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**Theoretical Framework**

** Premium Pension Funds in Sweden**

A fund is a collection of financial securities with ownership to the investor who possesses holdings in the fund. The purpose of having fund savings is to get an easy access to a portfolio of financial securities that possibly can implicate an increase in value. The funds available in the premium pension selection have requirements regarding the distribution of the funds risk through diversification, by containing a number of different financial securities. The holdings of the fund are determined by the fund’s manager, but it has to be within the specified requirements (Pensionsmyndigheten, 2018g). In order for a fund to be accessible in the premium pension selection, the fund must be approved by the Swedish pension authority and the fund’s manager must have a certification of operating in fund trading activities. The fund manager also needs to have a cooperation agreement with the Swedish pension authority and has to release specified information at requests. Furthermore, managers also need to undertake the requirements of not charging any withdrawal fees or other fees that has not been permitted by the agency (Riksdagen, 2018a).

The premium pension selection has different sorts of funds available, namely equity funds, interest-bearing funds, mixed funds, and generational funds. The motivation for using diverse funds is to make it easier for the investors to separate them and to be able to make comparisons. What kind of category the fund belongs to depends on what combination of financial securities the fund invests in (Pensionsmyndigheten, 2018g). There are two main types of funds, equity funds and interest-bearing funds. Interest-bearing funds only hold assets in debt securities and generally have a lower risk than equity funds. Equity funds, that is the most common fund alternative among Swedish investors and that will be the focus of this report, holds assets that are traded on the stock market (Avanza, 2018). Consequently, the fund will result in ownership of several companies. An equity fund contains ownership of at least 16 different companies, usually more, meaning that the risk will be diversified and not dependent on a single company (Pensionsmyndigheten, 2018g). An equity fund is therefore determined by the fluctuations on the stock market. Factors that may affect the degree of risk that the fund is exposed to are dependent on whether the fund manager invests in equities in Sweden or abroad, and in a stable or unstable market. An equity fund that is invested in markets and industries that has a high degree of fluctuations due to the world economy and currency movements, will ultimately result in a higher degree of risk (Avanza, 2018).

AP7’s equity fund functions as a normal equity fund but has particular directions from the Swedish pension authority. The fund cannot acquire more shares in a single company to the extent that it exceeds five percent of the total voting rights for all shares. The assets managed by the Seventh AP equity fund shall be divided into fund units and all fund units must be equal within the fund. The fund manager must also calculate the fund value every day and inform the pension authority about the value (Riksdagen, 2018b)

**Risk-Return Trade-Off**

In financial markets, the performance of monetary properties are measured by well-known mathematical models of risk and return. These statistical measurements are central to the finance community due to the reason that it enables the comparison of performance of various asset classes, thereby simplifying the selection of investment strategies (Higgins, 2015).

The financial concept, known as the risk-return trade-off, states that the return an investment will yield should increase as the risk associated with that investment increases. The risk-return trade-off is an important factor that affects the decision taken by the investor (Aslanidis, Christiansen and Savva, 2016). Investors have different levels of risk tolerance, meaning that some investors are willing to take on a low risk investment that can yield a potentially lower return whereas others are willing to take on higher risk that can yield a greater potential return. Subsequently, some investors are risk-averse whereas some are risk-takers. The level of risk tolerance can be affected by years to retirement, gender, and education (Bollen and Posavac, 2018). Furthermore, it is important to recognize that higher risk does not necessary imply a higher potential return. The risk-return trade-off only indicates that there is a relationship between a higher risk level and the likelihood of greater returns, however, higher risk can also cause problematic situations associated with major losses on an investment due to changes in the economic conditions (Aslanidis, Christiansen and Savva, 2016).

The risk-return trade-off is central to the field of finance, thereby making it an important tool for investors to use when assessing what level of risk they are willing to retain when making an investment. Moreover, it can be useful for investors to understand how risk and return are correlated in order to make a good investment decision (Lundblad, 2007)

**Financial Theory**

**Single Index Model**

The single-index model was introduced to the field of finance by William Sharpe (1963) and is an asset pricing model that measures both the risk and the return of a stock. According to the single-index model, the systematic risk that affects the returns of the stocks is caused by only one macroeconomic factor and that this factor can be characterized by the rate of return on an index, for example the S&P 500. This assumption has been made in order to make the analysing process easier (Sharpe, 1963). Furthermore, the model is based on the following assumptions (Sharpe, 2000):

- Nearly all stocks have a positive covariance, due to their similar response to macroeconomic factors.
- There are firms that are extra sensitive to macroeconomic factors and this firm-specific difference is usually symbolised by its beta.
- The covariance between stocks arise from different responses to macroeconomic factors. As a result, the covariance of the individual stock can be calculated by multiplying the beta of the stock with the market variance.

The single-index model equation, i.e. the mathematical expression, is stated as:

Where, represents the return of the stock and is the risk-free rate, symbolises the stocks abnormal return, represents the stocks sensitivity to the market return, is the market portfolio return and represents the residual returns of the stock which is caused by firm-specific factors. The single-index model was developed to simplify the portfolio analysis and to make the process of analysing the relationship amongst securities easier, today the model is widely used in the finance industry

**Modern portfolio theory**

The modern portfolio theory was introduced by Harry Markowitz (1952) and is a commonly used financial theory. Markowitz highlighted the importance of utilizing diversification and its effects on the portfolio to increase risk-adjusted returns. Markowitz emphasized that investors could through diversification reduce the risk, by allocating the holdings in different asset types, such as in different companies and industries. Risk is measured as the standard deviation and through risk spreading, investors can exploit the relationship between risk and return thereby improving their investment selections to maximize returns given a certain level of risk. Stated differently, it is significant to analyse how the selected assets in a portfolio relate to each other in terms of risk and return.

Markowitz theory is essentially built around the basis of two assumptions. Firstly, all investors are rational in their investment decision making. Secondly, all investors want to achieve the greatest return as possible given the lowest risk. In other words, investors are risk-averse, i.e. they want to avoid risk. Given two investment opportunities that yield the same return, the investor would choose the alternative with the lowest risk as there is no reason for the investor to choose the high-risk option given these assumptions.

The Modern Portfolio Theory contains several risk-measurements and risk-adjusted performance measurements, such as variance and standard deviation as well as Treynor ratio and Sharpe ratio.

**Variance**

The variance is a frequently used measure of dispersion, known as the squared expected deviation from the mean. The variance formula is stated as:

Where expresses the rate of return, ^{̅} is the average rate of return and denotes the number of assets (Berk and DeMarzo, 2014). In general, when calculating the variance, the mean rate of return is needed. However, the mean rate of return typically is an unknown factor and as a result of this the average realized rate of return can be used instead

**Standard Deviation**

The standard deviation, frequently referred to as the volatility, is calculated as the square root of the variance. The standard deviation is used to calculate the degree of dispersion when using a set of data. A low standard deviation specify that the data points are near the mean whereas a higher standard deviation express that the data points are further from the mean, resulting in a higher deviation. The standard deviation formula is expressed as (Berk and DeMarzo, 2014)

**Treynor Ratio and Beta**

The Treynor ratio, developed by Jack Treynor (1965), is a measure that exploits the correlation amongst annualized risk-adjusted return and risk in order to measure efficiency. In other words, the ratio tries to quantify to what extent an investment has compensated the investors given the degree of risk. The Treynor ratio is dependent on beta, which is a measurement of a stock’s sensitivity to market fluctuations. The principle idea of the Treynor ratio is that systematic risk, i.e. risk that is central to the whole market, must be penalized due to the fact that it cannot be reduced through diversification.

The Beta formula is expressed as: = ^{( , )}, where denotes the return of the asset and the return of the market (Hübner, 2005). A Beta equal to one indicates that the price of an asset moves with the market, whereas a Beta lower than one signifies that the asset theoretically is less volatile than the market and a Beta greater than one means that the asset theoretically is more volatile in comparison to the market. The Treynor ratio can be calculated as: = ^{−} , where represents the return of portfolio *i,* is the risk-free rate and denotes the portfolio beta (Hodges, Taylor and Yoder, 2003).

**Sharpe Ratio**

The Sharpe ratio, developed by William Sharpe (1966), measures the excess rate of return of a given asset and adjusts for risk. There are similarities between the Sharpe ratio and the Treynor ratio, but instead of using the beta factor, Sharpe ratio uses standard deviation. A higher value of the Sharpe ratio indicates that the security has superior performance, concluding in a greater risk-adjusted portfolio with a higher rate of return for every unit of risk (Sharpe, 1966). The Sharpe ratio is defined as (Berk and DeMarzo, 2014):

Where the [ ] is the expected portfolio return, denotes the risk-free rate and ( ) symbolises the standard deviation of the portfolio.

**Post-Modern portfolio theory**

The Post-modern Portfolio Theory is a further development of the well-known Modern Portfolio Theory. The basis for both theories is to explain how investors ought to use diversification to enhance their portfolios and how to price risky assets. In the early days, the Modern Portfolio Theory reformed the decision-making process of making investments within the field of finance by describing risk associated to investments and thus introducing a risk-return framework. The limitations of the Modern Portfolio Theory, such as assuming that variance is the correct risk measure and referring to all returns as normally distributed, steered the model to addressing all uncertainties the same. As a response to the inadequacies of the Modern Portfolio Theory, and the fact that the model penalizes both upside and downside deviation similarly, the creation of a new investment decision-making framework that would overcome the limitations of the Modern Portfolio Theory started. Today, this model is known as, The Post-modern Portfolio Theory (Rom and Ferguson, 1993).

**Sortino Ratio**

The Sortino ratio differentiates itself from the Sharpe ratio by measuring the risk-adjusted return of an asset, or portfolio, using the target rate of return. Although both ratios quantity an assets risk-adjusted return, the way of calculating is significantly different. The Sortino ratio use the so-called downside risk in the denominator, as a substitute for the standard deviation, meaning that the ratio only penalizes the return below the target rate of return (Sortino, 2010). The Sortino Ratio is expressed as (Sortino and Satchell, 2001):

Where signifies the rate of return, denotes the minimal acceptable rate of return and symbolizes the downside deviation

**Capital Asset Pricing Model**

A central question in the field of finance is how the risk affects the expected return of an asset. The first consistent framework that answered this question, was the Capital Asset Pricing Model (CAPM). The model was presented in the early 1960s by William Sharpe (1964), John Lintner (1965a and 1965b) and Jan Mossin (1966). The CAPM is built upon the notion that not all risks have an impact on asset prices. Particularly, a risk that is diversified away when united in a portfolio with other investments is, roughly considered, not a risk of any kind. The CAPM is an advancement of Markowitz’s (1952) portfolio theory which is founded by the concept that specific risk can be disregarded with diversification, on the other hand, systematic risk can only be reduced and not removed. Furthermore, the CAPM provides us with understandings about what type of risks that is correlated to return.

The Capital Asset Pricing Model is based on four assumptions:

- Investors are risk averse and assess their investment portfolios only in the matter of expected return and standard deviation of return
- Capital markets are perfect: all assets are endlessly divisible; transactions costs, short selling restrictions and taxes are non-existent; information is free and accessible by everyone; and investors can borrow and lend at the risk-free rate.
- All investors have the equivalent investment opportunities.
- Investors make similar estimations of expected returns, standard deviations of return and correlation concerning individual assets.

The assumptions above characterise a remarkably simple and faultless world, however, they are needed for the CAPM to be functional. Under the stated assumptions, every investor will determine a matching portfolio of risky assets with the highest Sharpe Ratio. Given their degree of risk tolerance, every investor will assign a part of their wealth to this collection of assets, known as the optimal portfolio, and the rest to risk-free borrowing or lending. In order to guarantee that the market is in equilibrium, the price, i.e. expected return, of every individual asset must be such that investors cooperatively agree to hold precisely the supply of the asset. If all investors hold the equivalent amount of risky assets, that amount must be equal to the amount of risky assets that are held in the market portfolio, which is the portfolio containing all existing shares of each risky asset. When in equilibrium, consequently, the portfolio of risky assets that have the highest Sharpe Ratio has to be the market portfolio

1. Introduction

1.1 Background

1.2 Problem description

1.3 Purpose

1.4 Delimitations

1.5 Definitions

2. Theoretical Framework

2.1 Premium Pension Funds in Sweden

2.2 Risk-Return Trade-Off

2.3 Financial Theory

2.4 Previous Research

3. Method

3.1 Choice of Method

3.2 Collection of Data

3.3 Research Design

3.4 Hypothesis Testing

3.5 Critical assessment

4. Empirical Results

4.1 Risk Exposure

4.2 Risk-Adjusted Performance

4.3 Portfolio Optimization

5. Analysis

5.1 Risk Exposure

5.2 Risk-Adjusted Performance

5.3 Portfolio Optimization

6. Conclusion

7. Contributions to the Research and Suggested Further Studies

8. References

8.1 References to theory sources

8.2 References to data sources

9. Appendix

9.1 The 51 equity funds included in the study

9.2 Start-of-month net asset values for AP7

9.3 Risk-free rate

9.4 Yearly average rate of return

9.5 Yearly risk measurements

9.6 Yearly risk measurement ranking

9.7 Yearly risk-adjusted return measurements

9.8 Yearly risk-adjusted return measurements ranking

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