How to create portfolios for different risk groups and what to consider

BscB, 6 semester Bachelor Thesis Department of business studies GROUP: S11-13,72 Authors: Anders G. Nielsen Gestur Z. Valdimarsson Supervisor: Michael Christensen How to create portfolios for different risk groups and what to consider Aarhus School of Business and Social Sciences Aarhus University Spring 2011

TABLE OF CONTENTS Abstract... 3 1. Introduction... 5 1.1. Problem statement... 5 1.2. Delimitations... 6 1.3. Research method... 6 1.4. Assumptions... 8 2. Markowitz Portfolio Selection Theory... 9 2.1. Weak point in Markowitz s theory... 11 2.2. Sub conclusion... 12 3. Diversification... 13 3.1 Expected return and risk for an asset... 13 3.2 Covariance and Correlation... 15 3.2.1 Covariance... 15 3.2.2 Coefficient of correlation... 16 3.3 Expected return and risk for a portfolio... 19 3.4 The efficient frontier... 20 3.4.1 Calculations of the efficient frontier without short selling... 21 3.5 Diversification strategy... 23 3.6 Expected utility; the reason for different risk groups... 25 3.7. Subconclusion... 27 4. Data... 29 4.1. Risk free return... 29 4.2. Test for normality... 29 Page 1

5. Performance evaluation... 32 5.1. Creating a Benchmark... 32 5.2. Performance indicators... 33 5.2.1. Sharpe... 33 5.2.2. Treynor... 34 5.2.3. Jensen... 34 5.3. Evaluating the performance... 34 5.3.1. Performance evaluation using benchmarking technique #1... 35 5.3.2. Performance evaluation using benchmarking technique #2... 37 5.4. Subconclusion... 39 6. Conclusion... 41 7. Discussion... 44 7.1. Black-Litterman model... 44 7.2. Regression analysis... 46 7.3. Forecasting problems for expected values... 48 8. References... 49 Appendix 1: Eview s output.... 50 Appendix 2: Jarque-Bera Table.... 60 Appendix 3: Fund information.... 61 Page 2

ABSTRACT This thesis will try to evaluate the usefulness of Harry Markowitz s theory and see if it is as relevant today as it was in 1952 when his article portfolio selection first came out. To do this the paper has been divided into two parts, a theoretical review of portfolio selection and an empirical study. The theoretical part aims to look at Markowitz s theory on portfolio optimization and go through the conditions that need to be fulfilled in order to apply this theory optimally. The paper shows that the condition of normal distribution on the returns is important, and also that incorrect expected returns may result in a far from optimal portfolio. Calculations of the covariance and correlation are performed to see what effect they have on a portfolios variance. These calculations show that the covariance and correlation have a large impact when trying to reduce the risk of a portfolio. It is because of these covariances that it is possible to construct a mathematical model for the actual portfolio variance. By the use of Markowitz mean-variance optimization, it is possible to create the efficient frontier, which is a line of optimal portfolios, where the expected returns are maximized for any given risk. This paper uses historical data for the calculation of expected returns, standard deviations and correlation for and between assets. For the expected returns, a geometric approach has been taken, as this paper will show the geometric returns are more accurate than the arithmetic returns, which tends to overestimate the return values. It will also be discussed why historical data is not very reliable and not an accurate measure of the future returns. Because investors have different preferences about risk, the empirical study will include portfolios with different risk. The thesis quickly goes through what it will say to be a risk-averse investor and what it will mean to be risk-affine. The empirical study will consist of a portfolio simulation which will show how well Markowitz s theory performs in a real-world example. Three different portfolios are created to test his theory. The simulation is running from 01.01.2008 to 31.12.2010 and is based on 26 Danish stocks and 3 Danish government bonds. A test for normality will have to be conducted before the simulation can be conducted, this is to see if the returns meet the requirement of Markowitz that they need to be normally distributed. This condition was however not met since none of the assets turned out to be normally distributed. After the portfolio simulation a performance evaluation will show how well the portfolios did. According to the performance indicators only one of the portfolios created is able to beat the benchmarks. Another Page 3

portfolio suffered major losses, mainly due to inaccurate return forecasts and partly lack of diversification. This outlines one of the main problems of Markowitz s theory. After the portfolio simulation and performance evaluation it seems like Markowitz s theory is as relevant today as it was more than 50 years ago. However it has some pitfalls as it puts too much faith on exact forecasting and has conditions that are hard to fulfil in reality. Finally it will be discussed how the expected returns could have been more accurately forecasted. This discussion will focus on Black-Litterman and regression analysis, which are being explained in a simple manner. It will also be discussed why expected returns can never be predicted with 100% accuracy. Page 4

1. INTRODUCTION Beating the market is something investors always have been trying to do using different financial techniques. For many companies it is essential to be able to make the best portfolio possible. Companies such as pension funds, hedge funds, banks and investment firms, all of which can lose customers and a lot of money on failed investments. If a company could find a best practise, that as a rule can beat the market every time and thereby create better returns with certainty, it could drastically improve their competitiveness within the financial markets. But is it possible to create such a best practise? In this thesis we will see if one of the most well-known portfolio selection theories can be used to create efficient portfolios which could beat the market. In March 1952 the article Portfolio selection by Harry Markowitz was published in the Journal of Finance. This article would later become a cornerstone in modern portfolio theory, mostly because of the introduction of mathematical models on how to spread the risk in a portfolio. According to Rubinstein (2002) Markowitz...can boast that he found the field of finance awash in the imprecision of English and left it with the scientific precision and insight made possible only by mathematics., a sentence that can only be considered high praise. But since then a lot has happened in the financial markets, things move faster and have become more dynamic. So is Markowitz s portfolio selection method still applicable today? Can his way of risk diversification still be used to pick an effective portfolio and can it beat the market? 1.1. PROBLEM STATEMENT The main objective of this assignment will be to find out if we can create three portfolios that can outperform the market over three years. This will be done using Markowitz s mean variance portfolio theory. By doing this we will try to see if Markowitz s theory is still applicable today. When dealing with Markowitz, we will also try to identify weak and strong points in his theory, this is important in order to find out what potential pitfalls should be considered when creating a portfolio. How can diversification help the investor create a low risk portfolio, which can provide the investor with a more reliable return? Another problem we are going to look into is the use of benchmark portfolios. Benchmarking is an important part of the performance evaluating process. A process that is crucial for the investor since it will tell him how well the applied investment strategy has done compared to others. It is therefore important for the investor to know what to consider when creating a benchmark. Page 5

1.2. DELIMITATIONS To be able to conduct our research we will have to establish a couple of delimitations: We will only use Danish stocks and bond. This is to simplify the simulation; we don t think that this choice will weaken our conclusion since the benchmark portfolios used when conducting our performance evaluation also will consist exclusively of Danish stocks and bonds. We are aware of the fact that only including Danish stocks and bonds will limit our possibility of diversification, which could have decreased the risk. However we avoid the foreign exchange risk by limiting our self s to domestic investments. We will not use active portfolio management strategies, these strategies are used to gain small pick-ups in the ROI. The only interaction we will have with the portfolio will be the monthly asset adjustment. Tax and trade costs are excluded from our calculations. In our portfolio simulation we will not short sell or have any leverage. There will be no cash inflow or outflow for the three years our simulation runs. This means that no funds will be taken out of the portfolio and no further investments will be made in the portfolio for the three years it runs. Dividend will however be reinvested in its asset, dividend calculations are included in the return index numbers we use for our portfolio. 1.3. RESEARCH METHOD To tackle the problem at hand we have chosen to split this thesis into two parts. The first part will be a theoretical review. This review will be made to see which assumptions and conditions have to be fulfilled in order to apply Markowitz s theory. We will then look at whether these assumptions and conditions can be fulfilled in the real world. Diversification theory will be evaluated and it will be shown under which conditions it is most effective. The second part of our thesis will be an empirical study, where we will apply the theory reviewed in part one to a portfolio simulation. Our main focus in this simulation will be on Markowitz s portfolio selection theory from 1952. We will use Markowitz theory to find the optimal portfolio for our different risk groups. As mentioned above we only chose Danish stocks and bonds. The stocks we are going to use are the stocks on OMXC20 and all large- and mid- cap stocks on the Copenhagen stock exchange. However we exclude Maersk A, this was done due to the high correlation between Maersk A and Maersk B. The same was the case for Rockwool A. B&O, Torm and Thrane where included to have stocks from all industries represented. Page 6

Industries were defined according to the definitions used on the Copenhagen stock exchange (www.nasdaqomxnordic.com). On the bond side we will use three government bonds one short, one medium and one long. All stock and bond prices are based on the reinvestment of dividend and interest (Also called the return index). Before using these assets in our portfolio selection process we will have to test if their returns are normally distributed. We will do this by conduction a Jarque-Bera test of our assets in Eviews. When all assets have been tested for normal distribution we can start putting the simulation together in Excel. To forecast the expected return and standard deviation (risk) we will use historical data one year before the investment date. For example on 01-10-2009 we will use data from 01-10-2008 to 30-09-2009 to forecast next year s expected return and standard deviation. By doing this we assume that the only factor determining future investment returns and risk, are historical returns and historical risk. This is however a wild assumption because it is common knowledge that historical return is not the best indicator for future return. We will however use this forecasting method in our simulation to simplify our calculations. Later in the paper we will discuss and explain how forecasting models can be used to create a better estimation of future returns. After completing the above steps we can begin to select the assets in our portfolio based on the Markowitz portfolio selection theory. The three portfolios will be the basis of our simulation. These portfolios will be updated once a month. The reason for this choice is because we need to change the portfolio constantly to fit the changes in the market, but it would be unwise to change it every day or every week because of the transaction cost that would be involved in the real world. We have set the duration of our simulation to be three years, running from 01.01.2008 to 31.12.2010. After the three years have passed we will conduct a performance evaluation to see if any of our portfolios were able to beat the market. In that part of our assignment we will also look at how performance evaluation should be conducted and discuss what makes a good benchmark portfolio. After the performance evaluation we will be able to draw some conclusions on the effectiveness of Markowitz s mean variance portfolio theory. Here both the theoretical review and the results gained from our empirical study will be taken into consideration. In the end we will discuss what we could have done differently and what effect it might have had on our results. Page 7

1.4. ASSUMPTIONS To conduct our simulation and apply the relevant theory the following assumptions have been made: In this example we have the role of price takers. This means that we assume that we will not trade a stock or a bond at a high enough volume to affect the pricing of that asset. Investors are rational. Investors see an increase in risk as a negative thing. Investors see an increase in return as a positive thing. We assume that returns are normally distributed. This assumption will be tested later. As mentioned above we assume that historical risk and return can be used to forecast future risk and return. The validity of this assumption will be discusses later. Risk and return are the only things considered when taking an investment decision. Page 8

2. MARKOWITZ PORTFOLIO SELECTION THEORY Markowitz is considered the father of modern portfolio theory, mainly because of his article Portfolio Selection from 1952 where he as the first person gave a mathematical model for portfolio optimization and diversification. He later gave a more detailed description of his theory in the book Portfolio Selection: Efficient Diversification of Investments from 1959. Before Markowitz s theory, diversification was well known, as he stated himself in his historical review of portfolio theory from 1999: Diversification of investments was a well-established practice long before I published my paper on portfolio selection in 1952. (Markowitz 1999). However no-one had made a scientific model explaining it. The thing that made Markowitz s theory revolutionizing was that it was the financial version of The whole is greater than the sum of its parts (Rubinstein 2002). In Markowitz article (1952) he stated that it is not the variance of the individual asset ( ) that is important, but the contribution that the assets variance makes to the variance of the entire portfolio ( ). This means that one of most important things to consider when creating a portfolio; is the correlation between the different assets, and the covariance of the portfolio (Markowitz 1952). The formula given by Markowitz to calculate the variance of the entire portfolio looks like this (Rubinstein, 2002): = + Where is the proportion of the portfolio held in asset j, and is the correlation between the returns of asset j and k. According to Rubinstein (2002) Markowitz s paper is the first to contain this equation in a published paper. We will have a closer look at this formula in a later chapter on risk diversification. Since we don t deal with short selling in our assignment, & 0. Markowitz s portfolio selection theory can however not stand alone when creating a portfolio. As the name of the theory indicates it focuses only on portfolio selection, in another way, what assets a portfolio should consist of. In his article Markowitz divides the process of portfolio creation into two stages, saying that the first step focuses on forecasting the expected return of assets and the second stage focuses on the choice of portfolio and how the assets should be combined. After describing this two-step process, he states that he will only be concerned with the second stage (Markowitz 1952). Another important aspect of Markowitz s theory is that risk, although reduced by diversification, cannot be completely removed by it (Markowitz 1952). In his article he states that: The portfolio with maximum expected returns is not necessarily the one with minimum variance. (Markowitz 1952). Here he also dismisses the validity of the law of large numbers in finance. The law of large numbers is that if you would Page 9

diversify your investment across a very large number of assets your risk will be close to zero. This does however not take the correlation of stocks into account, since it assumes that each asset is independent from each other. Markowitz says that the investor will at some point not will be able to gain a higher expected return by accepting higher risk. To illustrate the notion that an increased risk not always gives a higher return Markowitz uses Expected return Variance. This concept illustrates, in a simple manner, that the investor will not always gain a higher expected return by taking a higher risk. If we look at figure 2.1 it is clear to see that taking on additional risk in some cases decreases expected return. This will be illustrated in a more mathematical manner in the chapter on diversification. Figure 2.1.: Expected return Variance Source: Markowitz, 1952, Portfolio Selection. Michaud (1989) listed some of the strong points about Markowitz s theory in this article The Markowitz optimization Enigma: Is Optimized Optimal? Satisfaction of client objectives and constraints. Markowitz can provide a convenient framework for integrating a list of important client constraints. These could for example be a certain risk level or certain expected return. Control of portfolio risk. Markowitz s portfolio optimizer can be used to control the portfolio s exposure to various components of risk. Implementation of style objectives and market outlook. This means that it is easy for the investor to choose the appropriate exposure to different risk levels, the stocks of interests and the benchmark. Efficient use of investment information. Markowitz s model is designed to optimize portfolios from the investment information given. So information in a portfolio context will be used optimally. Page 10

Timely portfolio changes. Markowitz can process large amounts of information quickly. According to Michaud (1989) this is particularly important for large institutions which quickly need to determine the impact of new information on its entire portfolio. This aspect of Markowitz s theory could have become even more important, since the markets move a lot faster now than they did in 1989. 2.1. WEAK POINT IN MARKOWITZ S THEORY Although Markowitz s portfolio selection is broadly recognised it has also meet some criticism, which has proven that the selection method has its weak sides. One of the critics is Richard O. Michaud which in his 1989 article The Markowitz optimization Enigma: Is Optimized Optimal? pointed out some weaknesses in Markowitz s theory. Michaud (1989) states that the main reason Markowitz s theory is not being applied in investment firms is for political reasons. He argues that because portfolio optimization puts its emphasize on qualitative techniques, it would move the power of investment decisions from senior managers to qualitative experts and thereby changing the firm s organizational structure. This reasoning is however not perfect. We can easily assume that if these changes in organizational structure would lead to higher returns, they would probably be implemented. If not, new companies would be formed that applied this strategy. This argument by Michaud could however be obsolete since the article is from 1989. We will put more focus on the other critic points in his article. His other critic points were: Michaud (1989) state that Markowitz s theory is estimation-error maximization. He says this because of the importance of correctly estimated expected returns, variance and covariance play in Markowitz s theory. Even though these factors can be estimated with some precision they will never be 100% correct and an investor will therefore not be able to create an optimal portfolio using Markowitz s theory. He criticizes that the optimization process over weight s assets that have large estimated returns, negative correlations and small variances. While it underweight s assets with small estimated returns, positive correlations and large variances. Stating that these estimations are those which are most likely to have estimation errors (Michaud, 1989). To replace the mean with the expected return of a given asset, when using historical data, is according to Michaud (1989) also a drawback. More powerful statistical tools will have to be used to gain a better estimation of expected return. An aspect that is also quickly mentioned by Michaud (1989) is the low level of stability when using Markowitz s portfolio selection theory. A small change in the input can have a huge impact on the output (portfolio). For some investors this could reduce the profitability of Markowitz s model, due Page 11

to the costs of trading, and there by rule out some of its usefulness. Especially private investors will find that large changes in the portfolio on a monthly or weekly basis will lead to high transaction costs; they should therefore not use Markowitz s portfolio optimization, unless they invest very high amounts. Another concern that might arise when applying the Markowitz model is the possibility of a lack of diversification. The optimal portfolio will only take expected returns, variance and covariance into account when finding the best possible combination for a given expected return or a given expected risk. Therefore it could invest a very large part of the portfolio in a single asset not considering diversification. It would require a lot of trust in the accuracy of the expected returns and variances forecasted, since an estimation error in an over weighted asset could have devastating effect on the investment. To avoid this problem we have in our simulation put in some constrains that we believe will guarantee some level of diversification. The first constraint is that there cannot be invested more than 10% in a single stock, the second constraint is that the portfolio will have to consist of minimum 20% government bonds. We will have more on the constraints we used when running our simulation later. 2.2. SUB CONCLUSION In this chapter we have gone through the main aspects for Markowitz s theory, and also looked at some of the main critic points. Markowitz introduced the mathematical portfolio optimization process; he did this by putting focus on the correlation of assets and by creating a formula for the variance of a portfolio. This theory also dismissed the principal that a higher risk always meant a higher return. However some critic has been pointed at Markowitz s theory due to the high confidence put in exact estimates on future risk and return. If the estimates are off, the portfolio optimization process could lead to a portfolio that is far from optimal. Most of the critic is based around the fact that Markowitz s theory only looks at one part of the portfolio creating process and puts way to much faith in the first part being conducted to a precision that will ensure exact values for future risk and return. We will later in our assignment look into these forecasting problems and discuss the validity of Markowitz when historical values are used to predict future stock earnings and future risk. Page 12

3. DIVERSIFICATION 3.1 EXPECTED RETURN AND RISK FOR AN ASSET A portfolio is a combination of several assets; we are weighting these assets by a simulation to create our portfolios where, among other, their expected returns and standard deviations are being considered. According to Moffett, Stonehill and Eiteman (2009) the return of an asset can be calculated in one of two ways as arithmetic and geometric returns. There is no rule when to use which approach, but it is often seen in research projects that the geometric approach have been used. The arithmetic method is although the most intuitive approach, where you simply take the average of the historical returns. This is however not the most precise method, because if we had a stock giving 100% return one year, and -50% the next year, then the arithmetic return would be: )= 100% 50% 2 =25% But the actual return has been 0% and the arithmetic return has therefore overestimated the expected return. If the expected return should be accurate, then it requires that the investor will take out his winnings or replace his loses every year. In such a case he would double his investment the first year, but lose half of his original investment the second year, and hereby he would have made a profit of the 25%. Robert Ferguson et Al (2009), state that the expected arithmetic return is a poor long-term indicator of the return, and that the mean-variance efficient portfolios with arithmetic returns always will reward additional risk. Ferguson et Al on the other hand argues that a too aggressive investment will drive the expected return negative, opposite of what is happening with use of the arithmetic returns. Instead they are recommending the geometric approach, where the natural logarithm is being used to calculate the returns. If we take the same example as before, with 100% and -50% returns, then the geometric approach will give an expected return of zero: )= 1+1)+ 1 0,5)=0,693 0,693=0% The zero percentage return is equal to the expected return in the long run, if the two outcomes each have a probability of 50%. Page 13

In our simulation we have used the geometric approach to calculate our expected returns from historical data. The expected return and standard deviations have been calculated using daily returns one year before the investment date. We have calculated the daily return for a specific asset as: = r t is the return at date t, P t is the price at date t, and P t-1 is the price the day before P t. We have then used the assumption that there are 250 trading days in one year, and to get the yearly expected return, we have taken the average of the return for one year before investment date, and then multiplied with the amount of trading days (250): )=250 1 t is the number of observations one year before the investment date, r i is the return on the observation at date i, and the 250 is the number of trading days in one year. This is possible because the geometric return has a special feature to stretch over several periods. The return can over n periods be calculated as the sum of all the returns during this period, which is what we are doing when we take the average of the returns and then multiply with the amount of trading days. One concern about the geometric method is that for extreme negative values, the numbers will be inaccurate. Let us say that we have one stock which decreases with 70% in our observation period, in our case with daily returns this is though pretty unlikely, then the geometric return will be ln(1-0,7) = -120,4%, a return lower than what is actually possible, since a stock will never be able to drop below a value of zero, and therefore cannot lose more than 100% of its actual value. Another concern about our expected returns is that we assume that if an asset has increased 20% in value for a given year, it will also increase with an equal amount the next year. For the individual assets standard deviations, we are using the same values as we used to calculate our expected returns. The mathematical formula to calculate the standard deviation is: = 1 1 ) We skipped this calculation step and instead we simply used the stdev() formula in excel, which calculates the standard deviation for a sample. This calculation only gives us, as it also did with the expected return, the standard deviation for a single day, and what we need is the standard deviation for a year (250 days as Page 14

stated previously). Because we are dealing with the standard deviation and not the variation, we cannot just multiply with the 250 days, but we need to multiply the daily standard deviation with the square root of 250 to get the yearly standard deviation. 3.2 COVARIANCE AND CORRELATION Beside the risk of the assets independently, Markowitz is also taking the coefficient of correlation between the assets into account. By taking the correlation into account it will be possible to diversify the portfolio and hereby bring down the risk of the portfolio, it will even be possible for the portfolio to obtain a lower risk than the one of the least risky asset. We will talk about how it this possible to obtain such lower risk in section 3.2.2. 3.2.1 COVARIANCE The combined risk of the portfolio is not only calculated by the included assets standard deviation, but also by the use of the covariance between the assets that has been included in the portfolio. The covariance is a measure of how the assets variations are between each other. We can take A.P. Møller (APM) as an example, a company that both have an A and a B stock. This A and B stock vary much alike on the market, if the A is increasing with 1,5%, the B stock will most likely increase with a percentage close to that of A. The reason for this is that the A and B stock are affected by the same factors, all positive and negative factors that affect APM will have a positive/negative impact on both the A and B stock, because if it is going good for APM, then both stocks will go up, and the opposite if it is going bad. If we look at two different companies in the same industry, it could also be that a surprisingly good result for one of the companies could affect the other, because investors may see a positive trend in the industry. The covariance between two assets can be calculated with the following formula (Keller, 2005): = 1 ) Where n is the number of observations, x i is the return of asset x at day i, µ x and µ y are the mean values for the return of x and y respectively. We can from this formula see that the covariance can take any value from negative infinite to positive infinite. The two assets will have a positive covariance if both assets are either higher or lower than their respective mean value at a specific day. The assets would have a negative covariance if one of the assets had a higher return than its mean while the second asset had a lower return than its mean. The covariance does not give a good indication of how much the two assets actually Page 15

correlate with each other, because it is affected by both of the number of observations and the size of the gap between the actual return and the mean at the time of the observations. It is because of this difficult to say whether two assets have a strong relationship with each other, if their returns are varying together, or if their trends are just barely connected. 3.2.2 COEFFICIENT OF CORRELATION Another way to give a better impression of how much two assets correlate with each other, is by using the coefficient of correlation, which will have a value between -1 and 1. Two assets that have a correlation coefficient of 1 are perfect correlated, meaning that when one of them increases with 1% the other asset will increase with 1% as well. On the other hand if the coefficient is -1, then when one of the assets are increasing by 1% the other asset will decrease by the same amount, so they will go in opposite directions. The coefficients can be calculated using the following formula (Keller, 2005): = We took a more simple approach in our calculations, by using the excel formula Correl which automatically calculates the correlation coefficient between two assets. The table below shows how two assets correlate with each other with different correlation coefficients: if 0< 1 then the assets will have a positive correlation, which stocks generally do among themselves, and so does bonds. =0 the assets will not correlate with each other 1 <0 then the assets will correlate negative with each other, which is often the case between stocks and bonds. We will here show how different correlation coefficients between two assets will have their impact on a portfolio only consisting of the same two assets, the formula we use to calculate the variance of a weighted portfolio is the one used by Markowitz (1952): = +2 Page 16

And as we showed previously we have that: = = By inserting this in Markowitz formula we get that the variance of two weighted assets will be: = + +2 We will now show the calculations of the portfolio standard deviation for =1 = + +2 = + +2 = + = + We can from this formula see that if two assets are correlating perfectly with each other, then the risk of the portfolio will be a weighted average of the two assets standard deviations, and it will therefore not be possible to reduce the risk more than investing everything in the asset with the lowest standard deviation. This makes sense, because as we mentioned before, perfect correlation means that when one asset increases with 1% the other asset will increase with 1% as well, and there will be no risk reducing benefits from combining the two assets. Perfect correlation is of course basically impossible in the real world, where not only the company s performance is influencing the stock value, but also subjective views from investors and other factors have influence. For =0 The mathematical calculations will start from the same formula as previous with two assets in the portfolio, but will later be expanded to contain m assets. In these calculations we assume that all assets are weighted equally and also that their standard deviations are equal. = + +2 = + = + + + = 1 + 1 + + 1 Page 17

= 1 = = We can see from this formula that when we add 3 extra assets for each asset we already have in our portfolio (we multiply our assets with 4), our risk will be halved, if the assumptions mentioned above are fulfilled. If we keep adding assets the risk will almost completely be removed. This is the law of large numbers. However this is not possible due to the fact that assets will always have some level of correlation. This means that some things can affect the entire market and every asset in it. This is what is also called the systematic risk, the risk that cannot be removed by diversification. For = 1 = + +2 = + 2 = = Here we can see that the risk of the portfolio can be completely eliminated by diversification if: = = If we insert this in the equation for the portfolio standard deviation, we get; = = = =0 Where we with perfect correlation weren t able to reduce the risk of the portfolio to less than the asset with the lowest risk, then we can in this formula see that if two assets have a correlation coefficient of -1, then it is possible to completely eliminate the risk, if we choose a specific weight between the two assets. It is of course, as with the perfect correlation, not realistic to find to assets which have a correlation of exactly -1. The only two assets that will be perfectly uncorrelated will be a stock and the short selling of the same stock, which would give no risk, but the winnings/loses would also outtake each other, and you would therefore be sure to lose nothing more or less than the transaction costs. Page 18

Figure 3.1: Illustration of how the combined standard deviation between two assets can be reduced with different correlations. Source: http://invisibleeconomists.splinder.com/archive/2009-08 3.3 EXPECTED RETURN AND RISK FOR A PORTFOLIO The expected return for a portfolio, combined by multiple assets, can by the expected return of the individual assets be calculated using the following formula (Markowitz, 1952): = ) Where is the expected return of the portfolio combined with N assets available, is the weight of asset i, and ) is the expected return of asset i. From this formula we can see that the portfolios expected return is the weighted average of the expected returns for the individual assets. If all assets were weighted equally, then we could replace with, so if we had 10 assets, each asset would have a weight of =10%. The risk of a portfolio cannot be accumulated as easy as the expected return, this is due to the correlation between different assets. As mentioned earlier, the correlation also has an influence on the accumulated standard deviation and it will therefore be necessary to include the correlation in the calculation of a portfolios risk. As we are using excel to create our portfolios, a matrix formula will be very useful, we are going to use a matrix calculation of the variance for the portfolio (Markowitz, 1998, p. 172): Page 19

=, But as shown, we can instead of the covariance use the correlation coefficients: = = By inserting this in the previous matrix we get that: 0 =, 0 1 1 0 0 1 =, 1 And finally we will have to take the square root of the variance to obtain the portfolio risk/standard deviation. 3.4 THE EFFICIENT FRONTIER The efficient frontier is the optimal portfolio combination of the included assets, which provides the lowest possible risk for a given expected return, or as we use to create our portfolio, the highest possible expected return for a given risk. Figure 3.2: An example of how the efficient frontier and MVP could look like. 12,0% 10,0% 8,0% 6,0% 4,0% 2,0% MVP 0,0% 0,0% 5,0% 10,0% 15,0% 20,0% 25,0% 30,0% Source: Own creation Page 20

If you follow the curve of the efficient frontier, you will eventually reach the point with the lowest possible risk, which is called the minimum variance portfolio. All combinations which will give an expected return lower than the minimum variance portfolio, are not included in the efficient frontier. This is because it for these risks will be possible to gain a higher expected return. This is illustrated in figure 3.2. 3.4.1 CALCULATIONS OF THE EFFICIENT FRONTIER WITHOUT SHORT SELLING If we look at the lines in figure 3.2, then every point on the line could actually be different portfolios which we are able to create with the available assets. The way we created our efficient frontier, and hereby our portfolios, were to choose a static standard deviation for two of our portfolios, and then use the excel solver to find the maximum expected return for the given standard deviation. So in this case we wanted to optimize the following formula: )= We have some specific conditions we also want the solver to fulfill when solving this optimization problem: 1. =1 2. 1 = 1 1 3. 0, =1,2, 4. A fourth condition we use in our optimization problem is that none of the stocks can have a weight higher than 10%. The fifth and last condition is that we as a minimum for all our portfolios will have at least 20% 5. bonds. Condition 1 makes sure that the sum of our different asset weights adds up to 100%, so we have invested our full capacity in assets. Condition 2 will only be included in two of our portfolios, 5% and 15% risk portfolios, where we are able to set the standard deviation of our portfolio to a specific value, for which the solver will then find the highest possible expected return. Condition 3 states that the weight of an asset cannot be negative, which eliminates the possibility of short selling any of the assets. As stated in our Page 21

delimitations we have chosen not to include short selling, this is partly because there has been made new rules about short selling certain stocks (National Bank of Denmark, 2008) and partly due to our rather unreliable expected returns. Condition 4 makes sure that none of the stocks in our portfolios have a weight higher than 10%. Since our expected returns are not very reliable, we made this rule to make sure that we have some diversification in our portfolios. We are more interested in a broad diversification than an extremely high expected return we cannot be certain of. Condition 5 says that at least 20% of a given portfolio should be invested in the three included bonds. This is also because of our not completely reliable expected returns. Estimation errors could make the portfolio end up with 100% of the investment in stocks, given that the stock market had a very high increase the year before. As we can see our calculated expected returns for many stocks are higher than 50%. For our third and last portfolio we remove condition 2, and instead of optimizing our expected return, we are going to optimize the ratio between our expected return and our standard deviation. The reason we are choosing this third way to create a portfolio, is because our portfolio will then be able to have different risk and adapt to the market. If the stock market for example is experiencing a positive trend, then our portfolio could adapt to this trend, and hereby increase the risk so the ratio between stocks and bonds would increase on the stock side. Figure 3.3 illustrates where the third portfolio is located on the efficient frontier. It is located at the tangent point of a line going through 0.0 (no risk and no return) and the efficient frontier. Figure 3.3: shows how we find the third optimal portfolio (E(r)/ Std. Dev.). Source: Own creation Page 22

3.5 DIVERSIFICATION STRATEGY Diversification is an important aspect of portfolio theory because as we showed above, a diversification strategy can decrease the risk and uncertainty for investors. We also showed that a requirement for making diversification possible is that the assets do not have perfect correlation between each other. By reducing the risk, an investor will be able to rely much more on the expected return of his investments. When we are talking about risk on assets, then it s important to recognize between two different types of risk; Unsystematic and systematic risk (Christensen& Pedersen, 2003, p. 49). The systematic risk is a risk which does not occur for a single asset, for example due to internal disturbance, but is a market specific risk which has an impact on all stocks in the market. The systematic risk is not possible to reduce by diversification as it is with the unsystematic risk. The reason it is not possible to remove the systematic risk by diversification, is because no matter if you invest in one or ten stocks, then they will all be affected by the systematic risk. It is though possible to reduce the systematic risk by shortselling, this way if your stocks is decreasing/increasing due to some outcome which has an effect on the entire market, then the stocks you have borrowed and sold, will also have decreased/increased in value and therefore you buy them back cheaper/more expensive (Christensen& Pedersen, 2003). An example of systematic risk could for example be, as we are experiencing now, increasing oil prices. Since all companies are affected by the oil prices in some amount, either just by the increase in energy prices, or in worst case that the company is depending on fossil fuels for example for the shipping industry. In such a case the entire market would be hit, and therefore it wouldn t matter if you had invested in one company or another, some companies would of course be hit harder than others, but this extra loss is under the category of unsystematic risk. Unsystematic risk is a risk for a single company, it could be the dividend policy, or if essential raw material should either increase or decrease in prices. We can say that the unsystematic risk is a risk or chance that a certain event takes place, whether that event is negative or positive, which only affect one or more companies or an entire industry, but the entire market will not be affected by this event (Christensen & Pedersen, 2003). An example of this could be a company, whose managers are taking some very bad/good decisions that decreases/increases the value of the company and will therefore also affect the stock price for the company, but it will in general not have any direct effect on other companies in the market. A crucial difference between the systematic and unsystematic risk is, that the unsystematic risk can be almost completely eliminated by spreading your investments on many different assets, this have been showed Page 23

possible previously in this paper and is only an option for assets which do not have perfect correlation. Such a spread would of course minimize the impact of the managers decision in a single company and also the impact of how raw material may affect different industries. This risk spread is working because your investment in a company may now be only 10% instead of having invested 100%, so if the stock decreased with 10%, then you would have lost 10% of your investment if you had invested 100% in the company, but after the spread you will only lose 1% of your total investment on the same decrease. This also shows another assumption about the diversification strategy. You won t diversify your investment if you have for $100 shares in one company and then invest another $100 in another company. This would actually increase your risk, because you are now investing $200 which you risk to lose instead of $100. So you do not lower your risk by investing more money. The way the diversification strategy works is that if you have $100 worth of share in one company, then you could for example sell shares worth $50, which you would then spend on buying shares in another company. Figure 3.4 shows how the systematic and unsystematic risk changes as more assets (which are equally weighted) are added to the portfolio, where all assets have a standard deviation of 15, of which 10% is unsystematic risk and 5% is systematic risk, and the unsystematic risk has a correlation of zero between the assets: Figure 3.4, unsystematic and systematic risk with different amount of assets 16% 14% 12% 10% 8% 6% 4% 2% Total risk Systematic risk Systematic total risk 0% 0 10 20 30 40 Source: Own creation Number of assets This figure shows that if we assume that the assets all have the same expected return, we are able to reduce the risk on our portfolios, while maintaining the same high expected return. If all assets in the market for example gives 12% in return and had, as in this case, a 15% standard deviation (10% unsystematic, which have zero correlation between the assets, and 5% systematic), then we could by Page 24

including 32 different assets, be able to make a portfolio with an expected return of 12% and a standard deviation of: = % +5%=6,77%, which is much lower than if we only invested in a single asset. 3.6 EXPECTED UTILITY; THE REASON FOR DIFFERENT RISK GROUPS An investors view on risk is of course important to consider when creating a portfolio, some investors will settle for a low expected return, if they in return are given a low risk and hereby a higher probability of not losing their investment. Other investors would much rather have a portfolio with very high expected return and an equally high risk, than settle with the more certain low expected return, these different behaviors are described in the expected utility theory which talks about risk aversion (people trying to avoid risk) and risk affine (risk-seeking people). Christensen & Pedersen (2003) explains on page 66 that an investors risk profile determines how much he will invest in the risky portfolio and how much he will put in the risk-free asset. In our situation we do not include a risk-free asset in our model; we will create portfolios with different risk levels. Then it will be up to the investors to determine which portfolio matches their risk profile and how much of their wealth they will put in this portfolio. It will, for each investor, be possible to have a different risk profile, which should then be used to optimize the portfolio to match this exact investor. This risk profile can be quantified by a utility function, which calculates how much utility a certain winning or loss will give this investor. If we say we have two investors, they each have a different utility function: Investor 1: )= Investor 2: )=ln ) U is the utility function of x, which is the investor s wealth, and as we can see, x cannot be negative in this example. Figure 3.5 and 3.6 on the next page illustrate how the two investors utility functions behave. Page 25

Figure 3.5 Figure 3.6 x^2 ln(x) Source: Own creations As we can see investor 1 prefers to earn $1 extra rather than avoid losing $1, so in a game where he is able to bet $1 with a 50-50 percentage chance of winning $2 or lose his bet, he will take the bet, because if he should win the extra $1 it would give him more utility than he would lose if he lost the bet and the dollar. Investor 1 s utility, if he did not take the bet, would be: 1 2 = 1, if he on the other hand took the bet and lost, his utility would drop down to: 0 2 = 0, but if he won the bet his utility would increase to: 2 2 = 4. We can see that if the investor does not take the bet, he will have a utility of 1, but if he takes the bet, his expected utility will be )=0 0.5+4 0.5=2 so his expected utility will be higher if he takes the bet. He will actually be willing to pay more than $1 for the bet; the highest amount investor 1 would be willing to pay for the bet is 2=$1.41 which is $0.41 more than the expected return. From this we can see that investor 1 is risk-seeking and will be willing to take risk of losing more, if he is at least able to have the same chance of winning the same extra amount. Investor 1 would in this case be a candidate for our portfolio with a risk of 15%, since he will have the risk of losing more money, but he would at the same time be able to increase his return even more. Investor 2 is just opposite to investor 1; investor 2 will be happier not losing $1 than he will be if he wins $1. For mathematical reasons the same bet cannot be used in this example. If investor 2 is being offered a bet where he can either win $3 or $1 each with a probability of 50%, he will have an expected return of $2, but since this investor is risk averse, he will much rather just have the $2 with no bet, instead of having the risk that he is only getting $1. The utility for the investor will in each outcome of the bet be; 1)=ln 1)=0, 3)=ln 3) 1.1. Investor 2 will therefore have an expected utility of: )=0.5 0+0.5 1.1= 0.55, and from this we can calculate how much he will be willing to pay for taking the bet:. =$1.73, which is less than the expected return, so he is willing to take the bet if he is getting a risk premium of $2 $1.73=$0.27. So we can say that investor 2 will only take a bet if he statistically is going to make a Page 26