Arbitraj fiyatlandırma modeli

Abstract

ÖZET Sermaye piyasalarının gelişme sürecinin devam etmesi dolayısıyla ülkemizde de yatırımcılar giderek daha fazla bilinçlenmekte ve yatırım yaparken çok çeşitli kriterleri gözönüne almak zorundadırlar. Çünkü piyasaları etkileyen faktörler çok fazladır ve bunların iyi analiz edilmesi gerekmektedir. Bu çalışmada arbitraj fiyatlandırma modelini incelenmeye çalışılmıştır. Yeni bir model olan arbitraj fiyatlandırma modeli, menkul kıymet getirilerinin, çeşitli faktörlerden etkilendiğinden yola çıkan bir modeldir. Birinci bölümde yani giriş bölümünde, çalışmanın amacı ve yapılmak istenenler anlatılmıştır. İkinci bölümde, sermaye piyasası araçları yani yatırımlara konu olan enstrümanlardan bahsedilmiştir. Bu enstrümanların neler olduğu ve özelliklerinden bahsedilmiştir. Üçüncü bölümde, portföy analizi konusu ele alınmıştır. Finansal varlıkların getirilen ve riskleri, riskin çeşitleri, geleneksel portföy yaklaşımı, modern portföy yaklaşımı ve portföy yönetimi konulan ele alınmıştır. Dördüncü bölümde, mekul kıymet fiyatlandırma modeli konusu üzerinde durulmuştur. Modelin varsayımları, pazar portföyü ve sermaye pazarı doğrusu, finansal varlık pazar doğrusu ve beta katsayıları konulan incelenmiştir. Beşinci bölümde, arbitraj fiyatlandırma modeli incelenmiştir. Arbitrajın tanımı, özellikleri, modelin varsayımları, getiriler ve betalar, modelin oluşturulması, finansal varlıkların sayılarının sonsuz ya da sınırlı olduğu durumlarda modelin incelenmesi, modelin daha önce yapılmış deneysel testleri, menkul kıymet fiyatlandırma modeli ile karşılaştırılması ve model yardımıyla portföy performansının ölçülmesi konulan incelenmiştir. Altıncı bölümde, yani uygulama bölümünde İstanbul Menkul Kıymetler Borsası'nda işlem gören şirketler üzerinde modelin geçerli olup olmadığı test edilmeye çalışılmıştır. Uygulama çalışması iki bölümde yürütülmüştür. İlk bölümde seçilen hisse senetleri üzerinde makroekonomik faktörlerin açıklayıcı gücünün olup olmadığı araştırılmıştır. İkinci bölümde ise seçilen hisseler gruplara ayrılmış, bu gruplarda faktör analizi yapılmıştır. Faktör analizi sonrasında elde edilen faktör skorları ile o gruptaki hisseler arasında regresyon analizi yapılmıştır. Bu sayede hisse senedi getirilerini etkileyen faktörler bulunmaya çalışılmıştır. Yedinci bölümde, uygulama sonucunda elde edilen sonuçlar değerlendirilmeye çalışılmış ve önerilerde bulunulmuştur. vıı

SUMMARY The Arbitrage Pricing Theory is based on the law of one price which says that the same asset can't sell for two different prices. If the same asset does sell for different prices, arbitragers will buy the asset where it is cheap, and driving up the low price, and simultaneously sell the asset where its price is higher, thereby driving down the high price. Arbitragers will continue this activity untill all prices for the asset are equal. Let us assume the returns from assets a and b. The return generating equations (1) and (2); ra=E(ra) + e (1) rb=E(rb) + e (2) The random variable e is identical for the two assets in equations (1) and (2); it is assumed to have a mathematical expectation of zero, E(e) = 0. Equations (1) and (2) indicate that assets 1 and 2 have equally risky cashflows and equally risky rates of return. When trying the figure out the equilibrium prices of market assets, the law of one price is interpreted to mean that assets with identical risks are equivalent investments and therefore must have the same expected rates of return; in this case, E (ra ) = E (rb ). As long as the expected returns from assets a and b are equal, arbitrage between them will not be profitable. But, if the two expected rates of return are not equal, for example E(ra ) > E(rb ), in this case an investor can create arbirage profits by taking the proceeds of pb dollars from a short sale of asset b at time t=0 and investing the funds in a long position in asset a; I- pa 1= I pb I. This arbitrage portfolio requires zero initial investment since - pa + pb =0. The portfolio is also perfectly hedged to zero risk because any gains on the long position will be exactly offset by the simultaneous losses on the short position of equal size, and vice versa. The arbitrager can neverthless confidently expect to earn positive profits since {E (ra ) - E (rb )} > 0. Arbitrage opportunities like this are disequilibrium situations that will quickly be corrected by the first arbitrager that discovers them and finds a way to trade on them. Let us assume that one period rates of return for all assets are generated by a single risk factor denoted F in accordance with the linear models of equations (3) and (4). ra = a, + p, F (3) rb = a2 + p2F (4) Let F be a random variable with and expected value of zero, E(F)=0. The variable f might represent the unanticipated changes in inflation ; F=E{L -E(I)}. The Pi slope vmmight represent the unanticipated changes in inflation ; F=E{It -E(I)}. The Pi slope coefficients in equations (3) and (4) are measures of undiversifiable risk-they indicate how sensitive the asset returns are to the common source of variations F. It can be thought that equations (3) and (4) as being simplified characteristic lines that have no unexplained residual error terms so that the total risk is undiversifiable systematic risk. Since independent variable F was constructed so that it averages to zero, E(F)=0, it follows that equations (3) and (4) should have P E(F)= E(PF)=0. By taking this logic a step farther, we see that the two expected rates of return must also be equal to their intercept terms, as shown in equations (5) and (6). E(ra)=oti (5) E(rb)=a2 (6) Equqtions (5) and (6) show that E(rj)=a;. In addition, the law of one price tells us that since assets a and b are equally risky, they should have identical expected rates of return, E(ra)=E(rb). From these facts we can conclude that the two assets' expected rates of return and intercept terms should all be equal, E(ra)=ai=E(rb)=a2. Let x represent the weight of a two asset portfolio's total wealth that is invested in asset one, 0<x<l. For example an arbitrager wants to create a perfectly hedged portfolio by investing a fraction x of the portfolio's total wealth in asset a and the remainder (1-x) in asset b. Equation (7) defines the weighted average rate of return for the portfolio. rp=xra+ (l-x)rb (7) Substituting equations (3) and (4) into equation (7) produces equation (7i). rp=x (<xi + pi F )+(l-x) (a2 + p2 F) (7i) rp=x (ai - a2)+ a2+{x (pi - p2 )+ p2}F (7Ü) Let us select the proportion Xi ={p2 /( p2 -Pi)} that creates a perfectly hedged portfolio and then mathematically analyze the asset pricing implications of the resulting portfolio. The quantity x; can be substituted into equation (7ii) in place of x to obtain, after some rearranging, equation (8). rP=a2 +p2 (ai -a2 )/( p2 -pi) (8) It can be told that equation (8) represents a perfectly hedged riskless portfpolio since the systematic risk factor F, which is the only source of risk in this model, drops completely out of the equation. In equilibrium a riskless investment must yield the risk-free rate of return, denoted R. This allows us to say that rp =R and also permits us to substitute R in place of rp and to rewrite equation (8) equivalents as equation (9). IXR=a2 +p2 (a! -a2 )/( p2 -Pi) (9) Multiplying both sides of equation (9) by the quantity ( {32 -Pi) and rearranging the terms resultsin equation (10). (o^-Ryp^Co^-RyPi (10) From equations (5) and (6), E(rj)=oti. Substituting for a;. allows equation (10) to be restated as equations (10i) and (10ii). (oti-Ryp^EO-i-RVpi (10İ) (a* -R)/ Pi =?i=risk premium/risk measure ( 1 Oii) Equations (10), (10i) and ( 1 Oii) all define a constant term, denoted lambda X, that represents a factor risk premium. Equation (10i) can be rewritten as equations (11) and (1 li) to obtain the arbitrage pricing line. Combining equations (5), (6) and (10i) results in equation (1 1). E(ri) = R + pi(E(ri)-R)/pi (11) E(ri) = R + p;X (Hi) Equation (Hi) was by substituting from the definition of lambda in equation (lOii). The factor risk premium lambda (k) can be interpreted as the excess rate of return (E(rj)-R) for a risky asset with Pi =1. Equation series (11) is the essence of the APT. Equations (11) and (111) say that in the absence of profitable arbitrage opportunities, the expected rate of return from risky asset i equals the risk-free rate of return plus a risk premium that is proportional to the asset's sensitivity P; to the common risk factor F. This sensitivity is measured by the sensitivity coefficient, factor loading or factor beta, denoted Pi, for the ith asset. APT considers all assets that are in the same risk class to be perfect substitutes that should yield the same rate of return in equilibrium. There is a fundamental problem associated with the Capital Asset Pricing Model that it may not be possible to support or contradict the model. This problem has stimulated interest in an alternative model of asset pricing called the (APT), that first introduced by Ross (1976). Ross maintains that there can be a number of risk factors that are priced in the market. If these factors do not affect the expected return of a security, there will be arbitrage opportunities. Some of the proponents say that it has two major advantages over the CAPM. First, it makes assumptions regarding investors' preferences toward risk and return that some would argue are less restrictive. One of the assumptions of the CAPM is that investors could choose between alternative portfolio investments solely on the basis of expected return and standart deviation.The APT requires that bounds be placed on investors' utility functions, but the bounds are less restrictive. Second, some of the proponents argue that the model can be refused or verified empirically. This point of view has been the subject of much dispute, but to many the testability of the APT is, still an open question. The fundamental assumption of the APT is that security returns are generated by a process identical to the single or multi-index models. There is an assumption that the covariances that exist between security returns can be attributed to the fact that the securities respond, to one degree or another, to the pull of one or more factors. Another assumption is relationship between the security returns and the factors is linear, as in the case of a multi-index model. There need another assumption that there is an infinite number of securities and there are no restrictions on short selling to derive the approximate relationship between expected return and risk under the APT. Unless the relationship between factor risks and expected returns is approximately linear, unlimited arbitrage opprtunities may become available. This is the central message of the APT. If security returns are generated by a process equivalent to that of a linear multifactor model with n priced factors, the relationship between expected return and factor risk must be approximately linear. While the general relationship between factor risk and expected return will be linear, there may still be individual deviations from the relationship so long as there isn't a sufficient number of them to open up riskless arbitrage opportunities. In the context of APT it must be impossible to construct two different portfolios, both having zero variance, with two different expected rates of return. This will be the case if the relationship between the factor betas and the expected rates of return is linear. It will not be the case if the relationship is generally nonlinear. The absence of arbitrage opportunities doesn't ensure exact linear pricing. While the linear relationship prices most assets with negligible error, it can be highly inaccurate in pricing some of them. If there isn't unlimited number of securities to work with, the residual variance of our zero beta arbitrage portfolios can't be reduced to zero in any case. If the number of securities is very large, the residual variance can be reduced to a very small number, but there is still some risk in capturing our arbitrage profit. The deviation from the linear APT relationship is caused by the unavoidable residual variance in the arbitrage portfolios. This residual variance will be greater when the residual variance of the security is greater and when the portfolio weight assigned to the security is greater. If investors were risk neutral, they wouldn't care about unavoidable residual variance and it wouldn't affect risk. The greater their risk aversion, the greater the impact of unavoidable residual variance on expected return. In the APT, the number of factors and their identification is an empirical issue. Early results indicate that there may be three to five significant factors and work is proceeding in an attempt to associate these factors with recognizable economic variables. XIThe initial empirical test of the APT was conducted by Roll and Ross (RR) (1980). Their methodology is, similar to that used by Black, Jensen and Scholes in testing CAPM, since they first estimate the factor betas for securities, and then they estimate the cross- sectional relationship between security betas and average rate of return. RR estimate the factor betas using a statistical technique called factor analysis. The input the factor analysis is the covariance matrix between the returns to the securities in the sample. The factor analysis determines the factor betas which best explain the covariances exist between the securities in the sample. Each index can be thought of as consisting of the systematic portions of the returns to a differently weighted portfolio of the securitis in the sample. The analysis determines a set of index portfolios and index betas such that the covariances between the residual returns are as small as possible. The program continues add additional index portfolios until the probability that the next portfolio explains a significant fraction of the covariances between stocks goes below some predetermined level. Factor analysis makes the working assumption that the individual factor variances are equal to 1, and then it finds that set of factor betas for each stock that will make the covariance matrix, correspond as closely as possible to the sample covariance matrix, as computed directly from the the returns. After obtaining estimates of the factor betas, the next step is to estimate the value of the factor price associated with each factor. This is done by cross-sectionally relating the factor betas to average returns, using a procedure similar to that employed by BJS. Factor analysis can only be employed on a relatively small number of stocks at a time, because it is very complex. RR found that four or five different factors have significant explanatory power in their sample. They found that the residual variance of securities is unrelated to average returns. The APT would predict that the estimates of the intercept term and the values of the factor prices should be the same for each sample tested. Brown and Weinstein (1983) test this prediction and find ambigious results. Several authors have raised the issue of the testability of the APT. One proble is the necessity of conducting the factor analysis on relatively small samples of firms. In dividing up the overall sample, factors which explain covariances between securities in different groups may be ignored. Dhrymes, Friend and Gultekin (DFG) (1984) find, in a separate study, that as the number of securities included in the factor analysis increases from 15 to 60, th number of significant factors increases from 3 to7. As Roll and Ross (1984) point out, however, there are many reasons why it should be expected this to happen. They argue that this does not necessarily mean that conducting the tests on small samples is inappropriate, because unless the factors are pervasive, they can be diversified away and they will not be priced. As such they aren't of interest in XIItesting the theory. It is safe to say that there is no conclusive evidence either supporting or cotradicting the model. The Capital Asset Pricing Model and The Arbitrage Pricing Theory are not mutually exclusive. While the two theories are completely consistent with one another, it is not the case that the CAPM can be considered as a special case of the APT. The CAPM assumes nothing about the structure of security returns other than possibly that they are normally distributed. Normal distributions, however, do not necessarily imply the linear factor structure that is required for the APT. APT suggests how to price markets assets. Prior to the APT, the capital asset pricing model (CAPM) was the most prominent financial theory to explain the prices of market assets. It is natural to compare and contrast these two important theories. The APT requires fewer underlying assumptions admits more different variables into the analysis than the CAPM. Therefore, the APT is a more general theory than the CAPM. However, the APT is mathematically equivalent to the CAPM when only one risk factor is considered. Other similarities also show that the two theories do not contradict each other.Moreover, the two theories are similar because both delineate systematic communalities that form the basis for risk premiums in market prices and returns. Results for the initial published tests are moderately favorable, they tend to suggest that the APT might have more empirical explanatory power than the CAPM. Xlll