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The fundamental rejection of the CAPM Abstract The CAPM arguably is the pinnacle of our contemporary Risk and Return paradigm. This paper however claims the CAPM should be fundamentally rejected. In a world of two risky assets and one riskless asset it is demonstrated the CAPM invariably suggests an unchanged discount rate for the risky assets as a result of costless diversification benefits. An identical discount rate for both the total risk of an asset as well as for its undiversifiable risk only is irrational and inherently inconsistent with the basic principles of Risk and Return. The CAPM cannot reflect a model for efficiently pricing assets in equilibrium. Arguably the pinnacle of the theory of Risk and Return is the Capital Asset Pricing Model (Sharpe (1964), and Lintner (1965)). In spite of poor empirical evidence, the theoretical and practical significance cannot be understated: it allows us to objectively calculate the appropriate discount rate for individual securities, it is widely used on the stock markets in some shape or form and the CAPM simply is the central paradigm for understanding risk and return in equilibrium. The contemporary standard form of the general equilibrium relationship for assets, popularly referred to as Capital Asset Pricing Model (CAPM), is the so-called Sharpe-Lintner version of the model. In effect, this model was independently developed of one another by Treynor1 (1961, 1962), Sharpe (1964), Lintner (1965a,b) and Mossin (1966). Black (1972) later introduced another version of the model that does not require the assumption of unlimited borrowing and lending at the risk free rate, yet instead he introduces the assumption of unrestricted short sales of risky assets. In relation to the Sharpe-Lintner version of the model, Black’s version differs only in the treatment of the zero beta asset, the asset uncorrelated with the market. In the Sharpe- Lintner version this is the riskfree rate by default (RF), while in Black’s model the only condition is that this zero beta asset should be less than the expected market return, so that the premium for beta is positive. While relevant from an empirical angle because Black’s model can justify a flatter slope as a result of a higher zero beta asset, from a fundamental angle both models are highly consistent. Yet to be precise, unless otherwise stated, the version of the model specifically under review in this paper is the Sharpe-Lintner model; when referring to the CAPM going forward this shall thus be intended as being interchangeable with the Sharpe-Lintner version of the model. Early empirical evidence was still generally supportive of the model (notably Black, Jensen and Scholes (1972) & Fama and MacBeth (1973)), particularly with regards to the presumed linear relation between expected returns and beta, the positive beta premium and the finding that standard market proxies seem to be on the minimum variance frontier indeed. Yet since the late seventies empirical evidence against the CAPM began to mount. While not meant to be 1 Incidentally this refers to an unpublished draft, a copy of which can be found on the SSRN website (French & Treynor (2002)) exhaustive, among the empirical flaws identified is that in regressions the intercept is consistently higher than the proxy for the riskfree rate2, that the relation between beta and average return is too flat3, and not least that there is strong evidence that other variables capture variation in expected return missed by beta4. An extensive update of the evidence on the empirical failures of the CAPM is provided by Fama and French (1992). The status quo is that arguably the balance of scholars would regard the CAPM’s empirical record as poor enough to caution against a straightforward implementation in practice – in spite of most practitioners still embracing it. It is however not clear if the CAPM’s poor empirical record is a function of the difficulty of implementing valid tests for the model or if it is the effect of the many simplifying assumptions in the theory. From a more fundamental angle Roll published his famous critique (Roll (1977)) arguing that 1) it is impossible to observe the Market Portfolio and therefore impossible to test the validity of the CAPM and 2) that every efficient portfolio satisfies the CAPM equation exactly (so-called mean- variance tautology). Behaviorialists5, whose voice is getting stronger given poor equity returns over the past decade, question the relevance of the assumption set, such as rational investors and efficient markets6. And contending models such as the Intertemporal Capital Asset Pricing Model (Merton (ICAPM) (1973)) and the Arbitrage Pricing Theory (Ross (1976), have been introduced, attempting to address the major fundamental weakness of the CAPM, it being a one- factor model. This criticism has also just been reflected in the brief review of the existing empirical evidence. Regardless, within its own boundaries imposed by the underlying assumptions, the fundamental validity of the CAPM is still unscathed, and it still is the dominant model and paradigm for understanding Risk and Return in equilibrium. 2 For instance Douglas (1968), Black, Jensen and Scholes (1972), Miller and Scholes (1972), Blume and Friend (1973), and Fama and MacBeth (1973) 3 Notably Friend and Blume (1970), Black, Jensen, and Scholes (1972), and Stambaugh (1982) 4 Basu (1977) was the first among many to expose this empirical weakness of the model 5 Proponents hereof are for instance Miller (1977), DeBondt and Thaler (1985 & 1987), Gould (1986) and Lakonishok, Shleifer, and Vishny (1994) 6 An extension of this debate goes beyond the scope of this paper, although in the opinion of the author disappointing returns do not in itself undermine the essence of market efficiency In this article, contrary to Roll’s conclusion I shall nevertheless test the CAPM and outline why it should be fundamentally rejected as false even within its own assumption set. In part I a brief account of the contemporary interpretation of the CAPM paradigm will be given. In part II the CAPM shall be tested through a hypothetical thought experiment: in a hypothetical three asset universe it shall be demonstrated that based on the assumption that investors are rational and risk averse the model is actually irrational and therefore inherently inconsistent with the fundamentals of risk and return. Part III summarizes the conclusions. I. The CAPM Amongst the fundamental building blocks of the theory of Risk and Return is that there should be a relation between risk and return and that the standard deviation of expected returns is a measure of risk. The CAPM builds on Markowitz’s groundbreaking work (Markowitz (1952 & 1959)) on the model of portfolio choice. In this model an investor selects a portfolio at time t-1 that produces a stochastic return at t. Markowitz illustrates that through costless risk reduction efficient portfolios can be formed that dominate all other portfolios and assets, resulting in the so-called Markowitz efficient frontier. Following on from this work Tobin (1958) introduces the separation theorem demonstrating that all investor portfolios should lie on a linear function of the maximum utility portfolio (lying on Markowitz efficient frontier) and the risk free asset. This so- called capital market line7 (CML) intercepts at the risk free return at a standard deviation of zero and is tangent at Markowitz’ efficient frontier, dominating the remainder of Markowitz efficient frontier in terms of risk reward characteristics. The point of tangency is regarded as the market portfolio, i.e. the efficient portfolio of all risky assets held by all investors at equilibrium. The intercept of the CML is RF and its slope is (E(RM) – RF)/ σM; its equation: E(RP) = RF + [(E(RM) – RF)/ σM]σP (1) 7 Historically this expression has been used as a general description of market pricing, but over time it has become synonymous with the contemporary definition as outlined The implication of an efficient portfolio is that the risk of an individual security, its variance, can be broken down into two parts, undiversifiable, or systematic, and diversifiable or unsystematic: VAR(Rj) = b2VAR(RM) + VAR(ε) (2) Total risk = Undiversifiable risk + Diversifiable risk The coefficient slope, b, typically referred to as the beta, can be calculated exactly: b = COV(Rj,RM)/VAR(RM) (3) Next, based on the essential assumption that rational investors will only pay for the undiversifiable risk of individual assets, as the diversifiable part can be diversified to zero at no cost, the CAPM equation also known as the Security Market Line (SML) is derived: E(Rj) = RF + (E(RM) – RF)βj (4) Essential assumptions underlying the CAPM are: 1. Investors are risk averse by nature 2. Investors view the outcome of any investment in probabilistic terms 3. Investors assess the desirability of any investment in terms of only two parameters: the expected return and the standard deviation 4. Investors choose investments based on their utility maximization 5. Investors have homogeneous expectations 6. There is unlimited capital to borrow at the risk-free rate of return 7. Investments can be divided into unlimited pieces and sizes 8. There are no taxes, inflation or transactions costs Figure 1 graphs the CML and the SML, where it is illustrated that the assets A, B and X from panel a) all fall on point X on the SML in panel b), due to an identical amount of undiversifiable risk. M represents the market portfolio. Figure: 1a) Capital Market Line 1b) Security Market Line The significance of the CAPM cannot be overstated: any individual (equity) discount rate can be objectively determined based on the assumption of the market discount rate and the beta of the individual security in relation to the market. Regardless of its acclaim, I shall next outline why this paradigm should be rejected as false. II. Rejection of the CAPM In his famous critique, Roll (1977) eloquently argues that since every efficient portfolio satisfies the CAPM equation exactly (mean-variance tautology) and since the market portfolio is unobservable, it is impossible to test the validity of the CAPM. Yet in spite of the correct conclusions on mean-variance tautology and on the market portfolio being unobservable, there is a way to test the fundamental validity of the CAPM after all: by applying the CAPM to the following hypothetical thought experiment. While adhering to the essential assumptions underlying the CAPM as previously outlined, if we further assume, initially, that the market only exists of the riskless asset and of two risky assets with a significantly different standard deviation (read: total risk) and no diversifiable risk8, we can make the following observations. Under such conditions it is impossible to construct a more efficient portfolio than the risky assets in isolation provide; no Market Portfolio can be formed. Yet consistent with the basic principles of risk and return, at equilibrium the risky asset with the lower standard deviation must have a lower required return than the risky asset with the higher standard deviation9. In other words, due to the lack of an asset pricing model without the effects of portfolio theory, we cannot know how the fair pricing of the three assets relates exactly10, yet consistent with the fundamentals of Risk and Return theory, the (required) return profile should correlate positively with the risks involved. Next we assume a portion of diversifiable risk is allowed for both risky assets, while keeping the total amount of risk (the standard deviation) unchanged relative to the respective base states. It is then fair to assume that initially – that is, before the implications of portfolio theory (and market efficiency for that matter) are taken into consideration – the discount rates or required returns11 for both assets relative to the base state without diversifiable risks are unchanged. For while the CAPM derives the pricing of assets from Markowitz’ findings, the actual starting point for Markowitz, and thus the CAPM, is the initial pricing of assets before portfolio diversification benefits are priced in. Clarifying this position from a different angle: the fair discount rates for the two risky assets before the diversifiable risks are eliminated through the effects of the CAPM must be identical to those of two risky assets in a parallel three-asset universe that with the exception of the total risks of the risky assets being undiversifiable is identical in all respects to the universe where the (identical) total risks of the risky assets are diversifiable. While the context of the thought experiment described may be abstract, the testing of the CAPM in such an environment is in fact quite straightforward. We shall refrain from utilizing real equities with real returns as proxies, because this effectively provides us historic realized returns 8 I.e. the risky assets are perfectly correlated 9 Consistent with the basic principles of risk and return, this is true even in the absence of an exact equation for the pricing of risk. 10 For instance the risk reward relationship might not even be linear 11 Incidentally, while the discount rate and the required return may always be used interchangeably, there is a fine distinction between the expected return and the required return; the latter are necessarily identical at equilibrium only and the effect of the CAPM, if any, is then already incorporated. Instead we shall construct proxies where such distortions, inefficiencies and market imperfections are entirely eliminated: For simplicity we assume two hypothetical unleveraged zero growth companies A’ and B’, having identical assets, but being from a different risk class. Both companies pay out their profits as dividends annually and both companies are expected to be profitable in every state of the world. We assume there are annually three states of the world with respective expected profits (cash flows) for each company. The chance for each state occurring is randomly set, on the condition that the sum of the probabilities of each of the three states of the world occurring adds up to 100%. The expected (nominal) profits for each state of the world for each company is also randomly set, adhering to the above principle of profitability in every state of the world. In absence of an asset pricing model in a context of no diversifiable risks we cannot know which discount rates are fair. Regardless, we shall also randomly select a priori12 discount rates (kA/B) for both risky assets for which we shall assume investors agree those to be efficient for the total risks of the assets. That is, before the mechanics of the CAPM set in, that is, in case there were no diversifiable risks. Yet to not violate the fundamentals of Risk and Return, these discount rates are bound by the following conditions: 1) the a priori required returns for the risky assets are in excess of the riskfree rate and 2) the higher a priori required return is corresponded by a higher standard deviation. Furthermore, as previously indicated the correlation between A’ and B’ cannot be 1 (to allow for diversifiable risk). And lastly, in building efficient portfolios unrestricted short sales of the risky assets is allowed. Furthermore, while in principle the profits are uncertain, as is also reflected in the three possible states of the world, we assume consistent with the homogeneous expectations that investors agree on the expected nominal profit for each company for each state of the world and on the chances of each state of the world occurring. Also we assume that investors agree that such expectations are perpetual in nature, i.e. while the profits are as said uncertain and while the actual state of the world in each year is also uncertain, investors agree that the probabilities of the potential states of the world and the expectations for the corresponding profits are identical for all future years. 12 That is before taking the pricing effect of the CAPM into consideration Based on these assumptions for both companies the expected nominal cash flow can be calculated ( X ). Consistent with the contemporary paradigm on stock valuation, fair market values of the companies (V = X /k) can be calculated: the fair market value of the firm is calculated by discounting the expected annual nominal cash flow over the fair discount rate. The expected returns for each company for each state of the world can be calculated: the expected nominal cash flow in each state of the world divided by the fair value of the respective company (Xi/V), as well as the average expected return. Also the standard deviation and variance of expected returns for each company can be calculated, as well as the correlation of the expected returns between both companies. The resulting calculations in turn allow for the exact calculations of the Market Portfolio (M), in terms of the weights in each company and in terms of the expected returns, standard deviations and variance of M. And M ultimately allows for the calculation of the covariance risk, the resulting betas and (thus) not least, the resulting a posteriori13 required returns for both companies as a result of applying the CAPM. A numerical example of the effect of the CAPM on two such randomly selected perfect proxies is illustrated in Table 1. TABLE 1: The CAPM effect in a world of two risky assets and one riskless asset When comparing the a priori and the a posteriori discount rates for both companies it is immediately clear that the CAPM cannot reflect a method for pricing risky assets in equilibrium: 13 That is after taking the pricing effect of the CAPM into consideration the suggestion that the discount rates remain unchanged as a result of rational risk-averse investors pricing risk reduction is outright irrational. The exact amount of risk reduction in the above table is reflected in the difference between the variance and the covariance risk14. In other words, while the CAPM claims investors are not willing to pay for costless diversification benefits – suggesting markets would only price covariance risk – the implication of the CAPM is that rational risk-averse investors are willing to pay an identical yield for the total risk of a given asset, represented by its variance, as they would for its undiversifiable risk alone, represented by its covariance risk. The conditions on dividends and the profitability in every state of the world are arguably somewhat trivial, however given the heretical nature of this paper it is essential to take away any doubt on the justification of defining the expected return as the expected profit/ cash flow divided by the fair market value of the firm. Yet since we do not have a method for pricing assets before diversification effects are priced, critics might argue there is no certainty that the ‘randomly’ selected a priori discount rates in Table 1 are indeed efficiently reflecting the total risks of the assets, as if there were no diversifiable risks. While true, this is irrelevant; based on the general formula in a world of two risky assets and one riskless asset that provides the weights of assets A and B to obtain the resulting Market Portfolio15, it is a mathematical fact that the a posteriori discount rates will in all cases remain identical to any a priori quasi16 efficient discount rates. A detailed proof hereof goes beyond the scope of this paper, yet such is easily illustrated in Table 2 by replacing the a priori discount rates of Table 1 by randomly chosen other a priori (quasi efficient) discount rates. 14 Referring to formulae 1 and 2 from the previous chapter, the undiversifiable risk or covariance risk is: b2VAR(RM) 15 wA = [(E(RA) - RF) σ2B - (E(RB) - RF) σ2AB]/[(E(RA) - RF) σ2B + (E(RB) - RF) σ2A - ((E(RA) - RF + E(RB) - RF) σ2AB] wB = 1 - wA 16 That is, upholding the conditions described earlier. TABLE 2: The CAPM effect based on another set of a priori discount rates Clearly the same is true for varying the probabilities for each state of the world occurring, as illustrated in Table 3. TABLE 3: The CAPM effect based on a different probability set And clearly the same is true for using different values for the expected nominal cash flows in each state of the world for either company, as is illustrated in Table 4. TABLE 4: The CAPM effect based on different expected cash flows In other words, it is mathematically impossible to find (a priori) discount rates for the total risks of the assets that could result in lower (a posteriori) discount rates when applying the CAPM. Thus irrespective of the quasi efficient set of a priori discount rates selected for the total risks, the a posteriori discount rates for the covariance risk only, will always be identical to the a priori discount rates. Reversing the logic, to the extent that any set of a posteriori discount rates is considered efficient as function of applying the CAPM, then by implication the fair discount rate for the total risks of the assets must be higher. This is true, because the CAPM fair discount rates only ‘price’ the undiversifiable risk. However, it has just been found that in a tree-asset universe it is mathematically impossible to find a priori discount rates that satisfy the CAPM equation and that are not identical to the a posteriori discount rates and vice versa. While perhaps redundant, for reader comprehension the conclusions of the hypothetical thought experiment will also be presented from a different angle. Assume two parallel three-asset universes. Universe I consists of three assets, the riskfree rate (RF) and two risky assets A and B without any diversifiable risks. We assume that the universe is at equilibrium: the assets are fairly priced. Next we assume that in the parallel universe the risk free rate is identical to the one in universe I and that the risky assets A’ and B’ have identical total risks to assets A and B in universe I. However, in universe II there is a portion of diversifiable risks between A’ and B’. We further assume the investors in universe I are homogeneous to those in universe II. At least what is essential to this assumption is that any agreement on the fair value of assets in universe I based on a given dataset must result in an identical fair value in universe II based on an identical dataset. In other words, the investors in universe I are in agreement with those in universe II on the fair pricing of risk. Or to make the point more bluntly swapping the investors from one universe to the other has no influence whatsoever on the agreement of equilibrium pricing in either separate universe. As a result of non-diversifiable risks in universe I the CAPM cannot be applied and the assets will be priced according to their total risk. As a result of diversifiable risks in universe II between A’ and B’ the Market Portfolio can be formed and the CAPM can be applied. Figure 2 plots the efficient pricing of assets in either universe. Figure 2: The efficient pricing of assets in universe I and II The fundamental justification of the CAPM is embedded in the assumption that investors only ‘pay’ for the undiversifiable risks and not for the diversifiable risks, because diversifiable risks can be diversified at no cost. In other words, the CAPM claims that the model generates the efficient price for undiversifiable risk only, eliminating undiversifiable risks in full. However, as illustrated in figure 2 and as has just been mathematically proven: irrespective of the fair yields one selects for the total risks of two risky assets in a three-asset universe, the fair yields in a parallel universe for the diversifiable risks alone – according to the CAPM – remain unchanged. In other words, the investors in universe I require an identical yield for the total risks of the assets as the investors in universe II for just a fraction of these risks. Another, perhaps more telling way, to picture this conclusion is to plot the required return in universe I versus the total risk, the standard deviation, and to plot the required return in universe II against the covariance risk only. By default, given diversifiable risks, the covariance risk will be smaller than the total risk17. Yet while the investors in universe II, consistent with the CAPM, ‘price’ the covariance risk only, they still require an identical return for the (smaller) covariance risk than the investors in universe for the (larger) total risk. This is illustrated in Figure 3. Figure 3 Efficient pricing of assets for total risk and covariance risk An identical required return for lower risks is irrational behavior for rational risk averse investors. Thus, in spite of the apparent mathematical consistency of the CAPM, since investors in both universes invariably require an identical return for what is effectively a different amount of risk, the CAPM cannot be a model for efficiently pricing risk in equilibrium. Indeed, based on the proof provided earlier, it is mathematically impossible to have different required returns in either universe in spite of a difference in the amount of risk that investors are willing to price in either universe. Restating this conclusion in terms consistent with the initial approach: since all sets of a priori required returns yield identical sets of a posteriori required returns as a result of 17 As can also be seen in Table I &II risk reduction, the CAPM cannot be a model for efficiently pricing risk in equilibrium. The CAPM should be fundamentally rejected. Also the restrictive environment in which the test has been performed does not allow for a resurrection of the CAPM. While the assumptions are restrictive, if only to eliminate market imperfections, perhaps with the exception of allowing unrestricted short sales for the risky assets none of the assumptions are controversial in relation to the environment in which the CAPM itself was developed. That is to say, all other newly imposed assumptions are principally consistent with the default CAPM context. And if the CAPM works in a universe of many assets it should also work in a three asset universe, just like a rejection in a three asset universe invalidates the CAPM in a multi-asset universe. Yet while Black’s model allows for unrestricted short sales, the Sharpe-Lintner version of the model, which is the one that is explicitly being tested in this paper18, does not. In effect, it is highly debatable if this assumption is really controversial in a Sharpe-Lintner context, if only given the high fundamental consistency between both methods. Regardless, given the explicit difference in assumptions between either method, this issue cannot be denied. Moreover, when short sales are not allowed in cases where the Market Portfolio would otherwise have been built with a negative weight (short sale) for either asset, the aforementioned mathematical consistency with regards to unchanged discount rates breaks down. Irony has it however, that even without this mathematical fact of unchanged discount rates, there is anyway no hypothetical outcome for the CAPM that is sustainable. Both a posteriori discount rates increasing is irrational for risk-averse investors: increasing discount rates for risk reduction. One a posteriori discount rate increasing and the other decreasing, is for identical reasons at least partially irrational for risk-averse investors and also inconsistent given distinctive implications for either discount rate. And both a posteriori discount rates decreasing, implies zero discount rates at infinity. Indeed, since the expected nominal profits for both A’ and B’ for each state of the world remain unaffected by any CAPM ‘repricing’ effect: the correlation between both 18 Having said that, based on the findings in this paper, it should be clear that also the Black version of the model should be fundamentally rejected. The replacement in the test of the risk free rate by the zero beta portfolio would suffice companies (<1 from the start) remains unchanged. This is also reflected in the identical correlation in Table I and Table II in spite of different a priori discount rates in each. As a result, the decreased a posteriori discount rates would allow for a new Market Portfolio and hence again for (further) decreased discount rates. A decreasing repricing effect continuing indefinitely, implies zero yields at infinity. Thus even without allowing unrestricted short sales the CAPM should be fundamentally rejected19; there is no rational outcome. What is further ironic is that the CAPM cannot be based on the assumption that investors are not willing to pay for diversification benefits: risk-averse investors demanding an identical or higher yield for risk reduction is an irrational form of double counting and investors accepting a lower yield than the total risk would subscribe, implies investors are in fact paying for diversification benefits. III. Conclusion This paper may raise more questions than the answers it provides; particularly as it requires the paradigm on pricing risky assets to be rebuilt. Nevertheless, a fundamental flaw of the Capital Asset Pricing Model has been exposed: in a world of two risky assets and one riskless asset the CAPM invariably suggests that risk reduction through diversification results in an unchanged discount rate for both risky assets. An identical discount rate for the total risk of an asset as well as for its undiversifiable risk only (hence risk reduction) is irrational for risk-averse investors and thus inherently inconsistent with the basic principles of the paradigm on Risk and Return. It has further been demonstrated that there is anyway no (other) hypothetical outcome conceivable that would make the CAPM sustainable. In sum, while upholding the essence of Markowitz’ findings, this article claims the influential Capital Asset Pricing Model, in place for more than forty years, should fundamentally be rejected as false. 19 Incidentally, stress tests reveal a negative weight (short sale) of either asset occurs only if the discount rate of asset A is chosen such that it lies beneath a straight line connecting the riskless asset and risky asset B. One could argue this implies the a priori discount rate is inefficient and should therefore be rejected as a fair a priori discount rate. 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