capm - assessment and data analysis

CAPM – Assessment and Data Analysis

Table of Contents

Part A: Introduction 2

Early Components of CAPM 2

Evolution of CAPM 4

Empirical Performance of CAPM 5

Alternative Methods to CAPM 6

CAPM in the Modern Age 8

References 9

Part A: Appendix 10

Part B: Introduction 11

Price Evolution of Apple and S&P500 11

Statistical Analysis of Return 13

Explanation of Statistical Summary of Results 14

Risk-Return Analysis 17

Investigating the October Effect 19

Part B: Appendix 20

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Part A - Introduction

Within this report I will be providing a framework in which I will be discussing the

emergence of the Capital Asset Pricing Model (CAPM) and how the model has evolved over

time, while discussing its empirical performance. To conclude, I will be giving my opinion

on potential alternative models to the CAPM and laying across the usefulness of the CAPM

in modern finance applications.

Early Components of Capital Asset Pricing Model

The CAPM was built upon the foundations of the Modern Portfolio Theory which was

developed by Henry Markowitz. Which is essentially how risk-adverse investors can

construct portfolios to maximize expected return based on a given level of risk (Markowitz,

1952), thus highlighting that risk is a characteristic of higher reward. As Markowitz had

noted, there is a rate at which investors can gain expected return by taking on risk, or reduce

risk by giving up expected return (Markowitz, 1952). The CAPM is simply an example of an

equilibrium model in which asset prices are related to the exogenous data, the tastes and

endowments of investors (Brennan, 1989). The CAPM attempts to quantify the relationship

between the beta of an asset and its corresponding expected returns (Womack et al 2003).

The difference between the CAPM and alternative methods (which will be explained later) is

such that the CAPM incorporates the presence of a single risk (systematic risk) factor. While

alternative methods such as the Arbitrage Pricing Theory and the Fama French Model are

examples of multi-factor models. The CAPM is built around three main assumptions, which

are as follows (for a full list of CAPM assumptions see Part A- Appendix):

1. Investors’ concerns are only targeted towards expected return and the level of risk.

Thus assuming investors are completely rational when maximizing expected return

for a given level of risk (Womack et al 2003). In my opinion the idea of investors

being completely ‘’rational’’ when faced with the issue of wealth maximizing is not

necessarily true. As the field of behavioural finance provides an alternative way of

thinking, such that investors are not rational when faced with wealth maximizing.

Theories such as Gambler’s Fallacy, Over-Confidence or Mental Accounting have

disputed CAPM’s claim of investors acting completely rational.

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2. The second assumption is that all investors have the homogenous beliefs with

concerns about the risk and reward in the market.

3. The third assumption is that only one risk factor (systematic risk) is common to a

diversified market portfolio, which is in stark contrast to alternative methods.

Volatility and diversification have a strong role to play in the modern portfolio theory,

subsequently play an important role in defining the CAPM. Volatility is a measure of

dispersion of returns for a given asset or portfolio, with the general ruling being the higher

the volatility, the riskier the asset. Through the usage of diversification, a portfolio’s volatility

to some extent can be reduced. Since it has been established that investors are risk adverse

and prefer higher returns, such that through diversification (investors may opt for negative

covariance) they have adopted a strategy that allows them to decrease risk, while to some

extent not compromising expected return.

The CAPM starts out with two main ideas concerning the type of risk an investment is

subject to, namely, systematic risk and unsystematic risk. Unsystematic risk is reduced to

some extent by having a well-diversified portfolio, however systematic risk or market risk

cannot be diversified away due to the fact it’s a type of risk that affects the whole stock

market or industry. A practical example of market risk, is changes in interest rate. As I

mentioned earlier, CAPM is a one-factor model incorporating systematic risk, but it is indeed

beta which represents this risk within the CAPM. Beta is the measure of volatility of a

security into comparison with the market as a whole (BJS 1972). Within the CAPM a risk-

free rate of return is also present, which represents the level of interest an investor would

expect from a risk free investment over a specified period of time. The relationship between

the Beta and Risk Free Rate of Return is such that the CAPM predicts the expected return on

an asset above the risk-free rate is comparative to the non-diversifiable risk. In this sense the

higher the quantity of beta of a security, the higher is the expected return of that asset. It is

key to remember that the CAPM communicates to the investor by calculating the expected

return by taking into account the risk factor, such that if this expected return does not

adequately compensate the investor for taking greater risk with including the risk free rate,

investors should not invest. Such that when combining the following elements explained until

now, we are left with the CAPM formula. The CAPM lives by the fact that an asset is

expected to earn the risk-free rate plus a recompense for bearing risk as measured by that

asset’s beta.

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Evolution of the Capital Asset Pricing Model

Since the induction of the CAPM it had been subject to numerous additions, namely by

financial economists such as Jack Treynor, William Sharpe, John Lintner and Fischer Black.

Sharpe (1964) and Linter (1965) had published papers regarding the prices of securities under

different conditions of risk. Within these papers, they simply wanted to explore the

relationship between higher risk and expected return, furthermore how to distinguish the part

of the risk that the market values. Sharpe had noted the element of diversification, such that

some of risk present in an asset can be avoided so that its total risk is obviously not the

relevant influence on its price (Sharpe 1964). Sharpe and Linter extended assumptions of

Markowitz’ model with the notion that all investors can borrow and lend an unlimited amount

at an exogenously give a risk-free rate of interest. Such that Sharpe and Lintner added two

key assumptions to the Modern Portfolio Theory in order to identify a portfolio that account

for being a positive mean-variance relationship (which is simply weighing risk against

expected return). The first assumption is that all investors are assumed to choose mean-

variance efficient portfolios. The second assumption is that there is borrowing and lending at

a risk-free rate (Fama and French 2004), which does not depend on the amount borrowed or

lent. With the first key assumption complete agreement occurs, where investors agree on joint

distribution of asset returns (Fama and French, 2004). With this complete agreement investor

would see the same opportunity set, and they syndicate the same risk tangency portfolio with

risk-free lending or borrowing. Furthermore, by the equilibrium of asset market, since every

investor holds the same risky asset the tangency of the portfolio must be the value-weight

market portfolio of risky assets. With the second assumption of unlimited borrowing and

lending at risk-free rate Sharpe-Lintner had found where there is risk-free borrowing and

lending, the expected return on assets that are uncorrelated with the market return, must equal

the risk-free rate (Sharpe, 1964). Sharpe and Linter’s works laid the early foundations of the

CAPM. By combining the earlier components of the Markowitz’s Theory in this paper and

additions made by Sharpe and Lintner, we are left with the CAPM equation:

ERi=Rf +(ERm−Rf )βi

Concluding the two key assumptions added by Sharpe and Lintner, was such that there was

confusion surrounding risk-free borrowing and lending as an unrealistic assumption. This is

where financial economist Fischer Black (1972) developed another variety of the CAPM,

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known as the Black CAPM. Black (1972) developed a version of the CAPM without risk-free

borrowing or lending (Fama and French, 2004). Although, Black’s key results were such that

the market portfolio is mean-variance efficient (much like Sharpe-Lintner), instead Black

allowed for unrestricted short sales of risky assets.

Empirical performance of CAPM

During the past decades, there has been numerous empirical evidence that has

supported/targeted some of the basic assumptions of the CAPM, with many quarters calling it

to some extent practical but also unrealistic. One important study conducted on the CAPM

was that of Black, Jensen and Scholes (BJS, 1972). BJS study was conducted on sample size

of all securities listed on the NYSE for the period 1926-1966, using percentage monthly

returns. BJS came up with a strategy in which they created portfolios with very different

betas for use in empirical test (Jagannathan et al 1995). Methodology which was employed

by BJS was that of a ‘’Two Pass Methodology’’. The first pass involved estimating historical

beta using a time series regression (Jagannathan et al 1995). The equation below shows the

relationship between estimating the beta using a time regression in conjunction with excess

returns on the market returns.

For the first pass test BJS estimated the beta for individual security using monthly returns for

the 5-year period 1926-1930 (BJS, 1972). In which ten portfolios were ranked according to

the estimated beta (high to low). They then calculated the monthly return for each portfolio

for 1931. BJS repeated these phases numerous times, with the results of these process

exhibiting a series of monthly returns for 10 portfolios. Moreover, for the 35-year period BJS

calculated the mean monthly return and estimated the beta coefficient for each of the 10

portfolios. Lastly, BJS regressed the mean portfolio returns against the portfolios. The result

of the BJS test was such that they find the data is consistent with the predictions of the

CAPM, given the fact that the CAPM is an estimate to actuality just like any other model

(Jagannathan et al 1995). The results indicated that the relationship between average return

and beta is indeed close to a linear relationship, and that portfolios with a high (low) beta

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have high (low) average returns, this positive relationship is one in which the CAPM depicts.

Ultimately, the BJS study was in favour of the CAPM.

Fama and MacBeth (1973) study examined whether there is a positive linear relationship

between average return and beta and whether the squared value of beta and the volatility of

the return on an asset can explain the persistent variation in average returns across assets that

is not explained by risk factor alone. (Jagannathan et al 1995). Sample size for FM study was

all common stock traded on the NYSE for the period Jan 1926-June 1968. Like the BJS

study, the FM too was a ‘’Two Pass Methodology’’. Much like the BJS study, the results

from FM (1973) lent support to the CAPM. Mainly Fama and MacBeth (1973) highlighted

the evidence of a larger intercept term than the risk-free rate, the linear relationship between

the average return and the beta holds and lastly that the linear relationship holds well when

the data covers a long time. However, in my opinion, subsequent studies have provided

evidence of a weak relationship in the CAPM. For instance, Fama and French (1992) had

come to the conclusion that stock betas (risk) alone did not explain long term relationships,

although a combination of SMB and HML factors did. Furthermore, Ross (1976) published a

paper on the Arbitrage Pricing Theory (APT). With the APT it does not require restrictive

assumptions as does the CAPM. The CAPM model since its inception, as gone under the

scanner with reasonable success being found in its support through studies carried out by BJS

(1972) and Fama and MacBeth (1973). On the other hand, in my opinion academics realized

the limitations of the CAPM being a one-factor model, with numerous factors better able to

explain the relationship between risk and expected return. Thus, this saw the emergence of

competing models such as the Fama-French Three Factor Model and Arbitrage Pricing

Theory, which have to some extent redefined the very basic assumptions of the CAPM.

Alternative Methods to CAPM

Within this section I will be briefly explaining the fundamentals of the most widely thought

alternatives to the CAPM, known as the Fama and French Three Factor Model and the

Arbitrage Pricing Theory.

The Fama and French model was developed by financial economist Eugene Fama and Ken

French. The Fama and French Three Factor Model was developed in response to heavy

criticism the CAPM was receiving due to performing poorly in realised returns. In stark

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contrast to the CAPM which was a one-factor model (incorporating risk), the Fama and

French added two additional factors, namely value and size. To represent these two additional

risks, Fama and French had constructed SMB (Small Minus Big) to represent size and HML

(High Minus Low) to represent value (Womack et al 2003). Fama and French had come to

this conclusion of incorporating size and value due to research they had conducted, with

findings telling the story of small cap companies and value stocks outperform large cap and

growth stocks. SMB is designed to measure the additional returns investors have historically

received by investing in stocks with small cap. Such that by subtracting the average return of

the smallest 30% of stocks from the average return of the largest 30% of stocks, one would

get the SMB value. With a positive SMB highlighting small cap companies outperform large

cap companies. While the HML factor suggests higher risk exposure for typical ‘’value’’

stocks (High B/M) versus ‘’growth’’ stocks (Low B/M) (Womack et al 2003). The HML

factor is calculated much the same as SMB, the average return of 50% of stocks with the

highest B/M minus the average return of the 50% of stocks with the lowest B/M. With a

positive HML indicating that value stocks outperformed growth stocks in that given month,

HML was computed. Moreover, by combining the systematic risk factor of the CAPM, with

value and size risk we are left with the following equation:

Beta is still considered as a measurement to the market risk, while sA measures the level of

exposure to size risk, with hA measures the level of exposure to value risk. Within financial

academic works, the Fama and French Three Factor Model has seen considerable success, as

they are believed to have the greatest predictive power of any of two additional factors that

research has tested (Womack et al 2003). Furthermore, running a regression using the three

factor model often yields a R2 value of 0.95 if compared with CAPM only yielding usually

R2 value of 0.85. Principal uses of the Fama and French Model is to classify mutual funds

and separating the different funds to allow investors to weight their risk exposure. When

comparing the CAPM and Fama French Model it is important to put both models to an

empirical test. Such that a test was carried out in emerging markets by (Al-Mwalla et al 2012)

on the Amman Stock Market, during the period June 1999 – June 2010. When using the

CAPM the study did not find any evidence that support the ability of the single factor model

to provide a stable explanation to the variation in portfolios rates of returns (Al-Mwalla et al

2012). On the other hand, using the FF Model it explains a good part of variation in stock

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return, but not all of it which means that there are other variables to explain stock returns.

However, the FF model has more explanatory power than the CAPM.

The Arbitrage Pricing Theory (APT) was a model developed by Stephen Ross, which looks

to explain the approach to portfolio strategy decision which involves choosing the desirable

degree exposure to the central economic risks that effect both asset returns and organizations

(Roll and Ross, 1995). APT promotes the idea of returns being broken down into expected

and unexpected returns. The APT is seen to be a more relaxed model to the CAPM such that

it allows for multiple factors but in a more practical way in the sense that it does not assume

all investors implement Markowitz portfolio selection methods. The systematic factors his

research has identified as the most important are: (1) unanticipated inflation (2) changes in

expected level of industrial production (3) shifts in risk premiums (4) movements in interest

rates. (Ross and Roll, 1995), thus it makes the use of macro-economic variables. Using the

law of one price portfolio investors should be able to construct portfolios which are risk free,

as sensitivities in different stock cancel each other out.

CAMP in the Modern Age

Since the introduction of the CAPM decades ago, it has indeed gone under the scanner, with

additions being made, criticism, and alternative methods being proposed. Such that it begs the

question is the CAPM still useful in the modern age?

In my opinion the CAPM is still important to some extent, such that it is still widely taught

and used in the investment community. Using the CAPM an investor can still be assured that

a stock with a high/low beta will tell the story of how risk/less risk a particular asset is when

comparing it with the market. One obvious advantages of the CAPM is such that it is

generally seen as a much better model of calculating cost of equity than the dividend growth

model, such that it takes into account a stock’s level of market risk relative to the stock’s

market as a whole. More so the CAPM can still be used in the modern age for appliances

such as portfolio management, evaluating portfolio managers and cost of capital

determination. In my opinion, CAPM still have a place in modern finance applications

because of the model’s ability to quantify risk, as well as explaining risk measures into

estimates of expected returns. However, it should not be solely relied upon, because of the

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shortcomings as highlighted previously. As the famous Robert D. Arnott once said ‘’in

investing what is comfortable is rarely profitable’’.

References

Al-Mwalla, M., Al-Qudah, K. and Karasneh, M. (2012). Addtional Risk Factors that

can be used to Explain more Anomalies: Evidence from Emerging Market.

International Research Journal of Finance and Economics, -(99), pp64-74.

Black, F. (1972). Capital Market Equilibrium with Restricted Borrowing. The Journal

of Business, 45(3), p.444.

Black, F., Jensen, M. and Scholes, M. (1972). The Capital Asset Pricing Model: Some

Empirical Tests. Studies In The Theory of Capital Markets, Praeger Publishers, -(-),

pp.1-54.

Brennan, M.J., "capital asset pricing model", "The New Palgrave Dictionary of

Economics", Eds. Steven N. Durlauf and Lawrence E. Blume, Palgrave Macmillan,

2008, The New Palgrave Dictionary of Economics Online, Palgrave Macmillan. 06

April 2016, DOI:10.1057/9780230226203.0190

Fama, E. and French, K. (1992). The Cross-Section of Expected Returns. The Journal

of Finance, 47(2), pp.427-458.

Fama, E. and French, K. (2004). The Capital Asset Pricing Model: Theory and

Evidence. The Journal of Economic Perspectives, 18(3), pp25-46.

Fama, E. and MacBeth, J. (1973). Risk, Return, and Equilibrium: Empirical Tests.

Journal of Political Economy, 81(3), pp.607-636.

Jagannathan, R. and McGrattan, E. (1995). The CAPM Debate. Federal Reserve

Bank of Minneapolis Quarterly Review, 19(4), pp.pp2-17.

Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), pp.77-89.

Roll, R. and Ross, S. (1995). The Arbitrage Pricing Theory Approach to Strategic

Portfolio Planning. Financial Analysts Journal, 51(1), pp.1-7.

Ross, S. (1976). The arbitrage theory of capital asset pricing. Journal of Economic

Theory, 13(3), pp.341-360.

Sharpe, W. (1964). Capital Asset Prices: A Theory of Market Equilibrium under

Conditions of Risk. The Journal of Finance, 19(3), p.425-442

Womack, K. and Zhang, Y. (2003). Understanding Risk and Return, the CAPM, and

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the Fama-French Three-Factor Model. Tuck School of Business, 3(111), pp.1-14.

Part A -Appendix

Full list of CAPM assumptions

1. Investors are risk adverse 2. Investors seek maximizing expected return3. Investors have homogeneous expectations 4. Borrow or Lend freely a risk less rate of interest5. Market is perfect6. Quantity of risk securities in market is given 7. No transaction costs

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Part B - Introduction Apple Inc. (AAPL) was founded by Steve Jobs, Steve Wozniak and Ronald Wayne on April

1st 1979, with the company being involved within the computer industry. Apple underwent its

IPO on December 12, 1980 at $22.00 per share. Since the Apple listing the stock as split 4

times, with the first split taking place June 16, 1987 (2 for 1 split), June 21, 2000 (2 for 1

split), February 28, 2005 (2 for 1 split) and June 09, 2014 (7 for 1 split). Apple stock has

undergone ups and downs in terms of stock growth. Apple’s stock performance has a close

link with the relevance the general public sees with its products. Such that during the era of

the Macintosh, stock performance languished due to the general public not finding any

relevance with the product, which had a direct impact on share performance. When original

co-founder Steve Jobs returned to Apple, after originally being ousted implemented a new era

of innovation within their production lines. In the late 1990’s stock performance was very

respectable. Emergence of the iPhone and other related products completely transformed

Apple has a viable and profitable company, such that share price rallied and continued to

grow, even after Steve Jobs’ death. Currently, Apple’s share price can be described as steady

even though iPhone sales are sluggish. All data used in this analysis are lognormal returns.

Price Evolution of Apple and S&P500

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Figure 1

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S&P500 Apple

Above Figure 1 and Figure 2 represent the price evolution of both Apple’s stock price and the

S&P500 index. Figure 1 shows the trading volume of Apple (in millions) on its primary

vertical axis, while showing Apple’s stock price as well. Looking at the relationship between

the trading volume and stock performance of Apple it seems to be from 2000 onwards trading

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Figure 2

volume has exceeded Apple’s stock performance. Specifically, during the period form 2005-

late 2008, up until the financial crisis had hit. Although during the financial crisis, overall

trading volume declined, Apple’s share price performance improved, which concedes with

the emergence of innovative products such as the iPhone. Figure 2 represents the price

evolution of Apple and S&P500. Following the financial crisis in 2008, both Apple and the

S&P 500 took a sharp dip. In 2008, Apple shares fell more than 50%, however since as it can

be seen from Figure 2, Apple has been consistently risen 5% or more. Although Apple has

been consistently improving since the financial crisis, it was reported in last quarter of 2015

that Apple was on course for having its worst year since the financial crisis, even though it

has done better than the broader market. Like Apple, the S&P 500 took a significant hit from

the onset of the financial crisis. The index fell 56.8% from its peak on October 2007 to a low

point on March 2009. Unlike Apple, which had a consistent increase in stock performance,

the S&P 500 increased rapidly, ending 2013 with significant gains. Most recently, as oil

prices continue to slump among other aspects has had a significant impact on the S&P 500

performance for the close of 2015, with S&P 500 like Apple enduring its worst year since

2008. The S&P 500 ended the year down 0.73% after three-straight years of double-digit

gains.

Statistical Analysis of Returns

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Figure 3 Figure 4

Within this section I will be focusing on the statistical analysis of returns when using the

frequency of weekly returns, but also comparing how the statistical analysis of return changes

when using different frequencies. Referring to the Summary Results of Statistics (Figure 5)

when applying weekly returns Apple possesses a skewness value of -2.31 and S&P500 -0.82.

Both and Apple and S&P500 are negatively skewed (skewed to the left), with an

asymmetrical distribution (See Figure 3+4 above). Referring back to Figure 5 it is important

to point out that as the frequency of returns are increasing skewness for both Apple and

S&P500 are decreasing, becoming more symmetrical. Nevertheless, whatever frequency is

being applied the skewness value is below 0. Illustrating distributions that investors call a

long left tail (See Figure 5 Weekly Returns). Which allows for an investor for a greater

chance of extremely negative/positive outcomes. Using the descriptive statistics tool within

excel, I was able to compute a variety of summary results for both Apple and S&P500 across

all frequencies (See Part B - Appendix). Kurtosis for weekly frequencies was calculated as

being 25.39 for Apple and 6.91 for S&P500. However, not just for weekly returns, but across

all frequencies kurtosis for Apple is higher than the market index. In Apple’s case the

kurtosis is higher than three across all frequencies, which means they are said to be

leptokurtic. With kurtosis being leptokurtic, this fat tail means the distribution is more

clustered around the mean than in mesocratic distribution. Lastly adjusting the frequency of

returns brings along with it a reduction in kurtosis for both Apple and the market index.

Explanation of Statistical Summary of Results

Within this section I will be examining the returns estimated by CAPM and carrying out

linear regressions. I will also be showcasing how the changing of frequencies affect the beta

overtime.

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Figure 5

Referring to the table above, the beta has stayed fairly static with a small decrease when

changing frequencies from daily to weekly. However, the beta increased significantly when

applying monthly returns. Using the regression analysis tool in excel I was able to calculate

the beta of Apple in which I was able to assess how risky Apple is in line with the S&P 500.

With beta being a suitable measure of the volatility of an individual security or a portfolio in

comparison to the market as a whole. In the case of weekly returns of Apple, the beta

amounted to 1.10, which tells me that Apple is slightly more volatile than the S&P 500

(market beta of 1), furthermore the beta of 1.10 tells me Apple is 10% more volatile than the

S&P 500. Looking at the figures as an investor, I would quickly realize that a beta of over 1

is considered the norm among high tech companies listed on the S&P 500, secondly with a

higher beta provides me as the investor an opportunity for greater return by accepting this

higher element of risk. Using descriptive statistics which provided me with a summary of

results, concerning risk and return. Using the data results, I am able to use standard deviation

as another illustration of volatility. In the case of Apple had a standard deviation of

approximately 5.9% which is seen to be more volatile than the overall market, S&P 500, with

a standard deviation of 2.53%. Referring to Figure 1, Apple had high volatility in terms of

volume when the dot-com bubble reached its climax, although it seems Apple’s price was not

affected greatly by this speculation, on the other hand S&P 500 declined has dot-com bubble

reached its climax.

When taking beta into account as a measure volatility which is a central competent of the

CAPM, I investigated the results of the regression analysis. Firstly, regarding to the summary

results of the regression analysis of weekly returns (See Part B- Appendix), I found the

following. Interpreting the R2 Value of Apple, which amounted to 22%. In my

understanding, the R2 value explains the percentage of the risk-return relationship within

CAPM. Such that with 22% of the stock’ performance is explained by its risk exposure, as

measured by beta. The higher the R2 the better, as it simply tells the story that the CAPM

explains majority of risk exposure, however not in the case of Apple and S&P 500.

Additionally, corresponding to Linear Graph (see below) the trendline tells the story of how

accurate the CAPM is, with a perfect trendline being one that risk exposure points lie on the

trendline, which would exhibit a R2 of 1.0 or 100%. As you can see from the CAPM Graph

many points are dispersed which tells me in this case, the CAPM does not tell a significant

amount of the stock’s performance.

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-25.00% -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%

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To examine the mean, I will be applying the mean reversion theory to Apple, which simply

says that prices and returns eventually move back towards the mean. Such that investors can

utilize the mean to predict future return, in the sense that as prices deviate from the mean they

will eventually fall back to the mean, giving the investor a chance to either profit by buying

or profit/limit loss by selling before prices/returns retreat to the mean. Apple’s monthly return

mean during the period 2000-2015 amounted to approximately 1.79%. In the case of Apple,

monthly returns for 2009 was 8%, which saw a decrease to 4% by 2010, with returns

eventually retreating towards the mean of 2% in 2011 and 2012, before picking up again in

2014% with a mean of 4%. In terms of volatility, referring to the statistical table above

(Figure 5), it is clear Apple is indeed more volatile than the market overall, with an increase

volatility as frequency changes from daily to monthly. Apple experienced high amount of

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Figure 6

volatility during the recession years (2007-2010), with high/low swings of stock movement.

See Appendix for graphical representation.

Risk-Return Analysis

7 Year Analysis

Expected Return

Actual Return R2 Value

Percentage Difference Conclusion

2015 -0.06% -4.53% 50.24% 98.67% Overpriced

2014 10.17% 38.27% 23.03% 73.42%Underprice

d2013 9.99% 6.38% 1.88% 36.00% Overpriced

2012 16.64% 23.01% 27.52% 27.68%Underprice

d

2011 -1.00% 18.64% 39.94% 94.63%Underprice

d

2010 12.74% 41.98% 60.98% 69.65%Underprice

d

2009 22.17% 84.43% 53.69% 73.74%Underprice

d2008 -32.171% -64.3% 27.48% 48.44% Overpriced

In the first part of this section I will be covering the period 1st January, 2008 to 31st December

2015, of weekly Apple returns comparing CAPM results with actual stock return results. The

methodology which I undertook was based upon first firstly, the estimation of the systematic

risk beta of Apple relation to S&P 500; secondly, the estimation of market risk premium of

the model with regards to the market; and lastly, to test whether the model can explain the

relationship between individual stock return and systematic risk, beta. Based on my findings

majority of expected return produced by the CAPM seem to be under-priced when compared

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Figure 7

with actual returns of Apple. For example, during 2012 the actual stock return is 23.01%

while the model predicted a return of 16.64% with a beta of 1.2. Thus the actual return was

27.68% higher than predicted. Similarly, the actual returns exceeded expected returns (as

predicted by the CAPM) by 69.65%, 73.74% and 73.42% in 2010, 2009 and 2014

respectively. Such that if investors had contemplated to invest in these given years they

would have got a bargain since the stocks were undervalued in those years. Moreover,

investors would have been more than compensated for taking additional risk. Thus

highlighting investors are well compensated for this relatively high beta figures. Yet, 2013

the CAPM results indicated that the stock was overpriced. With the CAPM predicting an

expected return of 9.99% with an actual stock return of 6.38%. Moreover, in 2008 at the

height of the financial crisis, most stocks and markets overall were going downwards. In

2008, with a beta of 0.78 calculated an expected return of -32.17%, however it was grossly

inaccurate as Apple’s stock declined by -64.3%. From my findings, I conclude that the

CAPM cannot be used to statistically explain the observed differences in the actual and

expected return on the Apple stock. The implication is that, the observed differences in the

variables in the actual and the predicted returns are statistically insignificant and likely due to

chance or other factors and not due to the systematic risk factors as measured by beta of the

Apple stock under review.

7 Year Analysis CAPM Return

Beta

CAPM Expected Return

Apple Return Market Return

2015 1.3657 -0.06% -4.53% -0.04%2014 0.9181 10.17% 38.27% 11.08%2013 0.4574 9.99% 6.38% 21.84%2012 1.2086 16.64% 23.01% 13.77%2011 0.911138 -1.00% 18.64% -1.10%2010 1.357 12.74% 41.98% 9.38%2009 0.984822 22% 84% 23%2008 0.788817 -32% -64% -41%

Within the second part of this section I will be analysing how the beta has changed during the

last 7 years for Apple and what effect a change in beta had on the required rate of return as

predicted by the CAPM, to investigate if the formula holds up, while comparing CAPM

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Figure 8

results with market return. The cornerstone of the CAPM simply says that the only reasons

investors should earn more, is by taking additional risk. With the end result of the CAPM

formula giving investor a picture of the expected return they should expect for a given level

of risk. Such that if the expected return is not sufficient, investors should not invest. During

2012, Apple had a beta of 1.20 with a required rate of return of 16.64%. With this

considerable amount of risk an investor would expect Apple to outperform the market (which

has a market beta of 1). Indeed, Apple outperformed the market by achieving returns of

16.64% compared to 13.77% of that of the S&P500. However, in 2011 Apple had a beta of

0.91 which is still relatively high, in line with the market beta. With this beta in mind an

investor would expect Apple return to be in line with the market return. Nevertheless, Apple

achieved a return of 18.64% with the S&P500 achieving -1.10%, with CAPM estimates being

wide of the mark with a -1.00% expected return. This shows the inability of the CAPM to

account for other factors and shows the limitations of a one-factor model.

Investigating the ‘’October Effect’’

The October Effect is preconceived notion that stocks tend to decline during the month of

October. The October Effect is considered mainly to be a psychological anticipation rather

than an actual singularity. Such that I put this theory to the test when it comes to Apple.

Below Figure 9 showcases Apple’s September closing prices (October Opening) and October

closing prices (November Opening) during the period 2000-2015. As it can be seen from the

graph below October Prices tend to outperform during the month, thus dispelling the theory

of the October effect in Apple’s case.

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1998 2000 2002 2004 2006 2008 2010 2012 2014 20160

100

200

300

400

500

600

700

800

October Effect of Apple Disapproved 2000-2015

Sept Closing Oct Closing

Part B - Appendix

Monthly Returns Distribtuion

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Figure 9

Daily Returns Distribution

SUMMARY OUTPUT - Weekly Returns

Regression StatisticsMultiple R 0.470018368R Square 0.220917267Adjusted R Square 0.219979742Standard Error 0.052425996Observations 833

ANOVAdf SS MS F Significance F

Regression 1 0.647650164 0.647650164 235.6389645 5.2608E-47Residual 831 2.28399105 0.002748485Total 832 2.931641214

Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 0.003692052 0.001816702 2.032283005 0.042442576 0.000126188 0.007257916 0.000126188 0.007257916Beta 1.10067664 0.071702813 15.35053629 5.2608E-47 0.959936724 1.241416557 0.959936724 1.241416557

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SUMMARY OUTPUT - Monthly Returns




Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 0.014933303 0.00841286 1.775056628 0.07749804 -0.001661863 0.031528469 -0.001661863 0.031528469Beta 1.479622262 0.191023024 7.745779712 5.6306E-13 1.102811186 1.856433339 1.102811186 1.856433339

SUMMARY OUTPUT - Daily Returns




Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 0.000735334 0.00038916 1.889539908 0.058891379 -2.76359E-05 0.001498303 -2.76359E-05 0.001498303Beta 1.127649257 0.030726048 36.7001078 1.3602E-254 1.067409182 1.187889332 1.067409182 1.187889332

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-30.00% -20.00% -10.00% 0.00% 10.00% 20.00%

-100.00%

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

f(x) = 1.47962226235042 x + 0.0149333027261094R² = 0.240955025349552

Monthly Apple Returns

rApple Linear (rApple )

-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

f(x) = 1.12764925716497 x + 0.00073533371529803R² = 0.250870464417891

Daily Apple Returns

rAppleLinear (rApple)

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Weekly Returns Stats rApple rS&P500

Mean 0.004153485 Mean 0.000419227Standard Error 0.002056701 Standard Error 0.000878267Median 0.007510853 Median 0.001599362Standard Deviation 0.05935998 Standard Deviation 0.025348299Sample Variance 0.003523607 Sample Variance 0.000642536Kurtosis 25.39394458 Kurtosis 6.917077364Skewness -2.30843959 Skewness -0.81627775Range 0.942615312 Range 0.314396467Minimum -0.7064082 Minimum -0.20083751Maximum 0.236207111 Maximum 0.11355896Sum 3.459853233 Sum 0.349215883Count 833 Count 833

Monthly Returns Stats rApple rS&P500

Mean 0.01789543 Mean 0.002001948Standard Error 0.009620881 Standard Error 0.003191773Median 0.02907319 Median 0.008322837Standard Deviation 0.132963223 Standard Deviation 0.04411118Sample Variance 0.017679219 Sample Variance 0.001945796Kurtosis 10.15873283 Kurtosis 1.465655524Skewness -1.913894227 Skewness -0.726322813Range 1.235581828 Range 0.287943065Minimum -0.861414003 Minimum -0.185636474Maximum 0.374167825 Maximum 0.102306592Sum 3.418027053 Sum 0.382372073Count 191 Count 191

Daily Returns Stats rApple rS&P500

Mean 0.000830534 Mean 8.4424E-05Standard Error 0.000449559 Standard Error 0.000199681Median 0.000769827 Median 0.000535677Standard Deviation 0.028517753 Standard Deviation 0.012666774Sample Variance 0.000813262 Sample Variance 0.003523607Kurtosis 110.3880942 Kurtosis 8.021513126Skewness -4.322809213 Skewness -0.185966192Range 0.86144108 Range 0.204267093Minimum -0.73124689 Minimum -0.094695125Maximum 0.13019419 Maximum 0.109571968Sum 3.342070376 Sum 0.339722217Count 4024 Count 4024

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