be further evidence that the use of small stocks can lead to additional diversification benefits. The last hypothesis I will test therefore is the following:
Globally diversified portfolios containing both small cap and large cap stocks have a higher Sharpe ratio than portfolios containing only Large Cap stocks
3.2 Data
To test the hypotheses, monthly stock price data from January 2002 to December 2016 for ten different countries is used. Monthly stock return data for the countries Australia, Canada, France,
Germany, Hong Kong, Italy, Japan, the Netherlands and the U.K. is obtained from DataStream. US stock return data is obtained from the CRSP database. I also obtain end-of-year market cap information from these same sources to sort companies by size.
In this sample, firms that are incorporated outside of their home country and firms for which either market capitalization or stock price data is missing are excluded. DataStream sets prices and market capitalizations to a constant when a stock is no longer traded, which is dealt with by changing the constant value to a missing value in the dataset. To deal with recording errors in DataStream, monthly returns above 400% are changed to a missing value in the dataset.
Similarly to Eun et al. (2009), for all different countries three market cap-based funds (small-cap, mid-cap and large-mid-cap) are constructed at the start of each year, with all firms in each country being ranked based on their market cap at the end of the last year. For each country the top 20% of firms based on market capitalization are considered to be large-cap, the bottom 20% to be small-cap, and the remaining 60% to be mid-cap. Every stock is weighted according to market capitalization in each portfolio, meaning that the stock returns of a stock with a market capitalization of 10% of the total market capitalization of all firms in the same portfolio will have a weight of 10% in the portfolio.
After creating these funds, the monthly portfolio returns for every portfolio in every country are calculated using the following formula:
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑟𝑒𝑡𝑢𝑟𝑛 = ∑ 𝑤𝑖∗
𝑝𝑡,𝑖− 𝑝𝑡−1,𝑖
𝑝𝑡−1,𝑖
These portfolio returns are later used in calculating the VaR and dynamic correlations. Table 1 below contains some summary statistics for the portfolios created using this method.
TABLE 1
Summary statistics of size- and country based portfolios
Table 1 reports the average monthly return and standard deviation for all size- and country based portfolios from 2002 to 2016. For each country in every year of the sample, the top 20% of firms based on market capitalization are considered to be large-cap, the bottom 20% to be small-cap, and the remaining 60% to be mid-cap. For every portfolio, the average percentage monthly return and percentage monthly standard deviation are listed. The average monthly return and standard deviation by size is listed in the bottom row.
Country Small Cap Mid Cap Large Cap
Australia 3,00 | 7,20 0,25 | 5,27 0,32 |3,67 Canada 5,15 | 8,48 0,69 | 6,54 0,51 | 3,75 France 1,49 | 4,11 0,65 | 4,05 0,31 | 4,88 Germany 1,13 | 5,10 0,32 | 4,03 0,39 | 5,32 Hong Kong 3,26 | 10,26 0,73 | 7,84 0,56 | 5,79 Italy -0,38| 5,76 0,25 | 5,28 0,02 | 5,51 Japan 1,41 | 5,83 0,71 | 4,97 0,35 | 5,06 Netherlands 0,30 | 5,31 0,43 | 4,96 0,22 | 5,00 UK 0,38 | 4,24 0,27 | 4,28 0,28 | 3,97 USA 1,00 | 5,40 0,65 | 5,40 0,47 | 4,44 Average 1,75 | 6,25 0,48 | 5,25 0,33 | 4,77
As can be seen in Table 1, the Small Cap portfolios have considerably higher average monthly return in the sample period. This is consistent with the finding of Fama and French (1993) that small firms tend to have higher returns than large cap firms. The difference in average returns is very large, however, so small firms seemed to have performed exceptionally well during this sample period. It would be interesting to see if this difference persists in future periods. It is also apparent from Table 1 that the standard deviation is significantly higher for smaller firms compared to larger firms, which is in line with findings from existing literature.
I combine the US large-cap portfolio with all the other portfolios and calculate the returns, giving a weight of 50% to the US large-cap portfolio and 50% to the other portfolio and calculate the Value at Risk for every portfolio. To calculate the Value at Risk, the historical stock return distribution method as well as the Modified VaR are used to calculate the maximum loss that is not exceeded with a probability of 5% to compare the portfolios.
3.3 Dynamic Conditional Correlation
The assumption of correlations being constant over time has been found to be unrealistic (See for example Longin and Solnik, 1995; Longin and Solnik, 2001; Chua, Kritzman and Page, 2009). To calculate the time-varying correlations, this paper uses Engle’s DCC-GARCH (2002) model to measure how correlations change over time. This model builds on Bollerslev’s (1990) MVGARCH model, in which conditional correlations are constant, to allow conditional correlations to vary over time. The Dynamic Conditional Correlation GARCH model by Engle (2002) can be written as follows
𝑦𝑡 = 𝐶𝑥𝑡+ 𝜀𝑡 𝜀𝑡= 𝐻𝑡 1/2 𝑣𝑡 𝐻𝑡 = 𝐷𝑡 1/2 𝑅𝑡𝐷𝑡 1/2 𝑅𝑡 = 𝑑𝑖𝑎𝑔(𝑄𝑡) 𝑄𝑡 −12 𝑑𝑖𝑎𝑔(𝑄𝑡)−1/2 𝑄𝑡 = (1 − 𝜆1− 𝜆2)𝑅 + 𝜆1𝜀̃𝑡−1𝜀̃𝑡−1′ + 𝜆2𝑄𝑡−1
where 𝑦𝑡is an m x 1 vector of dependent variables, C is a m x k matrix of parameters, xt is a k x 1 vector of independent variables, 𝐻𝑡1/2 is the Cholesky factor of the conditional variance matrix that varies over time and 𝑣𝑡 is a m x 1 vector with mean zero, variance of 1 and is independent and
identically distributed, Dt is a diagonal matrix of conditional variances, Rt is a matrix of conditional quasicorrelations, 𝑒̃ is a m x 1 vector of standardized residuals
, D
t-1/2𝜀
𝑡; and λ1 and λ2 are nonnegative parameters that decide the dynamics of conditional quasicorrelations that summed together must be greater than or equal to zero and smaller than 1.Engle estimates the DCC using a two-stage procedure. For more details, regarding this procedure, see Engle (2002). For this paper, the time-varying correlations have been calculated using a DCC GARCH (1,1) model using the mgarch dcc command in Stata.
3.4
Value at Risk(VaR)
There are several methods available to calculate the Value-at-Risk for investments. For this paper, the historical distribution as well as the Modified VaR are used to calculate the 95% VaR of the monthly returns.
The historical method simply looks at actual realized historical returns and ranks them from worst to best. To calculate the 95% Value-at-Risk using this method, the value at the point that 95% of the returns are better and 5% of the returns are worse is used.
The Modified VaR, also known as the four-moment VaR, builds on the two-moment VaR, which is also known as the normal distribution method. The normal distribution method assumes that the investment returns are normally distributed. Given that this holds, the value at risk at a certain confidence can be calculated by applying the following formula:
𝑉𝑎𝑅 = −𝜇𝑝− 𝑧𝜎𝑝 ,
where 𝜇𝑝 is the mean of the monthly portfolio returns, 𝜎𝑝 is the standard deviation of the monthly
portfolio returns and z is the negative number of standard deviations specifying the probability level associated with the tail risk.
This formula is simple and easy to use. However, stock returns are not normally distributed. Indeed, as can be seen by looking at Table 4, stock returns usually are negatively skewed, meaning that most observations are smaller than the mean. Stock returns also have positive excess kurtosis, meaning the tails are fatter than in a normal distribution. Using the two-moment VaR for returns that are not
normally distributed would give inaccurate results. Favre and Galeano (2002) deal with this by developing a four-moment VaR that also takes kurtosis and skewness into account:
𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝑉𝑎𝑅 = −𝜇
𝑝− (𝑧
𝑐+
1 6(𝑧
𝑐 2− 1)𝑆
𝑝+
1 24(𝑧
𝑐 3− 3𝑧
𝑐)𝐾
𝑝−
1 36(2𝑧
𝑐 3− 5𝑧
𝑐)𝑆
𝑝2)𝜎
𝑝 ,where 𝜇𝑝 is the mean of the monthly portfolio returns, z is the negative number of standard
deviations specifying the probability level associated with the tail risk, 𝑆𝑝 is the portfolio skewness
and 𝐾𝑝 is the portfolio excess kurtosis(defined as kurtosis minus 3) and 𝜎𝑝 is the standard deviation
of the monthly portfolio returns.
To test whether the VaR is significantly higher for the large-cap portfolios I subtract the VaR of the small-cap portfolios in each country from the large-cap VaR of the same country and use the paired Student’s t-test. The paired Student’s t-test is calculated as follows:
𝑡 = 𝑥̅𝑠𝐷
𝐷
√𝑛𝐷
~𝑡 [𝑑𝑓 = 𝑛𝐷− 1] ,
where 𝑥̅𝐷 is the observed average difference between the monthly 95%-VaR of the small- and
large-cap portfolios, 𝑠𝐷is the sample standard deviation of the differences and 𝑛𝐷is the amount of
differences.
All t-statistics listed in this paper use this formula..
4. Results
4.1 Time-varying correlations
The summary statistics of the time-varying correlations calculated using DCC-GARCH are listed below in Table 2. These correlations are the time-varying correlations of the US Large Cap portfolio and the listed country/size combination portfolio. If including small stocks in a portfolio is to provide
to be significantly lower than those of large stocks. This is especially important during an economic bust: investors diversify to be less exposed to risk especially when times are bad. As both Table 2 and figure 1 show, correlations are indeed lower for small stocks, providing support for the hypothesis that diversifying using small stocks can be beneficial.
Over the sample period, the average correlation of the Small Cap portfolios is 0,4429. This is a lot lower than the average correlation of 0,7934 of the Large Cap portfolios. For every country, the Large Cap portfolio has a higher average correlation than the Small Cap portfolio, the smallest difference being 0,1770 for Italy. The largest difference between the average correlation of Small- and Large Cap portfolios is 0,5968 for Canada. Moreover, the difference between the time-varying correlations with Small- and Large Cap portfolios is so large that for eight out of nine countries, the highest Small Cap correlation over the sample period is lower than the lowest Large Cap correlation reached. Only for Japan is the maximum Small Cap correlation higher than the minimum Large Cap correlation. These findings confirm the hypothesis that time-varying correlations are lower for Small-Cap stocks than for Large-Cap stocks.
Interestingly, the standard deviation of the correlations for the Small Cap portfolios is significantly higher than the standard deviation of the correlations of the Large Cap portfolios. The average standard deviation of the time-varying correlations is 3,38% for the Small Cap portfolios, which is more than three times as high as the average standard deviation of the time-varying correlations for Large Cap portfolios, which is 1,09%. Furthermore, the difference between the minimum and maximum correlations is also higher for the Small-cap portfolios. These two findings are true for all countries in the sample. They indicate that the correlations of smaller stocks vary more over time than larger stocks do and are more affected by the general state of the economy and changes over time.
Figure 1 plots both the average Small- and Large-Cap correlations over time. From this figure it is clear that the average correlation is considerably higher at all times for Large-Cap only portfolios
compared to portfolios including Small-Cap stocks: at no point do the two lines come close to each other. Both average correlations fluctuate over time, but the average Small Cap correlation
fluctuates significantly more than the average Large Cap correlation. This is to be expected based on the higher correlation standard deviation for Small Cap portfolios reported in Table 2. The figure confirms the findings of existing literature (Longin and Solnik (2001); Chua, Kritzman and Page (2009)) that correlations increase in bear markets. This is especially noticeable from 2008 to early 2009. The correlations are higher in this period but when the economy improved, the correlations dropped again.
Interestingly, the correlations do not seem to have significantly increased over the sample period: they have remained at roughly the same level during this period. This implicates that markets have not become more integrated to the extent that correlations significantly increase over the sample period. It will be interesting to see how future developments affect global market integration and how it will affect international market correlations in the future.
The findings of Table 2 and Figure 1 confirm the hypothesis that correlations are lower for portfolios including Small Cap stocks compared to those using only Large Cap stocks during the entire sample period. The difference is substantial at every point in time, providing support to the notion that investors could benefit from the use of Small Cap stocks in their internationally diversified portfolio to reduce portfolio riskiness. Based on how much lower the correlations for small stocks are, it is very likely that the use of small stocks in internationally diversified portfolios can help to reduce portfolio risk. The correlations remaining at roughly the same level during this period also indicate that the benefits of international diversification have not significantly decreased from the start of the sample period. Therefore, it seems likely that the differences between small and large stocks will persist in the near future. These results show that the finding of Eun et al. (2009) that constant correlations of small stocks being lower still holds when time-varying correlations are considered.
To test whether the use of small caps in internationally diversified portfolios can indeed reduce portfolio riskiness I compare the tail risk of internationally diversified portfolios containing only large stocks with the tail risk of portfolios containing both small and large stocks, by measuring the VaR of both. The findings based on this measure are listed in Section 4.2
TABLE 2
Dynamic Conditional Correlations by country and size
Table 2 summarizes the time varying correlations by country and size. For each country, the correlation with the US Large cap portfolio is listed for both the Small Cap and the Large Cap portfolio from January 2002 to December 2016. Columns 2 to 4 report the summary statistics of the time-varying correlations for the Small Cap portfolios. Columns 5 to 7 report the summary statistics of the time varying correlations for the Large Cap portfolios. Columns 2 and 5 report the average correlations over this period and report the monthly standard deviation of the correlations in parentheses. Columns 3 and 6 report the minimum correlations over the sample period and columns 4 and 7 report the maximum correlations over the sample period. The last row lists the averages of each column.
Small Cap Large Cap
Country Mean
(SD)
Min Max Mean
(SD) Min Max Australia 0,5228 (0,0336) 0,3487 0,6640 0,7651 (0,0128) 0,7240 0,8166 Canada 0,2717 (0,0445) 0,0619 0,4768 0,8685 (0,0071) 0,8478 0,8959 France 0,4133 (0,0391) 0,2066 0,5962 0,8554 (0,0094) 0,7875 0,8758 Germany 0,4822 (0,0325) 0,3881 0,6064 0,8443 (0,0096) 0,7793 0,8734 Hong Kong 0,3964 (0,0362) 0,2647 0,5453 0,7512 (0,0130) 0,7037 0,7985 Italy 0,5735 (0,0324) 0,3811 0,6815 0,7505 (0,0129) 0,6851 0,7922 Japan 0,3817 (0,0412) 0,2435 0,5632 0,6048 (0,0184) 0,5509 0,6985 Netherlands 0,4290 (0,0320) 0,3114 0,5382 0,8280 (0,0080) 0,7845 0,8562 United Kingdom 0,5153 (0,0338) 0,3451 0,6802 0,8727 (0,0070) 0,8457 0,8923 Average 0,4429 (0,0361) 0,2835 0,5946 0,7934 (0,0109) 0,7454 0,8333
Figure 1.
4.2 Value-at-Risk
This section tests if including small stocks in an internationally diversified portfolio reduce portfolio risk by comparing the tail risk of internationally diversified portfolios containing only large stocks with the tail risk of portfolios containing both small and large stocks. Due to the correlations between large and small stocks being much lower as the correlations of large stocks with other large stocks as found in section 4.1, it is expected that the portfolios that include small stocks have a lower Value-at-Risk than portfolios that do not include small stocks and are therefore less risky.
As can be seen by observing tables 3, 4 and 5 below, international diversification using small-cap stocks does indeed perform better than international diversification using only large-cap stocks based on the 95% VaR of monthly returns for both measures used. Sections 4.2.1 and 4.2.2 will explain the results in greater detail.
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 2002 2004 2006 2008 2010 2012 2014 2016
Average correlation over time
Small Cap Large Cap
4.2.1 Historical VaR
Table 3 below present the 95%-VaR results using the historical method. As can be observed, using the historical distribution method, the small-cap portfolio performs best for eight out of nine countries, with the large-cap portfolio performing best only for Hong Kong. The difference in
performance between the small and large portfolios is less than 1% for Australia, Italy and Japan and is above 1% for the other countries. The difference is especially large in France and Germany, where the small portfolio has a 2,99% and 2,87% lower VaR than the large portfolio, respectively.
On average, the small-cap portfolios combined with the US large-cap portfolio have a 95%-VaR of 6,58% monthly, while the US large-cap portfolio combined with large-cap portfolios from other countries have a 7,78% monthly 95%-VaR. Based on this, the Small Cap portfolios have a significantly lower monthly 95%-VaR than the Large Cap portfolios at the 5% level. This difference is also
economically significant: a difference between the 95%-VaR of the large portfolios and the small portfolios of 1,20% on a monthly basis is substantial. This indicates that smaller stocks being less correlated with other stocks can indeed lead to a significant risk reduction when small stocks are used in internationally diversified portfolios.
TABLE 3
Monthly 95% - Value-at-Risk using the Historical Distribution method
Table 3 reports the calculated monthly 95%-Value-at-Risk. The table lists the VaR based on the historical returns from 2002 to 2016. The US-Large-Cap portfolio is given a weight of 50% in each of these portfolios, with the remaining 50% of the portfolio consisting of the listed country and size combination. The listed values are equal to the maximum monthly percentage loss of that is not exceeded more than 5% of the time. In the ‘Large – small’ column the difference between the VaR of the small-cap portfolio and the VaR of the large-cap portfolio is listed. The table lists the t-statistic of the average difference between the large and small portfolios in parentheses. *, ** and *** indicate significance at 10%, 5% and 1%, respectively.
Country Small Cap Mid Cap Large Cap Large – Small
Australia 7,01 7,36 7,49 0,48 Canada 5,30 7,25 6,66 1,36 France 5,12 7,04 8,11 2,99 Germany 5,81 7,01 8,68 2,87 Hong Kong 8,91 9,86 7,88 -1,03 Italy 8,17 8,22 8,39 0,22 Japan 6,90 6,59 7,35 0,45 Netherlands 6,78 7,88 8,19 1,41 United Kingdom 5,25 7,06 7,27 2,02 Average 6,58 7,59 7,78 1,20** (2,59)
4.2.2 Modified VaR and Sharpe ratios
Table 4 below reports the four moments used in the Modified VaR as well as the calculated values of the Modified VaR. The Sharpe Ratio performance measure, which is defined as average return divided by standard deviation is also listed. Average returns for the small-cap portfolios are higher on average, with the difference between the average monthly return being more than 1% for Australia, Canada and Hong Kong. For each portfolio, there is positive excess kurtosis and all portfolios but the Hong Kong and Netherlands small-cap portfolio are negatively skewed. Using the Modified Value at Risk, the small-cap portfolios have the lowest VaR for eight out of nine countries, with the large-cap portfolio performing best for Australian stocks based on the Modified VaR. Table 5 below
summarizes the results of Table 4 by country. As can be seen in table 5, the average monthly return for the small-cap portfolios is substantially higher. The standard deviation of the small portfolios and the excess kurtosis is slightly higher than those of the large-cap portfolios, though these differences are not statistically significant. The small-cap portfolios are also less negatively skewed on average.
Most interestingly, regardless of market capitalization, the average standard deviations of these portfolios that use international diversification are lower than the standard deviations of the individual country and size portfolios. This can be observed by comparing the standard deviations listed in Table 1 with the standard deviations listed in Table 4 and Table 5. This is strong evidence in favour of the idea that international diversification still has some benefits as it shows that the standard deviation of a portfolio containing stocks from two countries are lower than the average of the standard deviation of the individual country portfolios. Due to the lower correlations of the small stocks that we found earlier, the standard deviation is more significantly lower for the internationally diversified portfolios using small stocks. Due to this, the Sharpe ratios of the Small Cap portfolios are significantly higher than those of the Large Cap Portfolios, as the Small Cap portfolios retain their high average returns but due to the higher diversification benefits have a similar standard deviation than the Large Cap portfolios. The average Small-Cap portfolio Sharpe ratio is 0,231, which is more than double the average Large Cap portfolio Sharpe ratio of 0,093. This indicates that the Small-Cap portfolios offer a better risk-reward payoff than the Large-Cap portfolios. Small-Cap portfolios also perform significantly better on the Modified VaR measure. The small-cap portfolios combined with the US large-cap portfolio have an average monthly 95%-VaR of 6,45%. The US large-cap portfolio combined with large-cap portfolios from other countries have a 7,40% monthly 95%-VaR. This is a difference of 0,95%. Based on this, the portfolio containing only large firms has a significantly higher VaR than portfolios containing both US large-cap stocks and non-US small-cap stocks at the 1% level, confirming that investors can reduce portfolio riskiness by using small stocks to diversify their portfolios
TABLE 4
Monthly 95%-VaR using the Modified VaR
Table 4 reports the calculated monthly 95%-Value-at-Risk using the Modified VaR, based on stock price data from 2002 to 2016. The US-Large-Cap portfolio is given a weight of 50% in each of these portfolios, with the remaining 50% of the portfolio consisting of the listed country and size
combination. The listed VaR values are equal to the maximum monthly percentage loss of that is not exceeded more than 5% of the time. Columns 3 and 4 report the mean monthly return and standard deviation, which are determinants of the 95%-VaR and are used to calculate the Sharpe Ratio, which is listed in column 5. Columns 6 and 7 report the excess kurtosis and skewness while column 8 reports the calculated 95%-VaR. Excess Kurtosis is defined as Kurtosis minus three.
Country Size Mean
return(%) Standard Deviation(%) Sharpe Ratio Excess Kurtosis Skewness 95%-VaR Australia Small 1,74 5,08 0,343 1,36 -0,16 6,71 Mid 0,36 4,40 0,082 2,60 -0,73 7,52 Large 0,40 3,82 0,105 1,81 -0,86 6,63 Canada Small 2,81 5,28 0,532 1,73 -0,41 6,30 Mid 0,58 4,84 0,120 3,43 -1,01 8,35 Large 0,49 3,95 0,124 2,25 -0,71 6,58 France Small 0,98 3,57 0,275 2,33 -0,35 5,08 Mid 0,56 3,92 0,143 2,85 -0,98 6,68 Large 0,39 4,48 0,087 1,37 -0,52 7,49 Germany Small 0,80 4,08 0,196 1,43 -0,05 5,85 Mid 0,40 3,94 0,102 2,70 -1,06 6,97 Large 0,43 4,67 0,092 2,19 -0,64 7,87
Hong Kong Small 1,87 6,33 0,295 3,57 0,29 7,56
Mid 0,60 5,51 0,108 4,04 -0,61 8,90 Large 0,52 4,76 0,109 2,23 -0,71 8,00 Italy Small 0,05 4,52 0,011 1,00 -0,43 7,84 Mid 0,36 4,51 0,080 1,41 -0,69 7,77 Large 0,25 4,65 0,054 0,94 -0,42 7,85 Japan Small 0,94 4,31 0,218 1,30 -0,55 6,68 Mid 0,59 3,99 0,148 1,42 -0,76 6,68 Large 0,41 4,26 0,096 2,17 -0,90 7,44 Netherlands Small 0,39 4,09 0,095 1,71 0,07 6,12 Mid 0,45 4,39 0,103 1,57 -0,76 7,53 Large 0,35 4,51 0,078 1,95 -0,77 7,83 UK Small 0,43 3,74 0,115 2,39 -0,33 5,88 Mid 0,37 4,02 0,092 3,72 -0,75 6,76 Large 0,38 4,07 0,093 1,59 -0,66 6,91
TABLE 5
Average Modified VaR by size
Table 5 summarizes the results from Table 4 by size. The reported values are the averages of the reported values of Table 4, with each country having an equal weight. The listed VaR values are equal to the maximum monthly percentage loss of that is not exceeded more than 5% of the time. Columns 2, 3, 5 and 6 report the averages of the four determinants of the Modified VaR while column 7 reports the average 95%-VaR. Column 4 reports the average Sharpe ratio. Excess Kurtosis is defined as Kurtosis minus three. The table reports the t-statistic of the differences between the large and small portfolios in parentheses. *, ** and *** indicate significance at 10%, 5% and 1%, respectively.
Size Average Return(%) Standard Deviation(%) Average Sharpe Ratio Excess Kurtosis Skewness 95%-VaR(%) Small 1,11 4,56 0,231 1,87 -0,21 6,45 Mid 0,47 4,39 0,109 2,64 -0,82 7,46 Large 0,40 4,35 0,093 1,82 -0,69 7,40 Large – Small -0,71** (-2,62) -0,21 (-0,65) -0,138*** (-3,03) -0,05 (-0,17) -0,48*** (-4,30) 0,95*** (3,16)
5. Robustness
In this paper international diversification benefits have been evaluated from the perspective of an American investor with a portfolio with a weight of 50% US large cap stocks and a weight of 50% of stocks from another country. It is interesting to consider if changing the portfolio weights or considering the diversification benefits from the perspective of a non-US investor leads to different findings. Indeed, Driessen and Laeven (2007) find that the size of international diversification benefits when investing abroad differs greatly depending on the country that is being diversified from. They find that international diversification benefits from a US investor are limited and that developing countries enjoy larger diversification benefits than developed countries. They find that diversification benefits tend to increase when country risk increases. Based on this paper, it is expected that diversification benefits should increase when we consider international diversification from the perspective of an investor in a developing country, rather than from the perspective of a US
investor. It is interesting to see if the earlier findings of this paper hold if a different country is used to diversify from. Therefore, I calculate the Modified VaR from the perspective of a Japanese
investor by combining all country and size portfolio individually with the Japanese Large Cap portfolio as previously done with the US Large-Cap portfolio. The Japanese perspective is the most interesting to test for this purpose, because the Japanese Large-Cap portfolio is the Large-Cap portfolio that is least correlated with the US Large-Cap portfolio used earlier in this paper.
The Modified VaR for internationally diversified portfolios from a Japanese perspective and its determinants are reported below in Table 6. Similarly to my earlier findings, average returns for the small-cap portfolios are higher on average, with the difference between the average monthly return being more than 1% for Canada. The difference in average returns is smaller than the earlier findings from the perspective of a US investor. For each portfolio, excess kurtosis exceeds one, and all are negatively skewed. Using the Modified Value at Risk, the small-cap portfolios have the lowest VaR for all countries. The difference in VaR is more than 1% for France, Germany and the Netherlands. The Sharpe Ratio is also higher for the Small-Cap portfolio for all countries except for Italy.
Summary statistics for these findings are reported in Table 7 below. This shows that the average returns are almost twice as large for the Small-Cap portfolios compared to the Large-Cap portfolios, and the average returns are significantly higher for Small-Cap portfolios at the 5% level. Both the Standard Deviation and Excess Kurtosis do not differ significantly between the Small- and Large-Cap portfolios. The Large-Cap portfolios are significantly less negatively skewed than the Small-Cap portfolios. On Average, the Small-Cap portfolios have a monthly Modified VaR of 6,71% while the Large-Cap portfolios have a monthly Modified VaR of 7,35%. This difference is significant at the 1% level and shows that, as was the case when looking at international diversification from the perspective of a US investor, portfolios containing small stocks have lower tail risk than portfolios containing only large stocks. The Sharpe Ratio is also significantly higher at the 5% level, which
confirms the earlier findings from the perspective of an American investor that portfolios containing small stocks offer a better risk-reward payoff.
Based on these findings, changing the perspective from a US investor to a Japanese investor does not lead to different findings. Portfolios containing small stocks still lead to additional diversification benefits beyond those of portfolios containing only large stocks, based on both the Sharpe Ratio being higher and the tail risk being lower for the Small-Cap portfolios.
To test whether using different portfolio weights leads to different findings, I created additional portfolios for two additional portfolio weight combinations and compute the 95%-VaR of these portfolios using the same method used earlier. Earlier portfolios were created that had a weight of 50% of the US-Large Cap portfolio and 50% of all other size and country combinations. The average historical VaR over the sample period when the US-Large Cap portfolio is given a weight of 30% or 70% are listed below in Table 8. The historical VaR from the perspective of a Japanese investor and from the perspective of a American investor with portfolios with a weight of 50% on the domestic Large Cap portfolio are also listed for comparison.
For all different portfolio weights, the average VaR of the Small Cap portfolios is lower than the average VaR of the Large Cap portfolios. This difference is significant at the 5% level for all listed weights except for when the US Large Cap portfolio is given a weight of 30%, for this weight the difference between the Large and Small portfolios is only significant at the 10% level. While changing the composition of the internationally diversified portfolios changes the results slightly, the main result is the same: the portfolios containing both Small Cap and Large Cap stocks are less risky than the portfolios containing only Large Cap stocks, based on the Historical Value at Risk.
TABLE 6
Monthly 95% - Value-at-Risk using the Modified VaR from perspective of Japanese
investor
Table 6 reports the calculated monthly 95%-Value-at-Risk using the Modified VaR, based on stock price data from 2002 to 2016. The Japanese Large-Cap portfolio is given a weight of 50% in each of these portfolios, with the remaining 50% of the portfolio consisting of the listed country and size combination. The listed VaR values are equal to the maximum monthly percentage loss of that is not exceeded more than 5% of the time. Columns 3 to 6 report the four determinants of the four-moment VaR while column 7 reports the calculated 95%-VaR. Excess Kurtosis is defined as Kurtosis minus three.
Country Size Average
return(%) Standard Deviation(%) Sharpe Ratio Excess Kurtosis Skewness 95%-VaR Australia Small 1,23 4,58 0,269 1,85 -0,48 6,74 Mid 0,30 4,42 0,068 2,85 -0,97 7,86 Large 0,43 4,02 0,107 1,97 -0,80 6,89 Canada Small 1,88 4,51 0,417 2,66 -0,86 6,34 Mid 0,52 4,82 0,108 4,00 -1,28 8,63 Large 0,49 4,12 0,119 2,21 -0,70 6,88 France Small 0,78 3,77 0,207 2,42 -0,68 5,94 Mid 0,50 4,04 0,124 2,34 -0,99 7,02 Large 0,43 4,42 0,097 1,54 -0,59 7,41 Germany Small 0,67 4,05 0,165 1,49 -0,48 6,40 Mid 0,33 4,03 0,082 2,25 -1,02 7,20 Large 0,45 4,52 0,100 1,93 -0,66 7,62
Hong Kong Small 1,31 5,15 0,254 2,83 -0,38 7,42
Mid 0,54 5,64 0,096 3,83 -0,66 9,32 Large 0,50 4,53 0,110 2,28 -0,73 7,64 Italy Small 0,22 4,33 0,051 1,52 -0,62 7,50 Mid 0,30 4,60 0,065 1,39 -0,73 8,05 Large 0,34 4,48 0,076 1,38 -0,54 7,57 Netherlands Small 0,42 4,02 0,104 1,64 -0,42 6,52 Mid 0,39 4,48 0,087 1,27 -0,72 7,74 Large 0,40 4,43 0,090 1,94 -0,73 7,59 UK Small 0,45 3,89 0,116 2,37 -0,57 6,37 Mid 0,31 4,17 0,074 2,96 -0,86 7,26 Large 0,42 4,19 0,100 1,77 -0,67 7,08 USA Small 0,68 4,52 0,150 1,55 -0,49 7,22 Mid 0,50 4,67 0,107 1,90 -0,74 7,93 Large 0,41 4,26 0,096 2,17 -0,90 7,44
TABLE 7
Average Modified VaR by size from Japanese perspective
Table 7 summarizes the results from Table 6 by size. The reported values are the averages of the reported values of table 6, with each country having an equal weight. The listed VaR values are equal to the maximum monthly percentage loss of that is not exceeded more than 5% of the time. Columns 2 to 5 report the averages of the four determinants of the four-moment VaR while column 8 reports the average 95%-VaR. Excess Kurtosis is defined as Kurtosis minus three. The table reports the t-statistic of the differences between the large and small portfolios in parentheses.
*, ** and ***indicate significance at 10%, 5% and 1%, respectively
Size Average Return(%) Standard Deviation(%) Average Sharpe Ratio Excess Kurtosis Skewness 95%-VaR(%) Small 0,85 4,31 0,193 2,04 -0,55 6,71 Mid 0,41 4,54 0,090 2,53 -0,89 7,89 Large 0,43 4,33 0,100 1,91 -0,70 7,35 Large – Small -0,42** (-2,57) 0,02 (0,10) -0,093** (-2,83) -0,13 (-0,73) -0,15** (-2,04) 0,64*** (-3,65)
TABLE 8
Historical VaR portfolio comparison
Table 8 compares the calculated monthly 95%-Value-at-Risk using the Historical VaR method for portfolios given different weights, based on stock price data from 2002 to 2016. The values in columns 2 and 3 correspond to the VaR calculated by averaging the VaR of the listed main portfolio component combined with all nine other countries individually. The table reports the t-statistic of the differences between the large and small portfolios in parentheses. *, ** and *** indicate significance at 10%, 5% and 1%, respectively
Main component Average Small Cap VaR Average Large Cap VaR Large-Small
30% USA Large Cap 6,63 7,85 1,22*
(1,85)
50% USA Large Cap 6,58 7,78 1,20**
(2,59)
70% USA Large Cap 7,16 7,75 0,59**
(2,14)
50% Japan Large Cap 7,17 7,72 0,55**
6. Conclusion
This paper examined the potential diversification benefits of including Small Cap firms in
internationally diversified portfolios. This paper adds to the limited existing research on the use of small stocks to diversify by considering time-varying correlations instead of constant correlations, evaluating small stock diversification performance by considering tail risk and by calculating the Sharpe ratio of internationally diversified portfolios containing Small Cap stocks to those of
internationally diversified portfolios containing only Large Cap stocks. The sample period used in this paper spans from January 2002 to December 2016, which is more recent and therefore more
relevant to current investors than existing literature. This paper finds significant benefits to using small stocks to diversify internationally.
This paper used DCC-GARCH(1,1) to calculate the time-varying correlations, which is necessary as the assumption of constant correlations over time is not realistic. Using this method, time-varying correlations of international equity returns are found to be much lower when using firms with a small market capitalization during the entire sample period. While correlations for small cap equity
portfolios vary more over time and increase by more during a bear market, in every period during the sample period the correlations for Small Cap portfolios were substantially lower than their Large Cap counterparts.
Based on the monthly 95%- Value-at-Risk, using Small stocks for international diversification leads to significantly lower tail risk. This is true regardless of whether the historical distribution method or the four-moment VaR is used to calculate the Value-at-Risk. Based on this measure, including small stocks in a portfolio to internationally diversify reduces the downside risk of the portfolio, which is very important to investors.
Individual Small Cap stocks tend to have higher returns but they are also more volatile than Large Cap stocks. However, due to Small stocks being less correlated to other stocks, internationally
diversified portfolios containing both Small- and Large-Cap stocks are not riskier than portfolios containing only Large Cap stocks. Including Small Cap stocks does increase the average returns, which means that the Sharpe ratio is also significantly higher for portfolios that combine the use of Small Cap and Large Cap stocks compared to those using only Large Cap stocks.
These three main findings support the notion that international diversification can still be useful, when Small Cap stocks are used in internationally diversified portfolios. Investors looking to diversify should consider the use of Small Cap stocks in their portfolios, as including these stocks can lead to higher average returns without being exposed to more risk, due to their lower correlation with other stocks.
There are some limitations to my analysis, however. I did not consider the effect the reduced liquidity and higher transaction costs of small stocks has on the use of small stocks in international diversification. The advantages of the use of small stocks seem large enough in size to be robust to the higher transaction costs, but the true effect of these restrictions is hard to measure. The effects of the higher transaction costs and lower liquidity increase when annual turnover is high or when very large orders are placed, but the exact effect this has on the usefulness of using small stocks for diversification is hard to measure and should be considered in future research Furthermore, while this paper provides significant evidence that including small stocks in your portfolio reduces tail risk when considering monthly returns, the performance may differ when for example yearly returns are considered. For future research, it would be interesting to consider testing the performance of using Small Cap stocks for diversification over a longer time period. Testing the performance of small stocks for international diversification based on different measures of tail risk, such as testing the 99%-VaR or calculating the expected shortfall may also be interesting for future research, as this paper only considered the 95%-VaR tail risk measure.
References
Bollerslev, T. (1990). Modelling the Coherence in Short-Run Nominal Exchange Rates: A Multivariate Generalized Arch Model. The Review of Economics and Statistics, pp. 498-505.
Chiyachantana, C. N., Jain, P. K., Jiang, C., & Wood, R. A. (2004). International Evidence on Institutional Trading Behavior and Price Impact. Journal of Finance, pp. 869-898. Chua, D. B., Kritzman, M., & Page, S. (2009). The Myth of Diversification. Journal of Portfolio
Management, pp. 26-35.
Driessen, J., & Laeven, L. (2007). International portfolio diversification benefits: Cross-country evidence from a local perspective. Journal of Banking & Finance, pp. 1693-1712. Engle, R. (2002). Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized
Autoregressive Conditional Heteroskedasticity Models. Journal of Business & Economic
Statistics, 339-350.
Eun, C. S., Huang, W., & Lai, S. (2009, 4 1). International Diversification with Large- and Small-Cap Stocks. Journal of Financial and Quantitative Analysis, pp. 489-524.
Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal
of Financial Economics, pp. 3-56.
Favre, L., & Galeano, J.-A. (2002). Mean-Modified Value-at-Risk Optimization with Hedge Funds.
Journal of Alternative Investments, pp. 21-25.
Kahneman, D., & Tversky, A. (1979). Prospect Theory: An Analysis of Decision under Risk.
Econometrica, pp. 263-291.
Kang, J.-K., & Stulz, R. M. (1997). Why is there a home bias? An analysis of foreign portfolio equity ownership in Japan. Journal of Financial Economics, pp. 3-28.
Levy, H., & Sarnat, M. (1970, 9). International Diversification of Investment Portfolios. The American
Economic Review, pp. 668-675.
Longin, F., & Solnik, B. (1995). Is the correlation in international equity returns constant:1960-1990?
Journal of International Money and Finance, pp. 3-26.
Longin, F., & Solnik, B. (2001). Extreme Correlation of International Equity Markets. Journal of
Finance, pp. 649-676.
You, L., & Daigler, R. T. (2010, 1). Is international diversification really beneficial? Journal of Banking
