BACHELOR THESIS :
Do Sub-Saharan African Stock Markets Offer
Diversification Opportunities for International Investors?
Abstract:
In this paper, the potential diversification opportunities offered by Sub-Saharan stock markets to international investors is investigated. When regressed to a world market index, low correlations and low betas computed for the continent stock indices and country stock indices suggest that some stock markets in the region are yet far from being integrated into the world market. Furthermore, 2
portfolios are designed according to the Markowitz mean-variance optimization theory by
optimization of the Sharpe and the Sortino ratios, while rebalancing quarterly over periods of 6 and 8 years. The first portfolio, composed of a world index and 2 African continent indices, and the second portfolio, composed of a US index and 3 African country indices are tested against respectively the World index and the S&P 500. The Jobson-Korkie tests performed on the Sharpe and Sortino ratios provide enough statistical evidence that there are improvements in the performances of both portfolio compare to the reference indices, whereas the naïve portfolios perform poorly.
1. Introduction:
For decades, Sub-Saharan Africa has been facing huge economic development challenges. The region has been labeled the poorest and more unstable in the world. However, the last 2 decades have seen major changes occur and a dynamic of sustainable growth trigger. Indeed, according to the World Bank, the average GDP growth in Sub-Saharan Africa was as high as 4.5 % from 1995 to 2013. For instance, Rwanda and Angola, despite having both suffered devastating civil conflicts during the nineties are now growing at rates of respectively 8 and 7 % (World Bank, 2014). Furthermore the MSCI (Morgan Stanley Capital International) Frontier Market Index, which regroups stock indices of promising pre-emerging countries, is already listing 3 Sub-Saharan countries (i.e. Kenya, Mauritania and Nigeria), while Botswana, Ghana and Zimbabwe are being considered for future integration in this group (MSCI, 2014). Those strong economic prospects naturally awake the attention of international investors willing to benefit of the promising high returns accompanying high growth rates. Already net FDI inflows in the region increased 16% from 2012 to 2013 (World Bank, 2014).
Despite the Latin-American financial crisis in the eighties, the Asian financial crisis in the nineties and the more recent global financial crisis, it is commonly believed that well-structured financial markets and organized stock exchanges can foster growth in emerging economies. Developing such markets and stock exchanges can therefore be of crucial importance in order for African countries to attract a larger stake of international investment to finance local firms and projects (Yartey & Adjasi, 2007).
Moreover, international investors are typically looking into diversifying their assets in markets with low correlation with the home market in order to reduce overall risk (Coeurdacier & Guibaud, 2011). For many years Asian and Latin American emerging countries have offered such opportunities. Yet those markets have since grown to become increasingly more integrated into the world economy, driving away some of the benefit of diversification (Phylaktis & Ravazzolo, 2005, Lagoarde & Lucey and Driessen & Laeven, 2007). Sub-Saharan stock markets, with yet relatively smaller average sizes, have potentially less linkages to the major financial markets and to the world economy. In top of reaping higher returns, international investors could therefore efficiently diversify their portfolio by holding assets in the Sub-Saharan region. Besides, if one considers the home equity bias that characterizes developed countries’
investors, it appears clear that promoting the development of African stock exchanges could be strongly beneficial for international investors as well as for receiving country agents.
This paper will first look at some indicators of the level of integration of the Sub-Saharan stock markets to the world economy, compare to the level of integration of major economies such as the US and Europe. Next, the benefit of diversification will be investigated by constructing 2 portfolios composed of African stock indices, measuring their performances and comparing it to the world and the US indices. The first portfolio will be composed of the MSCI World index, the MSCI Frontier Markets Africa index and the S&P Pan Africa index. The second portfolio will contain the US S&P500 index, the MSCI South Africa index, the MSCI Nigeria index and the MSCI Kenya index. Both portfolios will be rebalanced quarterly over periods of respectively 6 years (i.e. 2008-2013, 23 periods) and 8 years (i.e. 2006-2013, 32 periods), by maximizing the Sharpe and the Sortino ratios. Those 2 ratios, as well as the variances and the average returns will be sampled and will be tested against the world and the US indicators using respectively a Jobson-Korkie test, an F-test and a t-test. It is expected that the existence of significant diversification opportunities in Sub-Saharan Africa will be brought to light.
2. Literature review
The development of Sub-Saharan African stock markets being relatively recent, their potential has yet to be explored thoroughly. On the other hand, emerging markets all over the globe have drawn
considerable attention from scholars. From those researches, interesting findings can be underlined as they could be used as a starting point when addressing Sub-Saharan stock exchanges.
In a paper investigating the effect of financial liberalization on economic growth in developed and emerging countries, Bekaert, Harvey & Lundbald (2005) have concluded that equity market liberalization on average increases the real economic growth of 1% annually. According to the same paper, the impact of liberalization is greater the more evolve the country’s legal institutions and global financial sector are. Pioneering in the analysis of emerging stock markets, Harvey (1995) studied 20 emerging countries. Amongst his conclusions, one can retained that those countries are characterized by high average return with high volatility and low correlation with developed countries returns. Based on those findings, Harvey suggested that including emerging markets assets in an efficient portfolio would increase expected returns while decreasing volatility, thus materializing a portfolio diversification opportunity. Additionally, Harvey noted that the predictability of asset returns in emerging market is not associated with correlation with the US market, as opposed to the predictability of asset returns in developed
markets; and local information is far more crucial in predicting asset returns in emerging markets than in their developed counterparts. Investigating the level of linkage between the US, Japan and emerging markets from the Pacific Basin, Phylaktis & Ravazzolo (2005) came to the conclusion that although linkages had increased in the years before the study, international investors could still make long-term gains through diversification opportunities by investing in the Pacific Basin countries. They explained this by the fact that the increasing linkages were due to a higher degree to local factors rather than to a common world growth factors. Similarly, Yu and Hassan (2008) have observed an increasing long-run relation between the US stock market and some of the Middle-East and North African countries (i.e. Egypt, Jordan, Turkey and Morocco). However, in the same paper, they found the stock markets of the Gulf Cooperation Council (GCC) group to be segmented from the world market, implying diversification opportunities for foreign investors. Hassan, Marroney, El-Sady and Telfah (2003) had previously
corroborated those findings in the context of the MEAF (Middle East and African) countries, noting that stock returns volatility and predictability were mostly determined by local political, financial and
economic risks, rather than global factors. As well as Lagoarde-Segot and Lucey (2007) who investigated diversification opportunities in the MENA countries, they have drawn the conclusion that there exist outstanding benefits in including emerging markets stocks into an efficient global portfolio. Focusing on the 7 Sub-Saharan countries, Enisan and Olufisayo (2009) have found consistent results. They have highlighted a long term cointegrating relationship between stock market development and economic growth for Egypt and South Africa, and causality for the seven countries concerned (i.e. Ivory Coast, Kenya, Morocco, Nigeria, Zimbabwe, Egypt and South Africa). They concluded that African policy makers should continue to encourage the development of stock markets by removing entry barriers and
designing policies aiming at enhancing financial efficiency.
All the aforementioned studies point to the evidence of substantial diversification opportunities for African stock markets. Nevertheless, the question of efficiency in those markets is one that is less in favor of investing in African stocks. Indeed, Appiah-Kusi and Menyah (2003) rejected that the Nigerian and South African stock exchanges, amongst others, were weak-form efficient, which might indicate the presence of market imperfections such as high transactions and information costs. This could represent an impediment for foreign as well as local investors. On the other hand, Alagidede (2011) examining the predictability of returns in African largest equity markets (i.e. those of South Africa, Nigeria, Kenya, Egypt, Morocco and Tunisia) found that all show evidence of returns predictability and long memory,
suggesting less than perfect arbitrage and compliance with the weak form of market efficiency hypothesis.
3. Data
First, some indicators of the level of integration of the Sub-Saharan stock markets to the world economy will be derived and compared to integration indicators of major economies such as the US and Europe. Afterwards, the benefit of diversification will be investigated thanks to 2 portfolios composed of African stock indices, and respectively a world index and an US index. The performances of those portfolios will be compared to the performances of the world and the US indices. The first portfolio will be composed of the MSCI World index, the MSCI Frontier Markets Africa index and the S&P Pan Africa index. The second portfolio will contain the US S&P500 index, the MSCI South Africa index, the MSCI Nigeria index and the MSCI Kenya index.
The MSCI World index will be used as a proxy for the world stock market. It is an index composed of 1610 listed companies from 23 developed countries. As its constituents cover 85% of the market capitalization of each country, it is seen as a fair representative of the global equity market (About.com, 2014). The MSCI Frontier Markets Africa index regroups large and mid-capitalization listed companies from 5 African Frontier markets (i.e. Kenya, Mauritius, Morocco, Nigeria and Tunisia). Obviously, it does not account for the entire African equity market. Instead, this index gives the opportunity to invest in high return and high volatility stock markets while mitigating country specific risk. Nigerian stocks represent 58% of this index though (MSCI, 2014). The S&P Pan Africa index is more comprehensive, being composed of stocks from 12 African countries, emerging and frontier, including South Africa and Egypt. The S&P Pan Africa is therefore seen as a benchmark for the entire African stock market (S&P Dow Jones indices, 2014). A portfolio composed of the 3 aforementioned indices would theoretically
outperform the MSCI world index alone. Indeed, both the MSCI Frontier Markets Africa and the S&P Pan Africa are expected to provide higher returns. In the same time country specific risk and the higher volatility characterizing emerging and frontier markets are reduced since stocks from several countries are pooled.
The MSCI South Africa, the MSCI Nigeria and the MSCI Kenya are country indices. All 3 represent 85% of the market capitalization of their respective country equity market which are the 3 largest in Sub-Saharan Africa (MSCI, 2014). One of the rules of thumb in international investment and diversification is to invest in priority in countries with larger market capitalization (Bodie et al.). Thus, an investor willing
to benefit of the strong economic growth in the Sub-Saharan region should look at those 3 markets in priority. The S&P 500 and the S&P Europe 350 are respectively indices for the US equity market, which is the world largest by market capitalization, and European equity market. The S&P 500 has such a
predominant role in the global economy that it is sometimes used as a proxy for the global efficient portfolio (Berk & DeMarzo, 2011). Because of the strong economic links between the US and Europe, we expect those indices to be significantly correlated with each other and with the world market. The opposite being expected for the Sub-Saharan stock markets indices, adding them to the S&P500 index (or alternatively to the S&P Europe 350 index) should procure meaningful diversification by increasing average return and reducing overall volatility.
For all indices mentioned, the weekly price level in dollar as found in DataStream will be collected in order to compute the weekly rate of returns, assuming reinvestment. The rate of return can be computed as such:
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝐼𝐼𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 𝑡𝑡− 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝐼𝐼𝑤𝑤𝑊𝑊𝑊𝑊𝑊𝑊 𝑡𝑡−1 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝐼𝐼𝑤𝑤𝑊𝑊𝑊𝑊𝑊𝑊 𝑡𝑡−1
Lagoarde and Lucey (2007) recommend using weekly index prices and return when dealing with less liquid securities in order to minimize the noise trading effect.
In order to compute the Sharpe and the Sortino ratio, we will need a measure of the risk free rate. As there is no such a thing as a global risk free rate, we will use the US 3-months Treasury Bills rate as a proxy, as it is commonly done in scholarly papers, academics books and by professionals (Berk & Demarzo, 2011).
4. Theoretical framework and Methodology
The idea of stock markets as a possible mean to boost real economic growth derives from the neo-classical theory which affirms that the cost of equity should decrease following equity market liberalization due to an amelioration of the risk sharing. Moreover, stock market liberalization makes foreign capital more easily accessible, therefore reducing financing constraints. As constraints are reduced, investment becomes less sensitive to cash flows and thus more consistent. Also, higher corporate governance standards could be achieved thanks to the involvement of foreign investors (Bekaert et al., 2005). Additionally, the attractiveness of international diversification is twofold for investors. First of all, as demonstrated by Markowitz (1952), the lower the correlation between the returns of the assets composing the portfolio, the lower the overall volatility of portfolio returns tend to be. Thus, risk-averse investors will be more likely to select securities with low correlation. Further, one
will expect the correlation between a domestic asset and a foreign asset to be somewhat lower than between 2 domestic assets. This follows from the differences in the industrial composition of the stock markets and the differences in monetary, fiscal and industrial policies across countries. Consequently, international diversification allows investors to benefit in terms of stability and long run yield of the differentials in country returns dynamics. In this context, emerging markets have theoretically the particularity to amplify the power of diversification. Indeed, in those markets, asset performances are more susceptible to be determined to a greater extent by country specific variables, such as political and currency risks, rather than global factors. Finally, the specific risks in emerging markets are normally compensated by higher returns due to faster capital accumulation and economic growth than in mature markets (Lagoarde-Segot & Lucey, 2007).
In the attempt to answer the question posed by this thesis, concepts such as the CAPM, the mean-variance optimization model, the Sharpe ratio and the Sortino ratio will be used as exposed further. The CAPM (Capital Asset Pricing Model) is commonly used by professionals in the finance industry in order to determine the expected return of a security or asset. A key component of this model is the beta of the asset, which is defined as the volatility of the asset’s return common with the volatility of the market’s excess return divided by the volatility of the market’s return:
𝛽𝛽𝐼𝐼𝛽𝛽𝛽𝛽
𝑎𝑎𝑎𝑎𝑎𝑎𝑊𝑊𝑡𝑡=
𝐶𝐶𝐶𝐶𝐶𝐶𝑎𝑎𝐶𝐶𝐶𝐶𝑎𝑎𝐶𝐶𝐶𝐶𝑊𝑊(𝑅𝑅𝑊𝑊𝑡𝑡𝑅𝑅𝐶𝐶𝐶𝐶𝑉𝑉𝑎𝑎𝐶𝐶𝐶𝐶𝑎𝑎𝐶𝐶𝐶𝐶𝑊𝑊(𝐸𝐸𝐸𝐸𝐶𝐶𝑊𝑊𝑎𝑎𝑎𝑎 𝑅𝑅𝑊𝑊𝑡𝑡𝑅𝑅𝐶𝐶𝐶𝐶𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎,𝐸𝐸𝐸𝐸𝐶𝐶𝑊𝑊𝑎𝑎𝑎𝑎 𝑅𝑅𝑊𝑊𝑡𝑡𝑅𝑅𝐶𝐶𝐶𝐶𝑀𝑀𝑎𝑎𝑀𝑀𝑀𝑀𝑎𝑎𝑎𝑎)𝑀𝑀𝑎𝑎𝑀𝑀𝑀𝑀𝑎𝑎𝑎𝑎),
where the market excess return corresponds to the market return minus the risk free rate, which is usually derived from government bonds. The beta of an asset measures its sensitivity to the market as a whole (Berk & Demarzo, 2011). Despite the validity of its international form being contested and despite some strong restrictions, the CAPM can still be used as a rule of thumb for international investment diversification. Indeed, Bodie, Kane and Marcus (2011) recommend to consider countries with lowest beta against the U.S. (or domestic market) in order to reduce the portfolio overall risk when diversifying assets by investing in higher-risk countries (e.g. emerging or frontier markets). Hence, the beta of a country index can provide 2 interesting indicators: the beta against the world index illustrates the degree of integration into the world market, while the beta against the domestic country points out the degree of diversification achievable. It can also be added that a lower correlation between 2 assets tends to reduce the risk of containing both into the same portfolio. As a consequence, investors will prefer to hold assets with low correlation level (Markowitz, 1952). The beta of an index can be computed by linear regression. When regressing the returns of a country index (dependent variable) on the excess returns of the world index (independentvariable or regressor), the slope of the variable gives us the beta (i.e. the sensitivity to the market). The closest the beta is to 1, the more closely the index moves with the market, suggesting a higher level of integration. To find out about the integration levels, the following regressions will be computed: the S&P 500, the S&P Europe 350, the MSCI South Africa, the MSCI Nigeria and the MSCI Kenya weekly returns on the MSCI World weekly excess returns. Using the linear regression function in Excel which will return the ANOVA table, it will be relevant to also look at the R² coefficient. Indeed, this coefficient measures how well the regressor explains the variation of the dependent variable and is computed from sum of the squares of the error terms as such:
𝑅𝑅
2=
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑎𝑎𝐶𝐶𝐶𝐶𝑊𝑊𝐸𝐸 𝑎𝑎𝑅𝑅𝑠𝑠 𝐶𝐶𝑜𝑜 𝑎𝑎𝑠𝑠𝑅𝑅𝑎𝑎𝐶𝐶𝑊𝑊𝑎𝑎𝑇𝑇𝐶𝐶𝑡𝑡𝑎𝑎𝐸𝐸 𝑎𝑎𝑅𝑅𝑠𝑠 𝐶𝐶𝑜𝑜 𝑎𝑎𝑠𝑠𝑅𝑅𝑎𝑎𝐶𝐶𝑊𝑊𝑎𝑎 , (Stock & Watson, 2012).
Thus a strong R² coupled with a beta close to 1 will suggest a high degree of integration of the country/region index into the world market. Another use of the linear regression, the beta and the R² will be done in order to preview the diversification potential of African stock markets. The S&P PAN Africa and the MSCI Frontier Markets Africa weekly returns will be regressed on the MSCI World weekly excess returns, and the MSCI South Africa, MSCI Nigeria, MSCI Kenya and S&P Europe 350 weekly returns will be regressed on the S&P500 weekly excess returns. A beta close to 1 and a strong R² coefficient will suggest less potential for diversification, as it could be interpreted as a sign of co-movement of the two indices concerned. Furthermore, the correlations level for each regression will be computed in an attempt to confirm the intuition given by the beta-R² couple. A higher correlation level will be expected when the beta is close to 1 and the R² coefficient is strong. Indeed, the correlation is a measure of dependence between 2 variables:
Correlation (index 1, index 2) =�𝑉𝑉𝑎𝑎𝐶𝐶𝐶𝐶𝑎𝑎𝐶𝐶𝐶𝐶𝑊𝑊(𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸 1).𝑉𝑉𝑎𝑎𝐶𝐶𝐶𝐶𝑎𝑎𝐶𝐶𝐶𝐶𝑊𝑊(𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸 2)𝐶𝐶𝐶𝐶𝐶𝐶𝑎𝑎𝐶𝐶𝐶𝐶𝑎𝑎𝐶𝐶𝐶𝐶𝑊𝑊 (𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸 1,𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸 2)
It is an important measure in portfolio theory since, as mentioned earlier in this paper, international investors should diversify their assets in markets with low correlation with the domestic market in order to reduce overall risk (Markowitz, 1952). The regressions and correlations will be computed using weekly dollar returns for a time period of 5 year (i.e. from 2009 to 2013, 260 observations) for all indices. The weekly returns will be computed from the indices price level as found in DataStream. The weekly rate of the 3-months US treasury bills will serve as a proxy for the risk free rate.
In order to test the diversification potential of Sub-Saharan stock markets, one must first of all derive the efficient portfolio comprising the different indices (note that this efficient portfolio is not the same as the
market portfolio aforementioned, but is labeled efficient in the sense that it minimizes the portfolio risk or a ratio measuring the portfolio performance). The efficient portfolio is defined as one that combines assets in such a way that the portfolio’s rate of return is maximized while the risk of its rate of return is minimized. The addition of an index with relatively low return but with less than perfect correlation with the home index will improve the efficiency of the portfolio. The vector of the index proportions within the portfolio that minimize the risk (as measure by the standard deviation of the rate of returns) can be found by the following Lagragian:
L: =𝑋𝑋′𝑠𝑠𝛴𝛴𝑋𝑋𝑠𝑠− 𝜆𝜆1(𝑋𝑋′𝑠𝑠𝜇𝜇 − 𝜇𝜇𝑃𝑃) − 𝜆𝜆2(𝑋𝑋′𝑠𝑠𝐼𝐼 − 1)
Where 𝜆𝜆1 𝛽𝛽𝐼𝐼𝐼𝐼 𝜆𝜆2 are the multipliers, e is the unit vector, 𝜇𝜇 is the return vector, 𝜇𝜇𝑃𝑃 is the portfolio return and Σ is the variance-covariance matrix. The proportions vector is then expressed as follows:
𝑋𝑋𝑠𝑠 = 𝛴𝛴 −1𝜇𝜇 𝐼𝐼′𝛴𝛴−1𝜇𝜇
The mean portfolio return and its standard deviation on the tangency portfolio are given by the following expressions:
𝜇𝜇𝑠𝑠 = 𝜇𝜇𝑋𝑋𝑠𝑠 and 𝜎𝜎𝑠𝑠2=𝑋𝑋′𝑠𝑠𝛴𝛴𝑋𝑋𝑠𝑠, (Jobson & Korkie, 1982).
The Sharpe ratio is obtained by dividing the portfolio excess return (i.e. the rate of return of the portfolio minus the risk free rate) by the risk associated with the return of the portfolio (i.e. the standard
deviation of the rate of returns): 𝑅𝑅𝐼𝐼𝛽𝛽𝑅𝑅𝑃𝑃𝐼𝐼𝑃𝑃𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜𝐶𝐶𝐸𝐸𝐶𝐶𝐶𝐶− 𝑅𝑅𝑃𝑃𝑅𝑅𝑅𝑅𝐹𝐹𝐶𝐶𝑊𝑊𝑊𝑊 𝑆𝑆𝛽𝛽𝛽𝛽𝐼𝐼𝐼𝐼𝛽𝛽𝑃𝑃𝐼𝐼 𝐼𝐼𝐼𝐼𝑑𝑑𝑃𝑃𝛽𝛽𝛽𝛽𝑃𝑃𝑑𝑑𝐼𝐼𝑃𝑃𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜𝐶𝐶𝐸𝐸𝐶𝐶𝐶𝐶
It is a measure of the reward to overall risk trade-off (Bodie et al., 2011). The Sortino ratio is similar to the Sharpe ratio with the difference that it divides the excess return by the downside standard deviation: 𝑅𝑅𝐼𝐼𝛽𝛽𝑅𝑅𝑃𝑃𝐼𝐼𝑃𝑃𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜𝐶𝐶𝐸𝐸𝐶𝐶𝐶𝐶− 𝑅𝑅𝑃𝑃𝑅𝑅𝑅𝑅𝐹𝐹𝐶𝐶𝑊𝑊𝑊𝑊
𝐷𝐷𝑑𝑑𝐷𝐷𝑅𝑅𝑃𝑃𝐼𝐼𝐼𝐼 𝑅𝑅𝑃𝑃𝑅𝑅𝑅𝑅𝑃𝑃𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜𝐶𝐶𝐸𝐸𝐶𝐶𝐶𝐶
The downside standard deviation can be obtained by computing the standard deviation of the negative rate of returns. The Sortino ratio more accurately captures the desire of investors to diversify their assets to reap off high returns while avoiding negative volatility and is therefore more relevant when studying emerging markets (Lagoarde-Segot & Lucey, 2007). Both the Sharpe ratio and the Sortino ratio can be used as the constraint to be optimized in order to find the proportions of the different assets (i.e. indices) within the portfolio. We will then obtain the assets proportions that maximize returns while minimizing risk.
The benefit of diversification will be investigated thanks to 2 portfolios composed of African stock indices, and respectively a world index and an US index. The performances of those portfolios will be compared to the performances of the world and the US indices. The first portfolio will be composed of the MSCI World index, the MSCI Frontier Markets Africa index and the S&P Pan Africa index. The second portfolio will contain the US S&P500 index, the MSCI South Africa index, the MSCI Nigeria index and the MSCI Kenya index. Those portfolios will be rebalanced quarterly using the solver tool in Excel. For each one of the portfolios the following procedure will be executed. First, the variance-covariance matrix of the returns of the indices composing the given portfolio will be computed quarterly using an Excel matrix multiplication (MMULT) function (see Appendix). Subsequently, equal weight will be assigned to these indices in order to design the naïve portfolio. This will allow deriving the expected return and the standard deviation on the naïve portfolio by means of another Excel MMULT function (see Appendix). Now, The Sharpe ratio of the naïve portfolio can be computed using the aforementioned formula. The same steps will be replicated starting from the negative indices returns in order to find the Sortino downside risk and compute the Sortino ratio for the portfolio. Finally, using the solver function in Excel, the Sharpe ratio (and alternatively the Sortino ratio) will be maximized by setting the indices’ weights within the portfolio as the variables. The sum of the weights will be set to be strictly equal to 1. This restriction means that short-sales are not allowed. This procedure will be repeated quarterly over a time period of 6 years for the first portfolio (i.e. 2008-2013, no data were available for the S&P Pan Africa earlier than February 2008, leaving us with a sample of 23 quarters), and 8 years for the second portfolio (i.e. 2006-2013, a sample of 32 quarters). The Sharpe and Sortino ratios, the standard deviations and the average portfolio returns thus derived and sampled will be tested against those of the MSCI World and US S&P 500 indices, also collected quarterly. The Jobson-Korkie test will be used to test the Sharpe and the Sortino ratios as recommended by Lagoarde-Segot and Lucey (2007). This test was developed by Jobson and Korkie (1981) to test the performance of a portfolio (as measure by the Sharpe ratio, or alternatively the Sortino ratio) against another investment opportunity. They found that the Sharpe ratio has an asymptotic normal distribution with a variance equal to:
𝜎𝜎2= (𝑅𝑅𝑃𝑃𝑅𝑅𝑅𝑅𝐼𝐼𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸2 𝑅𝑅𝑃𝑃𝑅𝑅𝑅𝑅
𝑃𝑃𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜2 − 𝑅𝑅𝑃𝑃𝑅𝑅𝑅𝑅𝐼𝐼𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸𝑅𝑅𝑃𝑃𝑅𝑅𝑅𝑅𝑃𝑃𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜𝐶𝐶𝑑𝑑𝑑𝑑𝛽𝛽𝑃𝑃𝑃𝑃𝛽𝛽𝐼𝐼𝑃𝑃𝐼𝐼𝐼𝐼𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸 𝑎𝑎𝐶𝐶𝐸𝐸 𝑃𝑃𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜).
An F test will be performed to test the difference in risk (F test for difference in variances), while a t-test will test the difference in the average returns. All tests will be performed for the 3 different strategies, namely the naïve portfolio, the portfolio maximizing the Sharpe ratio, and the portfolio maximizing the
Sortino ratio against the reference indices, namely the MSCI World and the S&P500 indices. The hypothesizes and statistics are defined as follows:
-Jobson-Korkie: 𝐻𝐻0: 𝑆𝑆ℎ𝛽𝛽𝑃𝑃𝑎𝑎𝐼𝐼𝐸𝐸𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜=𝑆𝑆ℎ𝛽𝛽𝑃𝑃𝑎𝑎𝐼𝐼𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸;𝐻𝐻1: 𝑆𝑆ℎ𝛽𝛽𝑃𝑃𝑎𝑎𝐼𝐼𝐸𝐸𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜 ≠ 𝑆𝑆ℎ𝛽𝛽𝑃𝑃𝑎𝑎𝐼𝐼𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸 -Jobson-Korkie: 𝐻𝐻0: 𝑆𝑆𝑑𝑑𝑃𝑃𝛽𝛽𝑃𝑃𝐼𝐼𝑑𝑑𝑃𝑃𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜= 𝑆𝑆𝑑𝑑𝑃𝑃𝛽𝛽𝑃𝑃𝐼𝐼𝑑𝑑𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸; 𝐻𝐻1: 𝑆𝑆𝑑𝑑𝑃𝑃𝛽𝛽𝑃𝑃𝐼𝐼𝑑𝑑𝐸𝐸𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜 ≠ 𝑆𝑆𝑑𝑑𝑃𝑃𝛽𝛽𝑃𝑃𝐼𝐼𝑑𝑑𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸
t=
𝑅𝑅𝐶𝐶𝑎𝑎𝑊𝑊𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃 𝑅𝑅𝑊𝑊𝑡𝑡𝑅𝑅𝐶𝐶𝐶𝐶𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖−𝑅𝑅𝐶𝐶𝑎𝑎𝑊𝑊𝐼𝐼𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖𝑅𝑅𝑊𝑊𝑡𝑡𝑅𝑅𝐶𝐶𝐶𝐶𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃 2𝑇𝑇�(𝑅𝑅𝐶𝐶𝑎𝑎𝑊𝑊𝐼𝐼𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖2 𝑅𝑅𝐶𝐶𝑎𝑎𝑊𝑊𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃2 −𝑅𝑅𝐶𝐶𝑎𝑎𝑊𝑊𝐼𝐼𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖𝑅𝑅𝐶𝐶𝑎𝑎𝑊𝑊𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃𝐶𝐶𝐶𝐶𝐶𝐶𝑎𝑎𝐶𝐶𝐶𝐶𝑎𝑎𝐶𝐶𝐶𝐶𝑊𝑊𝐼𝐼𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖 𝑎𝑎𝑖𝑖𝑖𝑖 𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃)
,
where the risk is the relevant risk measure for the asset, namely downside risk in case of the Sortino ratio, and total volatility or standard deviation for the Sharpe ratio (Lagoarde-Segot & Lucey, 2007). It might be interesting to note that a negative t result would infer that the portfolio outperforms the index, while a positive t result will infer the opposite.
-F-test: 𝐻𝐻0: 𝑆𝑆𝛽𝛽𝐼𝐼𝐼𝐼𝑑𝑑²𝐸𝐸𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜= 𝑆𝑆𝛽𝛽𝐼𝐼𝐼𝐼𝑑𝑑²𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸; 𝐻𝐻1: 𝑆𝑆𝛽𝛽𝐼𝐼𝐼𝐼𝑑𝑑²𝐸𝐸𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜 ≠ 𝑆𝑆𝛽𝛽𝐼𝐼𝐼𝐼𝑑𝑑²𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸
F=
𝑆𝑆𝑡𝑡𝐸𝐸𝑊𝑊𝐶𝐶𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃2 𝑆𝑆𝑡𝑡𝐸𝐸𝑊𝑊𝐶𝐶𝐼𝐼𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖2 -T-test: 𝐻𝐻0: 𝑀𝑀𝐼𝐼𝛽𝛽𝐼𝐼 𝑅𝑅𝐼𝐼𝛽𝛽𝑅𝑅𝑃𝑃𝐼𝐼𝐸𝐸𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜= 𝑀𝑀𝐼𝐼𝛽𝛽𝐼𝐼 𝑅𝑅𝐼𝐼𝛽𝛽𝑅𝑅𝑃𝑃𝐼𝐼𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸; 𝐻𝐻1: 𝑀𝑀𝐼𝐼𝛽𝛽𝐼𝐼 𝑅𝑅𝐼𝐼𝛽𝛽𝑅𝑅𝑃𝑃𝐼𝐼𝐸𝐸𝐶𝐶𝐶𝐶𝑡𝑡𝑜𝑜 ≠ 𝑀𝑀𝐼𝐼𝛽𝛽𝐼𝐼 𝑅𝑅𝐼𝐼𝛽𝛽𝑅𝑅𝑃𝑃𝐼𝐼𝐶𝐶𝐶𝐶𝐸𝐸𝑊𝑊𝐸𝐸t=
𝑅𝑅𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃−𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖 �𝑆𝑆𝑎𝑎𝑖𝑖𝑎𝑎𝑆𝑆𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃2 𝑖𝑖𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃 −𝑆𝑆𝑎𝑎𝑖𝑖𝑎𝑎𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖 2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖if the variances are not equal and
t=
𝑅𝑅𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃−𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖�𝑆𝑆𝑡𝑡𝐸𝐸𝑊𝑊𝐶𝐶2(𝑖𝑖𝑃𝑃𝑃𝑃𝑀𝑀𝑎𝑎𝑃𝑃1 −𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑎𝑎𝑖𝑖1 ) if the variances are equal, where n is the sample size and R the mean return (Keller, 2012).
5. Results
From table 1 it can be seen, as it was to be expected, that the correlation coefficients of the S&P500 and the S&P Europe 350 with the world index are relatively high. This tends to show that both the US stock market and the European stock market are significantly integrated to the world market. The betas of those markets to the world market confirm this intuition, both with R² significantly high. On the other hand, the correlation coefficients and betas of the African stock indices tend to be significantly lower, except for the MSCI South Africa index and the S&P Pan Africa index. There is no surprise in such findings since the Nigerian and the Kenya stock markets, as their status of frontier market indicates, are relatively small, characterized by high volatility and not expected to have strong linkages to the world stock
market. The same can be said of the MSCI Frontier Markets Africa index which is composed of stocks from African frontier markets such as Nigeria and Kenya. However, the high correlation levels and betas
of the MSCI South Africa and of the S&P Pan Africa might look surprising at the first glance. It might actually translate the fact that the South African stock market is considerably larger, more developed and thus globally more integrated than its Sub-Saharan counterparts. Moreover, the S&P Pan Africa index, unlike the MSCI Frontier Markets Africa, also contains stocks from more mature African markets such as Egypt, Morocco, Tunisia and South Africa, which all tend to be more globally integrated. Nevertheless, the R² coefficients for the MSCI South Africa and for the S&P Pan Africa are much lower than those of the S&P500 and the S&P Europe 350, suggesting that the later indices display stronger linkages to the world market.
Furthermore, in table 2, the correlations coefficients of the African country indices with the S&P 500 show a very low degree of co-movement. Again, the betas and respective R² confirm this intuition. The African index indicators contrast with those of the S&P Europe 350 index, which appears strongly linked to the US index as one could expect. Although it is once more an exception amongst African indices, the MSCI South Africa index has lower beta and correlation coefficient to the US index than the European index.
Those primary results tend to confirm that Sub Saharan stock markets can offer diversification opportunities to international investors.
Table 1: Correlation and Beta to the World Index (MSCI World)
Correlation Beta* R² for Beta
S&P500 95,47% 0,920 t-test: 51,71 => reject 𝐻𝐻0 0,912 S&PEURO350 93,46% 1,322 t-test: 37,80 => reject 𝐻𝐻0 0,847
MSCI Frontier Africa 15,50% 0,226
t-test: 3,127=> reject 𝐻𝐻0
0,037
S&P PAN Africa 81,35% 0,984
t-test: 22,32 => reject 𝐻𝐻0
0,659
MSCI South Africa 81,92% 1,238
t-test: 22,81 => reject 𝐻𝐻0 0,668 MSCI Nigeria 9,07% 0,149 t-test: 1,46 => do not reject 𝐻𝐻0 0,0081 MSCI Kenya 24,47% 0,303 t-test: 4,08=> reject 𝐻𝐻0 0,0607
*The values of the t-test are returned by the regression Anova table in Excel, for 𝐻𝐻0: 𝛽𝛽 = 0. The degree of freedom
Table 2: Correlation and beta to the US index (S&P 500)
Correlation Beta* R² for Beta
S&PEURO 350 83,50% 1,292
t-test: 24,27 => reject 𝐻𝐻0
0,695
MSCI South Africa 74,51% 1,158
t-test: 17,82 => reject 𝐻𝐻0 0,552 MSCI Nigeria 7,25% 0,122 t-test: 1,14=> do not reject 𝐻𝐻0 0,0050 MSCI Kenya 22,003% 0,283 t-test: 3,65 => reject 𝐻𝐻0 0,0491
*The values of the t-test are returned by the regression Anova table in Excel, for 𝐻𝐻0: 𝛽𝛽 = 0. The degree of freedom is 258 (number of observations – 2).The significance level is 5%. The critical values are between -1.960 and 1.960.
From table 3, it can be noted that both the Sharpe and the Sortino ratios of the portfolio 1 are
significantly greater than those of the MSCI World index. This indicates that diversifying the portfolio by including the African stock indices in proportions that maximize the Sharpe or the Sortino ratio is beneficial for the investor. On the other hand, the naïve portfolio that assigns equal weight to all assets provides a Sharpe ratio significantly lower and a Sortino ratio not significantly different than those of the MSCI world, tending to prove the superiority of the Markowitz optimization model. It can also be noted that although the standard deviations (i.e. the risk) are lower and the mean returns are higher when maximizing the ratios compare to the world index, those differences appear not to be statistically significant. Indeed the F test for different variances and the t test for different mean return for both strategies result in not rejecting the null hypothesizes that variances and returns are statistically equal. This might be due to the fact that the samples were composed of only 23 data (i.e. from Q2 2008 to Q4 2013). In the case of the variances, it might also indicate that the addition of highly volatile assets increases overall risk, even though the risk relative to return decreases.
Table 3: Results of the Jobson-Korkie tests, F-tests and t-tests: Portfolio 1 vs MSCI World
Sharpe* Sortino* Stdev** Average Return***
MSCI World 0,118 0,388 0,00796 0,0790% Portfolio maximizing Sharpe 0,278 𝛽𝛽𝐽𝐽−𝐾𝐾= -8,024 => reject 𝐻𝐻0 0,00939 F=0,719=> do not reject 𝐻𝐻0 0,278% t=-0,774 => do not reject 𝐻𝐻0 Portfolio maximizing Sortino 0,937 𝛽𝛽𝐽𝐽−𝐾𝐾= -2,425 => reject 𝐻𝐻0 0,00920 F= 0,748=> do not reject 𝐻𝐻0 0,303% t= -0,881 => do not reject 𝐻𝐻0 Naïve Portfolio 0,0874 𝛽𝛽𝐽𝐽−𝐾𝐾= 4,585=> reject 𝐻𝐻0 0,273 𝛽𝛽𝐽𝐽−𝐾𝐾= 0,910 => do not reject 𝐻𝐻0 0,00978 F= 0,662=> do not reject 𝐻𝐻0 1,6259E-03% t= 0,294438803=> do not reject 𝐻𝐻0
*The test is the Jobson-Korkie test for equality of the ratios. For 23 degrees of freedom, the critical values at 5% significance level are between -1,714 and 1,714. **The test is the F test for equality of the variances. For 22 degrees of freedom for the denominator and the numerator, the critical values at 5% significance level are between 0,573 and 1,744.*** The test is the t-test for equality of the ratios. The degree of freedom for equal variances is 44. The critical values at 5% significance level are between -1,680 and 1,680.
From table 4, it can be seen that adding the African country indices to the S&P 500 results in significantly higher Sharpe and Sortino ratios when those ratios are maximized. Moreover, the average returns are similarly found to be higher for those strategies. However, as it was the case with portfolio 1, the F test for equal variances indicates that the null hypothesizes should not be rejected at 5% significance level. One could explain the latter result by the fact that adding country indices with high volatility, although helping in diversifying away some risk relative to the portfolio return, do not significantly reduce the overall risk of the portfolio. As for portfolio 1, the naïve strategy performs poorly, confirming the
relevance of the Markowitz optimization theory. Indeed, selecting low correlated assets to be added to a portfolio is not enough per se. One must assign proportions within the portfolio by maximizing the Sharpe or the Sortino ratio using the mean-variance method in order to achieve higher efficiency level.
Table 4 Results of the Jobson-Korkie tests, F-tests and t-tests: Portfolio 2 vs S&P500
Sharpe* Sortino* Stdev** Average Return***
S&P 500 0,0964 0,338 0,00639 0,169% Portfolio maximizing Sharpe 0,427 𝛽𝛽𝐽𝐽−𝐾𝐾= -14,026=> reject 𝐻𝐻0 0,00809 F=0,624=> do not reject 𝐻𝐻0 0,703% t=-2,931=> reject 𝐻𝐻0 Portfolio maximizing Sortino 1,186 𝛽𝛽𝐽𝐽−𝐾𝐾= -6,935=> reject 𝐻𝐻0 0,00817 F= 0,611=>do not reject 𝐻𝐻0 0,714% t= -2,971=> reject 𝐻𝐻0 Naïve Portfolio 0,119 𝛽𝛽𝐽𝐽−𝐾𝐾= 1,744 => reject 𝐻𝐻0 0,336 𝛽𝛽𝐽𝐽−𝐾𝐾= -0,4199 => do not reject 𝐻𝐻0 0,00864 F= 0,547 => reject 𝐻𝐻0 0,176% t= -0,0386=> do not reject 𝐻𝐻0
*The test is the Jobson-Korkie test for equality of the ratios. For 32 degrees of freedom, the critical values at 5% significance level are between -1,694 and 1,694. **The test is the F test for equality of the variances. For 31 degrees of freedom for the denominator and the numerator, the critical values at 5% significance level are between 0,5488 and 1,822.*** The test is the t-test for equality of the ratios. The degree of freedom for equal variances is 62. The critical values at 5% significance level are between -1,670 and 1,670. The degree of freedom for unequal variances is 57. The critical values at 5% significance level are between -1,672 and 1,672.
Table 5 and table 6 show the average proportions of each index within the 2 portfolios. It is interesting to note that the S&P Pan Africa index received a lower weight than the MSCI Frontier Markets Africa index. This tends to corroborate our intuition since we identified the S&P Pan Africa index as being more correlated to the world index than the MSCI Frontier Markets Africa index due to the relatively more mature stock markets he is composed of. On the other hand, the MSCI South Africa, despite its higher correlation and beta to the S&P 500, is assigned a proportion of roughly the same order of magnitude as the other 2 indices. This suggests that the correlation level and the beta must be used carefully and only as an indicator when considering diversification opportunities. Finally, it can also be noted that the strategy maximizing the Sharpe ratio and the strategy maximizing the Sortino ratio return roughly similar proportions for both portfolios, with a slightly higher average return for the Sortino strategy.
Table 5 Average weights and returns Portfolio1
MSCI World MSCI Frontier Africa S&P Pan Africa Average Portfolio
Return Maximizing Sharpe ratio 45,58% 39,82% 14,60% 0,278% Maximizing Sortino ratio 50,06% 39,81% 10,13% 0,303%
Naïve portfolio 33,33% 33,33% 33,33% 1,62591E-03%
Table 6 Average weights and returns Portfolio2
S&P500 MSCI South Africa MSCI Nigeria MSCI Kenya Average Portfolio
Return Maximizing Sharpe ratio 28,44% 19,32% 26,26% 25,98% 0,703% Maximizing Sortino ratio 27,74% 24,79% 21,64% 25,83% 0,714% Naïve portfolio 25% 25% 25% 25% 0,176% 6. Conclusion
In this paper, the intuition that Sub-Saharan stock markets can provide diversification opportunities to international investors was confirmed. Indeed the relatively low correlations and low betas computed suggest that stock markets in the region are yet relatively segmented from the world market. Moreover, some measures of the opportunities of diversification were provided and tested through 2 portfolios designed according to the Markowitz mean-variance optimization theory. The proportions of the different indices within the portfolios were obtained by optimization of the Sharpe ratio and the Sortino ratio, while rebalancing quarterly over periods of 6 and 8 years. The first portfolio was composed of a world index and 2 African continent indices, while the second portfolio was composed of a US index and 3 African country indices. The portfolios were tested against respectively the World index and the S&P 500, as well as against the naïve portfolios of equal
proportions. Through Jobson-Korkie tests applied on the Sharpe and Sortino ratios, it was shown that both portfolios outperformed their respective reference indices when the optimization
strategies were followed, whereas the naïve portfolios performed poorly. These results suggest that including Sub-Saharan stocks into a diversified portfolio according to the mean-variance optimization theory is beneficial for international investors, therefore tending to strengthen the empirical findings of previous works such as those of Lagoarde-Segot and Lucey.
In the light of those findings, we can draw the conclusion that there is some evidence that Sub Saharan stock markets indeed provide significant diversification opportunities for international investors. Authorities in those countries should therefore continue their effort towards improving financial and legal institutions to further expand their markets in an attempt to benefit of the positive spillovers on real economic growth, while international investors should be encouraged to include Sub-Saharan equity stocks to their investment portfolios.
Taking this into consideration, the questions of the impact of the development of financial markets on the real economy, the dynamic of integration to the world market and regionally, country risks and transactions costs specific to the Sub Saharan countries might constitute relevant related subjects for further investigations.
7. References
About. Com US Economy. (2014). MSCI Index. Retrieved from http://useconomy.about.com/od/glossary/g/msci.htm
Alagidede, P., (2011), Return behaviour in Africa’s emerging equity markets, The Quarterly Review of Economics and Finance, 51, pp. 133-140.
Appiah-Kusi, J., & Menyah, K., (2003), Return predictability in African stock markets, Review of Financial Economics, 12, pp. 247-270.
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8. Appendix
a) Excel Formula used
Variance-Covariance Matrix:
MMULT(TRANSPOSE(𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝐶𝐶𝑊𝑊𝑡𝑡𝑅𝑅𝐶𝐶𝐶𝐶𝑎𝑎-𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝑎𝑎𝐶𝐶𝑊𝑊𝐶𝐶𝑎𝑎𝑎𝑎𝑊𝑊); 𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝐶𝐶𝑊𝑊𝑡𝑡𝑅𝑅𝐶𝐶𝐶𝐶𝑎𝑎-𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝑎𝑎𝐶𝐶𝑊𝑊𝐶𝐶𝑎𝑎𝑎𝑎𝑊𝑊)/n
Portfolio return:
MMULT(TRANSPOSE(𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝑤𝑤𝑊𝑊𝐶𝐶𝑎𝑎ℎ𝑡𝑡); 𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝑎𝑎𝐶𝐶𝑊𝑊𝐶𝐶𝑎𝑎𝑎𝑎𝑊𝑊)
Portfolio Standard deviation:
SQRT(MMULT(TRANSPOSE(𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝑤𝑤𝑊𝑊𝐶𝐶𝑎𝑎ℎ𝑡𝑡);MMULT(𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝑉𝑉𝑎𝑎𝐶𝐶−𝐶𝐶𝐶𝐶𝐶𝐶; 𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝑤𝑤𝑊𝑊𝐶𝐶𝑎𝑎ℎ𝑡𝑡)))
where 𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝐶𝐶𝑊𝑊𝑡𝑡𝑅𝑅𝐶𝐶𝐶𝐶𝑎𝑎 is the matrix of the individual indices weekly returns for the quarter considered, 𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝑎𝑎𝐶𝐶𝑊𝑊𝐶𝐶𝑎𝑎𝑎𝑎𝑊𝑊 is the matrix of the individual indices average returns for the quarter, n is the number of periods namely the number of weeks within the given quarter, 𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝑤𝑤𝑊𝑊𝐶𝐶𝑎𝑎ℎ𝑡𝑡 is the matrix of the indices weights within the portfolio and 𝑀𝑀𝛽𝛽𝛽𝛽𝑃𝑃𝑃𝑃𝐼𝐼𝑉𝑉𝑎𝑎𝐶𝐶−𝐶𝐶𝐶𝐶𝐶𝐶 is the Variance-Covariance matrix as computed using the aforementioned formula.
b) Summarizing tables Portfolio 1
Returns
MSCI World Portf1-Max-Sharpe Portf1-Max-Sortino Portf1-Naïve
Q2 2008 -0,182% -0,263% -0,701% -0,382% Q3 2008 -0,836% -1,466% -0,836% -1,203% Q4 2008 -2,174% -2,174% -2,174% -2,788% Q1 2009 -0,599% -0,599% -0,599% -1,391% Q2 2009 1,022% 1,546% 1,530% 1,736% Q3 2009 1,266% 1,286% 1,269% 0,790% Q4 2009 0,300% 0,300% 0,300% 0,052% Q1 2010 0,188% 1,114% 1,054% 0,600% Q2 2010 -0,680% -0,061% -0,061% -0,341% Q3 2010 0,648% 1,071% 1,071% 0,527% Q4 2010 0,622% 0,665% 0,681% 0,689% Q1 2011 0,362% 0,362% 0,362% -0,045% Q2 2011 -0,007% 0,092% 0,092% -0,001% Q3 2011 -1,188% -1,188% -1,188% -1,462% Q4 2011 0,391% 0,529% 0,541% 0,385% Q1 2012 0,801% 0,687% 0,736% 0,671% Q2 2012 -0,634% 0,177% 0,177% -0,344% Q3 2012 0,757% 1,648% 1,587% 1,073% Q4 2012 0,111% 0,580% 0,711% 0,397% Q1 2013 0,552% 0,932% 1,275% 0,483% Q2 2013 0,027% 0,027% 0,027% -0,284% Q3 2013 0,633% 0,640% 0,633% 0,532% Q4 2013 0,439% 0,478% 0,474% 0,345% Mean 0,079% 0,278% 0,303% 0,002% Stdev 0,00795877 0,009388818 0,009201394 0,009782169
Cov w/ MSCI W 6,88064E-05 6,72709E-05 7,31829E-05
Rat
ios
MS
CI W
Sha
rpe
MS
CI W
Sor
tino
Sha
rpe
-po
rtf1
Sor
tino-por
tf1
Naï
ve S
har
pe
Naïv
e So
rtin
o
MS
CI W
Dw
Ris
kP
ortf
1 Dw
Ris
k
Naïv
e D
w R
isk
Q2 2008
-0,124648806
-0,259854629
-0,160308093
-0,329030826
-0,256439474
-0,372829699
0,011081636
0,02454542
0,013097751
Q3 2008
-0,516617565
-0,593687892
-0,461469239
-0,593687892
-0,675924265
-1,247434942
0,016460204
0,016460204
0,010781493
Q4 2008
-0,25195692
-0,377175949
-0,25195692
-0,377175949
-0,467685434
-0,715051873
0,05934009
0,05934009
0,039889999
Q1 2009
-0,109362658
-0,181881397
-0,109362658
-0,181881397
-0,316447249
-0,474925963
0,033417839
0,033417839
0,029469093
Q2 2009
0,445156528
0,896186019
0,696051774
2,06287222
0,662951348
1,570137074
0,011215733
0,007334798
0,010951389
Q3 2009
0,465506539
1,459681156
0,492182016
1,462657682
0,295160302
0,577007924
0,008582511
0,008586395
0,013463522
Q4 2009
0,160157315
0,312714518
0,160157315
0,312714518
0,022981097
0,039943016
0,009340591
0,009340591
0,011166529
Q1 2010
0,081167265
0,13887821
0,551242495
1,304018074
0,324958765
0,737180638
0,013209064
0,008047384
0,008070813
Q2 2010
-0,198545807
-0,260500662
-0,02348948
-0,038105826
-0,126334512
-0,182013906
0,026574233
0,019289495
0,019413228
Q3 2010
0,234180595
0,398573226
0,401357153
0,74058393
0,232710762
0,380119161
0,015929167
0,014290056
0,013514015
Q4 2010
0,444565394
1,321244204
0,528872416
1,468264626
0,525125212
1,360428735
0,004611146
0,004551837
0,004975736
Q1 2011
0,178150083
0,308707285
0,178150083
0,308707285
-0,027061018
-0,05123539
0,011337252
0,011337252
0,010957406
Q2 2011
-0,006932944
-0,01380252
0,053574219
0,08708331
-0,003522524
-0,00667037
0,008964115
0,009947752
0,009585596
Q3 2011
-0,31665341
-0,442731274
-0,31665341
-0,442731274
-0,50051166
-0,68891337
0,026869999
0,026869999
0,021248676
Q4 2011
0,098043647
0,227331391
0,258278805
0,642247637
0,13192902
0,31880675
0,017144168
0,008392614
0,012037517
Q1 2012
0,763588124
2,261191035
1,010655542
2,553186168
0,64273208
1,366144866
0,00353617
0,002876147
0,004899642
Q2 2012
-0,40192717
-0,48451431
0,085070926
0,189319465
-0,215270221
-0,356633692
0,013214967
0,009036235
0,009819394
Q3 2012
0,359631677
0,853116141
1,072066211
4,772031343
0,560775078
1,494836875
0,008785536
0,00331054
0,007124256
Q4 2012
0,053856585
0,094533357
0,361003229
0,962770519
0,280623374
0,504029875
0,010961857
0,007310547
0,007733086
Q1 2013
0,587014716
1,23938017
0,913242756
4,383778102
0,488866863
1,046851097
0,004419792
0,002898615
0,004576864
Q2 2013
0,010182755
0,017013516
0,010182755
0,017013516
-0,17881856
-0,30340395
0,012293468
0,012293468
0,012012924
Q3 2013
0,425889017
1,230064641
0,426288245
1,230064641
0,321505694
0,726236671
0,005119211
0,005119211
0,007279074
Q4 2013
0,329062362
0,769141839
0,525562529
1,01105475
0,287333538
0,545352148
0,005690813
0,004672139
0,006293936
Me
an
0,117804666
0,387548177
0,278291246
0,93677194
0,087375575
0,272520073
0,014699981
0,013446462
0,012537476
Portfolio 2
Returns
S&P500 Portf2-Max-Sharpe Portf2-Max-Sortino Portf2-Naïve
Q1 2006 0,291% 1,290% 1,133% 0,477% Q2 2006 -0,137% 1,128% 0,977% 0,029% Q3 2006 0,400% 1,296% 1,208% 0,839% Q4 2006 0,465% 1,070% 1,829% 0,880% Q1 2007 0,031% 2,346% 2,337% 0,652% Q2 2007 0,443% 1,218% 1,254% 0,732% Q3 2007 0,141% 0,495% 0,495% 0,234% Q4 2007 -0,223% 1,147% 0,991% 0,490% Q1 2008 -0,854% -0,384% -0,384% -0,549% Q2 2008 -0,181% 1,140% 1,140% 0,056% Q3 2008 -0,389% -1,377% -1,276% -1,357% Q4 2008 -2,179% -1,217% -1,217% -2,318% Q1 2009 -0,523% 0,224% 0,224% -1,413% Q2 2009 0,791% 1,397% 1,686% 1,754% Q3 2009 1,060% 0,913% 1,069% 0,638% Q4 2009 0,441% 0,441% 0,441% 0,102% Q1 2010 0,389% 1,115% 1,179% 0,794% Q2 2010 -0,576% 0,559% 0,548% -0,112% Q3 2010 0,469% 1,352% 1,355% 0,367% Q4 2010 0,758% 0,856% 0,733% 0,740% Q1 2011 0,419% 0,419% 0,419% -0,125% Q2 2011 -0,018% 0,106% 0,106% -0,037% Q3 2011 0,672% -0,923% -0,923% -1,754% Q4 2011 0,704% 0,978% 0,946% 0,748% Q1 2012 0,817% 0,872% 0,899% 0,816% Q2 2012 -0,407% 0,481% 0,421% -0,086% Q3 2012 0,672% 1,926% 1,786% 1,063% Q4 2012 -0,136% 0,724% 0,788% 0,469% Q1 2013 0,787% 1,182% 1,011% 0,820% Q2 2013 0,230% 0,230% 0,230% -0,230% Q3 2013 0,410% 0,889% 0,820% 0,548% Q4 2013 0,633% 0,599% 0,615% 0,367% Mean 0,169% 0,703% 0,714% 0,176% Stdev 0,006390818 0,008090755 0,008172566 0,008640155
Rat ios SP 500 S ha rpe SP 500 S ort ino Sha rpe -po rtf 2 Sor tin o-p ort f2 Sh arp e N aïv e Sor tin o N aïv e SP 500 D w R isk Po rtf 2 Dw Ri sk Na ïve Dw Ri sk Q1 2006 -0, 010791541 -0, 023609271 0,296181747 0,748285697 0,107108273 0,242059844 0,006561463 0,011033192 0,007014014 Q2 2006 -0, 316202995 -0, 464620775 0,325659379 1,088733217 -0, 167194676 -0, 252869896 0,010473663 0,005763411 0,01268945 Q3 2006 0,011582947 0,027209722 0,605012095 1,231764625 0,245308491 0,486702928 0,006717566 0,00671204 0,009391947 Q4 2006 0,116105465 0,252018392 0,609335214 1,670298789 0,350434897 0,82175317 0,003926728 0,008756262 0,00625191 Q1 2007 -0, 180075012 -0, 288434532 0,506210691 2,302059613 0,094103982 0,164173934 0,011968797 0,008515713 0,01681315 Q2 2007 0,050685212 0,090443114 0,614119919 1,883733174 0,29265063 0,64263344 0,007180975 0,004653503 0,005511906 Q3 2007 -0, 106571507 -0, 165838167 0,025045053 0,040152686 -0, 077645913 -0, 126327342 0,013838168 0,031116752 0,010755256 Q4 2007 -0, 23238405 -0, 353650524 0,410822678 1,047806535 0,13755553 0,240841121 0,014649767 0,00664136 0,008122822 Q1 2008 -0, 365308817 -0, 549281312 -0, 10261883 -0, 178874704 -0, 318824109 -0, 467229792 0,020146181 0,035635877 0,017158594 Q2 2008 -0, 100957063 -0, 190722254 0,32032425 0,638728773 -0, 027635014 -0, 03960239 0,015060267 0,016191795 0,012584669 Q3 2008 -0, 303073903 -0, 437735186 -0, 346626498 -0, 47361502 -0, 859700004 -1, 390284383 0,012114206 0,029929705 0,010780329 Q4 2008 -0, 27279469 -0, 4161851 -0, 085675365 -0, 18980734 -0, 370393776 -0, 645912566 0,053920189 0,067513539 0,036884337 Q1 2009 -0, 092671735 -0, 156178858 0,030278405 0,053037344 -0, 300033453 -0, 528731596 0,03402528 0,040550135 0,026903251 Q2 2009 0,362986772 0,855787351 1,016429044 2,617360771 0,806329927 1,947950604 0,00905019 0,006376285 0,008916296 Q3 2009 0,37741874 1,02084647 0,493511375 1,443858548 0,251942917 0,507805194 0,010259908 0,007311912 0,012302632 Q4 2009 0,235070119 0,534607669 0,235368237 0,535285663 0,051444326 0,088803036 0,008104062 0,008104062 0,010632897 Q1 2010 0,191890293 0,333083204 0,678651672 1,663872622 0,47885699 1,178855594 0,011545813 0,007052965 0,006689122 Q2 2010 -0, 173159635 -0, 221671572 0,274420123 0,631679586 -0, 048600913 -0, 077097136 0,026536089 0,008484341 0,01618225 Q3 2010 0,173208419 0,297596747 0,406531721 0,775820148 0,160199006 0,258144698 0,015311942 0,01731302 0,013763271 Q4 2010 0,651022227 1,826502917 0,756136503 2,053410332 0,580292723 1,499996538 0,004080623 0,003506822 0,004847785 Q1 2011 0,236975131 0,397005535 0,236975131 0,397005535 -0, 059154682 -0, 111908688 0,010269451 0,010269451 0,012172617 Q2 2011 -0, 014381698 -0, 029449052 0,058016208 0,10587348 -0, 022131086 -0, 0375708 0,007940958 0,009487226 0,011188511 Q3 2011 0,177912194 0,257065048 -0, 245384171 -0, 354555201 -0, 631196397 -0, 835336633 0,026082073 0,026082073 0,021016624 Q4 2011 0,211726807 0,527752852 0,482409064 1,137014926 0,311242655 0,798163144 0,013308662 0,008302281 0,009355874 Q1 2012 0,863095305 3,341361938 1,848110188 4,289723743 0,839449506 1,712652062 0,002441785 0,002093279 0,004755011 Q2 2012 -0, 275564849 -0, 357852912 0,218999052 0,421068791 -0, 056048815 -0, 104894182 0,011538369 0,009847934 0,008813128 Q3 2012 0,3679935 0,902852342 1,079194277 5,204298295 0,617436379 2,020272188 0,00735807 0,003416738 0,00522297 Q4 2012 -0, 069922518 -0, 110356633 0,738810173 1,773692139 0,376357465 0,683954166 0,012984796 0,003416738 0,00675951 Q1 2013 0,707274471 1,764204281 0,944856217 2,854017007 0,586988925 1,108483664 0,004441706 0,003528573 0,007366638 Q2 2013 0,116384041 0,1858415 0,116384041 0,1858415 -0, 1096118 -0, 189715534 0,012055677 0,012055677 0,012455833 Q3 2013 0,254627737 0,584802703 0,416602109 0,85278113 0,314036382 0,671734541 0,006951181 0,009578629 0,00811773 Q4 2013 0,494633885 1,372082948 0,685405993 1,507177859 0,268467854 0,470353329 0,004599454 0,004072348 0,007777198 Me an 0,096460414 0,337671206 0,426546741 1,186172821 0,119438632 0,335557883 0,012982627 0,013541051 0,011537423
