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OVERVIEW OF FINANCIAL RISK ASSESSMENT A thesis submitted to the Kent State University Honors College in partial fulfillment of the requirements for University Honors by Bo Zhao May, 2014 ii
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Thesis written by Bo Zhao Approved by,advisor, Chair, Department of Mathematics Accepted by, Dean, Honors College iv
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TABLE OF CONTENTS LIST OF FIGURES.. vi LIST OF TABLES....vii ACKNOWLEDGEMENTS...viii CHAPTER I. INTRODUCTION.. 1 II. III. RISK PREMIUM AND SHARPE RATIO.7 RISK-FREE RATE...11 SHORT RATE IN DISCRETE AND CONTINUOUS TIME.14 IV. CAPITAL ASSET PRICING MODEL 18 SECTION ONE: ASSUMPTION 18 SECTION TWO: SECURITY MARKET LINE.26 SECTION THREE: TEST OF CAPM.35 V. VALUE AT R ISK.41 VI. SUMMARY.45 REFERENCES 46 APPENDIX 1. SEVEN DATA SETS..... 47 vi
LIST OF FIGURES Figure 1. Figure 2. Figure 3. Probability Densities of Stock A and B..5 Beta versus Expected Return...28 Fat-Tail Problem...43 vii
LIST OF TABLES Table 1. Table 2. Table 3. Table 4. Table 5. Table 6. Table 7. National CD and Investment Rates...11 Close Prices of IBM...21 Calculations of Price Differences and Rate of Return...22 Investor A s Portfolio...32 Dow Jones Industrial Average...33 IBM and Game Stop Corp Adjusted Close Prices...34 Six-Stock Chart...38 viii
ACKNOWLEDGEMENTS I would like to offer my special thanks to Dr. Richard Shoop for his patient guidance and valuable suggestions in this thesis process. I would also like to thank Mary Jessica for her support. ix
1 Introduction Financial risk assessment is one of the most important and difficult topics in finance. Risk assessment involves both financial theories and mathematical modeling. Students who have only basic finance and mathematics background usually find it difficult to integrate both conceptual and abstract elements when they study risk assessment methods. Therefore, this thesis serves the purpose of explaining basic risk assessment concepts and introducing several commonly used risk assessment methods in a simple manner. This thesis should help students of finance and of financial mathematic to understand the underlying concepts of risk assessment in a systematic way, making it easier for them to understand more advanced concepts they may encounter in their later studies. Readers can expect detailed explanations from this thesis about risk assessment related financial definitions and mathematical derivations supplemented with straightforward examples. After thoroughly reading this thesis, readers should expect to be able to answer questions such as how is financial risk defined from both financial and mathematical perspective? and Why can risk be evaluated by a specific risk assessment method? Therefore, this thesis is a tool to assist mathematics and finance students who are interested in studying financial risk assessment in depth. Financial risk assessment is one of the most important and difficult topics in
1 Finance. The most fundamental characteristic of risk is that higher risk is accompanied with higher potential reward. This means that the more risk an investor is willing to take, the more potential return from the investment this investor will receive. In other words, if investing in certain assets can give investors a higher return compared to other assets, these assets can also let investors lose more than other assets can do due to the higher risk level. Typically, returns are expressed as percentages, to the terminology rate of return is often used. In this paper, return will always mean rate of return. The risk of an asset can be divided into two portions: systematic risk and unsystematic risk. In order to comprehend the concepts of these two types of risk, it is important to understand the related idea of diversification. Diversification spreads risk across a range of assets, thereby balancing out the specific risks associated with each asset individually. One of the most important strategies about diversification is that investors have to choose assets which do not move together. For example, if an investor uses a thousand dollars to purchase stock A, his/her return is fully depended on how well stock A performs in the market. His/her risk of this investment is equal to the risk of stock A. If stock A had a 20% decrease in its price, this investor would lose two hundred dollars because 0.2 times 1000 is 200. However, if this investor divides these thousand dollars into two equal amounts and then invests them into stock A and stock B respectively, the investor invests five
2 hundred dollars in each stock and his/her return depends on how well both stocks perform in the market. The risk of this investment, which is called portfolio risk because the investment contains more than one security, is based on the combined risks of both stocks. The calculation of portfolio risk will be demonstrated in a later section of this thesis. Suppose price per share for both stock A and stock B are equal. If stock A decreases its price by 20% and Stock B, which has very little in common in terms of factors that influence its price, increases 20%, the investor will still have a thousand dollars and he/she does not lose any portion of his/her investments. Note that, in this case, although the investor did not earn any additional return from this investment, if the investor only invests in Stock A, he or she will lose two hundred dollars from this investment. Moreover, if both stocks drop 20% in their prices, the investor will lose two hundred dollars on the portfolio, but the amount he/she loses on stock A is a hundreds dollar less compared to when he/she only invests in stock A. This example shows that if investors put their money into different security products which are not very commonly related in terms of price fluctuation, then they are able to use the returns from some of their securities to cover the losses from other securities in their portfolio. Most investors use diversification to reduce their overall portfolio risk. Brigham and Daves state that, "almost half of the risk inherent in an average individual stock can be eliminated if the stock is held in a reasonably welldiversified portfolio, which is one containing 40 or more stocks in a number of different industries"(2009, p52).
3 Diversifiable risk, also known as unsystematic risk, is a type of risk that can be reduced by diversification. Diversifiable risk is caused by random events that affect the specific asset such as strikes and lawsuits. As Brigham and Daves mentioned before, half of the risk can be diversified if investors can create a well diversified portfolio (2009, p57). The "half of the risk" they mentioned is precisely the diversifiable risk. Systematic risk, however, cannot be reduced by using diversification. Systematic risk is also called market risk or non-diversifiable risk. It includes the risk of having inflation, recession and high interest rate. These factors impact the market as a whole and no individual or institution can simply manipulate these factors. Therefore, diversification does not work in terms of reducing market risk. It is this component of risk that financial institutions are most eager to manage. Financial institutions use various methods to evaluate market risks. The simplest method to determine the risk of an asset is to calculate standard deviation its price fluctuations over a specified time period. Standard deviation of the price for a share of stock, σ, is a weighted average of the deviations from its average value and it provides an idea of how far above or below the average value from actual value is(brigham & Daves, 2009, p39). The formula is defined as following: n 1 σ = N i i= 1 (x x) The concept of standard deviation related to risk can be explained in a simple example. For example, in Figure 1, graph A and graph B are the probability densities the market 2
4 prices of two different securities Figure 1. Probability Densities of Stock A and B products. Based on the definition of standard deviation, the actual values of A are located closer to average value 0 compared to the actual values of B so that stock B has a larger standard deviation than stock A. This shows that small standard deviation implies this stock has small volatility compared to the stocks which have large standard deviation. Therefore, small volatility means the stock fluctuate in a smaller range so that the risk of this stock is relatively smaller. The larger the fluctuation, the bigger the risk can be. Hence, stock B is riskier than stock A. In addition to the example, Brigham & Daves wrote that "the tighter the probability distribution, the lower the risk assigned to a stock"(2009, p37-38). This means lower risk stocks have narrower probability distributions of the actual values.
5 Standard deviation is very useful to obtain an overall picture of historical data and is a good estimator for future estimation of risk. However, there is a potentially serious disadvantage: standard deviation is extremely sensitive to outliers. This means that if several abnormal data values are included in the calculation, the result can provide a misleading estimation of risk. Hence, more advanced risk assessments are required to provide accurate estimations of risk. In this thesis, Capital Asset Pricing Model and Value at Risk will be discussed. These risk measurements are currently the most common risk assessment methods, and they are widely discussed in academic field. This thesis will provide a good understanding of all these two models for students and scholars who are interested in risk assessment methodologies.
6 Risk Premium and Sharpe Ratio Given that greater risk corresponds to potentially greater returns, an investor who takes a position in a risky asset should reasonably expect a bonus or premium to compensate him for this risk. This observation leads to the concept of risk premium. The risk premium measures the expected return in excess of that available through investment in risk-free securities over a specified time interval [, t t + dt]. It is mathematically defined as following: RP = E(r e ) r f where RP=risk premium E(r e ) = expected rate of return of an asset r f = risk-free rate Risk premium is a very simple method to measure the risk level of a security or a portfolio. Generally speaking, risk-free rate is rate of return an investor will gain with certainty. The 90-day Treasury bill rate is usually used as the risk-free rate in the market. Surprisingly, the concept of risk-free rate is more complicated than would first appear. Many subtle aspects of its definition require a careful explanation of how it is calculated and interpreted. We shall discuss the formal definition of risk-free rate in detail in the later section of this thesis. Risk premium is useful for investors to see how much extra return they are
7 likely to obtain compared to investment in risk-free assets. It can also be used to compare two or more assets or portfolios. Suppose that investor A purchased one share of stock of company X and investor B purchased one share of stock of company Y at same time. Based on historical data, the price of stock X is expected to increase by 15% and price of stock Y is expected increase by 13% over the next month. Assume that the risk-free rate is 3%. Then the risk premium will be 12% and 10% for stock X and Y respectively. From this example, stock X has a higher risk premium than stock Y does. This means that stock X has a higher expected excess return than stock Y. As mentioned before, the higher rate of return is accompanied with higher risk level. Risk premium gives a direct comparison between the different securities if their other factors are equal. This implies that risk premium cannot fully represent the risk level of an asset or a portfolio because it does not take standard deviation and other factors of the stock into consideration. This means that risk premium is not the most refined measure of risk that we can develop. In order to improve the accuracy of risk level measurement we introduce a new quantity, called the Sharpe ratio. The Sharpe ratio, also known as reward-to-volatility ratio or mean-standard deviation ratio, measures the tradeoff between risk and return of a portfolio (Bodie, Kane, Marcus, p125). The formula is defined as: S = E r p r f σ p
8 where E r p = expected rate of return of portfolio r f σ p = risk-free rate = standard deviation of portfolio excess return According to the formula, the numerator is the risk premium of the portfolio as explained previously. The denominator is the standard deviation of portfolio excess return which can also be defined as following: σ p = VAR(r p r f ) where r p = rate of return of portfolio and r f = risk-free rate since standard deviation is equal to the square root of variance. As mentioned earlier, standard deviation represents the volatility of a portfolio. Larger standard deviation implies that portfolio is more volatile so that the risk level of this portfolio is potentially higher. Therefore, Sharpe ratio shows the proportion between risk premium, or reward, of a portfolio and its volatility. Intuitively, investors should prefer a large value of the Sharpe ratio, since this would represent a higher expected return per expected risk level. From the last example, the risk premium is 12% and 10% respectively for stock X and Y. If the standard deviation of X is 6% and standard deviation of Y is 4%, the Sharpe ratios of X and Y will be 2 and 2.5 respectively. Hence, stock Y, although has lower expected rate of return, has a higher Sharpe ratio
9 compared to Sharpe ratio of X. This is because X has higher standard deviation than Y does which means that X is riskier than Y. If investors are risk aversion, they would choose portfolio A than B even though B has higher expected rate of return.
10 Risk-free rate Suppose that an investor has $1000 that he wants to invest, and that he is unwilling to experience any loss from this investment. The best option for him is to go to a local bank and open a savings account or purchase a certificate of deposit (CD). Investments such as depositing in savings account and buying CDs are insured by the Federal Deposit Insurance Corporation (FDIC), a branch of the United States Government. This means that the federal government guarantees both the safety and the specified return for such an investment. Hence, these types of investments are virtually risk-free. Although it is still possible that the federal government might become insolvent and that investors could experience partial or total loss of their funds, this is very unlikely to happen. Therefore, in general, such an investment is considered to be risk-free. The guaranteed return for this type of investment is called the risk-free interest rate, and is denoted by the symbol r. Some subtleties need to be addressed. First, the free rate is not a fixed rate but varies with the duration of the investment. The longer the duration of an investment, the higher the risk-free rate becomes. This is illustrated in the table below: Table 1. National CD and Investment Rates Today Last Week 6 month CD 0.4 0.39 1 year CD 0.58 0.58 2 year CD 0.72 0.71 5 year CD 1.24 1.25
11 Thus, the risk-free rate is a function of the duration of an investment so that a better notation would be r( T ) where T is the maturity date of the investment, measured in years. The table above also suggests that the risk-free rate for CD changes over time. Specifically, the risk-free rate can change from week to week, or even from day to day. Therefore, the even better notation for r is r( tt, T) +, where t is the time the investment is made, and T is the duration of the investment. By convention, t = 0 corresponds to the present time. These observations raise two questions: Why does the CD rate fluctuate over time and what are the consequences of this fluctuation? Suppose an investor decided to purchase a 5-year CD as suggested by his financial advisor and the CD rate at the time of purchasing was x%. Suppose that after this investment decision was made, the CD rate increased to y% for some y > x. Since the investor s assets are already tied up at the lower yield rate, he loses the opportunity to achieve a higher return. After he realizes that the rate has increased, he immediately draws another 1000 dollars from his checking account and purchases a 1- year CD instead of a 5-year CD to avoid being tied to the lower yield product if the risk-free rate rises again. However, after an additional year, the CD rate drops to z % for some z < yand the investor has lost the opportunity to realize the greater return he would have enjoyed had bought a longer term CD a year before. Clearly, there is risk with both decisions! Even though CD is risk-free, investors may lost opportunities based on the time-dependency of the CD rate. But why do CD rates fluctuate over time? The answer to this question lies with the
12 law of supply and demand caused by market forces. The law of supply and demand says that, assuming market efficiency, the market prices are adjusted depending on market supply and demand. If nobody is willing to buy cars at today s market price (low market demand), dealers will lower their prices to attract more customers to buy cars. On the other hand, if everyone is willing to buy a car (high market demand), then it may be difficult for dealers to restock their car storage to match the demand, and the car prices are likely to go up. The bond market is analogous to the automobile market. CD rates fluctuate based upon the yield rates of U.S. treasury bonds (T-bonds) which are sold on the open market. The value of a bond depends on how much investors are willing to pay. Note that, even though the risk-free rates are determined by bond values, the quoted rates can be lower than the actual risk-free rate and these quoted rates can vary from one bank to another. This is because banks are competing institutions whose purpose is to earn a profit. Therefore, quoted rates typically include a markup to account for bank s administrative costs and profits. The true risk-free rate is the rate before this markup. The formal definition is as follows: The risk-free rate over the time interval [ tt, T] + is defined to be the yield on a T-year treasury bond purchased at time t. It is given by the formula (, ) r tt = ( + ) ( ) P( t) P t T P t where P( t) is the market price at time t, and Ptt (, T) + is the face value of the bond at
13 maturity. It is customary to express bond values in terms of unit maturity value. That is, we assume that the value at maturity of the bond is $1, and our calculations are per dollar at maturity. Hence, we take P( t T) 1 + = and the defining formula riskfree rate over [ tt, T] + becomes (, ) r tt P( t) ( ) 1 = P t Short Rate in Discrete and Continuous Time For any asset (a share of stock, a bond, a portfolio, or a market index, for example) with value S( t ) at time t, the return on that asset over the time interval [ t, t dt] defined by (, ) R t dt ( + ) ( ) S( t) S t dt S t =. + is Where R is the rate of return and S( t ) is the price at time t. Unlike a bond which the return is specified at time of purchase, the return of a risky asset a share of stock, for example--is not fixed but random. Therefore, R( t, dt) is a random variable for any fixed value of t, and for any fixed value of 0, { R( t, t dt) : t 0} dt > the sets ( ) { : 0} S t t and + are stochastic processes. Mathematical models of the returns process usually involve using both the mean return and the volatility over a specified time period in estimating expected rate of return. To make the mathematics tractable, it is assumed that both the expected value and the variance of the return process are proportional to the length of the time interval over which returns are calculated. This means that there are constants µ andσ, withσ > 0, such that, for anyt > 0, dt > 0,
14 ( + ) = µ, and ( ) E R t dt dt 2 ( ) σ Var R t + dt = dt. E[ ] is the expected value and Var ( ) is the variance. The coefficient σ is the volatility of the return process. Historical data of an asset for a given time increment dt are typically used for estimating µ andσ. For example, suppose n-week weekly prices of a one-year treasury bond are used to estimate the expected return and volatility coefficient. Then, 1 dt = 52. Let t k = k dt for k 0,1,, n market price of a bond that matures at time t k + 1 and let =, let ( ) R k + 1 ( k+ 1) S( tk) S( t ) k S t be the S t = be the return on the bond over the time interval[ t, t + ]. The empirical estimates of µ and σ are then obtained statistically using k k 1 n 1 n 1 1 1 ˆ µ dt = R : = R, and σ dt = s : = R R n n 1 2 2 ˆ k+ 1 R ( k+ 1 ) k= 0 k= 0 The annualized expected return and volatility coefficient are then estimated by 1 1 ˆ µ = R = 52 R, and ˆ σ = sr = 52 s dt dt When time is measured on a discrete scale over a period of one year, the formulas listed above provide the risk-free short rate. It is common to think of 1 year as being the short time period but other specified time intervals can also be used. In the discrete case, calculation of risk-free short rate only requires elementary statistics. However, in continuous time, the definition of risk-free short rate requires calculus. Let (, ) t 1 2 P t1 t 2 be the price a bond at time 1 R t that pays $1 at timet 2, where t. Given t S T, suppose that an investor at time t short sells a bond that pays k 2
15 $ 1 at time S, and he immediately uses the proceeds from the sale to purchase (, ) (, ) PtS P tt bonds that pay $1 at time T. Notice that the portfolio value remains zero at this point. At time S, the investor has to pay $1 to the person from whom he borrowed the share and he will receive (, ) (, ) PtS P tt at time t to spend $1 at time S for an asset that pays dollars at time T. The net result is that he has contracted (, ) (, ) PtS P tt dollars at time T. This investment will provide a continuously compounded return RtST ( ;, ) for which e ( ;, )( ) (, ) P( tt, ) =. RtST T S PtS This return is called the continuously compounded forward rate, contracted at time t, for the interval[ ST, ]. This equation can be modified into the following form: ( ;, ) RtST When t= S, R( S; ST, ) or (, ) ( ) P( t S) ln P t, T ln, = T S R ST is called continuously compounded spot rate at time S for the interval [ ST, ]. Then the formula becomes: (, ) R ST ( ) ln P ST, = T S As S T in the formula for RtST ( ;, ), it is called instantaneous forward rate with maturity T, contracted at time t. Specifically, ( ) P( t S) ln P t, T ln, f( tt, ) : = lim S T T S = ln P( tt, ) T
16 Finally, whent = t, r( t ), the instantaneous short-rate in continuous time is given by ( ) : P( tt, ) r t =. T T= t We see that the concept of risk-free rate is actually a more complicated concept than it appears to be in different risk assessment models. The risk-free rate is not a constant but a random variable depending on the duration of investments and the time when the investment was contracted.
17 Capital Asset Pricing Model Section One: Assumptions The Capital Asset Pricing Model, also known as CAPM, is one of the risk measurements which many financial institutions commonly use. CAPM was first introduced by William F. Sharpe in his journal article Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk (Duffie, 2008, p.xiii) CAPM is based upon eight assumptions. Understanding all eight assumptions is crucial in order to comprehend CAPM itself. Therefore, each assumption will be explained in detail in this section. Harrington (1983) concluded in her book that assumptions for CAPM can be divided into two major categories, Efficient Market Assumptions and Capital Asset Pricing Model Assumptions (p.22). Efficient Market Assumptions contain the first five of the eight assumptions. The first assumption is that "The investor's objective is to maximize the utility of terminal wealth" (Harrington, 1983, p22). The most important part of this assumption is that investors maximize utility of wealth instead of maximizing return itself. The major difference between maximizing utility of wealth and maximizing return itself can be explained in two simple examples. Suppose worker A earns barely enough to pay his/her bills. Hence, the next dollar earned is very important to him/her. This means worker A has high utility. In contrast, worker B has a comfortable bank account and he/she can retire at anytime if he/she
18 wants to. However, worker B likes his/her job and would be willing to work that job even if he/she were paid less. Thus, the next dollar worker B earned is less important than that of worker A. In other words, worker B has lower utility. In addition to these examples, the example with worker B illustrates the concept of "diminishing positive marginal utility" that "Each increment of wealth is enjoyed less than the last because each increment is less important in satisfying the basic needs and desires of the individual"(harrington, 1983, p23). CAPM assumes investors will make their investment decisions based upon both their return from investment and their personal demand for future earnings. Hence, the first assumption specifies that investors maximize their utility of wealth and this assumption is the first step to define how investors behave under CAPM model. The second assumption is that investors make choices on the basis of risk and return, and risk is measured by the variance and the mean of the Portfolio's returns (Harrington, 1983, p23). This assumption has two portions. First, the assumption assumes investors are rational. They make decisions purely on available evidence of risk and return, and thus, do not make decisions based on any other factors. In reality, this part of the second assumption is not always true for all investors because investors usually make their decision by using more information than risk and return. Other considerations might include size of a firm and rates of diversification (Bon & Sinusi, 2011, p1). The second portion explains how risk and return are determined. In order to understand the second portion of this assumption, it is essential to understand
19 portfolio risk. Portfolio risk σ p is the risk of a group of individual assets held in combination (Brigham & Daves, 2009, p45). As stated in introductory section, diversification can be used to decrease the level of unsystematic risk. Therefore, portfolio risk cannot be reduced by diversification. Thus, portfolio risk is the total risk of a portfolio measured by its variance. Harrington states that "Portfolio variance is the sole factor determining the investors' perceptions of risk. Return, the only other influence on investors' choices, is the mean expected rate of return." (Harrington, 1983, p23-24). As mentioned before, in finance, return, also known as the rate of return, is not defined as an actual value in terms of the amount of currency but a ratio which is the difference between future price and current price of an asset divided by current price of that asset. The formula is defined as following: where R t = The rate of return at time t P t = The price of an asset at time t t = Time The mean value of expected return is the statistical average of all the rates of return. The rate of return of an asset is required in order to calculate the variance. For example, the following chart contains the stock prices of International Business Machines (IBM) in two-week period from September 10 th 2012 to September 21 st 2012. R t = P t P P t 1 t 1
20 Table 2: Close Prices of IBM September 10, 2012 September 21, 2012 Date Stock Price (dollars) Sep 10, 2012 200.95 Sep 11, 2012 203.27 Sep 12, 2012 203.77 Sep 13, 2012 206.36 Sep 14, 2012 206.81 Sep 17, 2012 207.15 Sep 18, 2012 207.07 Sep 19, 2012 206.43 Sep 20, 2012 206.18 Sep 21, 2012 205.98 Note: Adapted from Yahoo Finance. Retrieved February 24, 2013, from http://finance.yahoo.com By using the rate of return formula, it is simple to calculate the rate of return between each day within this two-week period. Table 3. Calculations of Price Differences and Rate of Return September 10, 2012 September 21, 2012
21 Date Price difference (dollars) Rate of return (%) Sep 10-11 2.32 1.1546 Sep 11-12 0.50 0.2460 Sep 12-13 2.59 1.2710 Sep 13-14 0.45 0.2181 Sep 14-17 0.34 0.1644 Sep 17-18 -0.08-0.0386 Sep 18-19 -0.64-0.3091 Sep 19-20 -0.25-0.1211 Sep 20-21 -0.02-0.0097 This chart provides the day-to-day rate of return for each day over a two-week period. The mean return is obtained by summing up all the daily returns and dividing by 9. The result shows that the mean value is 2.5756%, or in decimal form, 0.025756. Based on what Harrington stated, the variance of all rates of return of an asset from a certain period of time can used to estimate the risk of this asset. The variance of a set of data is equal to the square of its standard deviation. Applying the standard deviation formula defined in introductory section, the standard deviation of IBM's rate of return from September 10 th to September 21 st is 2.3483%. Therefore, the variance is the square of 2.3483% which is equal to 0.05515%. If the standard deviation of another firm A's stock from the same period is 0.1%, then IBM is less volatile compared to firm A so that IBM is less risky than firm A based on the variance calculation. Note that, one of the difficulties when investors use variance is to choose
22 how much data they want to include. In general, disregarding outliers, the larger the data set, the more accurate the information the variance can provide. In this example, only two weeks of stock prices are included, so the variance might not be able to fully represent the risk of IBM stock. The advantage of only using variance and mean is that it can be applied to determine any distribution of return. However, the disadvantage is that "The key problem in using variance is that it is an accurate description only for normal distributions" (Harrington, 1983, p24). For any non-normal distribution of returns, the portfolio variance may not fully reflect the measure of risk. Nonetheless, in CAPM, the distribution of return is considered to be a normal distribution. This assumption has resulted in criticism of CAPM from many quarters, since it is not always consistent with empirical data. Nonetheless it is made for a variety of reasons, including its ability to simplify the subsequent mathematical analysis. The third assumption of Efficient Market Assumptions is that "Investors have homogeneous expectations of risk and return" (Harrington, 1983, p 25). In the real market, investors have their own strategies regarding to risk analysis and expected rate of return. Therefore, investors do not have homogeneous expectations in the real world. The purpose of this assumption is to suggest that all investors evaluate risk and return in the same way so that the model can be generalized and easier to be applied. The fourth assumption states that investors form portfolios to achieve wealth at a single, common terminal date (Harrington, 1983, p27). This assumption assumes all
23 the data used in calculation of CAPM need to be evaluated in fix time range. For example, if an investor wants to calculate the rate of return of a stock and compares to the rate of return from the market, all the historical data from both stock and the market need to be taken from the same time period. If data were taken from different period, the comparison would not be accurate and useful to analyze the performance of this stock relative to the market. The last Efficient Market Assumption is that information is freely available (Harrington, 1983, p28). This assumption states that corporations and investors are not allowed to keep their information privately, instead, they have to share all of their information with everyone. In real markets, the majority of the present value of the stock market consists of stocks of the most carefully analyzed firms (Harrington, 1983, p29). Their financial information is readily available to all investors. However, information about stocks and bonds from smaller privately companies is not so commonly available. Therefore, this assumption may not be fully consistent with the real world. The next three assumptions are specific to the Capital Asset Pricing Model. The first is that there is a risk-free asset, and investors can borrow and lend at the risk-free rate (Harrington, 1983, p 29). The assumption suggests that investors can borrow or lend an unlimited amount at a given risk-free rate of interest (Brigham & Daves, 2009, p88). This is the most essential of the CAPM assumptions. In order to understand the risk-free rate assumption, it is necessary to know what the risk-free rate of interest is.
24 A risk-free asset is an asset whose return is known with certainty. This definition shows that a risk-free asset does not have any potential risk so that its return is already predetermined. Generally, short-term U.S Treasury bills, also known as T-bills, are recognized as risk-free assets, and the rate of return of Treasury bill is used as the definition of the risk-free rate of interest. The T-bill return is also called nominal riskfree rate of interest which is denoted asr RF. The nominal risk-free rate is the sum of the real risk-free rate of interest plus an inflation premium. The formula for the nominal risk-free rate is defined by where r* represents real risk-free rate of interest and IP is inflation premium. The real risk-free rate is the rate of a riskless security if no inflation is expected. The inflation premium is equal to the average expected inflation rate over the life of the security (Brigham & Daves, 2009, p140). It is important remember that "inflation rate built into interest rates is the inflation rate expected in the future, not the rate experienced in the past" (Brigham & Daves, 2009, p140). Therefore, instead of using the annual inflation rate for the past, investors should use the average expected inflation rate for the future. r RF = r* + The second assumption of Capital Asset Pricing Model assumptions is that "there are no taxes, transaction costs, or other Market Imperfections" (Harrington, 1983, p34). Since returns from different types of investments may be taxed at different rates, this assumption is not entirely realistic. In the real market, taxes and transaction costs IP
25 are actually important in investments. For example, if an investor in China wants to invest his/her money to purchase stocks of IBM Corp, this investor has to pay for transaction fee for currency exchanging from Chinese Yuan to US dollars. Also, if IBM pays dividend for its stockholders, the dividend is non tax-exempt income so that income tax is applied. Hence, this investor has to think about transaction costs and taxes in order to make his/her investment more profitable. Transaction costs and taxes are important in the real market. As is often the case when mathematical models are used to explain real-world situations, the main purpose of this assumption is to make the mathematics easier. The final assumption of all CAPM assumptions is that "total asset quantity is fixed, and all assets are marketable and divisible"(harrington, 1983, p35). This assumption suggests that CAPM ignores liquidity of securities. Liquidity of an asset represents how convenient it is for investors to buy or the asset in the market. In real markets, liquidity can vary among different assets. U.S Treasury bills are highly liquid. Compared to T-bills, stocks of companies which are not doing very well and even facing bankruptcy are less liquid. Therefore, CAPM neglects variations of asset liquidity, assuming that all are equally marketable. Section Two: Security Market Line The Security Market Line, also known as SML, is a representation of the Capital Asset Pricing model. The SML is defined as follows: r r r r = + β ( ) i RF m RF
26 Here, r i is the expected return for a specific asset or portfolio, r RF is the risk-free rate, r m is the overall market return, and β is a constant that measures the relationship between the overall market return and the return of the individual asset or portfolio under consideration. The expected rate of return is the minimum rate of return that investors want to have in order to bear a certain additional amount of risk (Sharpe, 1964, p1). For example, suppose that an investor originally decides to purchase a U.S Treasury bill and T-bill offers a return of 6%. Suppose also that the investor s friend wants to borrow money from this investor. The investor agrees to lend money to this friend if and only if this friend will pay 10% percent interest per year, then the expected rate of return is 10%. Note that, as mentioned before, the U.S Treasury bill is considered as a risk-free asset. Therefore, lending money to investor A's friend is riskier than purchasing U.S Treasury bill so that investor A requires a higher rate of return if he/she lends money to this friend. The difference rm-r f is called the risk premium. Risk premium is equal to the expected return from the market r m minus the risk-free rate. The expected return from the market is the return of purchasing securities from every single asset in the market proportionally. In other words, investors regard the whole market as a portfolio and the expected return of this portfolio is called the expected rate of return of the market. Risk premium measures "the additional compensation investors require for assuming the additional risk of assets (Brigham & Daves, 2009, p44). For example, as shown in Fig. 2, the Y- intercept of the graph is the risk-free rate of return which is 6 percent and expected
27 market return is at 10 percent. Hence, the risk premium equals 10 percent minus 6 percent or 4 percent. The risk premium tells investors how much additional return they can realize from a specific asset or portfolio if the market rate of return is at certain point. In the figure below, we think of the coefficient beta, which measures risk Figure 2. Beta versus Expected Return associated with the asset or portfolio as the independent variable, and expected return on the asset or portfolio as the dependent variable. Thus, the SML provides a direct link between risk and reward. The slope of the SML is the overall market risk premium. This slope of the SML measures the rate at which the expected return on the asset varies as a function of the overall market return, both in comparison to the riskfree rate.
28 The SML is based on a linear regression model. Linear regression represents the relationship between a response variable y and a single explanatory variable x. The formula of linear regression is then given by the linear equation y = a +, 0 ax 1 where the coefficients are determined by applying least-squares methods to actual data that relate historical values of x and y. Comparing the linear regression formula to the CAPM model, a is the nominal risk-free rate of return and x is the beta 0 coefficient, and a stands for the overall market risk premium which is the slope of 1 the linear regression model. Hence, in CAPM, market risk premium is the slope of Security Market Line. This means that when the expected market return increases or decreases, the graph of SML will become sleeper or flatter, respectively. The most important factor in Security Market Line is beta (β). In CAPM, "the market risk of a stock is measured by its beta coefficient, and beta is the proper measure of the stock's relevant risk" (Harrington, 1983, p59). Given a portfolio consisting of several stocks, a single stock's relevant risk is its contribution to the portfolio's risk" (Brigham & Daves, 2009, p52). This means that risks of individual stocks can impact the overall risk of portfolio. Based on Brigham and Daves's work (2009), the beta of a portfolio is equal to the weighted average of betas of securities in this portfolio (p60). The weighted average formula is defined as n β p = w β i=1 where β is the beta of portfolio, and the weights p w i are the proportional values of i i
29 the individual stocks in comparison to the overall portfolio value. We illustrate this formula with a simple example. Suppose investor A invests ten thousand dollars in 4 stocks. The information of these four stocks is presented in the chart: Table 4 Investor A's portfolio Stock Name Stock Price Number of Shares Total Amount Beta Stock A 10.00 500 5000 0.89 Stock B 15.00 200 3000 1.21 Stock C 5.00 100 500 1.34 Stock D 4.00 375 1500 0.99 In order to calculate the beta of investor A s portfolio, it is necessary calculate the weights the proportion of the value represented by each specific asset. Here, the relevant weights are 50%, 30%, 5% and 15% for Stocks A, B, C, and D, respectively. Therefore the portfolio beta is 0.89*0.5 + 1.21*0.3+1.34*0.05+0.99*0.15 = 1.0235. Having calculated the portfolio beta, it is logical to ask what it tells the investor about his possible investment decisions. CAPM defines the beta of overall market risk to be 1. This means that if the risk of an asset is greater than 1, this asset is riskier than the market as a whole, and if the risk of an asset is less than 1, this asset is less risky than the market as a whole. In the previous example, the portfolio beta is 1.0235 which is almost equal to the market beta. In this case, investor A's portfolio risk is
30 essentially equal to the market risk. In particular, his portfolio is well-diversified, nearly all of its risk being non-systematic. Notice that this example also shows the advantage of diversification. Stock B and Stock C have relative higher beta values than the market beta. However, by combining Stock A and Stock C which have betas that are lower than the market beta, the portfolio beta is close to the market beta that investor A's portfolio is as risky as the market. Two special cases are of particular interest. The first case is when beta is zero. If beta is equal to zero, then based on SML formula, it means the expected rate of return of an investor is equal to nominal risk-free rate. The second case is negative beta. Negative beta securities do not commonly exist in the market. They would be less risky than a risk-free asset. A negative beta coefficient is, however, mathematically possible. From regression theory, we know that the beta coefficient is mathematically defined to the correlation between return of individual stock and market return multiplied by the quotient of the standard deviation of return of individual stock divided by the standard deviation of the market's return (Brigham & Daves, 2009, p54). Symbolically, β i = ( σ i σ M The beta coefficient is denoted as β i. σ s the standard deviation of individual stock's i rate of return and σ is the standard deviation of the market's return. σ and σ are M i M calculated by using the same method as presented in the early section of this thesis. It ) ρ im
31 is interesting to note that, for fixed values of market volatility σ M and correlation ρ im, β is an increasing function of asset volatility σ. In simple language, the more i i volatile an asset, the greater the risk (and therefore, the greater the expected return) on the asset. Calculation of the standard deviations σ i and σ is based on historic data, M and requires that the investor identify a time period that determines how far back into the past he will go to obtain this historic data. This can be tricky. The investor needs to accumulate enough data to give accurate estimates of these standard deviations, but not go so far back into the past as to obtain data that no longer applies to the current behavior of the asset s returns. Recent history is good, but ancient history can be misleading. formula The correlation coefficient ρ im is also calculated from historic data using the ρ im = [ n 2 ( r i, t - r i, Avg) ][ ( r M, t -r M, Avg) t=1 n t=1 ( r i,t - r M, Avg )( r n t=1 j, t - r M, Avg ) 2 ] While this formula may appear to be complex, standard spreadsheet software has a built-in function for its calculation. More important than the method of calculation, however, is the interpretation of ρ im. Specifically, the correlation coefficient tells investors how a specific individual asset behaves compared to the market. When ρim is equal to 1, this asset is perfectly positive correlated with the market. In this case,
32 when the value of the market increases or decreases, this asset also increases or decreases at the same rate respectively. When ρim is equal to -1, this asset is perfectly negative correlated with the market. Perfectly negative correlation indicates this asset behaves totally opposite compared to the market. For example, if the market increases 10% in its value, this asset will have 10% decreases in its value. When ρ is equal to im 0, the asset is not correlated with the market. In this case no information about changes in the asset s return can be obtained from information about changes in overall market return. We illustrate these ideas with an example. The following chart contains the adjusted closing prices of Google Inc. and the Dow Jones Industrial Average from October 1 to October 9, 2012. Table 5. Dow Jones Industrial Average Date Adj Close of Google Inc. Adj Close of Dow Jones Oct 1, 2012 761.78 13515.11 Oct 2, 2012 756.99 13482.36 Oct 3, 2012 762.50 13494.61 Oct 4, 2012 768.05 13575.36 Oct 5, 2012 767.65 13610.15 Oct 8, 2012 757.84 13583.65 Oct 9, 2012 744.09 13473.53 Note: Adapted from Yahoo Finance. Retrieved February 24, 2013, from http://finance.yahoo.com
33 Using Microsoft Excel, the correlation coefficient is equal to 0.66498. In this example, the correlation coefficient is greater than 0 and less than one. This means that Google Inc. is positively correlated with the market. When returns on the Dow Jones index increase, we therefore expect returns on Google stock to increase as well, and when the Dow Jones returns fall, we expect a corresponding drop in Google returns. Since the correlation coefficient is not exactly 1, it is also possible for Google returns to fluctuate in part due to systematic characteristics of this specific stock that are not shared in common by the market as a whole. As a second example, the following chart contains the adjusted closing prices of International Business Machines Corporation (IBM) and Game Stop Corp from October 1, 2012 to October 9, 2012. Table 6. IBM and Game Stop Corp Adjusted Close Prices Adj Close of IBM Date Adj Close of Game Stop 210.47 Oct 1, 2012 21.24 209.84 Oct 2, 2012 21.33 210.51 Oct 3, 2012 21.30 210.39 Oct 4, 2012 22.37 210.59 Oct 5, 2012 23.08 209.82 Oct 8, 2012 23.25 207.99 Oct 9, 2012 23.99 Note: Adapted from Yahoo Finance. Retrieved February 24, 2013, from http://finance.yahoo.com
34 Based on the information of this chart and the closing prices of Dow Jones, the correlation coefficient for IBM is 0.13999 and the correlation coefficient of Game Stop Corp is -0.45974. This shows that IBM returns are positive correlated with the market, but that Game Stop returns are negatively correlated with the market. Note however, the value IBM s correlation coefficient is relatively closer to 0 compared to the correlation coefficient for Google. This means that the value of Google returns are more likely to fluctuate with the market than IBM returns. Since the correlation coefficient of Game Stop returns with market returns is negative, Game Stop s returns should opposite those of the market as a whole. If the Dow Jones index rises, we would expect Game Stop returns to fall. A final observation regarding these examples is in order. The data sets used in the calculations are very small, so the estimates we obtain from our calculations may be inaccurate. In practice larger data sets are used so that more reliable estimates of the correlation coefficients may be obtained. Section Three: Test of CAPM Capital Asset Pricing Model is created based on eight assumptions as stated in the first section. These assumptions allow investors to use CAPM to estimate the risk of an individual asset or a portfolio in a simplified way. However, in the real market, risk level can be affected by variables which are neglected by assumptions in CAPM. Therefore, the Capital Asset Pricing Model may not give investors accurate estimation of risk level of an asset or a portfolio. In this section, two experiments will be
35 presented to analyze the accuracy of CAPM. Data of Priceline.com (NASDAQ: PCLN) and Sonoco Products Company (NYSE: SON) are used in the first example. Data of PCLN and SON are from July 1, 2010 to Dec 31, 2010. Data of NASDAQ and NYSE from the same time period are used as the market data. Data sets are listed in the appendix section of this thesis. Recall that the SML formula requires the nominal risk-free rate, beta and the market risk premium in order to calculate the expected rate of return of an individual stock. Rate of ninety-day Treasury bill is used as the nominal risk-free rate in this example. The appendix lists risk-free rates of ninety-day T-bills from July 1 to Dec 31 2010 converted into daily risk-free rate, 0.0015. The market risk premium is calculated by subtracting daily risk-free rate from daily market rate of return. We begin by analyzing the PCLN data. The test for each stock is divided into two portions. The first part is to investigate how well CAPM performs to estimate the expected rate of return for a single asset.. The average expected daily returns are calculated by using SML. The average actual daily return in the appendix is equal to the average value of all daily rates of returns of the stock. After calculation, the average actual daily rate of return is 0.0033 or 0.33% and the average daily expected rate of return is 0.22%. The difference between the estimated rate of return and actual return is merely 0.11% which is a very small percentage. Returns of SON are calculated by the same method. The average actual daily return for SON is 0.0011 and the average daily expected rate