Rational Aprroaches on Stock Price Determination

To determine stock prices there are different approaches. The fundamental analysis (FA) is the approach that is used by most traditional investment analysts.  

Lev and Thiagarjan (1993) state that FA is an aid to define the value of corporate securities. One can do this by gently examination of key value-drivers, such as earnings, risk, growth and competitive position. According to Abarbanell and Bushee (1998) FA relies mainly on the analysis of actual and past data of financial statement to judge when underlying firm value varies from prevailing market prices. FA essentially tells us what the price of a stock should be. This can be considered as its intrinsic or fair value based on its future earnings and return on investment. However, the actual price of a stock is determined by the stock market and the stock market is driven by human emotion. So, what we really want to know is what price a stock will be on the stock market within a future time window.

Technical analysis (TA) is another way to determine the price of a stock. This approach applies statistical techniques to historical stock prices and volumes to determine future stock price fluctuations. Lo, Mamaysky and Wang (2000) state that TA is for many decades a part of financial practice. But this analysis has not the same level of academic acceptance and research as the more traditional disciplines like the FA. The highly subjective aspect of TA is one of the main obstacles. The actual stock prices consist of two components, the fair value price and a variance from the fair value due to dynamic environments and human emotion. The size of the variance depends on how volatile the environment and emotion is.

To make financial/investment decision there are several pricing models or theories that helps investors to decide. All these models rely on rationality.

 1. Modern Portfolio Theory (MPT) or Mean-Variance Portfolio Theory

Markowitz (1952) introduced Modern Portfolio Theory (MPT), began to formalize ideas of how a rational investor would invest in a set of assets by accepting risk to earn higher. This theory formed the foundation of financial economics for several decades and made many surprising and sharp predictions; for example, about how investors choose which stocks to hold and what market prices would result from these decisions. MPT is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk (Garcia, Bueno, & Oliver; 2015). It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. It uses the variance of asset prices as a proxy for risk.

            Becker, Gürtler, and Hibbeln (2013) explain how can an investor select best combination of risk & return to maximize wealth using MPT. They analyse the effects of portfolio rebalancing with and without restrictions on the weights of stocks included in the portfolio. Investors commonly have three possible attitudes toward risk; 1) a desire for risk: Risk seeker, 2) an indifference to risk, and 3) an aversion to risk: Risk averters. Investors are mostly risk averters. Given the two investment alternatives having the same expected returns, which of the following should be chosen.                                                                               X                    Y

                 Return, E(R)          10%                 10%

                 Risk,     SD(R)        3%                   8%

 

Mean-variance indifference curves

Utility Theory states that higher the indifference curve, higher would be the level of satisfaction. Different individuals have different indifference curves. Risk averse investor is indifferent between points X, Y& Z as there is same level of satisfaction (fig.5). Investor B requires higher return for the same risk (fig.6).

Fig.5. Mean-variance indifference curves/ Utility Theory

Fig.6. Family of indifference curves for individuals A & B

Mean & variance of single asset:

The expected return from a portfolio is the weighted average of the expected returns of individual stocks, given as:

Mean  E(R) =          

Var (R)=                

Hypothetical expected rates of return for two firms

Econ. conditions    Prob.       Sugar       Cement       Combined                                                                                                                                                 (50% each)

                               (P_i)           (R_s)         (R_C)                  (R_p)   

Bad                         0.2            0.1          0.5                   0.30

Average                 0.6            0.2          0.3                   0.25

Good                      0.2            0.3          0.1                   0.20

Total                      1.0                   

                                               

               P_i    R_s      R_C            R_p        E(R_s)    E(R_C)    E(R_P)     VAR(R_s)      VAR(R_C)      VAR(R_C)

Bad          0.2   0.1       0.5         0.30       0.02       0.10       0.06           0.002           0.008               0.0005  

Ave.         0.6   0.2       0.3         0.25       0.12       0.18       0.15           -                   0.000                0.0000

Good      0.2   0.3       0.1         0.20       0.06       0.02       0.04           0.002             0.008                0.0005

Total      1                                    E(R_s)=0.2  E(R_c)=0.3 E(R_p)=0.25    0.004          0.016                    0.001            

E(R_s) = 0.2 (0.1) + 0.6(0.2) + 0.2(0.3) = 0.2 or 20%.

VAR (R_s)  = Σ[{R_s-E(R_s)}² × p_i ] = 0.2(0.1 – 0.2)² + 0.6 (0.2 – 0.2)² + 0.2(0.3 – 0.2)² = 0.004

                                                                        SD = 0.063 (6.3 percent)

Similarly,

E(R_c) = 0.2 (0.5) + 0.6(0.3) + 0.2(0.1) = 0.3 or 30%.  

VAR (R_C) = Σ[{R_C-E(R_C)}² × p_i] = 0.016      SD = 0.126 (12.6 percent)

Similarly,

E(R_p) = 0.2 (0.3) + 0.6(0.25) + 0.2(0.2) = 0.25 or 25%.            

VAR (R_p) = Σ[{R_p-E(R_p)}² × p_i] = 0.001       SD = 0.0316 (3.2 percent)

 

                                    Sugar              Cement           Combined

                                                                                    (50% each)

Mean                           20%                 30%                 25%

Var                              0.004               0.016               0.001

SD                               6.3%               12.6%             3.16%

Mean & variance of portfolio of asset

Portfolios of assets usually offer an advantage of reducing risk through diversification. The s.d. of the returns on the portfolio of assets, s_p, is less than the s.d. of the returns from the individual assets. Portfolio theory deals with the selection of optimal portfolio. Optimal portfolio is the one that provides the highest possible return for any specified degree of risk or the lowest possible risk for any specified rate of return. 

Portfolio Risk: Variance of a Portfolio

The basic idea behind portfolio theory is that the riskiness inherent in any single asset held in a portfolio is different from the riskiness of that asset held in isolation. It is possible for a given asset to be quite risky when held in isolation, but not very risky if held in a portfolio.

Table 2: Portfolio mean-stdd. dev. opportunity set.

Portfolio    %Sugar %Cement     E(Rp)     s(Rp) 

A                0%            100%               30%     12.65%

B                20             80                    28        8.90    

C                40             60                    26        5.47    

D                50             50                    25        3.16    

E                 60             40                    24        -          

F                 80             20                    22        3.16    

G                100           0                      20        6.32

The portfolio risk is the lowest in the case of portfolio E (60% sugar & 40% cement). In fact, there are innumerable portfolios we can form. The set of all these mean-standard deviation choices is called the portfolio opportunity set because it is a list of all possible opportunities available to the investor. Analysis of this sample opportunity set is unusual because the rates of return for sugar & cement were negatively correlated. What will happen to E(R_p) & s(R_p) if the two assets are; a) Perfectly positively correlated, b) No correlation, or c) Perfectly negatively correlated.

Correlation and Covariance:

            COV_(s,c)                  -0.008
r_sc=  ---------------        = ------------------         = -1
            s_ss_c                        .063 x .126  

 

Table 3: Correlation, Expected Returns and Risk

%S %C    r_xy   =        +   1       r_xy  =    0             r_xy = - 1.0 

W  (1-w)  E(Rp)%  s(Rp)%   E(Rp) s(Rp)     E(Rp) s(Rp)

20  80          28             11.4       28      10.19        28        8.89

40  60          26             10.1       26        8.00        26        5.1 

50  50          25             9.5         25        7.07        25        3.16

60  40          24             8.8         24        6.32        24        1.4 

80  20          22             7.55       22        5.66        22        2.6 

 

Table 3, reveals the following:

1. Portfolio return, E(R_p) is not the function of correlation. It is a function of w.

2. Portfolio risk, s(R_p) is the function of correlation between the two assets.

3. We observe that risk measured by s(R_p) is the lowest when r_sc= –1.0. It implies that we should choose the assets whose returns are perfectly negative correlated.

 

 

 Fig. 7: Plot of Table 3 data in graph paper.

Fig. 8. General Shape of the Portfolio Opportunity Set (Boundaries)

General case occurs when risky assets are not perfectly correlated (line ANB), which is called minimum variance portfolio opportunity set. Since it is the general case that occurs most, maximum studies concentrate on this general shape only. There are infinite number of attainable points in the interior of the curve illustrated in Fig. 9. 

Fig.9. The Portfolio Opportunity Set & the Efficient Set: [Efficient Frontier. The parabola is sometimes referred to as the 'Markowitz Bullet', and is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficient frontier]

Optimal Choice or optimal portfolio

Fig. 10. illustrates portfolio choices for two individuals who have different indifference curves because they have differing attitudes toward risk. Efficient set is given by AC. All portfolios along the line are efficient portfolios because of the highest return for any specified level of risk.

Fig. 10. Selection of optimal portfolio for two individuals.

Pricing Inefficient Portfolios

CML guides investors to evaluate the risk-return combinations of the efficient assets, or the portfolios that lie on the CML.

If they are to choose among the Projects A, B, C, & D, they would choose A because it lies on the CML & hence this is an efficient asset. But, how do they choose among the inefficient assets, for example, Projects B, C, & D? Can they choose B because it has lower portfolio risk?

Total risk (the variance) of any inefficient asset or portfolio can be partitioned into two parts: diversifiable and undiversifiable risk (Unsystematic & systematic risks). Consider empirical study by Wagner and Lau (1971); When no. of securities in a portfolio goes on increasing:

 No. of Secs.                s_p            Correlation with

in a portfolio   (% per Month)   Market Index

1                             7.0%                     0.54    

2                             5.0                        0.63    

3                             4.8                        0.75    

4                             4.6                        0.77    

5                             4.6                        0.79    

10                           4.2                        0.85    

15                           4.0                        0.88    

20                           3.9                        0.89

-S.d. (risk) goes on decreasing but does not decline to zero & correlation does not increase to 1?

·       It is because of the presence of systematic or undiversifiable risk.

·       This kind of risk arises because of general market movements.

 2. Capital Asset Pricing Model

The CAPM enables to estimate undiversifiable risk of a single asset & compare it with undiversifiable risk of a well-diversified portfolio. The capital asset pricing model (CAPM) invented by Sharpe and Lintner in 1964 assumes rational expectations. Ferson and Harvey (1991) look at the issue of return predictability and rational pricing in a regression setting. They use a multi-beta capital asset pricing model to decompose the variance of the fitted values from a regression of returns on a set of instrumental variables into explained and unexplained components. According to Berk and DeMarzio (2008) the CAPM describes the relationship between risk and expected return and that is used in the pricing of risky securities. The CAPM equation or Security Market Line (SML), is usually written as,

E(R_j) = R_f + [E(R_M) – R_F] b_j       - (i)

E(R_j) = expected return on the risky asset, j.

E(R_M)= exp. return on the market portfolio

R_F = rate or return on a riskless asset

            COV(R_i, R_m)  

b = ---------------------

            Var (R_m)         

b_j = a measure of the undiversifiable risk of the security, j.

·       Since A & B have the same expected returns, can we choose the project on the basis of total risk?

·       Similarly, given the three projects, B, C, & D having the same total risk, can we choose the project on the basis of expected return?

·       Answer is negative. The only risk relevant for inefficient securities is the undiversifiable risk. 

Comparison of CAPM & SML:

If we are to choose among A, B, C, & D assets, we would be choosing A because it lies on CML & hence is an efficient asset. But how do we choose among B, C, & D because they are all inefficient assets. We cannot choose B over C & D simply on the ground that it has lower total risk.

·       CML may be used to determine required return on efficient portfolios that are perfectly correlated with the market portfolio. But SML may be used to explain required rate of return on all securities whether or not they are efficient.

·       The SML provides a unique relationship between undiversifiable risk (measured by b) & expected return while the CML shows the relationship between total risk & total expected return.

·       CML equation is given by,

{E(R_M)–R_F}               

E(R_p) = R_F  + ----------------  s(R_p) ----(ii)

·                                s_M

And the SML equation is given by,         

E(R_j) = R_F + [E(R_M) – R_F] b_j              ----(iii)

That leads to:

{E(R_M)–R_F}               

E(R_j) = R_F  + -------------------  r_j s_j                  (iv)

        s_M

The above equation shows that market price per unit of risk or slope is the same for SML & for CML.

Beta of the market is one. Why?

Because the covariance of the market with itself, COV(R_M, R_M) is the same as the variance of the market, VAR(R_M), & VAR(R_M)/VAR(R_M) = 1.

o   If beta = 1.0, stock is average risk.

o   If beta > 1.0, stock is riskier than average.

o   If beta < 1.0, stock is less risky than average.

o   Most stocks have betas in the range of 0.5 to 1.5.

The SML equation shows that required return on any investment is the risk-free return plus a risk adjustment factor. The risk adjustment factor is obtained by multiplying risk premium & riskiness of that asset, i.e., beta. Other things remaining the same, greater the value of beta, greater is the risk adjustment factor & greater would be the required rate of return. Thus required return on any investment depends on the size of its beta. The advantage of SML is that if we know expected market return & the riskless return, we can easily draw SML (as beta of the market =1).

After drawing the SML, we may compute the E(R_j) of our investment & see if it lies above or below the SML. Again we may compute b of our investment & see if it is less or more than b_M, which is one. If beta of the individual investment is greater than one, individual investment is more riskier than the market portfolio.

CAPM makes clear that investors need to be compensated in two ways: time value of money and risk. The time value of money is shown by the risk-free (R_f) rate in the formula. The other half of the formula represents risk and calculates the amount of compensations the investor needs for taking on additional risk. This can be calculated by taking a risk measure (beta) that compares the returns of the asset to the market over a period of time and to the market premium [E(R_m)-  R_f]. The CAPM defines that the expected return of a security or a portfolio must equal the rate on a risk-free security plus a risk premium. If this is not the case, then the investment should not be undertaken.

According to Black (1976) the demand side of the CAPM is based on the well-known portfolio model of Tobin and Markowitz. The seven basic assumptions they use are 1) a single holding period moving horizon for all investors, 2) no transaction costs or taxes on individuals, 3) the existence of a riskless asset with rate of return r*, 4) evaluation of the uncertain returns from investments in terms of expected return and variance of end of period wealth, 5) unlimited short sales or borrowing of the riskless asset (or borrow or lend unlimited amounts at the risk free rate), 6) All assets are perfectly divisible, 7) All investors are price takers, i.e., investors’ buying & selling won’t influence stock prices.  

Empirical tests of the CAPM

·       CAPM assumptions are clearly not completely correct.

·       Investors are not fully diversified, hence have not eliminated all diversifiable risk from their portfolios.

·       Thus beta would not be an adequate measure of risk.

·       SML would not fully explain how required returns are set.

·       And taxes & brokerage costs do exist & their presence may distort the CAPM relationships.

·       Hence CAPM is not completely valid & may not produce accurate estimates of required return.

·       CAPM must first be tested empirically & validated.

Potential tests that can be conducted to verify the CAPM

Literature dealing with empirical tests of the CAPM is quite extensive. Basically two potential tests were important: (1) Beta stability tests (whether βs are stable) and (2) Tests based on the slope of the SML (whether relationship between risk & return is linear).

Major findings from Tests of the SML

·       A more-or-less linear relationship between realized returns & market risk.

·       Slope is less than predicted.

·       Irrelevance of diversifiable risk specified in the CAPM model can be questioned.

·       βs of individual securities are not good estimators of future risk.

·       Portfolio betas of 10 or more randomly selected stocks are reasonably stable.

·       Past portfolio betas are good estimates of future portfolio volatility.

Critique on CAPM

The common difficulty with these analyses is that they imply a persistent irrationality on the part of investors. By these assumptions it is necessary that investors are rational because if they are influenced by behavioral factors the pricing model will not work and the wrong decisions will be made. According to Ross (1978) researchers have examined the rationality issue in capital markets paying special attention to the relationship between prices and beliefs. Further, he finds that only dividend uncertainty can exist in equilibrium, capital gains are sure. The natural solution to this conflict is to model equilibrium in terms of rational price functions.

            Roll (1977) questioned whether it was even conceptually possible to test the CAPM. He showed that the linear relationship that prior researchers have observed in graphs, resulted from the mathematical properties of the model being tested; therefore, a finding of linearity would prove nothing whatsoever about the CAPM’s validity. Roll’s work did not disprove the CAPM, but showed that it is virtually impossible to prove investors behave in accordance with CAPM theory.

Conclusions regarding the CAPM

·       It is impossible to verify. Recent studies have questioned its validity.

·       Investors seem to be concerned with both market risk (systematic) & stand-alone (unsystematic) risk.  Therefore, the SML may not produce a correct estimate of R_i.

·       CAPM/SML concepts are based on expectations, yet bs are calculated using historical data.  A company’s historical data may not reflect investors’ expectations about future riskiness.

·       Other models are being developed that will one day replace the CAPM.

ü  But CAPM still provides a good framework for thinking about risk & return.

ü  The CAPM is intuitively appealing and widely used because of its simplicity

ü  Final conclusion: Though market equilibration process is complex & that the CAPM cannot give a precise measurement of the required return, the CAPM is still a useful guide.

 

3 Fama and French three factor model (FFTF)

 

An expansion on the CAPM is the ‘Fama and French three factor model’. In this model size and value factors are added in addition to the market risk factor in CAPM. The model explains the fact that value and small cap stocks outperform markets on a regular basis (Fama & French, 1996). By including these two additional factors, the model adjusts for the outperformance tendency, which is thought to make it a better tool for evaluation manager performance. There are many discussions about whether the outperformance tendency is due to market efficiency or market inefficiency.  The efficiency side of the discussion says that outperformance is commonly declared by the excess risk that small cap stocks face resulting from their higher cost of capital and greater business risk. According to the inefficiency side, the outperformance is declared by market participants mispricing the value of these companies, which provides the excess return in the long run as the value adjusts. The model declares that the expected return on a portfolio in excess of the risk-free rate is explained by the sensitivity of its return to three factors: 1) the market premium (R_m - R_f), 2) the difference between the return on a portfolio of small stocks and the return on a portfolio of large stocks (SMB, small minus big) and 3) premium for relative distress, the difference between the return on a portfolio of high-book-to-market stocks and the return on a portfolio of low-book-to- market stocks (HML, high minus low). Specifically, the expected excess return on the portfolio is:

 

E(R_i) - R_f = b_i[E(R_m) - R_f] + siE(SMB) + hiE(HML)   --- (i)

 

Lakonishok, Shleifer and Vishny (1994), and Haugen (1995) argue that the premium for relative distress is too large to be explained by rational pricing. They conclude that the premium is almost always positive and so is close to an arbitrage opportunity. So, if the relative-distress premium is too high to be explained by rational asset pricing, one must also be suspicious of the market and size premiums.

4. Arbitrage pricing theory

 One of the problems with the CAPM is that only a single factor, the market portfolio, is used to explain security returns. Similarly, FFTF models has included only limited factors. APT is a multi-factor model. It allows us to use many factors, not just one, to explain security returns. For example, unexpected changes in interest rates are logical candidate for being a common factor that affects all securities at once. When interest rates rise, the market value of bonds, & stocks tends to fall. As the name implies, APT is based on the concept of arbitrage. Arbitrage means one purchase of and sale of the same, or essentially similar, security in two different markets of affordable different prices (Sharpe & Alexander, 1999). In this way it is possible to be profitable by using price differences of identical or similar financial products, on different markets or in different forms. Theory is based on the premise that security prices adjust as investors form portfolios in search of arbitrage profits.

Ross (1976) developed the APT as an alternative for the CAPM, since the APT has more flexible assumption requirements. It is based on the idea that an asset’s returns can be predicted using the relationship between the same asset and many common risk factors. It describes the price where a mispriced asset is expected to be. Arbitrageurs use the APT model to profit by taking advantage of mispriced securities. According to the APT there are two things that can explain the expected return on a financial asset: 1) macroeconomic or security-specific influences, and 2) the asset’s sensitivity to those influences. This relationship takes the form of the linear regression formula:

R_j = E(R_j) = R_F+ b₁F₁+ b₂F₂+ … .. + b_jF_k + e_j

where,

R_j = stochastic rate of return on the j^th asset.

E(R_j)= expected rate of return on the j^th asset.

b_jk = the sensitivity of the j^th asset’s returns to the k^th factor, i.e., sensitivities of an asset (security) return.

F_k = No. of factors that affect asset (security returns).

e_j = a random term for the j^th asset.

 

-The logic behind the APT is much the same as that for the CAPM. Diversifiable or idiosyncratic risk is not priced by the marketplace because it can be eliminated by including more assets in a portfolio. All that counts is systematic risk (Chen, Roll, & Ross; 1986).

Macroeconomic variables related to the APT

Chen, Roll, and Ross (1986) suggested five factor model where 5 macroeconomic variables are significant: a) Industrial production (or the market portfolio), b) Changes in a default risk premium (measured by the differences in promised YTM on AAA  vs. Baa corporate bonds), c) Twists in the yield curve (measured by the differences in promised YTM on long- and short-term government bonds), d) Unanticipated inflation, e) Changes in the real rate (measured by the Treasury bill rate minus the CPI). First variable, IP index is related to profitability, while the rest four are related to the discount rate.

According to the international arbitrage pricing theory (IAPT) posited by Solnik (1983), currency movements affect assets' factor loadings and associated risk premiums. The IAPT requires optimal capital markets.

 

Critique on APT

Its acceptance has been slow because the model does not specify what factors influence stock returns (Nazemi, Abbasi, & Omidi,2015). Factor analysis has been suggested to identify relevant factors. Identification of factors itself is a formidable task, so it is difficult to apply in practice. Moreover, tentative importance of the factors may change over time.

·       It is a one-period model that has a linear relation between the expected return and its covariance if arbitrage over static portfolios is excluded. However, this theory does not preclude arbitrage over dynamic portfolios.

·       APT is based on the assumption that an asset’s returns can be expected using the relationship between that same asset and many common risk factors. Ross (1976) finds that if there are no arbitrage opportunities in the equilibrium price, then the expected returns on assets have a linear relation to the factor loadings. The APT is a substitute for the CAPM, because both have a linear relation between assets’ expected returns and their covariance with other random variables. However, the ssuperiority of APT to CAPM has not been established empirically.

 5. Efficient Market Hypothesis (EMH) 

An efficient market is one where the market is an unbiased estimate of the true value of the investment. It is the degree to which stock prices reflect all available and relevant information. Market efficiency was introduced by Fama (1970), whose theory efficient market hypothesis (EMH) stated that is not possible for an investor to outperform the market because all available information is already built into stock prices.

The central proposition of finance theories, the EMH is based on the assumption that investors always act in a rational manner. Barberis and Shleifer (2003) opined that a security’s value price equals its fundamental value if the agents are rational and there are no frictions. This is the sum of expected future cash flow where investors take all the available information into account while making a decisions and where the discount rate is consistent. The hypothesis where actual prices reflect fundamental values is the EMH. Shleifer (2000) state that when investors are rational this implies the impossibility of earning risk-adjusted returns, just as Fama (1970) wrote. So the EMH is a consequence of equilibrium in markets which are composed with fully rational investors.

 

Discipline is the key: O’ Shaughnessy (2005) showed that the only way to beat the market over the long-term is to consistently use sensible investment strategies. 80% of the mutual funds covered by Morningstar fail to beat the S&P 500 because their managers lack the discipline to stick with one strategy through thick and thin. This lack of discipline devastates long-term performance.