Chapter 13 : The Arbitrage Pricing Theory
13.1
Introduction
We have made two first attempts (Chapters 10 to 12) at asset pricing from an arbitrage perspective, that is, without specifying a complete equilibrium structure. Here we try again from a different, more empirically based angle. Let us first collect a few thoughts as to the differences between an arbitrage approach and equilibrium modeling.
In the context of general equilibrium theory, we make hypotheses about agents – consumers, producers, investors; in particular, we start with some form of rationality hypothesis leading to the specification of maximization problems under constraints. We also make hypotheses about markets: Typically we as-sume that supply equals demand in all markets under consideration.
We have repeatedly used the fact that at general equilibrium with fully informed optimizing agents, there can be no arbitrage opportunities, in other words, no possibilities to make money risklessly at zero cost. An arbitrage opportunity indeed implies that at least one agent can reach a higher level of utility without violating his/her budget constraint (since there is no extra cost). In particular, our assertion that one can price any asset (income stream) from the knowledge of Arrow-Debreu prices relied implicitly on a no-arbitrage hypoth-esis: with a complete set of Arrow-Debreu securities, it is possible to replicate any given income stream and hence the value of a given income stream, the price paid on the market for the corresponding asset, cannot be different from the value of the replicating portfolio of Arrow-Debreu securities. Otherwise an arbitrageur could make arbitrarily large profits by short selling large quantities of the more expensive of the two and buying the cheaper in equivalent amount. Such an arbitrage would have zero cost and be riskless.
While general equilibrium implies the no-arbitrage condition, it is more re-strictive in the sense of imposing a heavier structure on modeling. And the reverse implication isnot true: No arbitrage opportunities1 – the fact that all
arbitrage opportunities have been exploited – does not imply that a general equilibrium in all markets has been obtained. Nevertheless, or precisely for that reason, it is interesting to see how far one can go in exploiting the less restrictive hypothesis that no arbitrage opportunities are left unexploited.
The underlying logic of the APT to be reviewed in this chapter is, in a sense, very similar to the fundamental logic of the Arrow-Debreu model and it is very much in the spirit of a complete market structure. It distinguishes itself in two major ways: First it replaces the underlying structure based on fundamental securities paying exclusively in a given state of nature with other fundamental securities exclusively remunerating some form of risk taking. More precisely, the APT abandons the analytically powerful, but empirically cumbersome, concept of states of nature as the basis for the definition of its primitive securities. It 1An arbitrage portfolio is a self-financing (zero net-investment) portfolio. An arbitrage
opportunity exists if an arbitrage portfolio exists that yields non-negative cash flows in all states of nature and positive cash flows in some states (Chapter 11).
replaces it with the hypothesis that there exists a (stable) set of factors that are essential and exhaustive determinants of all asset returns. The primitive secu-rity will then be defined as a secusecu-rity whose risk is exclusively determined by its association with one specific risk factor and totally immune from association with any other risk factor. The other difference with the Arrow-Debreu pricing of Chapter 8 is that the prices of the fundamental securities are not derived from primitives – supply and demand, themselves resulting from agents’ endowments and preferences – but will be deduced empirically from observed asset returns without attempting to explain them. Once the price of each fundamental se-curity has been inferred from observed return distributions, the usual arbitrage argument applied to complex securities will be made (in the spirit of Chapter 10).2
13.2
Factor Models
The main building block of the APT is a factor model, also known as a return-generating process. As discussed previously, this is the structure that is to replace the concept of states of nature. The motivation has been evoked before: States of nature are analytically convincing and powerful objects. In practice, however, they are difficult to work with and, moreover, often not verifiable, implying that contracts cannot necessarily be written contingent on a specific state of nature. We discussed these shortcomings of the Arrow-Debreu pric-ing theory in Chapter 8. The temptation is thus irresistible to attack the asset pricing problem from the opposite angle and build the concept of primitive secu-rities on an empirically more operational notion, abstracting from its potential theoretical credentials. This structure is what factor models are for.
The simplest conceivable factor model is a one-factor market model, usually labeled theMarket Model, which asserts that ex-post returns on individual assets can be entirely ascribed either to their own specific stochastic components or to their common association in a single factor, which in the CAPM world would naturally be selected as the return on the market portfolio. This simple factor model can thus be summarized by following the equation (or process):3
˜
rj =αj+βj˜rM + ˜εj, (13.1) withEε˜j= 0, c˜ov (˜rM,ε˜j) = 0,∀j,and cov (˜εj,ε˜k) = 0,∀j6=k.
This model states that there are three components in individual returns: (1) an asset-specific constant αj; (2) a common influence, in this case the unique factor, the return on the market, which affects all assets in varying degrees, withβj measuring the sensitivity of assetj’s return to fluctuations in the mar-ket return; and (3) an asset-specific stochastic term ˜εj summarizing all other stochastic components of ˜rj that are unique to assetj.
2The arbitrage pricing theory was first developed by Ross (1976), and substantially
inter-preted by Huberman (1982) and Conner (1984) among others. For a presentation emphasizing practical applications, see Burmeister et al. (1994).
3Factors are frequently measured as deviations from their mean. When this is the case,α
j
Equation (13.1) has no bite (such an equation can always be written) until one adds the hypothesis cov (˜εj,ε˜k) = 0, j 6=k, which signifies that all return characteristics common to different assets are subsumed in their link with the market return. If this were empirically verified, the CAPM would be the undis-puted end point of asset pricing. At an empirical level, one may say that it is quite unlikely that a single factor model will suffice.4 But the strength of the
APT is that it is agnostic as to the number of underlying factors (and their iden-tity). As we increase the number of factors, hoping that this will not require a number too large to be operational, a generalization of Equation (13.1) becomes more and more plausible. But let us for the moment maintain the hypothesis of one common factor for pedagogical purposes.5
13.2.1 About the Market Model
Besides serving as a potential basis for the APT, the Market Model, despite all its weaknesses, is also of interest on two grounds. First it produces estimates for theβ’s that play a central role in the CAPM. Note, however, that estimatingβ’s from past data alone is useful only to the extent that some degree of stationarity in the relationship between asset returns and the return on the market is present. Empirical observations suggest a fair amount of stationarity is plausible at the level of portfolios, but not of individual assets. On the other hand, estimating theβ’s does not require all the assumptions of the Market Model; in particular, a violation of the cov(˜εi,ε˜k) = 0, i6=khypothesis is not damaging.
The second source of interest in the Market Model, crucially dependent on the latter hypothesis being approximately valid, is that it permits economizing on the computation of the matrix of variances and covariances of asset returns at the heart of the MPT. Indeed, under the Market Model hypothesis, one can write (you are invited to prove these statements):
σ2j = βj2σ2M+σεj2, ∀j
σij = βiβjσM2
This effectively means that the information requirements for the implemen-tation of MPT can be substantially weakened. Suppose there areN risky as-sets under consideration. In that case the computation of the efficient fron-tier requires knowledge ofN expected returns,N variances, and N22−N covari-ance terms (N2 is the total number of entries in the matrix of variances and
covariances, take away the N variance/diagonal terms and divide by 2 since
σij =σji,∀i, j).
4Recall the difficulty in constructing the empirical counterpart ofM.
5Fama (1973), however, demonstrates that in its form (13.1) the Market Model is
inconsis-tent in the following sense: the fact that the market is, by definition, the collection of all indi-vidual assets implies an exact linear relationship between the disturbancesεj; in other words,
when the single factor is interpreted to be the market the hypothesis cov (˜εj,ε˜k) = 0,∀j6=k
cannot be strictly valid. While we ignore this criticism in view of our purely pedagogical objective, it is a fact that if a single factor model had a chance to be empirically verified (in the sense of all the assumptions in (13.1) being confirmed) the unique factor could not be the market.
Working via the Market Model, on the other hand, requires estimating Equa-tion (13.1) for the N risky returns producing estimations for the N βj’s and the N σ2
εj and estimating the variance of the market return, that is, 2N + 1 information items.
13.3
The APT: Statement and Proof
13.3.1 A Quasi-Complete Market Hypothesis
To a return-generating process such as the Market Model, the APT superposes a second major hypothesis that is akin to assuming that the markets are “quasi-complete”. What is needed is the existence of a rich market structure with a large number of assets with different characteristics and a minimum number of trading restrictions. This market structure, in particular, makes it possible to form a portfolioP with the following three properties:
Property 1: P has zero cost; in other words, it requires no investment. This is the first requirement of an arbitrage portfolio.
Let us denotexi as thevalue of the position in theith asset in portfolioP. PortfolioP is then fully described by the vectorxT = (x1, x2,..., x
N) and the zero cost condition becomes
N
X
i=1
xi= 0 =xT·1,
with 1the (column) vector of 1’s. (Positive positions in some assets must be financed by short sales of others.)
Property 2: P has zero sensitivity (zero beta) to the common factor:6
N
X
i
xiβi= 0 =xT ·β.
Property 3: P is a well-diversified portfolio. The specific risk ofP is (almost) totally eliminated: N X i x2 iσ 2 εi ∼= 0.
The APT builds on the assumed existence of such a portfolio, which requires a rich market structure.
6Remember that the beta of a portfolio is the weighted sum of the betas of the assets in
13.3.2 Statement and Proof of the APT
The APT relationship is the direct consequence of the factor structure hy-pothesis, the existence of a portfolio P satisfying these conditions, and the no-arbitrage assumption. Given that returns have the structure of Equation (13.1), Properties 2 and 3 imply thatP is riskless. The fact that P has zero cost (Property 1) then entails that an arbitrage opportunity will exist unless:
rP = 0 =xT ·r (13.2)
The APT theorem states, as a consequence of this succession of statements, that there must exist scalarsλ0,λ1, such that:
r = λ0·1+λ1β, or
ri = λ0+λ1βi for all assets i (13.3) This is the main equation of the APT.
Equation (13.3) and Properties 1 and 2 are statements about 4 vectors: x,
β,1, andr.
Property 1 states that x is orthogonal to 1. Property 2 asserts that x is orthogonal to β. Together these statements imply a geometric configuration that we can easily visualize if we fix the number of risky assets atN= 2, which implies that all vectors have dimension 2. This is illustrated in Figure 13.1.
Insert Figure 13.1
Equation (13.3) – no arbitrage – implies that xand ¯r are orthogonal. But this means that the vector ¯rmust lie in the plane formed by1andβ, or, that ¯
rcan be written as a linear combination of1andβ, as Equation (13.3) asserts. More generally, one can deduce from the triplet
N X i xi= N X i xiβi = N X i xir¯i=0 that there exist scalarsλ0, λ1, such that:
¯
ri=λ0+λ1βi for alli.
This is a consequence of the orthonormal projection of the vector ¯ri into the subspace spanned by the other two.
13.3.3 Meaning ofλ0 and λ1
Suppose that there exists a risk-free asset or, alternatively, that the sufficiently rich market structure hypothesis permits constructing a fully diversified portfo-lio with zero-sensitivity to the common factor (but positive investment). Then
¯
That is,λ0 is the return on the risk-free asset or the risk-free portfolio.
Now let us compose a portfolio Q with unitary sensitivity to the common factorβ = 1. Then applying the APT relation, one gets:
¯
rQ=rf+λ1·1
Thus,λ1= ¯rQ−rf, the excess-return on the pure-factor portfolioQ. It is now possible to rewrite equation (13.3) as:
¯
ri =rf+βi(¯rQ−rf). (13.4) If, as we have assumed, the unique common factor is the return on the market portfolio, in which caseQ=M and ˜rQ ≡r˜M , then Equation (13.4) is simply the CAPM equation:
¯
ri=rf +βi(¯rM−rf).
13.4
Multifactor Models and the APT
The APT approach is generalizable to any number of factors. It does not, however, provide any clue as to what these factors should be, or any particular indication as to how they should be selected. This is both its strength and its weakness. Suppose we can agree on a two-factor model:
˜ rj=aj+bj1F1˜ +bj2F2˜ + ˜ej (13.5) withE˜ej= 0, cov ³ ˜ F1,ε˜j ´ = cov ³ ˜ F2,ε˜j ´ = 0,∀j, and cov (˜εj,ε˜k) = 0,∀j6=k. As was the case for Equation (13.1), Equation (13.5) implies that one cannot reject, empirically, the hypothesis that the ex-post return on an assetj has two stochastic components: one specific, (˜ej), and one systematic, (bj1F1˜ +bj2F2˜ ).
What is new is that the systematic component is not viewed as the result of a single common factor influencing all assets. Common or systematic issues may now be traced to two fundamental factors affecting, in varying degrees, the returns on individual assets (and thus on portfolios as well). Without loss of generality we may assume that these factors are uncorrelated.
As before, an expression such as Equation (13.5) is useful only to the extent that it describes a relationship that is relatively stable over time. The two factorsF1 andF2must really summarizeall that is common in individual asset returns.
What could these fundamental factors be? In an important article, Chen, Roll, and Ross (1986) propose that the systematic forces influencing returns must be those affecting discount factors and expected cash flows. They then iso-late a set of candidates such as industrial production, expected and unexpected inflation, measures of the risk premium and the term structure, and even oil prices. At the end, they conclude that the most significant determinants of as-set returns are industrial production (affecting cash flow expectations), changes in the risk premium measured as the spread between the yields on low- and high-risk corporate bonds (witnessing changes in the market risk appetite), and
twists in the yield curve, as measured by the spread between short- and long-term interest rates (representing movements in the market rate of impatience). Measures of unanticipated inflation and changes in expected inflation also play a (less important) role.
Let us follow, in a simplified way, Chen, Roll, and Ross’s lead and decide that our two factors are industrial production (F1) and changes in the risk premium (F2). How would we go about implementing the APT? First we have to measure our two factors. LetIP(t) denote the rate of industrial production in montht; thenM P(t) = logIP(t)−logIP(t−1) is the monthly growth rate ofIP. This is our first explanatory variable.
To measure changes in the risk premium, let us define
U P R(t) = “Baa and under” bond portfolio return(t)−LGB(t) whereLGB(t) is the return on a portfolio of long-term government bonds. With these definitions we can rewrite Equation (13.5) as
˜
rjt=aj+bj1M P(t) +bj2U P R(t) + ˜ej
Thebjk, k = 1,2, are often called factor loadings. They can be estimated di-rectly by multivariate regression. Alternatively, one could constructpure factor portfolios – well-diversified portfolios mimicking the underlying factors – and compute their correlation with assetj. The pure factor portfolio P1 would be a portfolio with bP1 = 1 and bP2 = σeP1 = 0; portfolio P2 would be defined
similarly to track the stochastic behavior of U P R(t). Let us go on hypothe-sizing (wrongly according to Chen, Roll, and Ross) that this two-factor model satisfies the necessary assumptions (cov(˜ei,˜ej) = 0,∀i6=j) and further assume the existence of a risk-free portfolioPf with zero sensitivity to either of our two factors and zero specific risk. Then the APT states that there exist scalarsλ0,
λ1,λ2 such that:
rj=λ0+λ1bj1+λ2bj2.
That is, the expected return on an arbitrary assetjis perfectly and completely described by a linear function of asset j’s factor loadings bj1, bj2. This can
appropriately be viewed as a (two-factor) generalization of the SML. Furthermore the coefficients of the linear function are:
λ0 = rf
λ1 = rP1−rf
λ2 = rP2−rf whereP1 andP2are our pure factor portfolios.
The APT agrees with the CAPM that the risk premium on an asset,rj−λ0, is not a function of its specific or diversifiable risk. It potentially disagrees with the CAPM in the identification of the systematic risk. The APT decomposes the systematic risk into elements of risk associated with a particular asset’s sensitivity to a few fundamental common factors.
Note the parallelism with the Arrow-Debreu pricing approach. In both con-texts, every individual asset or portfolio can be viewed as a complex security, or a combination of primitive securities: Arrow-Debreu securities in one case, the pure factor portfolios in the other. Once the prices of the primitive securi-ties are known, it is a simple step to compose replicating portfolios and, by a no-arbitrage argument, price complex securities and arbitrary cash flows. The difference, of course, resides in the identification of the primitive security. While the Arrow-Debreu approach sticks to the conceptually clear notion of states of nature, the APT takes the position that there exist a few common and stable sources of risk and that they can be empirically identified. Once the correspond-ing risk premia are identified, by observcorrespond-ing the market-determined premia on the primitive securities (the portfolios with unit sensitivity to a particular factor and zero sensitivity to all others) the pricing machinery can be put to work.
Let us illustrate. In our two-factor examples, a securityjwith, say,bj1= 0.8
and bj2 = 0.4 is like a portfolio with proportions of 0.8 of the pure portfolio P1, 0.4 of pure portfolio P2, and consequently proportion −0.2 in the riskless asset. By our usual (no-arbitrage) argument, the expected rate of return on that security must be:
rj = −0.2rf+ 0.8rP1+ 0.4rP2
= −0.2rf+ 0.8rf+ 0.4rf+ 0.8 (rP1−rf) + 0.4 (rP2−rf)
= rf+ 0.8 (rP1−rf) + 0.4 (rP2−rf)
= λ0+bj1λ1+bj2λ2
The APT equation can thus be seen as the immediate consequence of the link-age between pure factor portfolios and complex securities in an arbitrlink-age-free context. The reasoning is directly analogous to our derivation of the value ad-ditivity theorem in Chapter 10 and leads to a similar result: Diversifiable risk is not priced in a complete (or quasi-complete) market world.
While potentially more general, the APT does not necessarily contradict the CAPM. That is, it may simply provide another, more disaggregated, way of writing the expected return premium associated with systematic risk, and thus a decomposition of the latter in terms of its fundamental elements. Clearly the two theories have the same implications if (keeping with our two-factor model, the generalization is trivial):
βj(rM−rf) =bj1(rP1−rf) +bj2(rP2−rf) (13.6)
Let βP1 be the (market) beta of the pure portfolio P1 and similarly for βP2.
Then if the CAPM is valid, not only is the LHS of Equation (13.6) the expected risk premium on assetj, but we also have:
rP1−rf = βP1(rM −rf) rP2−rf = βP2(rM −rf)
Thus the APT expected risk premium may be written as:
which is the CAPM equation provided:
βj =bj1βP1+bj2βP2
In other words, CAPM and APT have identical implications if the sensitivity of an arbitrary assetjwith the market portfolio fully summarizes its relationship with the two underlying common factors. In that case, the CAPM would be another, more synthetic, way of writing the APT.7
In reality, of course, there are reasons to think that the APT with an arbi-trary number of factors will always do at least as well in identifying the sources of systematic risk as the CAPM. And indeed Chen, Roll, and Ross observe that their five factorscoverthe market return in the sense that adding the return on the market to their preselected five factors does not help in explaining expected returns on individual assets.
13.5
Advantage of the APT for Stock or Portfolio
Selec-tion
The APT helps to identify the sources of systematic risk, or to split systematic risk into its fundamental components. It can thus serve as a tool for helping the portfolio manager modulate his risk exposure. For example, studies show that, among U.S. stocks, the stocks of chemical companies are much more sensitive to short-term inflation risk than stocks of electrical companies. This would be compatible with both having the same exposure to variations in the market return (same beta). Such information can be useful in at least two ways. When managing the portfolio of an economic agent whose natural position is very sensitive to short-term inflation risk, chemical stocks may be a lot less attractive than electricals, all other things equal (even though they may both have the same market beta). Second, conditional expectations, or accurate predictions, on short-term inflation may be a lot easier to achieve than predictions of the market’s return. Such a refining of the information requirements needed to take aggressive positions can, in that context, be of great use.
13.6
Conclusions
We have now completed our review of asset pricing theories. At this stage it may be useful to draw a final distinction between the equilibrium theories covered in Chapters 7, 8, and 9 and the theories based on arbitrage such as the Martingale pricing theory and the APT. Equilibrium theories aim at providing a complete theory of value on the basis ofprimitives: preferences, technology, and market structure. They are inevitablyheavier, but their weight is proportional to their ambition. By contrast, arbitrage-based theories can only provide a relative theory of value. With what may be viewed as a minimum of assumptions, they 7The observation in footnote 5, however, suggests this could be true as an approximation
• offer bounds on option values as a function of the price of the underly-ing asset, the stochastic behavior of the latter beunderly-ing taken as given (and unexplained);
• permit estimating the value of arbitrary cash flows or securities using risk-neutral measures extracted from the market prices of a set of fundamental securities, or in the same vein, using Arrow-Debreu prices extracted from a complete set of complex securities prices;
• explain expected returns on any asset or cash flow stream once the price of risk associated with pure factor portfolios has been estimated from market data on the basis of a postulated return-generating process.
Arbitrage-based theories currently have the upper hand in practitioners’ cir-cles where their popularity far outstrips the degree of acceptance of equilibrium theories. This, possibly temporary, state of affairs may be interpreted as a measure of our ignorance and the resulting need to restrain our ambitions.
References
Burmeister, E., Roll, R., Ross, S.A. (1994), “A Practitioner’s Guide to Arbi-trage Pricing Theory,” in A Practitioner’s Guide to Factor Models, Re-search Foundation of the Institute of Chartered Financial Analysts, Char-lottesville, VA.
Chen, N. F., Roll R., Ross, S.A. (1986), “Economic Forces and the Stock Market,”Journal of Business 59(3), 383-404.
Connor, G. (1984), “A Unified Beta Pricing Theory,” Journal of Economic Theory 34(1).
Fama, E.F. (1973), “A Note on the Market Model and the Two-Parameter Model,”Journal of Finance 28 (5), 1181-1185
Huberman, G. (1982), “A Simple Approach to Arbitrage Pricing,”Journal of Economic Theory,28 (1982): 183–191.
Ross, S. A. (1976), “The Arbitrage Pricing Theory,” Journal of Economic Theory,1, 341–360.