## Evaluation regarding the US fund market

*A comparison between different US fund risk * *classes and their performance *

*Author: Albert Svensson Kanstedt *
* Victor Sjöstrand *

*Supervisor: Christopher von Koch *

### Bachelor Thesis

### Abstract

The intent of this thesis is to investigate how US equity funds performance differ due to their standard deviation. In order to accomplish this study, we collected daily data for 99 US equity funds for the period 2011-2020 and divided the funds into three risk classification groups based on their standard deviation for the year 2011.

The collected data was used to perform an CAPM regression and to calculate returns on a three-, five- and ten-year basis. The results for the regression and the returns for the funds was later presented as average values for the

different risk classification groups. We then compared the average outcomes for the three risk classifications with each other and the index S&P 500.

Our result showed that the index S&P 500 outperformed the three risk classification groups average returns for every time period. We also noticed that the difference between the average returns and the index got greater by time.

We did not find any big differences between our risk classifications when it comes to their performance. Our regression analysis resulted in many negative alpha values indicating that S&P 500, as many previous studies claims, outperforms actively mutual funds. The conclusion is therefore that we could not show any evidence that the there is a major different in performance between our risk groups but also that it is difficult for fund managers to outperform index.

### Key words

Standard deviation, US Equity Funds, S&P 500, CAPM, Efficient Market Hypothesis, random walk.

### Acknowledgements

We would like to thank our supervisor Christopher von Koch and our examiner Håkan Locking for their guidance throughout this thesis. We are very grateful for the help and guidance you have provided for us.

### Table of contents

**Abstract ... 1**

**Key words ... 2**

**Acknowledgements ... 3**

**1. Introduction ... 6**

*1.1 Background ... 6*

*1.2 Problem discussion ... 7*

*1.3 Research questions ... 8*

*1.4 Purpose ... 9*

*1.5 Limitations ... 9*

**2. Theory and previous studies ... 10**

*2.1 Theory ... 10*

2.1.1 Efficient Market Hypothesis ... 10

2.1.2 Random walk ... 11

2.1.3 Equity Premium Puzzle ... 12

2.1.4 Loss aversion ... 13

2.1.5 CAPM ... 14

2.1.6 Fama-French Three-factor model ... 15

2.1.7 Carhart four factor model ... 17

*2.2 Previous studies ... 18*

2.2.1 Synthetic Risk Reward Indicator (SRRI) Funds Risk Classification ... 19

2.2.2 Different types of mutual funds ... 20

2.2.3 The US fund industry ... 22

2.2.4 S&P 500 ... 22

2.2.5 Statistical variables ... 23

**3. Methodology ... 25**

*3.1 Study Approach ... 26*

*3.2 Data collection ... 27*

*3.3 Returns ... 27*

*3.4 Regression analysis ... 28*

**4. Empirical results ... 30**

*4.1 Returns ... 31*

4.1.1 <25% Volatility Risk classification ... 31

4.1.2 25-30% Volatility Risk classification ... 32

4.1.3 >30% Risk classification ... 34

*4.2 Regression results ... 37*

4.2.1 2010 Standard deviation risk classification <25% ... 37

4.2.2 2010 Standard deviation risk classification 25-30% ... 40

4.2.3 2010 Standard deviation risk classification >30% ... 42

**5. Analysis ... 46**

*5.1 Which of the different US volatility risk classification funds will generate the *
*highest return over a three, five and a ten-year period? ... 47*

*5.2 Does index outperform mutual funds in the regression analysis? ... 48*

**6. Conclusion ... 51**

**7. Further research ... 52**

**References ... 54**

**Appendix ... 59**

### 1. Introduction

1.1 Background

Since 1980, the interest in investing in mutual funds in US households has increased by 714%, from 5,7% in 1980 to 46,4% by the year 2019 (Statista 2020a). And due to the digitalization, it is now easier for households to invest their money in mutual funds and the supply for regulated open-end funds has increased by 40%, worldwide, since 2011 to 2019 (Statista 2020b).

Gruber (2010) writes that investing in mutual funds has become one of America's fastest growing investment types. A possible explanation of this may be the fact that you can buy or sell an asset at their net asset value and that the price of the management ability may not be priced he further examine. If this is the case, the performance of the mutual fund would be predictable and thereby, if the investors are aware of the predictors, they should be able to benefit from it and the cash flow should be better than average.

Today, there are a bunch of different types of mutual funds, which investors
*has to make a decision in between, and according to Bodie et al´s book *
*Investments the most common classification of mutual funds is; money *
*market funds, equity funds, sector funds, bond funds, international funds, *
*balanced funds, asset allocation and flexible funds, and index funds. These *
different classifications divide the mutual funds based on their characteristics
which differs between the different classifications. Some classifications of
mutual funds are considered more or less risk-free due to what types of
investments the mutual fund consists of (2018, p.96). Therefore, it can be
interesting to analyze how the different investment types can affect how

investors choose to invest regarding their risk-aversion and how it can affect the returns.

This study will examine which of the low, medium or high volatility US- equity funds that are the most profitable during a three, five and a ten-year period. The data will consist of daily adjusted close return by analyzing US funds in different Standard deviation risk classifications. The CAPM model is afterwards used in order to more deeply analyze every individual fund and to be able to calculate and analyze the alpha, beta, Adjusted R-squared, level of significance and the expected return in order to answer our research questions. The purpose is to make a conclusion of the funds, collected and calculated by Datastream and see the impact during a three, five and ten year horizon. Initially three different risk classes will be divided in order to compare how different volatility funds perform. This is done by calculating the Standard deviation of all the funds collected in the year of 2011. The hypothesis the report will test is if any of the US funds risk classification is to be preferred during a three, five and a ten-year time horizon in relation to the S&P 500 index.

1.2 Problem discussion

Due to earlier reports and that mutual funds have become popular in US households; it is therefore of interest to examine the US fund market to contribute to the existing research in the field with results on how to invest on the current risk level of funds within different time horizons.

A big part of the previous research in this subject has examined that it is beneficial to invest more passively due to the Efficient Market Hypothesis in the long run. The theory states that when the financial markets are efficient, the price of an asset will reflect that all information is available and therefore that it is not possible to outperform the market. The only way to receive a

higher return further on is therefore by taking a higher risk. A major part of the studies that have been done today in the subject, both in the US and in other countries, often compares the performance between passive and index funds. Hence, it is less justified to contribute with a study within the same niche. This therefore makes it justified for the study to also analyze the returns from various risk-classified active mutual funds.

Malkiel (2003), claims that changes in the stock price depend on the new information and explains it as a “random walk” where tomorrow's prices do not necessarily need to differ from today's price changes. Therefore, the prices need to be unpredictable, and as the Efficient Market Hypothesis supports, also reflect on all information available. Even if a weak form of Efficient Market Hypothesis would occur, passively investing would still be to prefer due to factors such as transactions costs and fee to managers etc.

And due to the fact that many people today still invest in mutual funds, it is therefore interesting to examine which of the different volatility classes in the US investors should choose, in order to receive the highest returns.

1.3 Research questions

To be able to answer the purpose of this question, essential research

questions need to be investigated. This paper will compare different volatility US-equity funds, compared to the S&P 500 and leads us to the two

questions.

1. Which of the different US volatility risk classification funds will generate the highest return over a three, five and a ten-year period?

2. Does index outperform mutual funds in the regression analysis?

1.4 Purpose

The purpose of this paper is to examine which of the US-equity funds standard deviation risk classification is to prefer during different time intervals, compared to index and if the Efficient Market Hypothesis holds.

1.5 Limitations

In order to define this paper, we have chosen to focus on the mutual fund’s markets in the US. We have chosen to investigate our thesis only for the US funds market to limit this paper and not make it too wide. Therefore, we have only chosen US equity funds with a minimum of 70% US equity. It would have been a too detailed study, if we would choose to investigate the fund markets worldwide.

Regarding the time period, we motivate that a ten-year period is relevant to our study. If we instead would have chosen a longer time interval, we would have needed to contain a larger amount of data and domestic theories.

Therefore, we chose to just focus the study for the US mutual fund market.

The time interval for the study is between 2011-2020.

If the paper would have investigated a longer time period, we would receive a wider perspective of the funds returns in purpose of giving the investor a broader investment base. It would be interesting to investigate over a longer period of time and compare with other countries that have a weaker efficient market but that would make this paper too extensive.

### 2. Theory and previous studies

2.1 Theory

**2.1.1 Efficient Market Hypothesis **

The Efficient Market Hypothesis says that when the financial markets are efficient, the price of an asset should reflect that all information is available, which prevents these assets from being under or overvalued. The purpose is that everyone should have the same conditions and people can therefore not make arbitrage profits because some possess more information than others.

The hypothesis therefore demonstrates that the only way you will receive a higher return, is by taking a higher risk (Fama, 1970). This makes it justified to analyze how the different risk class funds performed, due to risk and return, and to examine if the Efficient Market hypothesis is as tangible as many reports today claim and if it's possible to make market profits without taking a bigger risk.

For the hypothesis to be applicable, Fama further examines that certain criteria’s must be met. Firstly, no transaction costs are included. Secondly that everyone should interpret the information in the same way and finally that the information is available to everyone.

According to Brealey et al. (2016) the Efficient Market Hypothesis can be divided into three different sections depending on how pronounced it is in different markets. The different forms are called; Strong, Semi - Strong and weak.

If a strong market hypothesis prevails, all the information is fully available, and investors can therefore not use historical information, annual reports, or insider information for example to make profits since the whole public has

the same information. (Fama, 1970), also expands the statements by saying that when new public information is presented, the price of an asset will be adjusted due to this information and therefore no arbitrage profits could be made in a Strong market Hypothesis.

In a Semi-strong market hypothesis, the investors can still not use historical return, annual or quarterly reports data to make profits, but have certain tendencies that could indicate a certain lack of information, according to Brealey et.al. (2016).

Brealey further explains if a Weak market would exist, experts and other types of investors can make a risk adjusted profit due to information problems. This means that, unlike above, investors can analyze and use historical information and look at annual reports to, what Brealey claims, use as an advantage to make a deeper technical analysis to receive a risk adjusted return.

**2.1.2 Random walk **

In the 1950's, a man named Maurice Kendall examined why stock prices are moving in the way they are doing. The result of his study showed that the changes in the stock prices were due to market psychology and the stock prices moved randomly without any pattern. Economists did at first find this result disturbing due to the fact that this result would mean that the financial market is not efficient and not logical.

After a while, economists changed their minds about Kendall’s result and they later thought that his result indicated that the market is efficient and well-functioning. The reason for their change of mind was that if all

information is available about a stock or an asset, the stock price will adjust

to that information. For example, if a stock is considered undervalued and there is information on the market, the stock price will adjust to a higher price immediately. When investors are aware of information of an undervalued stock for example, it is adjusted as they tend to buy, as the current stock owners will keep their holdings. All investors have different opinions about the available information on the market, and of course tend to be one step ahead all the time. Therefore, the random walk occurs.

The random walk is explained more concisely saying that stock prices can be moving randomly because of how the market reacts to the available

information. This is a sign that the market is efficient. If the market would not have a random walk, then it would be inefficient because of people's knowledge about how the stock's price will change (Bodie et al., 2018.

p.334).

**2.1.3 Equity Premium Puzzle **

The Equity Premium Puzzle on the other hand, is a phenomenon that tells us that high risk investments can outperform more risk-free investments

overtime. An example to explain this is the stock- and bond market where
stocks are considered a high-risk investment while bonds are considered low-
risk. Generally, even if there is a difference in low risk and high-risk assets,
high risk investments are generally the better investment in terms of the
financial return over time. Investors find it hard to explain why this
*phenomenon occurs, but one possible reason can be the Myopic Loss *
*Aversion. *

The Myopic Loss Aversion explains that if you follow your high-risk investment with a high volatility on a daily basis, investors can be afraid when their investments drop and can come to a stage where they panic sell.

This theory states that it is unpreferable to follow your investments on a daily basis and that it is preferable to survey your long-term investments less frequently. This is due to the fact that it is easier to make a more evaluative analysis of the stable change in your investments. Investors are then more likely to keep their assets for a longer period of time which according to the Equity Premium Puzzle generates more money, instead of investing in low- risk assets over time (Benartzi & Thaler, 1995) (Bodie et al., 2018, p.416- 418).

The Equity Premium Puzzle also describes that stocks, which are considered as a risky asset, for the most outperforms bonds over time. In Shiravnies et al. article (2020) they show data how the stock market has beaten the bond market on average between 1900 and 2018. The annual return of the S&P 500 was 6,81% while the annual yearly return on treasury bonds for the same period of time was 0,987%. This result shows that investing in the stock market ahead of the bond market is preferable over time, despite the risk differences, Shiravnies further claims.

**2.1.4 Loss aversion **

Loss aversion is a part of risk aversion which describes how an investor is considering a risk before making a decision. If an investor is classified as risk averse, it means that the investor prefers more risk-free investments ahead of high-risk investments (Bodie et al., 2018, p.159).

The definition of loss aversion is that the investor is more scared of losing money than earning money (Bodie et al., 2018, p.378).

This is a psychological concept that describes how humans behave which have an influence on the financial markets and on the private investments.

This can make people more prone to invest in less risky investments instead of investments with a greater risk (Schmidt & Zank, 2005).

According to the research question it is interesting furthermore, to consider how the Equity Premium Puzzle and Loss aversion can have an impact and be explained by the fund managers decisions, and on the return of

investment. It should, however, be noticed that many writers claim that it often is a matter of psychological aspects and therefore can be difficult to measure.

**2.1.5 CAPM **

*The Capital Asset Pricing Model, more known as CAPM describes the *
relationship between the market risk and expected return on an asset. This
model helps investors to find the expected returns of their investments and
set the price for the asset and is therefore frequently used as a base for their
investments. The result of the CAPM-formula indicates what a reasonable
return on your investment should be and is therefore a simple and useful tool
to compare your result with the reasonable outcome that CAPM provides.

(Bodie; Kane; Marcus 2018, p.277-300)

The Capital Asset Pricing Model consists of the Risk-free rate, beta and the expected return on the market. The formula can be written as follows.

*R**i**=R**f**+*𝛽**(R**m**-R**f**). *

As it sounds, the risk-free rate describes the rate that is considered to be risk free and the expected return on the market describes the return that is

expected on the market. The Beta describes how, for example, a stock is correlated with the market. If a stock has a beta value of 1, then the stock will follow the market at the same rate as the market index. If the stock had a beta

value of 1,2 instead, the stock is considered more volatile than the market and the stock changes 1,2 times the market index. The beta is also known as a measurement for the market risk or the systematic risk (Bodie et.al., 2018, p.250, 257).

*When a CAPM regression analysis has been made, we can use alpha to *
compare the result with the market. Alpha is a measure of the difference
between the security market line and the return of an asset. A positive alpha
indicates that an invested asset is performing better than the market index
which is the goal for active investors. A negative alpha shows that the
investment is performing worse than the security market line and therefore
worse than the market (Bodie et.al., 2018, p.286).

**2.1.6 Fama-French Three-factor model **

There are today several models that can be used to determine and analyze stock return, not only the regular Capital Asset Pricing Model. Based on CAPM, the Fama-French Three-factor model is an extension of this, which in comparison describes the return of stocks based on three different factors.

*These factors is The Market risk, the outperformance of the small cap *
*companies relative to the large-cap companies and lastly, the *

*outperformance of high book-to-market value companies versus low book-to-*
*market value companies. The model was developed by two professors at the *
University of Chicago, Euguene Fama and Kenneth French and where the
motive of the model is that companies who are smaller and companies which
have high values, tend to outperform the overall market more regularly.

The Fama-French Three-factor model is presented as;

*Expected rate of return = Risk free rate + Market Risk premium + SMB + *
*HML. The Market risk premium in model describes the difference of the *

expected market return and the risk-free rate and provides the investor with an compensation excess return for the extra risk over the returns and the risk- free rate.

The SMB, Small minus Big, describes the historic excess difference between small and high-cap companies and where the main motive, as mentioned above, is regarding the outperformance in return the small-cap companies has compared to the big-cap companies in the long run.

The Last Factor in the Fama-French model, the HML, High minus Low, represents the spread in returns between high value companies and low value companies due to the book-market ratio. The HML factor describes that companies with high book-market ratio have a tendency to result in higher returns compared to companies with low book-market ratio. And as both the SMB and the HML factors are determined, a linear regression can be made in order to estimate a Beta coefficient which either can result in positive or negative values.

The Fama-French Three Factor model is mathematically presented as follows.

The model, which is an extension of the CAPM including the two extra factors, makes it more flexible but is on the other hand also based on the same assumptions as CAPM, saying that it requires higher risk to receive a higher return (Corporate Finance Institute).

**2.1.7 Carhart four factor model **

As the Fama - French three factor model is an extension of the CAPM, the
Carhart four factor model is an extension of the Fama French model and is
often used in price securities. This model was developed in 1997 by Mark
Carhart and in comparison, with the Fama - French three factor model, it
*contains an extra cross sectional momentum factor that increases the power *
of the explanatory in these multifactor models which helps to examine funds’

performance further and more accurately.

The mathematical formula of the model is explained as.

The formula is very similar to the three-factor model but where the MOM variable, that is the difference, stands for the momentum factor. Since Carhart developed this model there have been more extensions of these models but which is not relevant for this paper’s purpose. (Breaking Down Finance).

We are well aware that some studies may motivate the use of Fama French or Carhart, as these models are a more developed method than the original CAPM. By using one of these methods in our paper, we could provide a more nuanced result because more variables are being taken into account. In Fama French three factor model, for example, this would have meant that we included variables that include the variance of high value companies and small companies since they tend to outperform the market more often, which could have been an advantage as the model becomes more flexible than CAPM. Nevertheless, the Momentum factor in the Carhart four factor model.

But since several papers today use CAPM as their estimating model, we motivate the paper to use this model to be able to narrow it down in this size

of work, but still be able to make it as comparable as possible to other papers in the same area.

2.2 Previous studies

Today, different types of studies and reports have been published in order to investigate how mutual funds risks have affected the return of assets. There are different factors that affect both risk and return for investments, for example which assets that are included in the mutual fund and if the mutual funds is actively or passive managed. These characteristics affect both the risk and return. Schizas (2014) have investigated and compared activity funds with passive funds where the result showed that the active funds, not is as active as it is considered, by participants. The results also showed that it is a correlation between the two but where the difference in risk and

performance is bigger in the active funds. It also shows that active structure can overperform funds when it comes to the terms of return.

In 1995, Malkiel published a study where he compared to performance between all the active mutual funds in the US, against S&P 500 and Wilshire 5000. During his investigation, he was able to establish that most of the active funds had negative alpha values. The conclusion he came to was that the active managed funds generally performed worse than the S&P 500.

In the discussion regarding how actively managed funds can outperform index, an early study was done in 1968 by Jensen, which today has become a study on which much research has been based. In his study, he analyzed 115 actively managed funds over a 20-year basis between 1945-1964. The result that Jensen presented was in the same line as Malkiel later claimed, that most funds underperformed index and resulted in a negative alpha value. He ended up with two different conclusions, partly that the return for most of the funds

could not cover for the management fee, and that the index outperformed the active managed funds.

Cuthbertson, Nitzche and O’Sullivan (2010) presented in their study that only 10% of actively managed funds in the US and Great Britain can predict a positively significant alpha value compared to index. The study was conducted during the period 1975-2002 and generally shows that actively managed funds have shown negative alpha values compared to index.

Unlike previous studies and research presented above, a Swedish study was presented by Engstrom in 2004. The study compared 112 Swedish actively managed funds and the results showed positive alpha values, and where the funds outperformed index, which contradicts previous studies in the field.

To summarize, over the years, many studies have been published in the area where the performance of funds often is measured. Many previous studies often come to the same conclusion, that the majority of active managed funds exhibit a negative alpha value compared to index, i.e., that index tends to outperform active managed funds. At the same time, studies that have been done have also shown other results, which motivates future studies in the subject.

**2.2.1 Synthetic Risk Reward Indicator (SRRI) Funds Risk Classification **
Synthetic Risk Reward Indicator, SRRI, is a stable and comparable index
tool which categorizes mutual funds risk and rewards and aims to give the
investor an indicator on the risk they are taking by investing in different
funds. The SRRI is based on volatility and analyses the fund's weekly return
during the past five years. Depending on available historical data and type of
the fund, SRRI index will be determined. A high SRRI rating means that the

mutual fund carries a high volatility and a low SRRI rating means that it has a low volatility (Ewen 2018). This is a common risk classification, among many others, that is usually determined on the basis of a fund's volatility.

Even though the SRRI rating was not taken into account in this paper while splitting our risk classes, it still provides investors with essential information about the risk they are taking. The purpose of our paper is to contribute with research in which of the volatility classes is the most profitable in different time horizons. We did not use the SRRI rating, but the risk classification is the base for this study and the SRRI rating inspired us to examine funds with different risk classification and measure their performance over time

compared to each other.

** 2.2.2 Different types of mutual funds **

Today, there are several different classifications of mutual funds that hold different characteristics. Some of the mutual funds carry a higher volatility, hence a higher risk for the investor, the risk depends on what the mutual fund consists of. Fund managers that offer investors their mutual funds portfolio, present a specified investment policy in their prospectus and where the purpose is to explain their investment strategy for the potential investors, for example which type of assets and which industries they are going to invest in.

A money market fund invests in money market securities like certificates of deposits, commercial papers and repurchase agreements and often, the maturity for this type of fund is about one month.

Equity funds mainly consist of stocks and sometimes also include other types of securities. To equip the liquidity that the fund needs to meet the potential redemption of shares, the equity funds usually hold money market securities at a level of 4-5% of the fund.

Sector funds means that the fund is only investing in a certain industry. An example of this is if a fund only consists of investment from the medical industry and the fund will then reflect how that specific industry is going.

Bond funds is a type of fund that is specialized and is focused against different types of bonds like treasury bonds, corporate bonds and mortgage- backed securities. The funds often have a specialization regarding if it is short- or long-term bonds or by the credit risk.

International funds, like the name tells, this type of fund is focusing on investments world-wide. It can be index funds that reflect on the whole world economy, a specific area in the world or a special type of a country's market.

Balanced funds are a mix of different types of funds which means that the risk is diversified. The purpose of this type of fund is that the balanced fund can contain an investor's total investments in one single fund.

Like the balanced fund, the asset allocation and flexible funds are also investing in more than one type of asset. The main difference between these two types of funds is that the asset allocation and flexible funds are using investment managers to allocate the investments to the speculated more beneficial sector which makes this type of fund riskier in comparison.

Index funds are trying to reflect a specific market's performance and

therefore consist of investment for the wanted indexes market. The index can for example reflect the global market where you include investments from sectors worldwide or for a specific country where you only invest in a

specific country and the goal is to try to match the performance of the market (Bodie et al., 2018, p.96).

**2.2.3 The US fund industry **

The fund market has increased dramatically since many years back, together with the increasing number of funds available. The information above also supports the fact that the number of studies in the subject of US mutual funds also has increased. Many of the previous studies have focused on the US stock market where the information is very readily available due to their publishing rules and the fact that the US market is in a mature financial phase. In comparison with the American stock market, other markets have not been studied to the same extent. The authors describe that a reason for this may be due to the institutional structure in which the United States is at the forefront. The first results shown in the study was that the European funds were still lagging the US mutual funds due to the size of assets, size of average funds and market importance. The most obvious result when Otten

& Schweitzer compared the performance between the European and US domestic funds was that the US funds performed rather poorly compared with the European funds and performed greater results than small company mutual funds (Otten & Schweitzer, 2002).

**2.2.4 S&P 500 **

Standard & Poor's Composite 500, more known as S&P 500, represents an index of some of the 500 biggest companies, listed in the US. Bodie et al

(p.47, 2018) describes the S&P 500 as a market-value-weighted index which means that the index measures the included company’s stock price changes and converts all the company’s stock price changes into a percentage increase or decrease for the aggregated firms. This results in the S&P 500 index, which often is used as a comparison for investors or fund managers trying to outperform the index.

There are at the moment 505 companies included in the S&P 500. The index includes different sectors where the sector “Information Technology” holds the biggest proportion of the index with 26,7%. Big firms like Apple, Tesla, Microsoft and Facebook are a part of the S&P 500 (S&P Dow Jones Indices, 2021).

**2.2.5 Statistical variables **

When studying previous studies in the same area, several common statistical variables are often used in order to explain funds or stock performance. Two of the most common variables when estimating a regression model are the Beta and the Alpha value, but also the Adjusted R-squared and the

significance levels.

*2.2.5.1 Beta *

In CAPM the Beta estimates a stock or funds return to volatility in relation to the whole market return. It is a risk measurement that indicates how volatile the security is. The higher Beta, the higher risk, but at the same time, higher expected returns. If a fund for example has a Beta of 1,5, then the return would have been 150% in relation to the whole market return in a given period (Corporate Finance Institute a).

*2.2.5.2 Alpha *

The alpha value is a performance measure that for example compares a fund's performance compared to the S&P 500 index, as in our paper. A daily alpha of 0,0001 for example, indicates that the fund outperformed the market return of 0,01% per day during a given time period. A negative alpha value indicates that the fund is underperforming, compared to the market. The ratio is often compared with the Beta value in a portfolio analysis and to be able to distinguish the performance and risk of different assets, stocks, funds and securities (Corporate Finance Institute b).

*2.2.5.3 R-squared *

The R-squared is also a common variable in portfolio analysis which

*explains the Goodness of fit. It is a statistical measure often correlated when *
estimating a regression model that shows in what grade the dependent
variable can be explained by the independent, by other words, how good the
data is fitted in the model. The R-squared is presented between 0-1, for
example if the R-squared is 80%, then 80% of the data fits in the model. It is
often a good predictor when a high R-squared value is presented, but the
article also mentions that there are more factors which also may be crucial in
a regression analysis (Corporate Finance Institute c).

*2.2.5.4 Adjusted R-squared *

As the name may reveal, the Adjusted R-squared is a modified tool that adjusts for factors that are not Significant in regression models. It is similar to the ordinary R-squared estimating due to the fact that it explains how well data fits in the regression model but is a modified version that is the most commonly used (Corporate Finance Institute d).

*2.2.5.5 P-value *

The P-value, probability value, is a measure to find statistically significant or more extreme results while doing a hypothesis test. While studying the

estimated P-value, the purpose is to test if the null hypothesis is rejected or not in different significance levels. The most common strategy today is to test on a 95% confidence interval which will be used and tested in the Empirical part of our paper (Corporate Finance Institute e).

### 3. Methodology

We have chosen to investigate the market return for the mutual funds for a ten-year period between 2011-2020 on US equity funds. To be able to compare the results in perspective comprehensive the efficient market hypothesis the funds that were chosen needed to contain at least 70% of US equity. The first step is to randomly collect funds containing at least 70% US equity and collect daily adjusted close price data of each fund during the time period of 2010-12-31 to 2020-12-31.

After the first collection is made, the daily fund prices are converted into HPR, as presented below, where we can estimate the daily price changes in percentages. In connection with our purpose and to be able to divide the funds in different risk classifications, we choose to calculate the Standard deviation of each fund during the first year, to be able to split the funds in different risk classifications. This in order to be able to analyze which of the low, medium or high-risk volatility classes that performs best. This is done by using the STDEVA in Excel, but to be able to make a market comparison, the daily funds standard deviation is SQRT by 252 days in excel which is the average trading days in the US. This provides us with an annualized Standard deviation, which is most commonly used.

When the results are presented, we are able to compare the different funds Standard deviation and split them into our three different risk classifications.

Due to this, the different risk classifications that were chosen was; <25%

Fund Standard deviation, 25-30% Fund Standard deviation and >30%

Standard deviation during the year of 2010. The purpose of this method is to be able to investigate which level of risk is to prefer if the investment was made in 2011. Totally 144 funds were collected, of which 99 of these funds were included in this paper, 33 in each risk class. Since a random selection of funds is made, we choose to remove funds that were irrevocable and that contributed with extreme values, in order to make our study as valid as possible.

3.1 Study Approach

In order to test which of the US funds Standard deviation is preferable in a three, five and a ten-year period, testing in time series will be preferred when comparing our research question with previous reports in our area and in order to calculate the return in our different time periods. Therefore, the following HPR formula will be used (French et al., 1987).

The first step, as mentioned above, is to collect a common fund factor, and randomly collect funds to test their annualized Standard Deviation, collected in DataStream and calculated in Excel. The funds data during the time period 2010-12-31 - 2020-12-31 was collected, and the Standard Deviation was calculated between 2010-12-31 - 2011-12-31.

To be able to present a Standard Deviation with minimized error, the HPR formula above obtains the daily return and estimates an average. Secondly, to be able to answer the paper´s hypothesis, a Capital Asset Pricing Model (CAPM) regression of each fund will be estimated and tested, in comparison with the S&P 500 and see if it is rejected or not. Lastly, the Data will be presented both visually and quantitatively and an analysis and a conclusion will be presented.

3.2 Data collection

Our data, as mentioned above, will be collected by DataStream and is collected with a daily return routine to give the paper as best possible credibility while analyzing our funds. We have chosen the time period between 2011-2020 and have chosen to estimate 99 funds in total, 33 in each risk classification.

We have gathered our funds looking at Morningstar's site where they listed the US funds with 70% US equity. This selection is made randomly, but of these we have chosen 33 of each risk classification to receive such a high credibility that the paper has room for. Since the paper is testing the US fund market, the requirement is that US domestic funds are collected with 70%

US Equity as mentioned. This is in order to avoid foreign impact in our study as much as possible.

3.3 Returns

The first step in our Empirical part is to estimate a geometric return. A fund return is the value an investor will receive from an investment during a period of time. Today, according to Bodie et al. (2011 s.129-131), there are two common ways to calculate the return. These are the arithmetic and the geometric return. The arithmetic return method is estimated by adding

several returns for different time periods to calculate a total return. This method is according to Bodie et al. most appropriate when the purpose is to forecast a fund's future expected return, based on the funds returns from previous years. In this paper, the purpose is to compare the difference between our funds return, which makes it important to calculate a more accurate value of how the fund has performed during a certain period. The geometric return estimates an actual return for the given time period in this report, which makes it justified to use the geometric return in our study.

(Bodie et al., 2011, p.129-131)

Our return is calculated by taking the daily return in percentage, and by using
*excel typing; PRODUCT(1+time interval)-1, estimating our total actual *
return of all our funds in a three, five and ten year period of time which will
be visually demonstrated in our empirical part.

3.4 Regression analysis

To try to predict and compare the outcome of the mutual funds depend on the risk and return we are going to run a regression analysis. We will do this by using CAPM in Stata which will provide the paper with necessary variables such as Beta, Alpha and Significance levels to be able to analyze all our funds’ performance.

The CAPM formula consists of three different variables that affect the outcome and therefore it is important to collect proper data.

*When collecting the risk-free rate, R**f **, Damadoran claims that there are two *
basic conditions that need to be fulfilled, there cannot be a default risk or a
*reinvestment risk (Damodaran, What is the risk-free rate). With this in *
consideration we have therefore chosen that the risk-free rate in our

regression analysis further down, will contain a daily 3-month US treasury bill, that is lately divided by 360 days to get the average risk free rate per day and that has been collected for 10 years between 2011-2020. The US treasury bill is collected from DataStream.

The beta value will be estimated by the regression model in Stata. According to Damodoran, the beta value has two basic characteristics which are

essential to remember when estimating or using beta. The first is that the beta measures the added risk on the diversified portfolio, and not the total risk.

The second is that beta measures the relative risk of an asset. (Damodaran,
*Estimating Risk Parameters) *

For the market return variable, we will be using the average return for the index S&P 500 which will provide us with numbers that are relevant to the study as we are focusing on the US market. The data for the annual return for the S&P 500 index will once again be retrieved from DataStream with a daily return. When looking at previous reports that have studied the same area, the S&P 500 is the most commonly used.

When all data for these variables are collected, we can start performing the regression analysis on the US mutual funds within the different risk

classifications we have chosen. We will run the CAPM regression analysis in Stata with the data for the variables that we retrieved earlier. When the regression analysis has been made, we can start our comparison of the mutual funds due to their Standard deviation risk classification over the different time horizons.

When the regression analysis is made, an analysis of the results will be presented in relation to the Betas, Alphas, the R-squared and the level of significance.

### 4. Empirical results

The first primitive test that was made according to our research question is comparing the different funds return over different time periods compared to the S&P 500 index. This is presented in three diagrams with the different risk classifications below.

4.1 Returns

**4.1.1 <25% Volatility Risk classification **

The first risk classification visually presented above underperformed compared to the S&P 500 index. For three year, the return was not excessively lower but looking in the long run it is clear that the S&P 500 outperformed the <25% Standard deviation risk class. 15 out of the 33 funds overperformed the S&P 500 in a three year period.

After five years, the difference between the average return for the funds starts to fall behind more noticeable compared to the S&P 500. We can notice that the average return gets lower after five years compared with three years for the funds while the S&P 500 continues to grow. 2 of the 33 funds managed to overperform the S&P 500 in a five-year period. These funds were BRLGX ending at 87,76% and LOMAX ending at 68,91%.

The average return grows for the funds after ten years and ends at 100,77%

compared to 39,48% after five years and 44,45 % after the first three years.

The final result for the S&P 500 is 193,43% after ten years, 62,23% after five years and 44,75% after three years. There were also 2 out of 33 in a ten year period that outperformed the index but these were not the same funds as the two in the five year period. The funds that outperformed the S&P 500 were GCEGX ending at 206,22% and SAIFX ending at 193,68%.

**4.1.2 25-30% Volatility Risk classification **

While studying the medium risk fund class the result follows a homogeneous pattern compared to the first risk class. The average fund returns

underperform the S&P 500 index but have a general lower return the three years compared to the <25% Standard deviation risk class. Like the first risk classification, there is not a big difference between the average return for the funds compared to the S&P 500 for the first three years.

We can see the same phenomena after five years for the middle risk

classification as for the low-risk classification where the average return for the funds is lower after five years.

After three years, 13 out of 33 funds outperformed the index. After five years the pattern continues showing that comprehensively 4 out of 33 funds

outperformed the index and for ten years 2 out of 33 funds outperformed the index. TPLGX outperformed the index over the three time periods with a return of 410,7% over 10 years.

The similar pattern continues in this volatility class due to return as for the first risk classification, the return recovered after ten years and delivered an average return of 103,45%. The average return for the funds was 39,44%

after five years and 43,13% after three years.

**4.1.3 >30% Risk classification **

The results for the funds with a standard deviation greater than 30% shows the same pattern as for the two earlier risk groups as presented above. The S&P 500 outperformed the fund's average return for every time period even in this risk class.

Three years in, the average return for the funds was 41,35% in comparison with 44,75% for the index. This is the lowest result of the three risk

classification groups. The middle risk group had an average of 43,13% and the low-risk group produced an average of 44,45%. This means that the

funds with a standard deviation between 25-30% had the highest average return of the three groups.

The high volatility risk class that is lastly presented underperformed the other risk classes and in a five-year interval the general return is 32,03% compared with the other risk classes that has a five-year return of 39,48% respectively 39,44%. In the three-year period, 12 out of 33 funds outperformed the index, in a five-year period, 1 out of 33 funds outperformed the index, which was the NEFJX with 65,26%. In a ten-year period 2 out of 33 funds outperformed the index, NEFJX ending at 196,5% and HISCX at 211,43%.

After ten years did the high-risk classification group provide an average return of 101,96% which is higher than the low risk group but lower than the medium risk group.

To summarize the first part of the results above, we can see that the results between the risk classes for all three time periods are very similar. We can observe that the return after three years is higher than in a five-year period.

When observing the result in a longer time, we can see that the trend reverses and the return increases again based on a ten-year interval. We also see that the funds generally have a similar return as the S&P 500 after three years where several funds outperform the index, but the longer the time passes, the clearer the difference becomes, that the S&P 500 delivers a higher result in the longer term. Nor can we, according to the results, credibly prove that some risk classes perform better than others, based on calculating our geometric return.

4.2 Regression results

As mentioned above and to be able to estimate whether the different funds over or underperform compared to index, a CAPM regression was examined using Stata. In order to be able to analyze the funds’ performance and in correlation with the index, Alpha, Beta, R-squared and the level of

significance was tested in order to make a deeper analysis of the results. The test was estimated with a total of 2517 observations. When estimating the statistical variables and to be able to answer the research question the regression analysis was estimated from the different risk classification, visually presented as follows below.

The first variable Beta shows a correlation between the fund and the S&P 500 index and the Alpha on the other hand explains the daily performance of the fund compared to the S&P 500 index. And as mentioned above, the R- squared presents how good our data fits in the regression model and lastly the level of significance.

**4.2.1 2010 Standard deviation risk classification <25% **

As presented above, the Beta and the Alpha follow a slightly homogenous distribution with some deviations. The average estimation of the first <25%

risk classification alpha resulted in -0,000155 and the Beta in 0,99. The Adj R-squared in the table is also presented in the result due to the fact that it represents variations in different funds returns, in other words, how good the data was fitted in the model, and resulted in an average of 0,82 for this risk class.

The fund with the highest Beta had 1,13 which is the TISCX fund. The alpha of this fund resulted in -0,0001262, the Adj R-squared 0,81 and the alpha is Significant on a 5% significance level. The fund with the lowest Beta had a value of 0,70 which FEAFX contributed with also had an alpha of -0,000176, an Adj R-squared of 0,37 and the alpha is Significant on a 5% significance level.

As seen in the table above, GCEQX had the highest alpha of 0,0000112 which also had a Beta value of 0,99, Adj R-squared of 0,98 and the alpha is not significant on a 5% significance level. The fund with the lowest alpha is the PXTIX fund with a result of -0,0003085, the Beta of this fund is 0,98, Adj R-squared of 0,83 and the alpha is Significant on a 5% significance level.

The only fund in this risk class that did not result in a beta relatively close to one was the FEAFX with a Beta of 0,70 with an alpha of -0,000176. Totally 15 out of 33 funds are significant on alpha on a 5% significance level, and 31 out of 33 has a negative alpha value. The lowest Adj R-squared fund is the

FEAFX with 0,37 and the highest R-squared fund is GCEQX with a R- squared of 0,98.

**4.2.2 2010 Standard deviation risk classification 25-30% **

The second risk class follows the homogeneous pattern with a Beta value close to one, more specifically an average of 1,04, an average alpha of - 0,0001693 and an average Adj R-squared resulting in 0,82. Compared to the first risk class, it has an average Beta value over one. Totally 14 out of 33 funds were Significant on a 5% significance level on alpha, and 32 out of 33 had a negative alpha value as seen in the table above.

The fund with the highest Beta is the DFTCX in this case that resulted in 1,13, which is quite high compared to the average and the alpha for this fund gave us a value of -0,0000141. The R-square for that fund is 0,76 and the alpha is ot significant on a 5% significance level. The fund with the lowest beta was PTWZX, resulting in a value of 0,95 with an alpha of -0,0003465 while the R-squared gave the result of 0,69 and alpha is significant on a 5%

significance level.

The highest alpha is 0,0002296 and was produced by the TPLGX fund, which has a Beta of 1,02, Adj R-squared of 0,72. The alpha is not significant on a 5% significance level for this fund. Moving on to the fund with the lowest alpha, PXWGX which produced a result of -0,0003385, a beta of 0,96, Adj R-squared of 0,74 and alpha is significant on a 5% significance level.

**4.2.3 2010 Standard deviation risk classification >30% **

When looking at the third and last regression, the Beta value in this risk class, compared to the other, was increasing with an average of 1,10

compared to 1,04 for the middle risk classification and 0,99 for the low-risk classification. This means that the beta increases with the higher standard deviation for our regressions. The result for this risk classifications average alpha is -0,0001612 which is lower than the middle standard deviation risk classification but higher than the low standard deviation risk classification.

The Adj R-squared average resulted in 0,78 which is the lowest when comparing and analyzing the three risk classes. In this risk class, 11 out of the 33 funds alpha were Significant and 32 out of 33 funds resulted in a negative alpha. This compared with 15 of 33 significant funds and two of 33 funds with a positive alpha for the low-risk classification and 14 of 33 significance and one of 33 positive alpha for the middle risk classification.

As shown in the table above, DSCVX got the highest beta with a value of 1,18, the presented alpha value was estimated to -0,0002465 for the fund and a Adj R-squared of 0,71 and the alpha is significant on a 5% significance level. VALUX had the lowest beta value for this category with a result of 0,95, an alpha of -0,0001703, Adj R-squared of 0,92 and the alpha is Significant on a 5% Significance level.

The fund with the highest alpha in this risk class was HISCX with an alpha of 0,0000819, a Beta of 1,13, Adj R-squared of 0,69 and alpha is Not Significant on a 5% significance level. HSLYX is the fund with the lowest estimated alpha in this risk class, resulting in -0,0003731, a Beta of 1,11, an

Adj R-squared of 0,81 and the alpha is significant on a 5% significance level.

To summarize the second part of the results for the regression analysis, we can see a small pattern where the beta gets larger as the standard deviation gets higher. The average beta for the <25% standard deviation is 0,99 and it is growing to 1,04 for the 25-30% risk classification and 1,09 for the funds with a standard deviation over 30%. As we can tell from the results of the beta, the differences between the average betas are not large but there is a slight difference between the risk classifications. The result of the average alpha values does not show the same distinct pattern as for the beta results.

We found the highest average alpha in the middle risk classification and the lowest average alpha in the low standard deviation class.

We found a pattern regarding the numbers of significant funds where the amount of significant funds decreased as the standard deviation got higher.

Regarding the numbers of positive alpha values, there were just two for the first classification and one each for the two other classifications.

The Adjusted R-squared that describes the goodness of fit to the security market variable, is 0,82 for the first two risk classifications and 0,78 for the high-risk classification. Even though the difference is not big, the result tells that independent variables are more correlated to the dependent variable for the low and middle risk classification compared to the high-risk

classification.

### 5. Analysis

In the following chapter, an analysis will be made regarding how the US equity funds in different risk classes will perform compared to each other and to the S&P 500 index. Firstly, an analysis of the return in the different risk classes will be made, compared to the index. Secondly, the regression analysis will be extended with an analysis of the different variables, Beta, Alpha and R-squared and how they interact and is comparable to the research question.

As mentioned earlier in the paper, and due the Efficient Market Hypothesis, the only way to receive a higher return is by taking a higher risk. And even if the market is not efficient, the passive funds would still perform better due to transaction cost. While analysing the total 99 funds and the different risk classes, in general, the actively managed mutual funds underperformed the index, as the hypothesis also claims in terms of returns. While studying a three-year period, the 99 fund's returns were quite similar to the index as presented above. In the long run though, there is a larger difference between the actively managed funds and the index with a few exceptions. We are aware that some of the funds are responsible for the low average return which can be for many several reasons, for example the fund managers risk aversion, which industries that have performed better and diversification and the spread. In this chapter, we have analysed potential reasons why the index in the long run performed better in most cases than the actively mutual funds.

It should be clarified, however, that all the funds are chosen randomly and have been active with the condition that all funds have a minimum of 70%

US equity. We are aware that the results could have been different if other

funds with higher or lower returns would have been included in the calculation.

5.1 Which of the different US volatility risk classification funds will generate the highest return over a three, five and a ten-year period?

As presented in the result above, the estimated return for the different risk classes were closely related. The medium volatility risk class generally performed higher compared with the low and high-risk class, but with a very small margin. This makes it difficult to draw any conclusions that one risk class performed better against another while estimating the geometric return.

If 33 more funds had been included in each risk class, perhaps the result could have looked different. It should also be mentioned that when risk class division was divided, it was done randomly based on collected US funds with at least 70% equity. But the majority of all funds had a volatility of 2011 between 22-35%. The results over three years in all of the risk classes showed that almost 50% of the funds in each risk class overperformed index.

But in a five- and ten-year period of time there were only a handful of funds succeeding to outperform index. In comparison with previous studies in the same area, our result supports the effective market hypothesis, saying that it is hard to outperform the index due to all available information. It should be noted that investors seem to have a bigger opportunity to outperform index in a shorter period of time where some of the funds have almost twice the return compared to index, which may contradict the effective market hypothesis a bit. On the other hand, these results are in line with the part in the efficiency market hypothesis claiming that there can be deviation to the fair market value when looking at a shorter period of time. Also, these results and arguments support that the effective market hypothesis is valid and that it is difficult to outperform index in the long run. Most previous studies in the

field, which also support our results, show negative alpha values. In a study conducted on Swedish funds, on the other hand, the majority of the funds showed positive alpha values. Therefore, it can be discussed how efficient the US market is versus the Swedish market. An interesting question that can be derived, that could have been further examined, is why US fund managers continue their actively managed funds and why mutual funds have become so popular in the US households when several reports showing similar results.

The prices can also, due to the result that the index performs better in the long run, be assumed to be random, linked to the theory that random walk lasts, which can be an explanation for the prices. Many of the funds also had a weaker closing compared to the index, which may be the result of the fund manager finding stocks that performed weaker in a final stage. The further interpretation, however, is that we do not have evidence that it is systematic with the fund managers. Instead that it is a result of coincidence, and where the price variations and our random fund selection may have had an impact on the outcome. Another aspect that would have been more interesting to examine further is the Equity Premium Puzzle and Loss aversion and how these more psychological effects, may have had an impact on the

performance of the funds and how the fund managers are affected by their risk aversion.

5.2 Does index outperform mutual funds in the regression analysis?

To be able to make a deeper analysis, comprehensive based on previous studies and to our research question, CAPM was tested. The regression analysis of CAPM was made in order to contribute with a further explanation of the ordinary returns and performance, comparing statistical variables. This

to make the report as valid as possible. We are also aware that if the Fama French model or the Carhart model would be estimated instead, we would have included a more flexible regression with variables as the SML and HML which maybe could have resulted in a different outcome.

The results from the test of CAPM also showed a homogenous distributio as the geometric estimated return. The average alpha of the three risk classes was closely related to each other, with an average around -0,00016. Two funds in the lowest risk class had a positive alpha, and only one in the other risk-classes resulted in a positive alpha over a ten-year period. The results in the regression model also state that the index outperforms mutual funds in a ten-year period, while looking at the alpha. Comparing our estimated alpha values with previous studies in the area, we can see that our result shows the same pattern, the majority contributing with a negative alpha. While

estimating the Betas, a slightly minor difference was discovered. As the risk increases, the average Beta for each risk class also increases. The average Beta for the first risk class is below 1, and the medium and high-risk classes resulted in a Beta over 1. This indicates that the higher volatility, the higher Beta which is a logical value based on the Efficient Market Hypothesis saying that the only way to receive a higher return is by taking a higher risk.

While analyzing the Adjusted R-squared value, estimating how well our funds correlate with the S&P 500 index, the first two risk classes resulted in 82%. This shows that our funds correspond with S&P 500 in a reliable way, increasing the validity of our study. In the high-risk class, the Adj R-squared value dropped to 77% which indicates in what grade the independent

variable can be explained by the dependent. Though this result also has a good fit, it should be noted that the higher risk class contributes with a higher grade of uncertainty in the model which can be explained by bigger

variations in the active funds in the higher volatility risk class.

While the coefficients have been estimated, we can see that there are very small alpha values and a Beta close to 1 for most of the funds, still as there is a bigger difference while looking at the return over a ten-year period. At the same time, it can be a coincidence that alpha is not significant. Throughout the period, only weak evidence of excess returns of the US mutual funds exists and only a few funds demonstrate positive alpha values. However, no positive significant alpha value was abducted, meaning that we have not been able to strongly demonstrate that any of the funds, according to our

regression analysis, certainly outperform the S&P 500 during a ten-year period.

It could also be motivated to discuss the CAPM explanatory degree. The regression analysis shows that CAPM has a fairly low degree of explanation of the outcome when observing the significance of our funds. If Fama French or Carhart had been used in the study, the results might have looked

different. Furthermore, several previous studies in the area that have been done before have included models as Fama French and Carhart, but the conclusion is usually the same; Index outperforms actively managed funds, especially in the long run.

### 6. Conclusion

When analyzing the geometric returns, and the regression analysis we can conclude that the results support the Efficient Market Hypothesis claiming that index outperform actively managed funds in a longer time interval. It should be clarified also that in a shorter period of time, some active funds have a bigger tendency to perform better than index. As many further studies have shown before, the USA is a very open financial information market, supporting the thesis that index outperform the funds. Our first method also showed that there is no major difference between the funds returns between the different risk classes while calculating the geometric return, but when doing the regression analysis, Beta continuously increases with the risk in between the risk classes. The negative alphas presented in the report are correlated with previous studies in the area, showing that it is very difficult for fund managers to outperform index.

To conclude, no bigger difference between the different Standard deviation fund risk classes was noticed in terms of return. This leads us to a similar conclusion as many previous studies, when the market is efficient, as demonstrated by the efficient US market in this case, the index outperforms actively mutual funds in the long run. We cannot with certainty determine through this report whether one of our risk classes performs better than any other.

### 7. Further research

As mentioned in the text above, a previous study that was presented above showed that actively managed funds could overperform index. The study that presented positive alpha values on index was done on Swedish funds and the other reports that had the same conclusion as this paper, were done on American equities. Comparing the efficient market between Sweden, the USA and how strong it is, is a topic that would have been more interesting to study further. If the study would have had room for more time, it would have been interesting to further include more performance measures such as the Sharpe ratio, Traynor etc. to be able to give the paper more depth and to be able to draw conclusions about which performance measures would have the best degree of explanation for our hypotheses. Furthermore, it would also have been interesting to do a regression analysis through the Fama French model and Carhart to include more variables and to further compare and test CAPM explanatory degree compared to the models developed afterwards.

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