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This paper applies the mean-variance portfolio optimization (PO) approach and the stochastic dominance (SD) test to examine preferences for international diversification
Despite greater integration of international capital markets, investors continue to hold portfolios largely dominated by domestic assets. International investors’ preference for domestic stocks remains a subject of controversy, since many studies indicate that greater profits can be made by diversifying internationally. This paper applies the mean-variance portfolio optimization (PO) approach and the stochastic dominance (SD) test to examine preferences between domestic and international diversification strategies. We also examine whether there is an optimal investment strategy for American investors according to their risk level.
Our study is based on daily data consisting of the prices of the 30 highest capitalization US stocks and 20 international market indices from Latin American and Asian financial markets and the G6. The purpose of this paper is to identify empirically preferences for international diversification
We first apply the PO technique to obtain efficient portfolios for both domestic and international diversification and study the preference for international
Since most of the portfolios including both DOD and IND portfolios are rejected to be normally distributed, the results drawn from the PO rule may be misleading. To circumvent this limitation, in this paper we also apply the SD test to examine preferences between the domestic and international diversification strategies, and we check whether there is an arbitrage opportunity between international and domestic stock markets, whether these markets are efficient, and whether investors are rational.
Our SD analysis shows that there is no arbitrage opportunity between international and domestic stock markets; domestically diversified portfolios with smaller risk dominate internationally diversified portfolios with larger risk and
The remainder of this paper is organized as follows.
The classical mean-variance portfolio optimization (PO) model introduced by Markowitz [
Most of the work on portfolio theory over the past decade has been based on the principle of utility maximization, where either the investor’s utility function is assumed to be a second-degree polynomial with a positive first derivative and a negative second derivative, or the probability functions are assumed to be normal [
To extend the MV model, Leung and Wong [
To circumvent the limitation of the MV approach, academics suggest adopting the SD rules. The primary advantage of using the SD approach is that it provides a very general framework for assessing portfolio choice without the need for asset-pricing benchmarks. It does not need any assumption on the distribution of the assets being examined, and it satisfies the general utility function and takes into consideration all the distributional moments in the comparison [
The SD theory has been continually developed for more than half a century, and many SD comparisons have been carried out empirically. For example, Hodges and Yoder [
Recently, Abhyankar, Ho, and Zhao [
Many studies document the benefits of the diversification strategy. For example, Solnik [
In addition, Driessen and Laeven [
On the other hand, French and Poterba [
To analyze preferences between international and domestic diversification, in this paper we use daily arithmetic returns of the closing prices for the 30 highest capitalization US stocks and 20 international market indices from 1 January 1993 to 31 December 2012. The data are obtained from Datastream. We use the daily closing prices of the 30 highest capitalization US stocks (
List of selected U.S. stocks.
Apple (AAPL) | Citigroup (C) | ||
Exxon Mobil (XOM) | Merck (MRK) | ||
Microsoft (MSFT) | Verizon Communications (VZ) | ||
Johnson & Johnson (JNJ) | Cisco Systems (CSCO) | ||
General Electric (GE) | PepsiCo (PEP) | ||
Wal-Mart (WMT) | Schlumberger (SLB) | ||
Chevron (CVX) | Disney (DIS) | ||
Wells Fargo (WFC) | JPMorgan Chase (JPM) | ||
Procter & Gamble (PG) | Intel (INTC) | ||
IBM (IBM) | Home Depot (HD) | ||
Pfizer (PFE) | United Technologies (UTX) | ||
AT&T (T) | McDonald’s (MCD) | ||
Coca-Cola (KO) | Boeing (BA) | ||
Bank of America (BAC) | ConocoPhillips (COP) | ||
Oracle (ORCL) | Amgen (AMGN) |
From the perspective of US investors, in this paper we adopt both PO and SD approaches to compare the performance of DOD and IND portfolios. We discuss the methodologies used in the paper in the following subsections. The selected stocks are listed by market capitalization.
We first adopt the classical portfolio optimization (PO) model [
Bai, Liu, and Wong [
To overcome the limitations of the traditional MV criteria, the SD approach developed by Hadar and Russell [
The most commonly used SD rules are first-, second- and third-order SD, denoted as FSD, SSD and TSD, respectively. All investors are non-satiated (that is, prefer higher return to less) under FSD, non-satiated and risk-averse under SSD, and non-satiated, risk-averse, and possessing decreasing absolute risk aversion (DARA) under TSD. The SD rules [
X dominates Y by FSD (SSD, TSD), denoted by X ≽_{1 }Y (X ≽_{2 }Y, X ≽_{3} Y) if and only if F_{1}(x) ≤ G_{1}(x) (F_{2}(x) ≤ G_{2}(x), F_{3}(x) ≤ G_{3}(x)) for all possible returns x in [a,b]. In addition, if the strict inequality holds for at least one value of x, X dominates Y strictly by FSD (SSD, TSD), denoted by X ≻_{1 }Y (X ≻_{2 }Y, X ≻_{3 }Y).
The theory of SD is important since it is related to utility maximization [
We note that under certain regularity conditions,
There are two broad classes of SD tests: one is the minimum/maximum statistic [
For any two domestically and internationally diversified portfolios
It is empirically impossible to test the null hypothesis for the full support of the distributions. We test the null hypothesis for a pre-designed finite numbers of values
Bai, Li, Liu, and Wong [
We note that in the above hypotheses,
We first adopt the PO approach to examine the preferences of different DOD and IND portfolios for risk-averse investors. From the MV efficient portfolios derived, we construct the efficient MV frontiers for both IND and DOD strategies and display them in
Mean-variance (MV) efficient frontiers of international and domestic diversification strategies.
Note: IND is the efficient frontier of internationally diversified portfolios and DOD is the efficient frontiers of domestically diversified portoflios.
We first summarize in
We turn now to comparing the efficient frontiers of the DOD and IND from
Summary statistics of the domestic MV efficient diversified portfolios.
Mean (µ) | Std Dev (σ) | CV (σ/µ) | Skewness | Kurtosis | |
---|---|---|---|---|---|
0.00048 | 0.00973 | 20.26 | 0.19096 | 9.21767 | |
0.00055 | 0.00997 | 18.01 | 0.15212 | 8.61416 | |
0.00063 | 0.01059 | 16.88 | 0.10630 | 7.39123 | |
0.00070 | 0.01159 | 16.53 | 0.07693 | 6.09004 | |
0.00077 | 0.01295 | 16.72 | 0.06230 | 5.29170 | |
0.00085 | 0.01459 | 17.19 | 0.03572 | 4.87478 | |
0.00092 | 0.01645 | 17.84 | 0.02482 | 4.73178 | |
0.00010 | 0.01851 | 18.58 | 0.03847 | 4.94001 | |
0.00107 | 0.02093 | 19.57 | 0.03114 | 5.46898 | |
0.00114 | 0.02371 | 20.73 | 0.02005 | 5.95474 |
Note: This table reports the summary statistics of the 10 domestic MV efficient diversified (DOD) portfolios, DOD1 to DOD10, including mean (µ), standard deviation (σ), the coefficient of variation (σ/µ), skewness, and kurtosis coefficients. The construction of DOD1 to DOD10 is described in
Summary statistics of the international MV efficient diversified portfolios.
Mean (µ) | Std Dev (σ) | CV (σ/µ) | Skewness | Kurtosis | |
---|---|---|---|---|---|
0.00032 | 0.00684 | 21.53 | −0.21007 | 5.04703 | |
0.00038 | 0.00712 | 18.92 | −0.18726 | 4.45829 | |
0.00044 | 0.00799 | 18.36 | −0.15911 | 3.76602 | |
0.00050 | 0.00973 | 19.34 | −0.12344 | 3.49837 | |
0.00051 | 0.00997 | 19.54 | −0.11918 | 3.54673 | |
0.00052 | 0.01059 | 20.04 | −0.10654 | 3.72651 | |
0.00055 | 0.01159 | 20.93 | −0.05806 | 3.88917 | |
0.00059 | 0.01294 | 22.00 | −0.04853 | 4.43614 | |
0.00063 | 0.01459 | 23.30 | −0.00687 | 4.79284 | |
0.00066 | 0.01645 | 24.74 | 0.01599 | 5.32968 | |
0.00071 | 0.01851 | 2609.17 | 0.07545 | 5.83010 | |
0.00076 | 0.02093 | 27.69 | 0.10862 | 6.26873 | |
0.00081 | 0.02371 | 29.39 | 0.13647 | 6.54289 |
Note: This table reports the summary statistics of the 13 international MV efficient diversified (IND) portfolios, IND1 to IND13, including mean (µ), standard deviation (σ), the coefficient of variation (σ/µ), skewness, and kurtosis coefficients. The construction of IND1 to IND13 is described in
The results from the PO approach discussed in the previous subsection show that the international diversification strategy dominates the domestic diversification strategy in a risk-return range but the dominance relationship is reversed in another risk-return range. In addition, from
To get a better picture of the results of the DD test, we plot the DD test and corresponding CDFs for each DOD and IND for two pairs of observations in
Plot of the cumulative distribution function (CDF) of the daily returns of the fourth domestic diversified portfolio (DOD 4) and the first international diversified portfolio (IND 1) and their DD statistics.
Plot of the CDF of the daily returns of the third domestic diversified portfolio (DOD3) and the 13 international diversified portfolios (IND13) and their DD statistics.
Stochastic dominance test between domestically and internationally diversified portfolios.
Portfolios | IND1 | IND2 | IND3 | IND4 | IND5 | IND6 | IND7 | IND8 | IND9 | IND10 | IND11 | IND12 | IND13 | SSD |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
ND | ND | SSD | SSD | SSD | SSD | SSD | SSD | SSD | SSD | 8 | ||||
ND | ND | SSD | SSD | SSD | SSD | SSD | SSD | SSD | SSD | 8 | ||||
ND | ND | ND | SSD | SSD | SSD | SSD | SSD | SSD | SSD | 7 | ||||
ND | ND | SSD | SSD | SSD | SSD | SSD | SSD | 6 | ||||||
ND | SSD | SSD | SSD | SSD | SSD | 5 | ||||||||
ND | SSD | SSD | SSD | SSD | 4 | |||||||||
ND | SSD | SSD | SSD | 3 | ||||||||||
ND | SSD | SSD | 2 | |||||||||||
ND | SSD | 1 | ||||||||||||
ND | 0 | |||||||||||||
0 | 0 | 0 | 0 | 0 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Note: This table reports the stochastic dominance results to test whether domestically diversified (DOD) portfolios strictly dominate internationally diversified (IND) portfolios in the sense of the
Stochastic dominance test between global internationally and domestically diversified portfolios.
Portfolios | DOD1 | DOD2 | DOD3 | DOD4 | DOD5 | DOD6 | DOD7 | DOD8 | DOD9 | DOD10 | SSD | TSD | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
SSD | SSD | SSD | SSD | SSD | SSD | TSD | SSD | TSD | TSD | 7 | 3 | 10 | |
SSD | SSD | SSD | SSD | SSD | SSD | SSD | SSD | SSD | SSD | 10 | 0 | 10 | |
SSD | SSD | SSD | SSD | SSD | SSD | SSD | SSD | SSD | SSD | 10 | 0 | 10 | |
ND | ND | ND | SSD | SSD | SSD | SSD | SSD | SSD | SSD | 7 | 0 | 7 | |
ND | ND | ND | SSD | SSD | SSD | SSD | SSD | SSD | SSD | 7 | 0 | 7 | |
ND | ND | SSD | SSD | SSD | SSD | SSD | SSD | 6 | 0 | 6 | |||
ND | SSD | SSD | SSD | SSD | SSD | SSD | 6 | 0 | 6 | ||||
ND | SSD | SSD | SSD | SSD | SSD | 5 | 0 | 5 | |||||
ND | SSD | SSD | SSD | SSD | 4 | 0 | 4 | ||||||
ND | SSD | SSD | SSD | 3 | 0 | 3 | |||||||
ND | SSD | SSD | 2 | 0 | 2 | ||||||||
ND | SSD | 1 | 0 | 1 | |||||||||
ND | 0 | 0 | 0 | ||||||||||
3 | 3 | 3 | 5 | 7 | 8 | 8 | 10 | 10 | 11 | ||||
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | ||||
3 | 3 | 3 | 5 | 7 | 8 | 9 | 10 | 11 | 12 |
From
On the other hand, from
In addition, from
Moreover, from
Moreover, from
Now, we use two examples displayed in
The results of the stochastic dominance (SD) statistics for the first domestic diversified portfolio (DOD1) and the second international diversified portfolios (IND2).
T_{1} > 0 | T_{1} < 0 | |
Total (%) | 18.3 | 20.8 |
Positive Domain (%) | 18.3 | 0 |
Negative Domain (%) | 0 | 20.8 |
Max (|T_{j}|) | 8.31 | 9.92 |
T_{2} > 0 | T_{2} < 0 | |
Total (%) | 39.9 | 0 |
Positive Domain (%) | 39.9 | 0 |
Negative Domain (%) | 0 | 0 |
Max (|T_{j}|) | 8.53 | 0.63 |
T_{3} > 0 | T_{3} < 0 | |
Total (%) | 27.9 | 0 |
Positive Domain (%) | 27.9 | 0 |
Negative Domain (%) | 0 | 0 |
Max (|T_{j}|) | 6.97 | NA |
Notes: Readers may refer to equation (3) for the formula of T_{j} for j=1,2,3 with F = DOD1 and G = IND2. The period is from 1 January 1993 to 31 December 2012.
The results of the stochastic dominance (SD) statistics for the first domestic diversified portfolio (DOD1) and the fifth international diversified portfolios (IND5).
T_{1} > 0 | T_{1} < 0 | |
Total (%) | 0 | 0 |
Positive Domain (%) | 0 | 0 |
Negative Domain (%) | 0 | 0 |
Max (|T_{j}|) | 2.80 | 2.99 |
T_{2} > 0 | T_{2} < 0 | |
Total (%) | 0 | 0 |
Positive Domain (%) | 0 | 0 |
Negative Domain (%) | 0 | 0 |
Max (|T_{j}|) | 1.88 | 2.12 |
T_{3} > 0 | T_{3} < 0 | |
Total (%) | 0 | 0 |
Positive Domain (%) | 0 | 0 |
Negative Domain (%) | 0 | 0 |
Max (|T_{j}|) | 1.75 | 1.21 |
In this paper, we compare the performance of the efficient domestically and internationally diversified portfolios by applying both the mean-variance portfolio optimization (PO) and the stochastic dominance (SD) approaches to analyze the Latin American and Asian financial markets and the G6 from 1 January 1993 to 31 December 2012. Comparing the MV efficient frontiers, we find that one efficient strategy could dominate the other over a range of risk and return but the dominance relationship could be reversed over another range of risk and return. Our PO results show that for investors who are willing to accept a higher risk level, a domestic diversification strategy is better for them. On the other hand, if they are only willing to accept a lower risk level, an international diversification strategy is a better choice.
Since the normality hypothesis is rejected for most of the portfolios, the results drawn from the PO rule may be misleading. To circumvent this limitation, we use the Davidson and Duclos test to examine investors’ preferences between the domestic and international diversification strategies. Our SD analysis shows that there is no arbitrage opportunity between international and domestic stock markets. We find that some domestically diversified portfolios SSD-dominate some internationally diversified portfolios, supporting those who claim that domestic diversification is better. On the other hand, we observe that some internationally diversified portfolios SSD- and TSD-dominate some domestically diversified portfolios, supporting the argument of those who claim that international diversification is better. In addition, we find that some domestically and internationally diversified portfolios do not dominate each other, supporting the argument that there is no difference between domestic and international diversification. Nevertheless, we find that domestically diversified portfolios with smaller risk SSD-dominate internationally diversified portfolios with larger risk and vice versa. This implies that for a lower risk level in domestic markets, the domestic diversification strategy stochastically dominates the international diversification strategy with higher risk, which, in turn, supports the finding that there are benefits from domestic diversification. On the other hand, for a low risk level in foreign markets, an international diversification strategy stochastically dominates a domestic diversification strategy with higher risk, which, in turn, supports the finding that there are benefits from international diversification. Nonetheless, we cannot find any domestically diversified portfolio that stochastically dominates all internationally diversified portfolios but we find that the internationally diversified portfolios with a risk smaller than that of any domestically diversified portfolio SSD- or TSD-dominate all the domestically diversified portfolios. This supports the claim that international diversification is better. However, although our findings imply that international diversification is better at a lower risk level, the markets for the domestic and international diversification strategies are efficient when they have the same risk level.
One may argue that our findings could fit any argument. So what can investors learn from our findings? Our answer is that yes, it is true that our analysis provides grounds to support any argument. However, all the arguments are true only under some conditions. We summarize our main findings as follows: (1) risk-averse investors are indifferent between the domestic and international diversification strategies if both strategies have the same risk; (2) when comparing the domestically diversified portfolios with lower risk and the internationally diversified portfolios with higher risk, risk averters would prefer to invest in the domestically diversified portfolios; (3) when comparing the internationally diversified portfolios with lower risk with the domestically diversified portfolios with higher risk, risk averters would prefer to invest in the internationally diversified portfolios; and (4) the risk of some internationally diversified portfolios is smaller than that of all domestically diversified portfolios, implying that risk averters will prefer only international diversification if they can invest in these smaller-risk internationally diversified portfolios.
Finally, one may wonder whether there is any contradiction for our PO and SD findings because from our PO finding, we conclude that the domestic diversification strategy is better for investors with a higher risk level, while the international diversification strategy is the better choice for investors with a lower risk level. This seems to contradict the SD findings of (1) to (4) above. We note that there is no contradiction. Our SD finding is the refinement of our PO finding. Our PO finding shows that for a higher risk, the mean of the DOD is higher than that of the IND, and thus, the results from our PO analysis conclude that the domestic diversification strategy is better. However, our SD analysis shows that this is not true. Our SD analysis shows that for a higher risk, the mean of the DOD is higher than that of the IND, but the DOD does not stochastically dominate any IND portfolio, and thus, investors in these 2 markets are indifferent between the IND and DOD portfolios with the same risk.
An extension could include examining the preferences of other types of investors, such as risk seekers [
The research is partially supported by the University of Sfax, The Chinese University of Hong Kong, Hong Kong Baptist University, and the Research Grants Council (RGC) of Hong Kong.
All authors made substantial contributions for this paper.
The authors declare no conflict of interest.
See [
The SD theory could be extended further to satisfy non-expected utilities; see, for example, [
See [