# Modern Portfolio Theory

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## Risk and Return

Investors take on risk in order to achieve return. Due to this, it is important to be aware of the risk exposure related to a certain investment. Risk is generally known as uncertainty of expected return and this uncertainty is better described as variability. Investors want a higher expected return with increased variability, meaning that risk and variability are correlated. A well known measure of variability is standard deviation, which measures the volatility of a security. (Simons, 1998).
The return of mutual fund investments is defined as income in form of dividends or interest payments, and capital gains or losses (Simons, 1998). The average return, or mean, is used to calculate the standard deviation of the fund. This can be achieved by calculating an arithmetic or geometric mean. The arithmetic mean measures the return of an investment that is held constant at the initial level whilst the geometric mean measures return of an investment as in each period will grow at precisely the rate of the portfolio (Modigliani, F., Modigliani, L. 1997).

Modern Portfolio Theory

The modern portfolio theory has its origin in the article ”Portfolio selection” which was published in The Journal of Finance in 1952 by Harry Markowitz. He developed the theory while he was studying the effects of risk, return, correlation and diversification on portfolio returns. Markowitz (1952) was reflecting over that investors had one thing they were interested in and that was the value of the portfolio, and in order to maximize that value, the only thing the investor needed to do was to invest in one single security – the one with the best return. But he concluded that this is not the way investors invest. Investors deposit their money in different securities since they are interested in minimizing risk as well as in maximizing return. This means that if one security yields a negative return, everything will not be lost as you still have investments in a number of other securities. This is a way to spread the risk in the portfolio, which is in line with the assumptions of the modern portfolio theory, i.e. how to maximize the expected return of the portfolio for a given amount of risk. This is referred to describing diversification, which means reducing risk by investing in different assets. In other words, do not lodge all of your eggs in one basket. (Fabozzi, Gupta and Markowitz, 2002).
Modern portfolio theory demands an understanding of three factors; the expected return of the portfolio, the risk of each element of the portfolio and the way each behaves in relation to the other (Shipway, I., 2009).
The modern portfolio is not only about mixing the portfolio with different securities. There are two key assumptions for the theory:
• Investors are rational and want to achieve a return in proportion to the given risk.
• All investors are risk averse. i.e. investors are willing to take any risk, but given two assets that offer the same expected return, they prefer the one with lowest risk (Shipway, I., 2009)
When investors fulfil the key assumptions, managers can determine an optimal portfolio for their clients. First, they select a set of asset categories as large- or small cap for instance. Then they generally start with historical performance of the index representing these classes to obtain estimates of returns, volatilities and correlations. The estimates are then used as inputs in the mean-variance optimization, which can be explained as the efficient frontier (see Figure 2-1). Then the portfolio is implemented using either index or actively managed funds strategy (Fabozzi, Gupta and Markowitz, 2002).
According to modern portfolio theory, the most efficient portfolio is achieved by investing at a point where the risk free and borrowing rate is tangent to the efficient frontier. This is where one will have the highest expected return given risk. In Figure 2-1 Rf represents the risk free rate, MVP is the minimum variance portfolio. The y-axis is represented by expected return and the x-axis by standard deviation (Fabozzi et. al, 2002)

### CAPM

The Capital Asset Pricing Model (CAPM) was introduced in the mid-1960s by William Sharpe and John Lintner, which resulted in Sharpe receiving Nobel Prize in 1990. Today, the CAPM is widely used in investment applications, such as risk evaluation, performance assessment of managed portfolios and asset pricing. The CAPM offers predictions about risk measurements and the relation between expected return and risk, but the model´s record is unfortunately poor. Its empirical problem probably reflects both the difficulties in its implementation and its theoretical failings, due to many simplifying assumptions (Hirschey, M., Nofsinger, J., 2008).
CAPM is considering the portfolio total risk consisting of two types of risk; systematic and unsystematic risk. The former involves risk between portfolio and market return, market risk, whilst the latter is the internal risk of the portfolio return. In a diversified portfolio the unsystematic risk is small, thus the total risk will mostly consist of the systematic risk (De Ridder, 2003).
The security market line (SML) illustrates the results from the capital asset pricing model where the x-axis represents the beta value and the y-axis represents the expected return. SML intercept the y-axis at the risk free rate and the risk premium gives the slope of the line. If a security is plotted above the SML then the security will yield high return in relation to its risk, the stock is undervalued. The security market line is based on that all securities will have the same amount of risk premium at a certain point in time. (Green, C. R., 1986)

Beta

The systematic risk can be defined as β (beta) of the portfolio and implies risk between the portfolio and the market. Since the beta covers market related risk it cannot be removed through diversification. β is an important part of the CAPM and the calculations of Jensen’s alpha (De Ridder, 2003). Beta is a measurement of sensitivity since it evaluates how the portfolio reacts to fluctuations in the market portfolio. A β-value of 1 implies that the portfolio exactly follows the fluctuations of the market. If the β-value is 1,5 it implies that the portfolio is reacting stronger than the market to fluctuations, i.e. if the market increases by 10% then the portfolio would increase by 10%*1,5=15%. Similarly a decrease in the market of 10% would result in a decrease of the portfolio by 15%. If the β-value is less than 1, say 0,5 then the portfolio is less sensitive to changes in the market. If the market would increase by 10% then a portfolio with a β of 0,5 would increase by 10%*0,5=5% (Morningstar, 2012a)

Jensen´s Alpha

Michael Jensen first introduced Jensen´s alpha also known as the Jensen´s Performance Index in the 1960s. He introduced alpha to evaluate performance of fund managers, to see if there was a difference between the actual return of the portfolio during a period and expected return of the portfolio by using the Capital to asset pricing model (CAPM). The definition of alpha is the actual return compared to the expected return based on the systematic risk (beta). A positive alpha indicates that the development of the fund has been better than the given beta. Let us say a fund has an alpha value of -1.2, a beta of 0.97. Then the negative alpha says that the fund performed 1.2% poorer than the beta had predicted. On the other hand, if the alpha would have been 1.2, then it tells us that the fund has performed 1.2% better than the beta had predicted. The higher the alpha, the better the fund has performed relative to its beta. (Morningstar, 2012b)

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Sharpe Ratio

There are numerous approaches of measuring portfolio performance. Nobel price winner William F. Sharpe introduced the Sharpe-Ratio, which is measuring risk-adjusted performance of a portfolio (Sharpe, 1966). This performance ratio has become one of the most frequently used ratios when measuring portfolio performance (Andrew W. Lo, 2002). The ratio is constructed through a risk and performance relationship measurement by James Tobin. Sharpe constructed the ratio as a reward-to-variability ratio, i.e. a ratio that describes reward per unit of variability. In 1994 Sharpe modified the ratio to make it measure what it does today; expected return per unit of risk.
To clarify the use of the measurement, when comparing two portfolios considering return as the only factor, the result is considered biased, therefore it is important to take risk into consideration (Simons, 1998). Because of the fact that a rational investor seeks the highest return for exposed level of risk, the risk cannot be ignored (Morningstar, 2012c). For a comparison between two different portfolios to be possible, adjustments for risk have to be made. When using the Sharpe ratio it is possible to compare different portfolios carrying completely different risk levels but also with different correlation to an index benchmark (Simons, 1998). An assumption that needs to be fulfilled in order for two different Sharpe ratios to be comparable is that the funds in subject of the comparison need to invest in the same market. The reason for this is the fact that what may be considered as a good Sharpe ratio differs from market to market (Morningstar, 2012c). A positive Sharpe ratio implies that the portfolio generates greater return than the risk-free asset. On the other hand, a negative Sharpe ratio implies that the return of the portfolio is lower than the return of the risk free asset. Comparing two different portfolios, the one with the highest Sharpe ratio is preferred (Morningstar, 2012c).

When interpreting the Sharpe ratio there can be some difficulties when trying to analyse return differentials between portfolios. This is the case both for the Sharpe ratio and the Treynor ratio, and can be seen as a disadvantage in using these models (Scholz, H., Wilkens, M., 2005).
Modigliani and Modigliani propose a different measurement, compensating for this problem. In 1997 they presented the Risk-Adjusted Performance measurement, which is being expressed in basis points for the different portfolios. The risk-adjusted performance of portfolio i (RAPi) can be compared to the market average return rm in order to show whether the portfolio has out- or underperformed the market expressed in basis points (Modigliani, F., Modigliani, L. 1997).
The RAP is computed through increasing or reducing the amount of leverage of the portfolio in interest for the analysis. Levering a portfolio implies that the investment in the portfolio is being increased through borrowing. This will both result in an increase of portfolio risk and an increase in expected return. To clarify, if we invest an additional amount of d% in the portfolio then the risk and expected return will increase by d%. Reducing the leverage of the portfolio on the other hand implies selling parts of the portfolio and using the additional earnings to invest in a risk free security. This will decrease the portfolio risk but also the expected return by d% (Modigliani, F., Modigliani, L. 1997).
The reason for increasing or reducing the leverage of the portfolio is to make it risk-equivalent to the market, i.e. making the risk of the portfolio matches the risk of the market. This can also be made by transforming the Sharpe ratio into terms of absolute performance (Scholz, H., Wilkens, M., 2005).

Efficient Market Hypothesis

A well-debated theory within discussions of finance is the Efficient Market Hypothesis. In short, the theory is based upon the assumption that the price of a security is a reflection of known information about that specific security. I.e. if new information becomes available about a company, the price of that company’s securities will be directly and rationally adjust according to the new information. If the new information is positive, then the price of the security will increase. If the information is of negative nature it will decrease. (Fama, 1970)
This implies that in an efficient market, prices of the securities would be a reflection of all already known information. The only way for prices to change is when new information about the company is released. New information is considered impossible to predict, and therefore the future price of a security is impossible to predict. This is where the theory of “Random Walks” is being introduced which is built on the assumption that all future prices are completely random. (Malkiel, 2003)
Fama (1970) divides the market into three different efficiency levels; weak, semi-strong and, strong form.
1. In the weak form of the efficient market hypothesis, prices are reflected only on historically available information (Fama, 1970). Hence it does not support the usage of technical analysis to add further value to an investment.
2. The semi-strong form states that prices reflect all available information and that new information immediately changes prices (Fama, 1970). This means that public information that is being released will have an immediate impact on prices that implies that fundamental analysis adds no value to the investment.
3. For the strong form of the market hypothesis, prices are reflected by all available information, including insider information (Fama, 1970). Because of the fact that all information is included in the price, fundamental analysts´ work is not a value adding process (Malkiel, 1999)

1 Introduction
1.1 Background
1.2 Problem
1.3 Purpose
2 Frame of Reference
2.1 Risk and Return
2.1.1 Modern Portfolio Theory
2.1.2 CAPM
2.1.3 Beta
2.1.4 Jensen´s Alpha
2.1.5 Sharpe Ratio
2.2 Efficient Market Hypothesis
3 Method
3.1 Research Approach
3.2 Selection of Fund Market
3.2.1 Laws & Regulations
3.3 Index as a Benchmark
3.4 Selection of Funds
3.5 Fund Categories
3.6 Selection of Time Span
3.7 Data Collection
3.8 Performance Measurements – Return
3.8.1 Security Market Line
3.9 Risk-Free Rate of Return
3.10 Hypothesis testing
4 Results
4.1 Accumulated Return
4.1.1 Significance test – Accumulated Return
4.2 Standard Deviation
4.2.1 Significance test – Standard Deviation
4.3 Sharpe Ratio
4.3.1 Significance Test – Sharpe Ratio
4.4 RAP – Risk Adjusted Performance
4.4.1 Significance Test – Risk Adjusted Performance
4.5 Jensen´s Alpha
4.5.1 Significance Test – Jensen’s Alpha
4.6 Beta
4.6.1 Significance Test – Beta
4.7 T-test Summary
4.8 Performance 2007-2009
4.9 Security Market Line
5 Analysis
6 Conclusions
7 Suggestions for Further Research
References
Appendices

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