Finance is defined as the most powerful technology of 20th century. As every complex technology it needs continuous research and development. In nowadays world of globalization and securitization, governments and multinational corporations are stressing out of the importance of effective financial security which in most of the cases means to lower possible risk as much as possible, however they are not willing to pay high price. This thesis shows possible alternatives of assessing and dealing with financial risks.
The main focus of this paper is the Modern Portfolio Theory. Harry M. Markowitz is well known father of portfolio investments. He firstly introduced Modern Portfolio Theory in 1952, which became quickly a foundation for any further development in this field. For achievements of his studies in field of finance he was awarded The Sveringes Riksbank Prize in Economic Sciences in Memory of Alfred Nobel in 1990. This thesis explains the basic elements of statistics related to finance, correlation coefficient and covariances. Furthermore it deals with concepts of legitimate and efficient portfolio which are essential in his theory.
Moreover the Capital Asset Pricing Model (CAPM) is introduced. William F. Sharpe, Professor of Finance at Stanford University, set the basis of CAPM in 1964. He structurally developed a model which deals with assets pricing based on several types of risks. The main risk factor is Beta, which is discussed in details.
However, in 1976 economist Stephen Ross initiated so-called Arbitrage Pricing Theory as an alternative to CAPM, which takes into account various macroeconomic factors influencing prices of assets. All these form a basis for any profit seeking investment decisions. Risk factors as confidence in the market, investment time horizon, GPD fluctuation or inflation risks are specified. Furthermore an example of such APT model is part of last chapter.
Diminishing Marginal Utility of Wealth
According to law of marginal utility of wealth every human being perceives one's utility as a function of one's wealth. Utility of wealth can be defined as personal perception of his emotional well being as the level of his wealth. Every investor, as human being, perceives his utility differently and it is very difficult to compare two investors in measurable concept. However it is essential to understand why most of investors are risk-averse, because risk as an uncertainty is a salient element of investing (Blake 1990).
There are 3 different types of investors according to risk perception:
Risk-averse investor
Risk-averse investor is the one who always chooses the less risky possibility while choosing between securities offering the same return on investment. The main reason for most investors being risk-averse is that they have diminishing marginal utility of wealth. The diminishing marginal utility of wealth of risk-averse investor is a concave function of his wealth (Figure 1). In principle, it states that as the level of any individual investor's wealth increases so does the level of satisfaction that is derived from owning more wealth. Furthermore, additional growth (a marginal increase) in wealth increases utility of investor by smaller percentage amount. There is another important connection between diminishing marginal utility of wealth and investing behavior- for any level of wealth, a fall in wealth results into a higher percentage fall in utility compare to the increase in one's utility while having the same increase in wealth (Blake 1990).
Risk-neutral investor
If there is a choice between two securities offering equal return on investment, risk-neutral investor is indifferent to either of investments. He does not take risk into account while making investing decisions and therefore it is obvious that his utility function is linear- straight line. Constant marginal utility of wealth means that any positive or negative change in wealth brings exactly the same change in his utility level. Therefore his ∆U2= ∆U1 (Blake 1990).
Risk-loving investor
Risk-loving investor can be defined as the one who enjoys taking risk. While offered two investment possibilities which offer exactly same level of return on investment he always chooses the more risk possibility. His utility function is defined as convex and therefore his ∆U2= ∆U1. However, in real world there are not many risk-loving investors (Blake 1990).
Since neither utility nor expected utility can be objectively measured and compared, another concept has been developed by simply modification of utility theory. The different method of measuring the choice between risk and return can be achieved by taking into account Taylor's expansion of expected utility function. This expansion rather consider utility as a function of the return on investment into the risky portfolio than a utility function defined on the level of wealth invested in risky assets as does the standard theory of diminishing marginal utility of wealth. In other words it is the difference between wealth in the portfolio at the end of investing period and wealth invested in that portfolio at the beginning of period (Blake 1990).
Portfolio Selection
A good portfolio is much more than only a long list of various stocks and bonds. It should be well balanced, providing the investor protections as well as opportunities. Each portfolio is unique and suits different needs of individual investor. Each investor has always particular needs. An analysis of portfolio begins with information gathering concerning individual securities. On the other end there is the conclusion concerning the portfolio. As one source of information is the past performance of securities. The choice based solely on past performances assumes that the average returns are the best estimates of returns in the future. A second important determinant is the beliefs concerning future performances. No analyst can be expected to predict with certainty whether a given security will increase or decrease in value. Therefore only carefully, professionally and diligently formed judgments about the potentialities and possible vulnerabilities form the best basis upon which the portfolio analysis can be performed. The proper choice of criteria for selection of securities among portfolio depends on the nature of the investor. However, it is assumed that all investors are risk-averse. The proper choice among efficient portfolios is conditioned by the willingness to bear risk. If the priority of investor is the safety, return must be sacrifice and logically the higher degree of risk taken necessarily means higher level of return obtained. The relationships between securities and portfolios are mathematical in nature. They are based on the definitions of properties of numbers (Markowitz 1959).
Elements of Probability
The basic relevant element of probability is random variable. It is a number generated by a "chance device", which is for example a coin. Random variable of a flip of coin is one of the two sides represented by each side of coin- head or tail. We cannot predict with any confidence the result of flipping a single coin. However it is easily estimated when flipping a large number of coins that the heads will appear approximately on half of them. Another example of chance device is a dice, with random variables represented by numbers 1-6 on each side of dice. The expected value is, by definition, the weighted average of all possible outcomes (Markowitz 1959).
The expected value of a single roll of a dice is therefore 3.5, however it is not possible to obtain such outcome. The theorem of expected value is used to describe probability of returns. Historical data are composed in order to approximate outcomes. Expected value is one of the possible measures of central tendency, which shows where the center of distribution is located. The mean is in fact the expected value of the random variable and therefore it is assumed it is the more appropriate measure of central tendency compare to mode or median of a distribution, since there is only 1 mean but there might be more modes or medians. Mean is the essential element for conduction any further security analysis, since it is used for conducting the deviations of returns from the average returns. The sum of all the deviations from average of series is always equal to 0 (Markowitz 1959).
The variance of a past series (for example- returns on single security) is the averaged squared deviations from the expected value. The variance of a random variable is equal to the expected value of squared deviation from the expected value:
If there is a random variable r and A is any number and w is equal to product of r and A (r*A), then:
VAR (w) = A2* VAR(r)
The standard deviation is the simple the square root of the variance:
The standard deviation is an effective measure of variability, since it takes into account more than one or two extreme returns. It measures the level of uncertainty associated with the future event. It became preferable measure of risk and variability of given portfolio or single security. Since it is easy to interpret, standard deviation is used for a direct comparison of past series or a set of probabilities rather than variance. However the relationship between securities and portfolios are, in general, simpler to express in terms of variance. Even in computing the efficient portfolios, variance is used to very last step when it is transmitted to standard deviation (Markowitz 1959).
Covariance and Correlation
Covariance is used for the measurement of the extent to which two different set of numbers move either up or down together. The covariance between random variables q and r can be defined in the terms of their expected values q' and r' respectively. q'=q - expt(q) and r'=r- expt(r). If both q and r > expt(q) and expt(r) respectively, or both q and r < expt(q) and expt (r) respectively, q'* r' is positive. If one random variable of given pair (q, r) is above and second one is below it's expected value, the product of q' and r' becomes negative (Markowitz 1959).
Let's assume A,B to be any numbers, then the general formula for covariance between variables Aq and Br can be written as following:
Cov(Aq,Br)= A*B* expt(q'*r')
Correlation coefficient is a very similar measure of the relationship between random variables. It is the tendency for random variables to vary together. The correlation coefficient is easily expressed in terms of covariance and standard deviation of two random variables. In fact it is a ratio:
Correlation coefficient (=
The covariance proves to be more adaptable measure in formulae and proofs. It is much easier to interpret compare to the covariance, which is as well the measure of movement of two random variables. The correlation coefficient is always defined within boundaries of [-1, 1] (-1 and 1 are included) (Markowitz 1959).
Three main types of correlation between two variables are: positive, negative or uncorrelated. Positive correlation is the indicator of movement of one variable in the same direction as the second at the same time. Negative correlation between two variables occurs when one variable moves the opposite way to second variable at the same time. If variables move independently of each other they are characterized as uncorrelated (Fabozzi 2003).
Figure 1(a,b,c): Correlation Coeficients
Furthermore, two extremes of correlation might occur. If one variable is the exact positive multiple of another one, the correlation coefficient of these two variables is equal to 1, they are perfectly positively correlated- Figure 2a). In opposite to that, if one variable is the exact negative multiple of another one, the correlation coefficient of these two variables is equal to -1, they are perfectly negatively correlated- Figure 2b). Furthermore, Figure 2c) is the representing returns of two uncorrelated securities (Fabozzi 2003).
Two-asset portfolio
When considering two securities are perfectly positively correlated, , two expected returns (r1,r2) always increase and decrease together. The expected return (rp) and the standard deviation (σp) are equal to:
rp= 01r1+ 02r2
σp= 01σ1+ 02σ2
01and 02 are the proportion of total wealth invested in security 1 and 2, respectively. 01=(1- 02) (Blake 1990)
Figure 3: Perfectly Positively Correlated Securities
Figure 3 is representing two-asset portfolio. At point A, the 01=1, therefore 100% of wealth is concentrated in security r1, and 0% in security r2. Point B simply represents the exact opposite. The different portfolio options while lies on the linear line AB. Point P represents any random portfolio made up of security r1 and security r2. There are no benefits obtained by diversification - it is not possible to sacrifice risk without sacrificing some return (Blake 1990).
When correlation between two assets in given portfolio is perfectly negative, , their returns are always moving together in opposite direction. Even though the return of such a portfolio is calculated the same way as with positive correlation, the variance (σ2p) and standard deviation (σp) is calculated by following formulae:
(σ2p) = 01σ1+ 02σ2-2 01σ102σ2
(σp)= 01σ1- 02σ2
Figure 4: Perfectly Negatively Correlated Securities
Figure 4 represents perfectly negatively correlated two-asset portfolio. Point A represents 01=1, point B 01=0. The relationship 01= (1- 02) implies. All portfolio opportunities are lying on linear segments PA and PB. Starting at point B and moving towards point P ensures increasing returns and decreasing risk. No investor chooses any point on segment PB, because point P dominates all possibilities on PB. Linear segment PA serves as a possible portfolio option for any risk-averse investor. Any increase in risk is compensated with adequate increase in return. Point P' is representation of possible portfolio with opposite correlation of assets. The benefits from diversification when securities are perfectly negatively correlated are obvious (Blake 1990).
Significance of diversification
The correlation among securities is one of the most significant features of investing. The returns on different securities tend to move together, however this correlation is not perfect. In case of zero correlations between securities, diversification could eliminate risk to very minimum. Opposite to that if the correlation is be perfect, diversification would do nothing to eliminate risk. The fact of securities are non-perfectly correlated implies the diversification is able to reduce risk however not eliminate it completely. In order for reduction to be as high as possible it is necessary to avoid portfolio whose securities are highly correlated. (Markowitz 1959).
Insurance companies are based on so-called risk pooling. The insurance principle is based on wide diversification of uncorrelated risky events. In that case the diversification becomes extremely powerful tool which practically eliminates the uncertainty of final output. However in portfolio of securities only limited reduction of risk can be achieved by increasing the number of securities into certain portfolio. A security might add much or little variability not according to the value of its own variance. It is the sum of all its covariances between the security and other securities of the given portfolio (http://academicearth.org).
Diversification example
Assumptions for Table 1:
Hypothetical Portfolio A consists of securities with 0 correlation and variance of returns of 0.5.
Hypothetical Portfolio B consists of securities with 0.5 correlation and variance of returns of 0.1.
Even though the Variance (σ2) of Portfolio A is 50 times greater compare to the Variance (σ2) of Portfolio B, the zero correlation among 10 000 securities in Portfolio A cause the Standard deviation (σ) of Portfolio A to approach 0, on the other hand, diversification beyond 100 securities in Portfolio B has apparently no effect on the Standard deviation (σ) of Portfolio B. This unwanted outcome is caused by correlation of 0.5 among securities representing in Portfolio B. The reduction of 9.2% in Standard deviation (σ) of Portfolio B has been achieved - the risk has been reduced however not eliminated (Markowitz 1959).
Number of Securities
σ2 , Portfolio A
σ, Portfolio A
σ2, Portfolio B
σ, Portfolio B
1
5
2.236
0.1
0.316
10
0.5
0.707
0.055
0.235
25
0.25
0.5
0.052
0.228
50
0.1
0.316
0.051
0.226
100
0.05
0.224
0.0505
0.225
250
0.02
0.141
0.0502
0.224
500
0.01
0.1
0.0501
0.224
1 000
0.005
0.071
0.05005
0.224
10 000
0.0005
0.022
0.05005
0.224
Table 1: Diversification and Correlation
The returns on a security might be correlated with different extent with different securities. The returns on another security might have a partially overlapping pattern of high or low correlation. The most challenging task is to pick securities in a way that their average covariance is small. Since different amount of total investment can be allocated into certain securities, the goal is to keep the weighted average of covariances as small as possible. Since portfolio consisting of 100 securities gives 5000 covariances, computer programs are used as supplant for human analyst. The analyst, or rather analyst team, must only decide the model upon which software derives covariances. The machine takes over the routine and leaves the team of analysts free to concentrate on induction and judgment process (Markowitz 1959).
Efficient Portfolios
" A portfolio is inefficient is it is possible to obtain higher expected return with no greater variability of return, or obtain greater certainty of return with no less average or expected return" (Markowitz 1959)
Three-security definition
Figure 5: 3-Security Legitimate PortfolioFirstly it is necessary to define the term of set of legitimate portfolios. The three-security standard portfolio analysis requires all X1, X2 X3 0 and at the same time X1+ X2 +X3 = 1. By substituting X3= 1- X1+ X2 we can geometrically exhibit the set of legitimate portfolios (Figure 5) in two dimensions. The area on and in this triangle abc is referred as the set of legitimate portfolio. Point P inside of the triangle is example of legitimate portfolio (Markowitz 1959).
Iso-mean Lines form a system of lines, which are parallel to each other. They are loci of points with the same expected return. Increasing expected return (E) causes Iso-mean line to shift. The example of Iso-mean Lines is exhibit on Figure 6a).
= fractions of the portfolio invested in security 1, 2 and 3, respectively.
expected returns on security 1, 2 and 3, respectively.
Iso-variance Curves form a system of ellipses, with common center (C), then same orientation and exactly equal ratio of the longest diameter to shortest diameter. The point C shows where the minimum variance of set of portfolio is located. There exists an Iso-variance Curve for every value of variance (V) higher than C. Increasing V causes Iso-variance ellipse to expand outwards C- Figure 6a). As the variance increases, there are shits to ellipse system- starting at point C, shifting to V1, V2, V3 and so on (Markowitz 1959).
V=
= fractions of the portfolio invested in security 1, 2 and 3, respectively.
= covariance between the returns of security x and y.
Another very important element of efficient portfolio theorem is so-called critical line. It is a straight line, usually labeled by l (see Figure 6a). It connects all the tangencies between Iso-mean Lines and Iso-variance Curves. It links all the points minimizing the level of variance among portfolios with the equal expected return. It always passes through the center point C (Markowitz 1959).
Figure 6(a,b): 3-Security Efficient Portfolio
Figure 6b shows where the set of efficient portfolios is located. Since point C is the minimum variance point (and it is part of legitimate set of portfolios) it must be a part of efficient portfolio. Moving along critical line in the direction of increasing E until reaching border of legitimate portfolio set ensures increasing expected return for an exchange for minimum variance. Although point b in Figure 6b has higher expected return than point a, it cannot be a part of efficient portfolio since it is not legitimate. The reaching point 1 at X1 axis means concentrating 100% of investor's wealth into security X1, which therefore must dominate in terms of highest expected return (Markowitz 1959).
Four-security definition
Figure 2: Four- security legitimate portfolio Four-security case requires 3-dimensional geometry. The standard set of legitimate portfolio is illustrated by Figure 7. Points a, b, c, d are corners of tetrahedron representing legitimate set by points on and in it. In four-security analysis the Iso-variance Curves became sets of ellipsoids, flattened spheres, which are in fact 3-D generalization of ellipse. The term of subspace is needed to be introduced. Subspace s1,2,3,4 means that X1, X2, X3, X4 > 0. On the other hand s1 means that X2, X3, X4 = 0. Several critical lines, also called critical set l, are needed for the purpose of separating efficient from inefficient portfolio set. However every efficient portfolio lies on some critical line. Critical set l1,2,3,4 is associated with the subset s1,2,3,4. In Figure 8, Point X is the legitimate portfolio with the smallest variance than any other portfolio. To find all efficient portfolios, one must move along critical lines towards increasing returns. As soon as intersection with another critical line is reached, transfer of critical lines is required. This pattern is repeated until the point
Figure 8: 4-Security Legitimate Portfolioof ultimate return is reached. In Figure 8, such point is located on axis X2 at s2 (meaning X2=1; X1, X3, X4 = 0). The line along the efficient portfolio is created is called set of efficient portfolios (Markowitz 1959).
N-security definition
A portfolio X can be expressed by column vector X:
Where X1 is the fraction of total portfolio invested in first security. The expected return on a portfolio X consisting of N numbers of different securities can be expressed by following formula:
Or simply by:
Where, is the expected return on ith security. The column vector µ is the representation of µ. The transpose (change from column to row matrix) is equal to:
(Markowitz 1959)
The expected return on portfolio is then equal to inner product:
E=
Furthermore the total variance of return on portfolio X consisting of N numbers of securities can be expressed by following formula:
V=X'C X
Where X' is transpose of X, and C is the covariances matrixes (Markowitz 1959).
Capital Asset Pricing Model
Assumptions for CAPM
1. All investors are seeking for points on the efficient frontier. The specific point on the efficient frontier depends on the individual investor's risk-return utility function.
2. Investors can borrow or lend any amount of money at the risk-free rate of return (RFR). If assumed the government T-bills to be risk-free rate it is clear, it is possible to lend money at such a rate. However it is also assumed there is the possibility to borrow at this riskfree.
3. All investors have homogeneous expectations about future rates of return.
4. The model is developed for a single hypothetical period- all investors have the same one-period time horizon, for example: one month, six months, or one year.
5. All investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio.
6. No taxes or transaction costs involved in buying or selling assets.
7. There is no inflation or any change in interest rates, or inflation is fully anticipated.
8. Capital markets are in equilibrium. Proper level of return is associated with each level of risk.
(Fabozzi and Peterson 2003)
Risk and Return
William F. Sharpe, Professor of Finance at Stanford University, firstly introduced the CAPM in 1964. He structurally developed a model which deals with assets pricing based evaluation of two basic elements- risk and return associated with investing into certain financial assets. (www.investopedia.com)
Sharpe set forth the term market portfolio, which is the portfolio which is composed of all the assets in the given market. The ratio of assets' market value to the total market value is the weight by which each of the assets is corresponded in the market portfolio. There exists one and only one such a portfolio in specified market. The market portfolio is the representation of the most diversified portfolio. The only risk in portfolio consisting of all the assets in the market is call market risk. Since market risk is associated with market portfolio, each of the assets possesses some degree of market risk. Market risk is systematic across all assets in given portfolio. The opposite of market risk is the company-specific risk, when referring to stocks, while it relates to specific company's own situation and it does not pervade all other securities and therefore it is not associated with the market. By definition it is independent of the market-wide risk (Bodie, Kane and Marcus 2001).
Figure 9: Diversifiable and Undiversifiable Risks
According to Sharpe, all profit seeking investors are risk-averse, meaning they are willing to take risk only if there is adequate compensation for doing so. The lowest risk assets in the market are called risk-free assets- payments received from such securities are accurately predicted. Neither time horizon nor the total value received is uncertain. Nowadays this term often refers to US government bond, since it has never happen that US government had to default its debt, therefore from historical statistics it has 0 credit default risk. The return on such asset is the compensation for the time value of money. (Alexander 2008)
The risk premium is the compensation for additional risk bearing (Bodie, Kane and Marcus 2001).
E(r1)= Expected return on assets= Expected return on a risk-free asset + Risk premium
If a portfolio is consisting only from two different assets- risky and riskless, expected return on portfolio is:
E(rp)= 01E(r1)+02rf
E(rp)= expected return on portfolio p
01 = proportion of risky asset in the portfolio p
E(r1)= expected return on risky asset
02= (1- 01)= proportion of riskless asset in portfolio p
rf= return on riskless asset
The standard deviation of portfolio is very simple. There is zero default risk associated with investing into riskless asset which means there is no standard deviation (Reilly and Brown 2002).
σp= 01 σ1
σp= standard deviation of portfolio
01= proportion of risky asset in the portfolio p
σ1= standard deviation of risky security
Since σ of riskless security is equal to 0, variance of riskless security (σ2f)= 0, and therefore by definition is uncorrelated with return on risky asset (Reilly and Brown 2002).
Beta risk
Another very important element of CAPM is so-called beta- . This is the measure of asset's return sensitivity to the market's return. Beta serves to fine-tune the level of risk premium for individual asset, since the return on security i is defined as following (Blake 1990):
ri= rf+(rm-rf)
, where:
= Beta of security i
COV(I,M)=Covariance between security i and market portfolio m
VAR(M)=Variance of market portfolio m
rf= return on risk free asset f
rm= return on market portfolio m
rf= return on risk free asset f
The graphical view of Beta can be obtained by composing Security Market Line (SML). SML is plotting expected return as a function of Beta. In equilibrium all securities are priced in a way they lie on SML. Underpriced security are situated above SML, on the other hand, overpriced securities are plotted under the actual SML. SML demonstrates that undiversifiable risk consists of two elements- security risk and its correlation with market (Blake 1990).
"It provides a unique relationship between the required rate of return and on a security and the amount of undiversifiable risk (measured by) contained in it"
(Blake 1990)
According to D. Blake we distinguish two different types of securities (stocks) based on their Beta:
Aggressive stock- Point A in Figure 10 represents security with Beta= 1,5. It is known as an aggressive stock, since its price is more volatile compare to aggregate market. During bull phase it raises more than market and during bear phase it falls more than the market. Since aggressive stock carries more undiversifiable risk than the market, higher required rate of return is necessary- in Figure 10 it is 18% (Blake 1990).
Figure 10: Stocks according Beta
2. Defensive stock- Point D in Figure 10 shows where the defensive stock is located. It requires only slightly higher (+3%) rate of return than risk free asset. It is less volatile than the entire market in both bull and bear phases, since it carries less undiversifiable risk than M (Blake 1990).
Capital Market Line
The return an investor is expecting for given level of risk is specified by capital market line (CML). CML is the tangent to the efficient frontier. The returns on risky assets (return on risk-free asset+ risk premium) are relative to that of the market portfolio (M). All efficient portfolios are valued in the way they lie on the CML. The slope of CML is the indicator of market price of risk.
Figure 11 shows all the efficient risky portfolios- set AMB, CML and two sets of indifference curves of two investors. The utility function of investor 1 is represented by indifference curves u-11 and u-10. Investor 1 demands higher compensation of risk, which is signaled by the steeper slope of his indifference curves compare to investor 2, whose indifference curves are u-21 and u-20. Without possibility of lending or borrowing at riskless rate, investor 1 would maximize his utility at point P10 and investor 2 at point P20 which are both part of the efficient portfolio. Both investors are able to increase their utilities by either lending or borrowing at riskless rate. Investor 1 will swap P10 for M and after that trades-in part of M in order to lend out the riskless asset, ending with holding the portfolio P11. Investor 2 prefers P21 to P20 since it increases his utility. He obtains market portfolio M by switching from P20 to M, and by borrowing at riskless rate, he invest even more to M, ending with portfolio P21 (Alexander. 2008).
Figure 3: CLM
According to David Blake, the general version of CAPM is an equilibrium model which deals with asset pricing. In market equilibrium, every asset is held voluntarily. There are two basic elements on which CAPM is based- utility maximization and given portfolio opportunity set. In other words, there exists price equilibrium of each asset, which is determined in the market by the forces of supply and demand for a given asset (Blake 1990).
The expected utility of investor h is defined as a function of expected return E(r) on portfolio p held by investor h and the variance of the return σ2 on portfolio p held by investor h:
E(uh)= E(uh)[E(rph),σ2ph]
1. (rph) can be defined as following:
E(rph)= () * ()
= proportion of investor h's wealth in total wealth
= proportion of total market invested by investor h in security i
E(ri)- expected return on security i
= proportion of the total market taken up by investor h in riskless debt
rf= return on riskless asset (Blake 1990)
2. ( σ2ph)can be defined as following:
σ2ph=( )2 * ( )
= proportion of investor h's wealth in total wealth
= proportion of total market invested by investor h in security i
= proportion of total market invested by investor h in security j
= [ri-E(ri)]*[rj-E(rj)]= covariance between the returns on security i and security j
(Blake 1990)
Arbitrage pricing theory
The Arbitrage pricing theory (APT) was developed by Stephen Ross in 1976 as an alternative to the CAPM, which was criticized for the lack of empirical benefits in portfolio management since it holds too many assumptions. Therefore the APT was introduced with only 3 main assumptions:
"1. Capital markets are perfectly competitive.
2. Investors always prefer more wealth to less wealth with certainty.
3. The stochastic process generating asset returns can be expressed as a linear function of a set of K risk factors (or indexes)." (Bodie, Kane and Marcus 2001)
The general formula for the return on single security defined by APT is following:
= actual return on security i
= expected return on security i
reaction of return on security i to movements in a common risk factor j
= systematic risk of security i
= unsystematic risk of security j
(Reilly and Brown 2002)
One of the most important features of APT is the multiple risk factors (), which effects all the securities in the portfolio. Comparing to CAPM, where only β determines the relationship of market risk and individual security risk, APT takes into account more common return factors. Given these factors measures the reaction (or so called loading) of each individual asset to the particular risk factor. Similarly to CAMP, APT assumes the unsystematic risk () is independent and therefore in well diversified portfolio it is possible to minimize it (Reilly and Brown 2002).
Although there is the same risk-compensation principle, higher risk means higher compensation, among both models APT is more complex in matter of compensation calculations, since it takes into account more risk factors than CAPM (Sharpe 2007).
= expected return on a security with 0 systematic risk
= risk premium related to common risk factor j
= correlation between risk factor j and security i
(Sharpe 2007)
Risk Factors
The essential step in APT model is selecting the appropriate set of risk factors. One might argue it involves as much art as does science. Factors must be easy to interpret, robust over the investment period and should explain much of the fluctuations of securities' returns. There are five main risks meeting these criteria (Burmeister, Roll and Ross 2003):
1. Confidence risk
Confidence risk measures the unanticipated changes in investors' attitude towards undertaking relatively risky investments. It is defined as the difference between the rates of return on relatively risky corporate bond and government bond. Both bonds have twenty-year maturities and are adjusted, in a way that the mean of the difference is zero over a long period of time. In case when the return on corporate bond is higher than return on government bond by more than the long period mean it is the indicator of positive Confidence risk. The positive return difference reflects an increased investors' confidence, since the required yield on risky corporate bond has decreased relatively to safer option- government bond. Securities which hold a positive correlation between confidence risk and security itself (> 0) will then rise in price (Burmeister, Roll and Ross 2003).
2. Time horizon risk
Time horizon risk is the measure of unanticipated changes in investors' desired payout time horizons. It is calculated as the difference between returns on twenty-year government bond and thirty-day T-bill, adjusted to be mean zero over a long period of time. If the Time horizon risk is positive, it means the price of long term bonds is higher than short term ones. It signals that investors demand a lower compensation for holding investment with higher maturities. Prices of securities positively exposed to this risk will rise (Burmeister, Roll and Ross 2003).
3. Inflation risk
Inflation risk combines both unexpected components of short run and long run inflation rates. The expected future inflation rate is calculated from historical inflation rates, interest rates and other economic factors. While considering Inflation risk factor, the term positive or negative inflation surprise is used instead, since it is computed at the end of period as the difference between actual inflation rate and the expected rate at the beginning of the period. Most stocks have a negative exposure to Inflation risk, therefore a positive inflation surprise produces a negative contribution to return. In case of negative inflation surprise (deflation shock) a positive contribution of return occurs(Burmeister, Roll and Ross 2003).
Generally, luxury-product industries are the most sensitive to Inflation risk. Consumer demand for such products falls when real income is decreases due to inflation. In opposite, necessity-product industries such as food, cosmetics, drugs or rubber goods are relatively insensitive to fluctuation of real income (Brealey and S. Myers. 2003).
4. Business cycle (GDP) risk
Business cycle risk is determined by unanticipated changes in the real business activity. The expected values of GDP are computed by using available information both at the beginning and the end of period. The Business cycle risk is measured as the difference between the end- and the beginning-of-month value. An increased growth of economy is indicated by a positive realization of Business cycle risk. Industries with the highest correlation to fluctuation of GDP, for example retail stores, will outperform companies which do not respond to changes in level of business activity (for example utility companies) in times of positive growth of GDP (Burmeister, Roll and Ross 2003).
5. Market timing risk
In case when all other macro-economic factors fail to explain additional fluctuation of any security, Market timing risk is computed. In very unlikely conditions of all other risk factors being zero, Market timing risk would be proportional to, for example, S&P 500 total return. Under these circumstances () Market timing risk would be the same as the CAPM beta (Burmeister, Roll and Ross 2003).
Although Market timing risk is not necessary in APT model, which already includes all the relevant risk factors. However in practice, some factors might be difficult to observe and measure, therefore Market timing risk is used as a buffer to absorb all the unobserved risk factors (http://freecoolarticles.com).
APT Model example
Assumptions for following example:
Zero- systematic asset risk premium: = 4%
Factor 1 Beta- unanticipated changes in inflation: = 2%
Factor 2 Beta- unexpected changes in real GDP: = 3%
There exist 2 different investing vehicles x and y
= the response of asset x to changes in the inflation factor is 0.5
= the response of asset x to changes in the GDP factor is 1.5
= the response of asset y to changes in the inflation factor is 2.
= the response of asset y to changes in the GDP factor is 1.75
By substituting into the general formula we can obtain the expected return on each asset.
For asset x:
For asset y:
Figure 4
Figure 12: 2-Asset APT Model
Figure 12 is the graphical demonstration of ATP 2- risk factor model. It is simple to exhibit this example in graphic basis, since it is only two risk factor model. In general, APT is K+1 dimensional model, where K is the number of risk factor input and +1 is the determinant of Expected return (Reilly and Brown 2002).
Conclusion
The general conclusion of this paper is that risk is the most important factor of any investment decision. Every investor, a living human being, must decide how much risk is he or she willing to undertake and must assess whether an appropriate return is associated with each individual investment. I believe understanding Modern Portfolio Theory is the essential step for building up structural understanding of this interesting topic. Furthermore linking all the relevant elements of statistics, such as standard deviation, variance and covariance is the important stage in development of portfolio. Input of the right data is a crucial step in creation of healthy portfolio. In my opinion well diversified portfolio of investment vehicles is conducted by the least correlated securities. It is not the performance of each individual asset represented in a portfolio which matters, however the correlation among all the securities differs the excellent portfolio apart from unsatisfactory one. Finding such securities is therefore the most influential task of any portfolio management.
In my opinion, APT model is more adaptive compare to CAPM "in the real world". The assumptions for CAMP are too unrealistically complex and therefore I see lack of empirical applicability of the model. I am aware of the importance of the linkage between security and the market, however I think there exist much more factors influencing the volatility of prices of investment vehicles. From my point of view, in the long run rises or falls of the prices are connected to macro- economic factors such as inflation, GDP fluctuations, unemployment and many others. The fatal short term shocks might be caused by herd effect of contagious confidence of investors or wrong market timing. I believe taking right investment decisions requires to combine both deep knowledge in finance and art of perception of the big picture.
