A probabilistic model

There are \(n\) risky assets with random returns \(R_1, \dots, R_n\). In addition, the return of the market portfolio is denoted by \(R_M\). The real-life portfolio that corresponds best to the market portfolio does not have to be specifically defined at this point, but it is at least assumed that it contains each asset \(i\) (or just a strict subset of the assets) in proportion \(\omega_i > 0\) with \(\sum \omega_i = 1\). Finally, the dynamics of the random returns are given by:

for some real numbers \((\alpha_i, \beta_i)_{1 \leq i \leq n}\), and \((\varepsilon_i)_{1 \leq i \leq n}\) being a family of random residuals verifying

The residual standard deviation \(\sigma(\varepsilon_i) = \sqrt{\text{Var}(\varepsilon_i)}\) is also called the idiosyncratic volatility of asset \(i\). The first two conditions in \(\eqref{eq:hypothesis}\) make the model equivalent to \(n\) individual linear regressions where observations of \(R_i\) are regressed onto observations of \(R_M\).

      By linearity, the notations naturally extend to any given portfolio formed from a subset \(J \subseteq \{1,\cdots,n\}\) of the \(n\) assets with positive weights \((\psi_j)_{j \in J}\) verifying \(\sum \psi_j = 1\). The beta of the portfolio is given by \(\sum \psi_j \beta_j\), its alpha by \(\sum \psi_j \alpha_j\), and its residual by \(\sum \psi_j \varepsilon_j\). We can illustrate the power of diversification within the model by observing that, because the \((\varepsilon_j)\) are uncorrelated, the idiosyncratic variance of such a portfolio verifies

and concluding that as the cardinal of \(J\) increases, \(\sum \psi_j^2\) is typically much smaller than \(1\) because of the relation \(\sum \psi_j = 1\). This argument is used in Advanced Portfolio Management to justify that the idiosyncratic volatility of the market portfolio itself is comparatively small². However, another argument could be made for the market portfolio: since we are regressing asset returns onto the market portfolio return, the beta of the market portfolio is one and its residual is zero by definition, therefore its idiosyncratic volatility is trivially zero. We can observe more rigorously that

then notice that for a fixed \(j \in \{1, \cdots, n\}\), applying the second condition of \(\eqref{eq:hypothesis}\) and substituting \(R_M\) with the expression from \(\eqref{eq:circular}\) yields

where I used the fact that the covariance with a constant is zero, the second condition of \(\eqref{eq:hypothesis}\) once more, and the uncorrelation of the \((\varepsilon_i)\). Hence, we obtain \(0 = \mathop{\text{Var}}(\varepsilon_j) = \mathop{\mathbf{E}}(\varepsilon_j^2)\) because of the first condition in \(\eqref{eq:hypothesis}\), which is only possible if \(\varepsilon_j = 0\) (almost surely), and this result holds for any \(j \in \{1, \cdots, n\}\).

      Surely, a trivial model where every asset return is an affine combination of the others was not what Sharpe had in mind when he wrote his seminal papers, yet this inconsistency was present from the beginning. From there, how can we be sure whether anything we derive from the model is not contradictory? In fact, empirical applications tend to show that most of the familiar consequences, such as additivity of idiosyncratic variances, still hold approximately: as we will see, the theoretical issue really lies in the imprudent approximation of “small” residual cross-correlations with “exactly zero” in \(\eqref{eq:hypothesis}\).

The CAPM and the origins of the model

The Single Index Model is often amalgamated with the CAPM published as an independent article one year later³: Sharpe initially wrote a first version of the CAPM in terms of what he also called the Diagonal Model in his 1961 PhD dissertation⁴, but it was rewritten completely in the 1964 article such that the two models are now in fact independent and use different sets of assumptions. I will briefly highlight the differences between them.

      Fundamentally, the CAPM is a solution to the Mean Variance Optimization problem introduced by Markowitz, under certain economic assumptions such as asset prices being in equilibrium. MVO really only relates to expectations of returns rather than their dynamics, and as such the CAPM formulation is also given in terms of expectations:

where this time \(R_M\) is the return of a precisely defined portfolio containing all risky assets in the universe. Note that I have omitted the risk-free rate from \(\eqref{eq:capm}\) as a simplification, but feel free to assume that we are talking about excess returns if that triggers you. Of course it is straightforward to go from expectations to dynamics, just define

and you fall back on the familiar \(R_i = \beta_i R_M + \varepsilon_i\) with \(\mathbf{E}(\varepsilon_i)= 0\). Also notice that

such that in the CAPM, \(\varepsilon_i\) is indeed uncorrelated with \(R_M\). However, this is a consequence of the value of \(\beta_i\) rather than the other way around: as shown in a footnote of the 1964 paper (almost reminds me of something else), the expression for \(\beta_i\) directly stems from the CAPM framework without ever talking about \(\varepsilon_i\). If you are more used to Bourbaki-style mathematics like I am, you might find this proof to be more readable. Hence, the CAPM verifies the first two conditions of \(\eqref{eq:hypothesis}\), which enables deriving the usual risk decomposition

with \(\beta_i^2 \mathop{\text{Var}}(R_M)\) being the systematic risk component common to all returns and \(\text{Var}(\varepsilon_i)\) the unsystematic component.

      From the equation of returns of the CAPM and its consequences in terms of risk decomposition, it really takes two leaps to arrive to the Diagonal Model:

• abandon the market equilibrium idea and instead focus on an abstract probabilistic model describing the dynamics of returns
• embrace the systematic vs idiosyncratic decomposition by adding the third condition of \(\eqref{eq:hypothesis}\).

The Diagonal Model now becomes a completely independent model that simply attempts to encode the empirical observation that market risk is a common driver of returns, free of any other economic assumptions. In fact, \(R_M\) does not have to be the return of the exact portfolio from the CAPM, but rather any portfolio or factor that is believed to drive returns in a given universe of assets: Sharpe does leave this door open in his 1963 paper (even though he only proceeds to apply it with actual portfolios), and this interpretation will help us fix the soundness issue described earlier. Still, the model remains in a sense compatible with the CAPM: this time as a consequence of the uncorrelatedness of \(\varepsilon_i\) and \(R_M\), we obtain again

so that if the market portfolio is chosen to be that of the CAPM, both models assign the same value to \(\beta_i\). It is clear to me that this feature may have appealed to Sharpe.

Fama to the rescue

The internet is pretty sparse when it comes to acknowledging the inconsistency exposed in the first section. For example at the time of writing, Wikipedia defines the model more or less exactly as I do. The Wikipedia page does acknowledge that empirically, the residual cross-correlations are not exactly zero, but apparently without realizing that using \(\text{Cov}(\varepsilon_i, \varepsilon_j) = 0\) as an approximation of reality makes the model so constrained as to render it trivial. I even asked ChatGPT but needless to say, the result was disappointing (every time I ask something to ChatGPT, I secretly hope that it gives me an incorrect answer: it makes me feel a bit less concerned about the imminent AI takeover prophetized by X/Twitter experts).

      Trying my luck at old papers instead, I finally found that Fama first described the issue in a 1968 article⁵, where he also notices that the three conditions in \(\eqref{eq:hypothesis}\) cannot hold all at once, and proposes a revised model that stays in the spirit of Sharpe’s. Jensen also reviewed Fama’s new model in an exhaustive study published one year later⁶, and gave some justifications as for why the Diagonal Model is a decent approximation. Instead of regressing the \(R_i\) directly onto \(R_M\), Fama postulates the existence of an unobservable random factor \(\widetilde{R}\) and a new family of residuals \((\widetilde{\varepsilon}_i)\) such that

where the new residuals satisfy three conditions similar to \(\eqref{eq:hypothesis}\):

How is it different from the Diagonal Model? The key point is that the regressor \(\widetilde{R}\) is not defined in terms of the \(R_i\) anymore, it is instead an external force that drives returns rather than an actual portfolio: it is an early example of a factor model where the underlying factor cannot be observed directly in the market. Within Fama’s model, \(R_M\) now has a beta a priori different from one, as well as non-zero alpha and residual:

However, it can be noted that applying an affine transformation to \(\widetilde{R}\) gives another equivalent formulation of Fama’s model, so that we can choose to have \(\alpha_M = 0\) and \(\beta_M = 1\) without loss of generality and simplify the previous relation to \(R_M = \widetilde{R} + \widetilde{\varepsilon}_M\). Using the same argument as in \(\eqref{eq:idio_ptf}\), we can expect the idiosyncratic variance \(\text{Var}(\widetilde{\varepsilon}_M)\) to be small relative to \(\text{Var}(\widetilde{R})\), such that \(R_M \approx \widetilde{R}\) illustrates the concept of factor-mimicking portfolios, which are tradeable portfolios designed to have a performance approximating that of an unobservable factor. It then makes sense to translate \(\eqref{eq:fama}\) into a regression onto \(R_M\) as in Sharpe’s model:

after having defined

      With the formulation from \(\eqref{eq:new_regression}\), we can in theory estimate \(\beta_i\) exactly as in the original model by regressing observations of \(R_i\) onto \(R_M\). However, this regression is no longer well-posed in the sense that \(\varepsilon_i\) and \(R_M\) are not necessarily uncorrelated. Hence, the OLS estimator of \(\beta_i\) is no longer consistent and instead converges to

Assuming \(\text{Var}(\widetilde{\varepsilon}_M) \ll \mathop{\text{Var}(\widetilde{R})}\) and \(\text{Var}(\widetilde{\varepsilon}_i) \approx \mathop{\text{Var}(\widetilde{R})}\), the OLS estimator would in general still converge to a value close to the real \(\beta_i\), but would nevertheless tend to overshoot for assets having a large weight in the chosen portfolio. We can also give a few other properties of the “proxy” residuals \((\varepsilon_i)\), for example the non-diagonal elements of the covariance matrix are given by

which is indeed different from zero: assets with a large beta and a large weight would typically have their proxy residuals be more correlated to other proxy residuals. And when it comes to reformulating the risk decomposition of asset returns, we obtain

which is a decomposition similar to \(\eqref{eq:decomposition}\), but with a slightly decreased contribution of the systematic component and an increased contribution of the unsystematic one. Other elementary properties can be derived in the same way, and can be used to assess whether some approximations are likely to hold empirically or not.

On the meaning of consistency

One question that may arise is why even bother with this theoretical issue if the consequences of the Single Index Model still hold approximately in an empirical setting. The most important aspect lies in the precise definition of the word “consequences”, especially in the context of a contradictory model, as I will illustrate with simplified concepts drawn from formal logic.

      In mathematical logic, the concept of inconsistent or contradictory theory is used to describe a set of assumptions or axioms from which we can derive both one thing and its contrary. It can easily be shown that an inconsistent theory also proves every single sentence: not only can you prove one thing and its contrary, but you can in fact literally prove anything you want. Our model as laid out in the first section was not inconsistent in the strict sense, we just proved that it is equivalent to a much simpler one where \(\varepsilon_i\) is always zero. However if we start assuming that \(\varepsilon_i \neq 0\), then it becomes truly inconsistent, because it proves both \(\varepsilon_i = 0\) and \(\varepsilon_i \neq 0\). And in general, it is very easy to implicitly add simple assumptions like \(\varepsilon_i \neq 0\) in a reasoning: for example, just observe that I assumed \(\text{Var}(R_M) \neq 0\) throughouht a majority of the previous sections without ever stating it. This ties back to the question asked as a conclusion of the first section: we can’t trust any consequence of the model when it is inconsistent. The reason why some of the consequences are still reasonable is because they stem from another model which happens to be consistent: for example Fama’s revised model, but it is impossible to know that until we uncover said model.

      Is there a way to make sure that Fama’s model is not contradictory either though? One of the areas of mathematical logic contains a theorem stating that it is sufficient (and necessary) to produce a concrete example that verifies all the axioms in order to have consistency: this is known as Gödel’s completeness theorem. We can apply this theorem in the context of Fama’s model by letting \((\widetilde{\varepsilon}_1, \cdots, \widetilde{\varepsilon}_n, \widetilde{R})\) follow a \((n+1)\)-variate normal distribution with mean vector \((0, \cdots, 0, \widetilde{r})\) and diagonal covariance matrix

then defining \(R_i \triangleq \alpha_i + \beta_i \widetilde{R} + \widetilde{\varepsilon}_i\) for \(i \in \{1, \cdots, n\}\). It is easily verified that the \((\widetilde{\varepsilon}_i\)) and \(\widetilde{R}\) satisfy the three conditions of \(\eqref{eq:hypothesis_fama}\), while each \(R_i\) obviously satifies \(\eqref{eq:fama}\). In what precedes, \(r \in \mathbb{R}\) and \(\sigma_1, \cdots, \sigma_{n+1} > 0\) are arbitrary parameters, so that we really produced a collection of examples for which Fama’s axioms are true. A theory being consistent does not mean that is is universally true though, i.e. there may still be interpretations where the axioms do not hold: it would be very easy to produce an example of random returns for which Fama’s axioms are false, and we also have empirical evidence that the axioms do not really hold in the real world either, providing us with a very concrete example.

      What about the CAPM, is it consistent? Making abstraction of the economic arguments that lead to its formulation, we need to produce an example of random returns that verify \(\eqref{eq:capm}\), and we also saw that \(\beta_i = \mathop{\text{Cov}}(R_i, R_M) / \mathop{\text{Var}}(R_M)\) is equivalent to \(\text{Cov}(\varepsilon_i, R_M) = 0\). Fama believed this last condition to be contradictory just by itself⁷, because \(\varepsilon_i\) appears in the expression of \(R_i\) which is itself a term in the expression of \(R_M\), hence the two could not be uncorrelated. However, he made a mistake in that what really appears in \(R_M\) is \(\sum \omega_i \varepsilon_i\) and this sum may be equal to zero. If we start from random returns that satisfy Fama’s axioms, such as the example we produced in the previous paragraph, then it is indeed the case since

It can be easily verified that \(\eqref{eq:capm}\) holds if we set \(\alpha_i = 0\) and

So while Fama’s model is consistent for arbitrary values of \(\alpha_i\) and \(\beta_i\), the CAPM needs to restrict the weights of the market portfolio in order to be consistent for arbitrary values of \(\beta_i\), which I suppose ties back into the economic assumptions that impose a certain form to the market portfolio, although I did not take the time to run the numbers. Still, this shows that only the third condition of \(\eqref{eq:hypothesis}\) crosses the frontier between consistency and inconsistency. Models written in terms of unobservable factors, beyond their increased empirical power, are a necessity for theoretical soundness.

References