Excerpt

## Inhaltsverzeichnis

1. Introduction

2.a Basic figures of the firms

2.b Statistical moments of stock market returns

2.c The minimum variance portfolio

2.d The tangential portfolio

3. Turn-of-the-month anomaly

4. References

## List of figures

Graph 1 – The MVP

Graph 1 – The Market Line

Graph 2 – Graphical overview of the ToM

## List of abbreviations

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List of tables

Table 1 – Overview of the different portfolios, MVP

Table 2 – Overview of the different portfolios, Market portfolio

Table 3 – The 50:50-Portfolio

Table 4 – Oversight of ToM of the three shares and the S&P 500

Table 5 – Monthly data of the ToM anomaly of the S& P 500………………………………………..11

## 1. Introduction

The purpose of this paper is to do empirical research on the capital asset pricing model. The bases of our research are the returns of three stocks, the *S&P 500* index which represents the market and the *LIBOR* as a proxy for the risk-free interest rate. The three companies that were chosen in this paper were *Kellogg Company*, *KB Financial Group Inc.* and *Kate Spade & Company* and all of them in combination represent our fictive market. The reason why we chose the *LIBOR* instead of the *EURIBOR* was that all three stocks we used note on the New York stock exchange (NYSE) and thus using American interest rates seems to be more appropriate than using European ones.

### 2.a Basic figures of the firms

First of all, some key statistics and characteristic numbers shall give a quick overview of the companies. The following data was last updated on December 1st, 2014. The *Kellogg Company* has the highest market capitalization with 23.8 billion dollars, further it has a book value of about 31.1 billion. In comparison: *KB Financial Group Inc*. has a market cap of 13.7 billion and a book value of 40 billion. *Kate Spade & Company’* s market cap is 3.7 billion and its book value is around 4 billion. The market cap is calculated by the stock price times all the shares outstanding (not including the shares held by the company itself), while the book value can be calculated out of the assets on the balance sheet (Google Finance online, 2015). The market/book ratio of a firm tells if a share is overvalued or undervalued. If the book value is higher than the market cap, then the share is undervalued and the price is expected to rise. Therefore a market/book-ratio lower than one indicates an undervalued share (Peavler, 2015). The *Kellogg Company* as well as *KB Financial Group Inc.* and *Kate Spade & Company* have undervalued shares with ratios of 0.765 for *Kellogg’s*, 0.338 for *KB Financial* and 0.934 for *Kate Spade*. Another important characteristic number is the earnings per share (EPS). It results from the net income of the last year divided by the shares outstanding and it is an indicator for the profitability of a company. The *Kellogg Company* has an EPS of 4.9, *KB Financial’s* EPS is 3.47 and *Kate Spade’s* EPS is 0.67. To figure out the return on equity (ROE), the EPS are divided by the stock price, if the market value serves as a basis. If the basis is the book value, the EPS is divided by the book value per share. The ROE measures the efficiency of a company at generating profits for investors. A ROE of 15-20% is considered to be good. *Kellogg’s* has a ROE (based on market value) of 7.31%, *KB* has 9.9% and *Kate Spade* has 2.25%. The ROE based on book value is 5.59%, 3.32% and 2.16% for *Kellogg’s*, *KB* and *Kate Spade* (Google Finance online, 2015). The EPS is needed for another important number, the P/E ratio (price/earnings ratio). It is calculated from the stock price divided by the EPS and it can be interpreted as how much investors are willing to pay per dollar of earnings. Historically, there is an average ratio of approximately 15-25 (Investopedia online, 2015). The P/E ratio of *Kellogg’s* is 13.67, *KB’s* is 10.11 and *Kate Spade’s* ratio is the highest with 44.54. One of the most important numbers for an investor is the dividend yield. It tells the investor how much the firm pays out in dividends per year relative to the stock price. The basis of the calculation can either be the market value or the book value. For *Kellogg’s* there is a dividend yield of 1.96/67.02 = 2.92 % and, based on book value, 1.96/87.63 = 2.24 %. In comparison, *KB* has a dividend yield of 1.37 % or 0.46 % and *Kate Spade’s* dividend yield apparently is zero due to a non-existent annual payout (Google Finance online, 2015). In order to calculate the growth rate (plowback ratio * ROE), the plowback ratio (PR) is needed. The formula for the PR is 1-(annual payout/EPS) and it signifies the amount of earnings a firm retains after dividends are paid out. *Kellogg* retains 60 %, *KB* 86 % and *Kate Spade* 100 %. Finally, the estimated growth rate can be calculated. Since there were two possibilities to figure out the ROE (market value or book value), it is necessary to continue in the same way and calculate two growth rates. *Kellogg’s* estimated growth rate is, based on the market value, 4.39% or, based on book value, 3.36 %. *KB* grows at 8.53 % or 2.86 % and *Kate Spade* has a growth rate of 2.25 % or 2.16 % (Google Finance online, 2015).

### 2.b Statistical moments of stock market returns

The basis of the following calculations is the monthly returns from 12/2008 until 11/2014 of *Kellogg*, *KB*, *Kate Spade* and the *S&P 500* index (data from Yahoo Finance). First of all, the focus lies on the mean returns, variances and standard deviations of the stocks as well as the index. *Kellogg* has a mean return of 0.97 %, *KB’s* mean return is at 1.4 % and *Kate Spade’s* is at 5.1 %. Compared to the *S&P 500*, which represents the market and has a mean return of 1.2 %, only *Kate Spade* has a significantly higher mean return. In the enhanced excel sheet price charts and return plots are visualized. A closer look at the standard deviation says that *Kate Spade* also has the highest standard deviation (0.21), followed by *KB* (0.11), *Kellogg* (0.046) and the index (0.042). In addition to that, it is always useful to have a closer look at the covariance and the correlation coefficient. Both numbers try to predict dependencies between two stocks. If the covariance is positive, both stocks tend to move in the same direction. If it is negative, the stocks’ returns are likely to move in the opposite direction. In the created fictive market there are only positive covariances. To testify whether they are weakly, moderately or strongly correlated with each other, the correlation coefficient is the crucial number. All three stocks correlate weakly with each other with a coefficient always slightly higher than 0.25. All of them are correlating moderately or even strongly with the *S&P 500* (*Kellogg* 0.45, *KB* 0.7, *Kate Spade* 0.45). To define a beta for the stocks, the covariance between the stock and the market portfolio is divided by the variance of the market portfolio (Palan, 2014). The beta coefficient can be interpreted as a measurement of the volatility of a portfolio compared to the market as a whole. In other words beta can be seen as a tendency of a portfolio’s return reacting to changes in the market (Investopedia online 2015). *Kellogg* has a beta of 0.39, *KB’s* is 0.61 and *Kate Spade’s* beta is 2.49. The beta coefficient of the *S&P 500* is 0.33, and the beta for the market portfolio is 1, since it is efficient (more on the following pages). After that, it is important to diversify systematic and unsystematic risk. Investors can calculate the systematic risk, but there will always be some risk that can’t be calculated. Therefore it has to be estimated. Under the assumption that the variance represents total risk of an asset, it is possible to say that the beta of a stock times the variance of the market portfolio is the systematic risk of this stock and the unsystematic risk is just what is missing to complete the total risk. Now, having created a huge basis with a lot of characteristic numbers, defining a minimum variance portfolio will be the next step.

### 2.c The minimum variance portfolio

The minimum variance portfolio (MVP) is a certain portfolio, which either can consist of any possible combination of all available stocks (in this case three stocks) or which can as well consist of just one single stock. Using all conceivable combinations/portfolios of stocks, the MVP is the portfolio which has the lowest standard deviation. Therefore the MVP is the portfolio which offers the lowest achievable risk to the investor out of all possible stock combinations.

Creating a plot including all these combinations on a graph depicting the return of the portfolio on the ordinate and its standard deviation on the abscissa, the MVP will be the most left one of all points on this graph. Although it seems to be intuitive, it is necessary to mention that the MVP has not the highest return of all portfolios. The portfolio which offers the highest possible return to its investors is called the maximum return portfolio and is located at the highest point on the graph mentioned before. These two considerations lead us to a conclusion which is as simple as intuitive: There must be a trade-off between the expected return and the riskiness of a portfolio. To visualize this fact, there are eight hundred simulated random portfolios consisting of this paper’s three stocks enhanced to the excel sheet as well as the MVP out of these portfolios.

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Graph 3 – The MVP

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Table 6 – Overview of the different portfolios, MVP

As you can see the MVP is the most left point on the graph (highlighted square) and it has a standard deviation (the strength with which the real return differs from the expected return) of 0.04578 the lowest risk of all portfolios. Rounded 95 % of the MVP consists of the *Kellogg Company’s* stocks and 5 % of the MVP consists of the *KB Financial Group Inc*.’s stocks. Certainly the MVP does not include any stock of *Kate Spade & Company*. Furthermore you can see that the MVP only offers an expected return of 0.9907 % which is much lower than any random or the equally weighted portfolio’s expected return. Therefore the expected return is quite save because due to the low standard deviation the real return of the portfolio varies just around 4.5 % and for example the real return of the equally weighted portfolio will vary around 9.3 %. This proves that there is a trade-off between a high expected return and the riskiness of a portfolio. Furthermore the expected return of the equally weighted portfolio is about 2.5158% and the expected return of the MVP only about 0.9907%.

Another important thing that can be seen in this calculation is that the beta-factor and the standard deviation of portfolios are likely to correlate. If there is a high standard deviation there is also a high beta-factor which shows how strong the portfolio reacts to market movements. The beta-factor below one says that the MVP reacts lower to market movements than the market portfolio which includes all stocks on the market or the market on its own. This is important information because on the one hand the risk of a stock or a portfolio consists of the specific risk which can be reduced by diversification and on the other hand of the market risk which cannot be reduced by diversification. The market risk simply is the risk of a bad market development. Since a beta-factor of one means that a stock’s or a portfolio’s performance one-to-one imitates the market movement it is possible to say that all beta-factors below one are reducing riskiness and thus come with a low standard deviation. Beta-factors above one are only useful if the whole market has a heyday. They increase the riskiness of a portfolio or stock.

### 2.d The tangential portfolio

Next to portfolio construction there is a possibility to achieve lower risks for any certain level of expected return and to achieve higher expected returns for any certain level of risk compared to simple construction of a portfolio. This works by introducing lending and borrowing at the risk free interest rate. The *LIBOR,* which is the used interest rate for inter-bank credits, will serve as the risk free interest rate. It is obvious that the risk free interest rate must have a standard deviation of zero, because it is risk free. Depicting the risk free rate on the graph of the capital asset pricing model (CAPM) it must be on the left side on the ordinate. To achieve the goal of further reducing risk or further increasing expected return beyond the given risk and return through simple portfolio construction, investors have to invest in a combination between the *risk free interest* rate and a certain portfolio which is called the *tangential portfolio*. Graphically this consideration builds a straight line between the risk free interest rate on the left side and the tangential portfolio. This line is called the *capital market line*.

**[...]**

- Quote paper
- Alexander Moßhammer (Author)Elias Danzl (Author)Kilian Altenberger (Author), 2015, Capital Asset Pricing Model (CAPM). A Case Study, Munich, GRIN Verlag, https://www.grin.com/document/288267

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