Quantitative Analysis of Sub- Saharan Stock Markets, from a Danish Perspective

Transcription

1 Date: 1 st of May 2014 Quantitative Analysis of Sub- Saharan Stock Markets, from a Danish Perspective Author Sofus Asboe Jørgensen Bachelor student Institute of Economics Aarhus University (DK) Student- id Academic supervisor Kim Christensen Associate Professor Institute of Economics Aarhus University (DK) Number of characters:

2 Abstract This study examines if Sub- Saharan stock markets could be a good investment for a Danish investor, and it also tries to determine which Sub- Saharan stock market will make the best investment candidate. The theoretical part of the study is based on Hillier, Grinblatt & Titman s textbook about the subject of portfolio theory and management. The analysis in the study is mainly based on empirical data from Bloomberg, which is derived from a Bloomberg Terminal at Aarhus University. The study selects four Sub- Saharan stock markets based on size and liquidity. The four biggest and most liquid markets in the Sub- Saharan region are: South Africa, Nigeria, Zimbabwe and Kenya. The study also concludes that the average Sub- Saharan stock market is small and illiquid. A theoretical framework about portfolio theory is explained, and empirical data is applied to evaluate the stock markets under different assumptions. All investigations pointed towards South Africa, Zimbabwe and Nigeria being beneficial investment candidates for a Danish- investor, even after adjusting for foreign- exchange rate risk. The study shows that Zimbabwe and Nigeria recorded very low market- correlation with other markets, and consequently delivers a possible diversification- option, but they also experienced liquidity issues. The final conclusion is that at the present time, the South African stock market is the best investment- option in Sub- Saharan Africa. Keywords: Sub- Saharan stock markets, Modern Portfolio Theory, Tangency Portfolio, Foreign- Exchange rate risk, Bloomberg.

3 Indholdsfortegnelse 1. INTRODUCTION STRUCTURE OF THE STUDY METHOD DELIMITATIONS AFRICAN STOCK MARKETS AN INTRODUCTION CHOICE OF MARKET TO INVESTIGATE MARKET CAPITALIZATION SIZE STOCKS TRADED LIQUIDITY PORTFOLIO MANAGEMENT AND THEORY THE EFFICIENT FRONTIER THE CAPITAL MARKET LINE AND THE TANGENCY PORTFOLIO THE OPTIMAL PORTFOLIO THE MINIMUM VARIANCE PORTFOLIO AND DIVERSIFICATION Theoretical example of the minimum variance portfolio Diversification EMPIRICAL DATA ANALYSIS OF SUB- SAHARAN STOCK MARKETS ANNUAL RETURNS STANDARD DEVIATION / RISK REWARD- TO- RISK RATIO ANALYSIS CORRELATION MODERN PORTFOLIO THEORY APPLIED TO THE SUB- SAHARAN AND DANISH MARKET The Efficient Frontier The Capital Market Line and the Tangency Portfolio LIMITATIONS OF SUB- SAHARAN STOCK MARKETS FOREIGN- EXCHANGE RATE RISK ZAR/DKK NGN/DKK USD/DKK KES/DKK The optimal portfolio Adjusting for the foreign- exchange market Individual Tangency Portfolios LIQUIDITY OF MARKETS CONCLUSION REFERENCES

4 1. Introduction Today s economy enables all investors to branch out and invest in almost every foreign stock market, including stock markets in Sub- Sahara. New markets present new opportunities within fields such as return and correlation, among others. Still some markets are more beneficial than others. The degree of the beneficial effect depends on the performance of a given market, but also on a market s relationship with the investor s home market. This study will examine if Sub- Saharan stock markets can be beneficial for a Danish stock- investor. The actual overall research question is: Does the Sub- Saharan region contain stock markets that would be a good investment for a Danish investor? And which markets would be the best option? These questions will be the focal point throughout the study, and a quantitative analysis will be applied in order to answer these questions. 1.1 Structure of the study The study will begin in section 2, by giving a quick overview of African stock markets. The rest of the study will be split into four parts: A choice of Sub- Saharan markets, a presentation of portfolio theory, a quantitative analysis of the chosen stock markets and limitations of Sub- Sahara stock markets. Section 3 will examine the size and liquidity of all Sub- Saharan markets, in order to select the best candidates in the region. Size and liquidity will be based on market capitalization and turnover ratio, respectively. Section 4 will present a theory about general portfolio management. The theoretical framework applied in this study is the Modern Portfolio Theory. The section will explain the theory and model, but also present the formulas that are needed to calculate the different elements in the model. An explanation of the Efficient Frontier, the Tangency Portfolio and Diversification are all included in this section. Section 5 is an examination of empirical data from the Danish market and the four chosen Sub- Saharan markets. The empirical analysis will also involve a comparison with well- 2

5 established and emerging markets, in the form of the U.S. based S&P500, and the Brazilian market. The section will include an analysis of annual returns, standard deviation, reward- to- risk ratio and correlation. All this will lead up to the construction of the Efficient Frontier and the Tangency Portfolio. Every calculation, in section 5, is under the assumption that foreign- exchange risk is excluded, in order to illustrate the performance of the stock markets. Section 6 will examine some of the specific limitations in the Sub- Saharan markets. It will look at two different issues. The first- one is an examination of the foreign- exchange market s impact on the theoretical framework and the optimal portfolio. The second- one is an examination of the liquidity- degree in Sub- Saharan markets compared with the Danish, Brazilian and the American stock markets. All the analyzed elements will lead to a final conclusion that will try to deliver a conclusive answer to the research questions. 1.2 Method The analytical parts of the study are based on two main elements: Market- data and portfolio theory. Market- data is used when analyzing the different stock markets, and for the calculation of the optimal portfolios. The data is gathered from two different databases. A set of prices for different markets- indexes is gathered from Bloomberg Terminals. The prices are used to compute the elements in the theoretical framework, such as the Efficient Frontier and Tangency Portfolio. The used prices are the price of the index on the last annual trading day, and it is collected over a period of 10 years. These prices are used to compute the annual returns, which work as the basis for the rest of the analysis. Bloomberg also delivers the data for the analysis of the foreign- exchange market. The data is solely conversion spot rate between the Danish Kroner and the currencies used for trading in the four Sub- Saharan markets. The data is recorded at the last annual trading day over a 10- year period. From this data the annual appreciation of the DKK, against different currencies, can be computed. Bloomberg is applied because it is the world- leader in delivering financial data ( The other data source is S&P Global Stock Market Factbook via Indexmundi.com. This source is used to gather data about market capitalization and turnover ratio. 3

6 The section about portfolio theory is mainly based on specific chapters in an economical textbook called Financial Markets and Corporate Strategy (Reference 3). The theory is explained and applied to the specific case. At the beginning of each section, the study will deliver a main reference for the following section. More direct references will be posted as brackets inside the text. 1.3 Delimitations The study will not aim to analyze individual stocks from the different stock markets. Instead it will concentrate on analyzing stock markets as a whole. The study will not try to make an in- depth analysis of why stock markets performed as they did. Instead it will focus on examining, which markets performed best based on the data. The study will only examine stock markets in Sub- Sahara Africa. Sub- Sahara is defined as African countries south of the Saharan dessert. Further more the study will limit its in- depth analysis to only four Sub- Saharan markets, in order to have a reasonable scope of investigation. 4

7 2. African stock markets an introduction This section will deliver an overview of the number and functions of African stock markets. This section will take a theoretical approach, and not examine the empirical macro- economical data. Later on, in section 3, the study will examine the Sub- Saharan region and select the most well- functioning for further analysis based on empirical data. The section is mainly based on: Amo Yartey & Komla Adjasi (2007). Almost every African country has established a stock exchange to list domestic companies. Most African countries have their own domestic exchange, but some countries have enrolled in collaboration with other countries, and listed domestic companies on a common stock exchange. The best example of this is the West African regional stock exchange, called BRVM, created by the West African Monetary Union. The West African Monetary Union is a union between African countries that are all French speaking, and more important share the same currency. The eight member states of BRVM are Togo, Senegal, Niger, Mali, Ivory Coast, Guinea- Bissau, Burkina Faso and Benin. Table 1 displays African stock exchanges ranged from date of founding, while figure 1 show the cumulative development in number of African stock exchanges. Country Founded Country Founded Country Founded Egypt 1883 Swaziland 1990 Tanzania 1998 South Africa 1887 Namibia 1992 Mozambique 1999 Morocco 1929 Zimbabwe 1993 Cameroon 2001 Kenya 1954 Zambia 1994 Cape Verde 2005 Nigeria 1960 Sudan 1994 Libya 2007 Tunisia 1969 Malawi 1995 Rwanda 2008 Mauritius 1988 Uganda 1997 Seychelles 2012 Botswana 1989 Algeria 1997 Somalia 2012 Ghana 1990 BRVM 1998 Table 1:The date of founding of African stock exchanges. Source: Wikipedia.com 5

8 Number of Stock exchanges Year Figure 1: The cumulative development in the number of African stock exchange since the first African stock exchange was founded in 1883 (Egypt). Based on data from table 1. Stock exchanges requires time to obtain a liquid and well- functioning stock market, and table 1 shows that a number of African countries have had a stock exchange for a substantial period of time, indicating that it should be possible to locate a well- functioning and attractive stock market in Africa. Figure 1 displays a large increase of African countries with stock exchanges between 1980 and But what are the general functions of stock exchanges? One of the clearest functions of a stock market is that it enables better capital mobilization. Capital mobilization is simply the allocation mobility of capital across boundaries. Stock markets enable private investors to invest/allocate capital in African countries via acquisitions of stocks or bonds. Consequently stock markets help the domestic listed firms by ensuring a lowering of the cost of mobilizing financial resources, and also allocating risk by sharing the risk in the market place. This will make it easier for listed firms to ensure funding for different value- creation projects, promoting growth at the listed firm level, which again leads to domestic economic growth of the given African country. Another function of a stock market is that it endorses private savings, by allowing the individual saver to locate and use their specific return- to- risk profile. Traditional- investing, investing in projects around the stock market, could result in a liquidity problem, because the investor would have to wait until the funded project pays off. The stock 6

9 market allows for a more liquidated investment, meaning that the investor is able to liquidate the investment at any time. This will allow the investor to better smooth out their consumption plan, and promote investing. Consequently stock market contra traditional investing can lead to a potential increase in investing and economic growth. Another function of stock markets is that they act as corporate governance, meaning that listed companies and management are monitored closer, due to the external pressure of investors. This will potentially result in a more stable and successful business, and thereby have a positive impact on the domestic economy. The above listed functions of establishing a stock market is valid under the assumption of a well- functioning and liquid stock market. This is not the case for all African stock markets, as will be shown later on. The conclusion is that a well- functioning stock market will promote growth in the national economy. Consequently an investment in African stock markets can be seen as a form of development assistants, which could present an added- value for investors in African markets. Also it is concluded that stock markets contra traditional investments, generally allow for more liquid and secure investments in Africa. 3. Choice of market to investigate Africa consists of multiple stock markets, as displayed in table 1. In order to facilitate an in- depth analysis throughout the study we will limit the scope of investigating to only four Sub- Saharan stock markets. The evaluation will be based on two criteria s: Market Capitalization and Stocks Traded. The study will limit its research to the Sub- Saharan market and also exclude stock market founded after 2000, because the Sub- Saharan markets established after 2000 are very small. The section found inspiration by the findings in: Moss & Thuotte (2013). 3.1 Market capitalization Size. Market Capitalization, also referred to as Market Cap, is a calculation that aims to display the size of a given firm or stock exchange. Market capitalization is calculated by multiplying the stock price and the number of shares at a given time ( exchanges.org). This 7

10 study will use market capitalization as indicator of stock market size, because market capitalization is a simple and assessable estimate. Sub- Saharan stock markets, in general, have a problem being small in size compared to other regions around the world. The average Sub- Saharan stock market had a market capitalization of approximately 55,8 billion USD in By comparison the Danish market capitalization was approximately 224,8 billion USD in 2012 ( - Reference 24). Table 2, below, displays the market capitalization from for each of the sub- Saharan stock markets. Mozambique is not included, due to insufficient data. In million USD Average 2012 Share of total market capitalization South Africa ,42% Nigeria ,77% Kenya ,04% Ghana ,48% Zambia ,41% Malawi ,10% Zimbabwe ,63% Mauritius ,98% Botswana ,63% Swaziland ,03% Namibia ,18% BRVM ,08% Tanzania ,25% Total ,00% Table 2: The market capitalization in million USD for each Sub- Saharan country. Also the table calculates the average market capitalization from Last the table shows each country s share of the total market capitalization in Source: S&P global stock market factbook via Indexmundi.com (Reference 11-23) Table 2 shows that the, by far, largest market in Sub- Sahara Africa is South Africa, which stands for 84,42% of the total Sub- Saharan market capitalization in Also the Nigerian stock market is clearly larger than the rest of the Sub- Saharan stock market, and it accounted for 7,77% of the total market capitalization in Consequently the South African and Nigerian markets account for approximately 92% of the total market capitalization in The rest of the Sub- Saharan countries are much closer, when looking at their share of the total 1 Calculated using the data given in table 2. 8

11 market capitalization in Figure 2, below, displays the share of total market capitalization for Sub- Saharan countries, when South Africa and Nigeria are excluded. % of total market cap 2.50% 2.00% Kenya Zimbabwe 1.50% 1.00% Mauritius Botswana BRVM 0.50% 0.00% Ghana Zambia Malawi Namibia Swaziland Tanzania Figure 2: Each country s share of the total market capitalization in South Africa and Nigeria are excluded. Source: S&P global stock market factbook via Indexmundi.com (Reference 11-23) Figure 2 show that the Sub- Sahara region contains a large number of very small markets compared to the total market capitalization of the region. Consequently the study will choose some of the largest markets in the region. The 3 largest markets, when excluding South Africa and Nigeria, are Kenya, Zimbabwe and the union of BRVM, which is clearly shown in figure 2. These are the three only markets with a share above 1% of total market capitalization in The 5 largest markets in 2012: South Africa, Nigeria, Kenya, Zimbabwe and BRVM accounted for approximately 97% of the total market capitalization in The conclusion is that South Africa, Nigeria, Kenya, Zimbabwe and the BRVM union were the largest market in the Sub- Saharan region in 2012, and they combined for an extremely large proportion of the total market capitalization. However the study will also evaluate the liquidity degree in each market, before choosing which markets to investigate further. 9

12 3.2 Stocks traded Liquidity Liquidity is an essential measure of stock market efficiency. Liquidity of stock markets refers to the degree of which an asset can be sold or bought in the marketplace. So in a stock market it refers to the situation surrounding the sale and purchase of a stock. A high level of liquidity is a good element in a stock market, since it will ensure both seller and purchaser that they are able to make their transaction. Consequently it is safer for investors to allocate their capital towards market with high liquidity, because the investor will be relatively sure of being able to convert the investment into capital in the end. Liquidity is a particularly large concern in Sub- Saharan stock markets, as they are generally very illiquid. So it will be important to choose the most liquid markets within this illiquid region of stock markets. As a result of this, the study will examine the liquidity degree of the Sub- Saharan stock markets, in order to compare and choose the most liquid markets. Liquidity does not have a direct measure, but different measures are used as indicators of liquidity within a marketplace. This study will use stocks traded as an indicator of liquidity. Stocks traded refer to the total value of all shares traded in a given period, for example This is only a nominal value, so it is difficult to compare the different Sub- Saharan markets using this measure, because it doesn t consider the size of each market. Consequently the study will use a measure that reflects the liquidity of each market, relative to the size of the market. So a new measure called Turnover ratio is constructed by dividing stocks traded with the market capitalization of each market. This is done to show the liquidity of the market, relative to its size. All this is displayed in table 3 below. 10

13 Stocks traded Market capitalization Stock traded / Market cap (million USD) (million USD) (%) South Africa ,92% Nigeria ,46% Kenya ,81% Ghana ,56% Zambia ,49% Malawi ,12% Zimbabwe ,63% Mauritius ,17% Botswana ,46% Namibia ,61% BRVM ,08% Tanzania ,50% Table 3: A table showing 3 measures. First, stocks traded refer to the total value of share traded in million USD. Secondly, Market Capitalization imported from table 1. Thirdly, Turnover ratio is an indicator of market liquidity relative to the market size. Swaziland is excluded due to insufficient data. Source: S&P global stock market factbook via Indexmundi.com (Reference 28-39) Before interpreting the data from table 3, it should be stated that stocks turnover ratio in Denmark, UK and USA in 2012, were 47,32%, 81,09% and 114,5% respectively ( - Reference and 40-42). Based on this it is clear that most of the Sub- Saharan markets are relative illiquid. The only market which stands out from the rest, is South Africa with a turnover ratio of 50,92%. The rest of the Sub- Saharan markets are more evenly liquid, and figure 3 delivers a graphical overview of the liquidity in Sub- Saharan stock markets, when South Africa is excluded. 11

14 Turnover ratio 16.00% 14.00% Zimbabwe 12.00% 10.00% 8.00% Nigeria Kenya Zambia 6.00% 4.00% 2.00% Ghana Malawi Mauritius Botswana Namibia BRVM Tanzania 0.00% Figure 3: The turnover ratio of Sub- Saharan markets, excluding the South African market. Source: S&P global stock market factbook via Indexmundi.com (Reference and 28-39) All markets in figure 3 are relative illiquid compared to the Danish, UK and US markets. The figure shows that Zimbabwe is notable more liquid than the rest of the countries, and that the rest of the top 5 liquid markets in the region are Nigeria, Kenya and Zambia. The rest of the region all have a turnover ratio below 5%, and it is remarkable that the fifth largest market, the BRVM union, only has a turnover ratio of 2,08%. The conclusion of liquidity is that the most liquid markets in the region are South Africa, Zimbabwe, Nigeria, Kenya and Zambia. This is almost coherent with the five largest markets in the region by market capitalization. The only difference is that the BRVM union was the fifth largest in size, while Zambia was the fifth most liquid market in the region. Both markets are excluded for further analysis, due to ether a lack in size or lack in liquidity. Consequently South Africa, Nigeria, Zimbabwe and Kenya are selected for further analysis. 4. Portfolio management and theory This section will examine a theoretical way to manage a stock portfolio. The model and information given in this section are needed to fully explain and understand the following sections, which examine the possible benefits of investing in Sub- Saharan stock markets. 12

15 The study will only focus on one specific portfolio theory, Modern Portfolio Theory also known as MPT. This section of the study will mainly be based on: Hillier, Grinblatt & Titman (2008). The MPT is based on the work of Nobel price winner, Harry Markowitz in The traditional way to evaluate a portfolio was to examine and evaluate each investment/stock individually, and not as a whole. So by stock- picking each stock individually you were able to create a portfolio of stocks, but not knowing how well they would perform as a whole. The new element introduced in MPT was to evaluate the portfolio as a whole and not each stock individually. The way this is done in MPT is by examining, and applying, the correlation that exists between the different markets or stocks in a portfolio, because the right correlation can lead to diversification of the portfolio and a reduction of risk. This will be examined and explained in greater detail in 4.3. The final goal of MPT is to construct an optimal portfolio in relation to the reward- to- risk combination. A key assumption of the MPT is that variance or standard deviation is used as a direct indicator of the risk of the stock, and that every investor is risk adverse. The result is that an investor will always choose the stock with lowest risk, given the same expected return. Consequently an investor, who is interested in higher expected returns, must be willing to take on additional risk, to compensate for the higher expected return. So each investor must consider how much risk to undertake. In relation to this, MPT applies a mathematically approach to derive multiple sets of possible portfolio allocations, which maximize return for a given risk. This is known as the Efficient Frontier. 4.1 The Efficient Frontier The Efficient Frontier is basically an illustration of different portfolio allocations in a mean- standard deviation diagram. So it displays the expected return and standard deviation/risk for different portfolio allocations containing a number of stocks/assets. Figure 4, below, shows the Efficient Frontier for a set of theoretical stocks: a,b,c and d. 13

16 Expected return X Ef*icient/Frontier c Z b d a Y Standard deviation Figure 4: A mean- standard deviation diagram displaying the efficient frontier, the minimum variance portfolio (Z) and the individual stocks (a,b,c,d). The figure shows a theoretical mean- standard deviation diagram of different portfolios, not all optimal, but they are all constructed using individual stocks: a, b, c and d. The red part of the graph is the Efficient Frontier, which displays all optimal portfolio allocations. They are optimal because they maximize the expected return for a given risk- level. This can be illustrated by looking at the two portfolios: X and Y. They both have the same standard deviation, but X has a higher expected return than Y. Consequently a risk adverse investor will always choose portfolio- X, and the X- portfolio is located on the Efficient Frontier, meaning that no portfolio that delivers a higher expected return for this given risk- level can be constructed. The Efficient Frontier consists exclusively of optimal portfolio, like the X- portfolio, but they all differentiate in expected return and standard deviation. The slope of the Efficient Frontier shows a trade- off between expected return and risk, meaning that by taking on more risk, an investor is able to get a higher return. The black part of the graph only 14

17 consists of none optimal portfolios, meaning that there will always be a better allocation alternative. The specific portfolio allocation that separates the two lines is called the Minimum Variance Portfolio, denoted Z in figure 4. As can be observed from the figure this portfolio allocation delivers the lowest variance or standard deviation. All portfolios placed between the two lines are also feasible portfolios. So every investor should choose a portfolio on the Efficient Frontier, but the final choice could depend on each investor s preferences with respect to expected return and willingness to take on risk. The Efficient Frontier and its specific formulas will be applied in the analysis of the Sub- Saharan markets later in the study, and will therefore be presented here. These calculations are presented in a general case with n- number of stocks. First, the introduction of the A- matrix, which is applied in the calculation of the portfolio weights. Where µ is the expected return matrix, while is the covariance- matrix, and ɩ is an n*1 matrix only consisting of values equal to 1. The general formula for the portfolio weight can be derived and rewritten as: Where µp is the return of the portfolio and the variance of the portfolio can be written as: Now the efficient frontier can be calculated, given relevant data, like the covariance- matrix and expected return matrix. It is done by selecting an expected return of a portfolio and calculating the weights and variance of this portfolio. This can be repeated for different µp s in order to archive the Efficient Frontier. This will be done later in the study with the relevant market data. 15

18 4.2 The Capital Market Line and the Tangency Portfolio The optimal portfolio The capital market line is the introduction of a risk- free asset in the model. A risk- free asset has zero variance, and an example of a risk- free asset could be a short- term government bond. The Capital Market Line presents possible optimal combinations of the portfolio of risky assets and the risk- free asset. So in many ways it is like the Efficient Frontier, but with a risk- free asset in the mix. This introduction of the risk- free asset results in a linear line, instead of a hyperbola. The Capital Market Line can then be expressed as this linear function: Where RT and σt are the mean and standard deviation of the tangency portfolio. The Tangency Portfolio is a portfolio located on the Efficient Frontier and the Capital Market Line. Figure 5 illustrates the Capital Market Line, the Efficient Frontier and the Tangency Portfolio. Expected return Capital*Market*Line Ef$icient*Frontier T r f Standard deviation Figure 5: The Capital market line, the Efficient Frontier and the Tangency portfolio (T). The figure illustrates that the Tangency Portfolio is in reality the portfolio that is tangent to the Efficient Frontier. This implies that the Capital Market Line, going through the Tangency Portfolio, will have the steepest possible slope, given an intersection with the Efficient Frontier. 16

19 Since the Tangency Portfolio is located on the Efficient Frontier, it is investing only in risky assets and nothing in the risk- free asset. So when looking at the Capital Market Line, above the tangency portfolio, it implies that an investor can go short in the risk- free asset, and allocate more towards the risky assets. Hereby the investor will experience higher expected returns, but also higher standard deviation, as he moves up the line. Below the Tangency Portfolio, an investor will allocate more towards the risk- free assets, resulting in a lower expected return and lower standard deviation, as he moves down the line. So the Capital Market Line represents the maximum return for each risk- level, when combining both risky and risk- free assets. On the other hand, the Efficient Frontier represents the maximum return for a portfolio of risky assets at each risk- level. So the intersect between the two represents the portfolio of risky assets that still maximizes the return for a given risk- level, after the introduction of a risk- free assets. The slope of the Capital Market Line, also called Sharpe s ratio, is a measure of reward- to- risk. So the Tangency or optimal portfolio is the portfolio of risky assets, on the Efficient Frontier, which maximizes the Sharpe s ratio. Sharpe s ratio is defined as: The study will use this information, along with the relevant market data to compute the Tangency Portfolio in section The Minimum Variance Portfolio and diversification The Minimum Variance Portfolio, or MVP, is based on using diversification to develop a portfolio of stocks, which delivers the lowest possible risk/standard deviation. Consequently extremely risk adverse investors can use the MVP, in order to hedge for what they see as high uncertainty or risk. The MVP is constructed in a way that tries to use diversification to eliminate unsystematic risk. The following part of the study will derive the MVP, and try to explain the benefits of diversification and low market correlation. This section will be based on both: Hillier, Grinblatt & Titman (2008) and de Blas (2006). 17

20 In order to derive the basic elements in the MVP, this section will assume a portfolio of two different stocks, which could be a Danish and an African stock, denotes D and A respectively. The end goal of this section is to derive a mathematical formula for the two specific stock- weights in the portfolio that minimizes the variance of the portfolio, the MVP. These weights will in this instance be denoted as wd and wa, to symbolize allocations in Danish and African stocks. The two main elements concerning any investor, when evaluating investments, are risk/variance and return on investment. The return on investment of a standard portfolio containing two stocks is simply: The returns of the individual stocks are denoted rd and ra, for Danish and African stock respectively. The overall return of the portfolio is denoted rp, and it is simply the sum of the weighted returns of the two individual stocks. The expected return can be written as: The expected return of a portfolio is the expected value of equation 1, but the expected value of a constant, like wd and wa, is simply the constant itself. Equation 2 will be needed in order to derive the variance of a portfolio. The other crucial element when evaluating an investment/stock is variance or standard deviation. Standard deviation and variance both measure the spread of data compared to the mean or expected value. In relation to stocks, it measures how much each stock deviates from the mean in a given time period. Consequently standard deviation and variance of stocks are often used as an estimate of the risk involved with a given stock. In order to derive the variance of a standard portfolio of the two stocks A and D, an application of basic statistical formulas and knowledge will be needed. A standard statistical formula for variance is: 18

21 This can be applied for the overall stock portfolio, by inserting the rp instead of x, in order to derive a variance formula for the return of the portfolio. The variance of the portfolio is derived below, and each step is explained. Step 1: rp is inserted into the basic formula of variance, and rearrange. Step 2: Insert the formulas for rp and E(rP), from respectively equation 1 and 2. The consequence is that the variance formula now consists of 2 different quadratic functions. Step 3: Calculating each of the two quadratic functions. It also has to be noted that the weights are constant, and that the expected value of a constant is simply the constant itself. Step 4: A simple rearranging of the formula, in order to make it easier to manipulate in the next step. Step 5: A mathematical conversion of the formula was done in order to obtain some specific statistical formulas within the variance formula of the portfolio. These formulas are: Step 6: The short and understandable equation for portfolio variance is the result of inserting the statistical formulas for individual variance and covariance. Equation 3 is the final equation for portfolio variance, and it will be used to derive the weights of the minimum variance portfolio. As the name suggests the MVP is a specific allocation within a portfolio of assets, which results in the minimum variance of the whole portfolio. So in order to derive the MVP, one would have to minimize the variance of the portfolio. The simple way to find a minimum value, or equation, is by differentiate with respect to the preferred variable. So in order to derive the 19

22 weights of the MVP, one would have to differentiate with respect to only one of the weights. Due to the fact that the portfolio weights have to sum to 1 (100%), then the weight that is not derived will simply be equal to 1- wderived. In this case we will differentiate with respect to wd, and therefore derive a general formula for wd in the MVP. The next problem is that the portfolio variance (equation 3) consists of both weights, but it should only contain the weight that will be derived, wd. This can be solved by remembering that the portfolio weights have to sum to 1, and therefore the weight in African stock market can be expressed as: wa=1- wd By inserting this into equation 3 and rearranging, we achieve an equation for portfolio variance, which only contain one weight, wd, as displayed below. The next step will be to differentiate equation 4 with respect to wd, in order to derive the weight of the MVP. So now just isolate wd and rearrange. Equation 5 is a general equation for one of the weights in the minimum variance portfolio. The other weight, in this case wa, will then be defined as: wa=1- wd. The last step is to incorporate correlation into equation 5, because the study will examine market- correlation later on. Correlation coefficient can be defined and rewritten as: Appling this to equation 5, to achieve the following: 20

23 Equation 6 displays that the minimum variance portfolio doesn t consider returns when deciding on the allocation between stocks. The only aim of the portfolio is to minimize the variance/risk of the portfolio Theoretical example of the minimum variance portfolio Diversification This example is based on a extremely risk- adverse Danish investor, who wants to establish a overall portfolio containing a Danish stock portfolio and a African stock portfolio. In this example the standard deviation of the Danish portfolio is 0.2(20%), while the standard deviation of the African portfolio is 0.3(30%). The correlation coefficient between the two portfolios is 0.5. It is possible to calculate the weights of the MVP by using the given data and equation 6. In this case wd=0.8571(85.71%) and wa=0.1429(14.29%) in the MVP. The actual variance and standard deviation, of the portfolio, can be calculated by inserting values into equation 3. The standard deviation for this specific portfolio is: σ(wd=0.8571, wa=0.1429)=0.197 (19,7%) Assume the same portfolio and same standard deviation, but that the correlation coefficient between the Danish and African portfolios changes to 0. What effect will this have on the MVP? Firstly the weights will change and have to be recalculated. The new weights are: wd=0.6923(69.23%) and wa=0.3077(30.77%). The new standard deviation of the portfolio will be: σ(wd=0.6923, wa=0.3077)=0.167 (16,7%) This example shows that a lower correlation coefficient between the individual assets in the MVP will result in an increase in the allocation toward the assets with highest volatility, in this case the African portfolio. More important this adjustment to the allocation, will lead to a lower variance of the whole portfolio. So even though the new portfolio weights will increase allocation toward the more risky assets, this will still result in a less risky portfolio as a whole. This is due to an increase of the diversification of the risk, meaning that because the two markets do not covariate as much as before, we are now able to diversify our investment and risk. 21

24 This is only an example, but the general relationship between correlation coefficient and the allocation within the minimum variance portfolio can also be showed mathematically. This can be shown by evaluating equation 6 under different correlation coefficient scenarios. The first scenario is perfect positive correlation between the two assets, meaning that the correlation coefficient is 1. By using this information in equation 6, it is possible to derive a new equation for wd in the MVP. The second scenario is perfect negative correlation between the two assets, meaning that the correlation coefficient is - 1. Again by using this information in equation 6, it is possible to derive a new equation for wd in the MVP. By comparing of the two scenarios it is clear that under perfect negative correlation the denominator is larger than under perfect positive correlation. Since the numerator is the same in both scenarios, it can be concluded that a perfect positive correlation will lead to a larger allocation towards wd. The conclusion is that a low correlation is a good element in a portfolio, because it allows us to spread our investments. This spread will lead to a greater diversification of the risk, which results in a lower portfolio risk- level. 5. Empirical data analysis of Sub- Saharan stock markets This part of the study will examine the market data from multiple stock markets. The focus will primarily be on examining the Danish market and the four chosen Sub- Saharan markets: South Africa, Nigeria, Zimbabwe and Kenya, which were chosen based on size and liquidity, earlier in the study. 22

25 The elements that will be analyzed in this part of the study are elements, which are crucial for investors, and are necessary to determine in order to establish an optimal portfolio of African and Danish stocks. The elements that will be analyzed are:! Annual Return! Standard deviation / Risk! Reward- to- risk ratio! Market Correlation This part of the study will also include a comparison between the Sub- Saharan markets and other markets, in order to show the relative value of the Sub- Saharan markets. The study will compare with the US- based S&P500 and the Brazilian stock exchange, in order to compare with a well- established stock market, S&P500, and Brazil, one of the largest emerging markets. The final output of this part of the study will be to construct an optimal portfolio, based on the theory presented earlier in the study. The portfolio will contain shares in the four Sub- Saharan countries and Denmark. The data, which are needed for this process, will also be presented and constructed in this part of the study. The data needed are: Mean Return matrix Covariance matrix All the data presented in this part are based upon historical prices of different indexes. All prices are imported via Bloomberg Terminals. The time- period used throughout is the last ten years, because ten years is a relevant time interval for a stock investment. Notice that it was not possible to obtain all market prices from for the Zimbabwean stock market. So the Zimbabwean data will only present the values from The following analysis is mainly based on data from Bloomberg, but is also inspired by the elements in the Modern Portfolio Theory (Hillier, Grinblatt & Titman. 2008) The index that is used to represent South Africa is the JALSH index, which is an all- share index on Johannesburg stock exchange. Nigeria is represented by the NGSEINDX, which is the all share index in Nigeria. Zimbabwe is represented by the ZHINDUSD index, which is the industrial index in Zimbabwe. Kenya is represented by the KNSMIDX index, which contains 20 selected shares from Kenya. Denmark is represented by the KFX index, or C20, which is the 20 23

26 most actively traded shares in Denmark. The United States are represented by the S&P500 index, which is 500 U.S. shares selected by S&P. Brazil is represented by IBOV index, which consist of shares selected based on liquidity. Throughout the analysis the study will assume a constant value of the DKK against the currencies used for traded in the Sub- Saharan market. So the foreign- exchange market is excluded for now, in order to show the general performance of each stock market. The assumption is valid because an investor could use financial derivatives to hedge for the foreign- exchange rate risk. 5.1 Annual Returns The study will use the percentage increase/decrease in the price of an index, to illustrate the annual return of each index. Annual returns are one of the main interests of any investor, as it displays the possible annual profit/loss for a stock if it were to be sold. Stocks can also provide profit in the form of paid dividend, but this will not be considered as a part of the annual return in this study. So the annual returns only illustrate the profit/loss associated with the annual price change in each stock market. The table below shows the computed annual returns in the Danish stock market, along with the Sub- Saharan markets. All in % South Africa Nigeria Zimbabwe Kenya Denmark ,85% 15,91% - 6,97% 17,32% ,98% 2,70% - 35,68% 37,28% ,68% 38,74% - 42,10% 12,19% ,23% 74,73% - - 3,56% 5,13% ,72% - 45,77% ,33% - 46,63% ,63% - 33,77% - - 7,76% 35,92% ,09% 18,93% - 0,47% 36,48% 35,90% ,41% - 16,31% - 3,58% - 27,69% - 14,78% ,71% 35,45% 4,09% 28,95% 27,24% ,85% 47,19% 32,41% 19,21% 24,05% Mean 17,79% 13,78% 8,11% 9,50% 13,36% Table 4: The annual returns of five different stock markets, based solely on the annual price change in each index. It also shows the computed mean- value of each index. Source: Bloomberg. Table 4 show that all markets felt the impact of the global financial crisis in 2008, while they differentiate in their ability to bounce- back and record positive growth. The table also shows that Zimbabwe have had a hard time generating large growth in the index, by only recording a 24

27 growth in prices of above 10% in one year, which were The simplest way to evaluate the different annual returns is by looking at the mean of each market. The mean- value is important because it illustrates the annual return that can be expected for a given year in the future, based on the data. Consequently the mean- value will also be used as the measure of expected return, when construction the optimal portfolio later on. Figure 6, below, is a mean- diagram of the Danish and Sub- Saharan markets, but also the S&P500 and the Brazilian market. Mean- values 20.00% 18.00% 16.00% 14.00% 12.00% 10.00% 8.00% 6.00% 4.00% 2.00% 0.00% South Africa Nigeria Zimbabwe Kenya Denmark S&P500 Brazil Figure 6: A diagram illustrating the mean- value of the annual return for seven different stock markets. Nominal values are listed in table 4. Source: Bloomberg. By looking at the mean- values in table 4 and figure 6 it is clear that the South African market generates the largest annual return. It is also clear that S&P500 delivers the lowest annual return, but remember that the standard deviation/risk have not been examined yet. Meaning that all market, which will have to be included in the final portfolio, all have larger annual returns than the S&P500. The Danish and Nigerian markets are both at the approximately same level as the annual return in the emerging market of Brazil, while Kenya and Zimbabwe are dragging behind. It should be stated that the Zimbabwean mean of approximately 8% is almost solely generated through the large return in 2013, as can be seen in table 4. The conclusion is that only South Africa outperformed the other markets, along with the emerging market benchmark of Brazil. Another observation is that Zimbabwe recorded the lowest mean- value of the markets that are included in the optimal portfolio. From the value of S&P500 it can also be concluded that smaller and less well- established markets were able to 25

28 record larger expected annual returns. This is coherent with the economic theory of Size- effect or growth option (Banz, 1980). 5.2 Standard deviation / Risk The study already examined the returns of the different markets, and will now look into the volatility of each market, by observing standard deviation. As stated earlier, the study will use standard deviation as a measure of riskiness of a given stock market, because standard deviation measures the spread of data from the mean. So risk is another of the crucial elements that every investor is interested in before investing. Risk is crucial, because every investor is basically risk- adverse, and would consequently always minimize risk for a given level expected return. More importantly the risk is crucial because it displays the possible deviation of the expected return. The standard deviations are calculated from the annual returns shown in table 4. So this standard deviation represents the risk over a period of 10 years. The calculated standard deviation values for the Danish and the Sub- Saharan markets are displayed in table 5, below. All in % South Africa Nigeria Zimbabwe Kenya Denmark Standard deviation 19,49% 37,70% 16,50% 27,52% 26,61% Table 5: The standard deviation of five different stock markets. They are all calculated using the data in table 4. Source: Bloomberg. Table 5 shows that the Zimbabwean stock market generated the lowest standard deviation, and could be considered as the least volatile market of the five markets. A reason for this could very well be the lack of data from the Zimbabwean stock market. The Zimbabwean data for annual returns only dates back to 2010, meaning that the instant effect of the global financial crisis is not included in the data. As seen from table 4, the crisis in 2008 made all markets record a large negative return, causing an increase in standard deviation. This effect is excluded in the Zimbabwean standard deviation and could be a reason for the low volatility. Table 5 also displays that South Africa, Kenya and Denmark all experienced a normal level of volatility, with standard deviations ranging from 19-27%. Nigeria, on the other hand, shows a high level of volatility with a standard deviation of 37,7%. 26

29 Figure 7, below illustrates standard deviations of the Danish and Sub- Saharan markets, along with the benchmarks of S&P500 and Brazilian market. Standard deviation 40.00% 35.00% Nigeria Brazil 30.00% Kenya Denmark 25.00% 20.00% South Africa Zimbabwe S&P % 10.00% 5.00% 0.00% Figure 7: A diagram of the standard deviation of 7 different stock indexes. Source: Bloomberg. S&P500 has a standard deviation of 18,41%, and figure 7 shows that the well- established index of S&P500 is the most stabile, if Zimbabwe is excluded. The Brazilian index has a standard deviation of 35,64%, and is almost as volatile as the most volatile market, which is Nigeria. This indicates that large and well- established markets, like S&P500, are generally more stable than emerging markets, like the Brazilian market. It should be noticed that South Africa almost has a standard deviation at the same level of the S&P500 index, and is by far the least volatile market in Sub- Sahara, if Zimbabwe is excluded. The conclusion is that the data could indicate that the average Sub- Saharan market is more volatile than more well- established markets, like S&P500. This is also supported by the emerging market benchmark, Brazil, which has a large level of volatility. 5.3 Reward- to- risk ratio analysis The study has now examined both returns and risk in the Danish and Sub- Saharan markets. This section will look to examine the relation between return and risk. This is done by computing the reward- to- risk ratio for each market. The reward- to- risk ratio is better known as Sharpe s ratio, and is defined as: 27