Chapter 6 Equilibrium, efficiency and the ability to “beat the market”
Introduction
There is a -strong- temptation to jump into financial markets and try to make easy money. There are structural reasons why this might be a deceiving attempt. Even if there are many intermediaries e.g. through brokers, exchanges, clearing house etc., between you and you counterparts you are confronting a wealth of informed and smart persons.
In the case of used cars or more generally, used goods, people are naturally warry about information asymmetry: the car seller might have some information I do not have. Differently stated is a cheap used car a good deal or the indication that its quality is flawed. This example, as well as several others, has been first introduced by Akerlof (1970) and gave to the literature related to information asymmetry.
6.1 The efficiency debate
Recommended readings: Bernstein (1993) and Bernstein (2009).
Jean-Philippe Bouchaud:
Fama’s Efficient Market Hypothesis (EMH) was awarded a Nobel Prize in economics in 2013. Admittedly, Robert Shiller, one of its most persuasive opponent, got it as well the same year. It sounded a bit like giving the Nobel Prize simultaneously to Galileo and to the Church. In my opinion, EMH is a great example what Richard Feynman called “Cargo Cult Science” (one of the most inspiring – and fun ! – text on the nature of scientific theories).
EMH offers no explanation whatsoever for the slew of interesting and non-trivial phenomena happening in financial markets. In fact, these phenomena are waved away as “anomalies”, even when they reveal the deep nature of the underlying mechanisms driving price changes.
Is EMH-bashing like flogging a dead horse? Until EMH ceases to be taught as the pillar of modern financial economics, I think it is worth keeping pummeling it with empirical facts and arguments. Imagine the theory of epicycles still being taught to physics freshmen.
The phenomenology of financial markets is incredibly rich and complex, and EMH is only, at best, a medieval `flat earth’ approximation.
Peter Tanous’ interview of E. Fama:
Peter J. Tanous: How did you first get interested in stocks?
Fama: As an undergraduate, I worked for a professor at Tufts University. He had a “Beat the Market” service. He figured out trading rules to beat the market, and they always did!
I beg your pardon?
They always did, in the old data. They never did in the new data [laughter].
I see. Are you saying that when you back-tested the trading rules on the historic data, the rules always worked, but once you applied them to a real trading program, they stopped working?
Right. That’s when I became an efficient markets person.
Bachelier (1900)
I1 semble que le marché c’est-à-dire l’ensemble des spéculateurs, ne doit croire à un instant donné ni à la hausse, ni à la baisse puisque, pour chaque cours coté, il y a autant d’acheteurs que de vendeurs. En réalité, le marché croit à la hausse provenant de la différence entre les coupons, et les reports; les “vendeurs font un léger sacrifice qu’ils considèrent comme compensé. On peut ne pas tenir compte de cette différence, à la condition de considérer les cours vrais correspondant à la liquidation, mais les opérations se réglant sur les cours cotés, le vendeur paye la différence. Par la considération des cours vrais on peut dire : Le marché ne croit, à un instant donné, ni à la hausse, ni à la baisse du cours vrai.
Mais, si le marché ne croit ni à la hausse» ni à la baisse du cours vrai, il peut supposer plus ou moins probables des mouvements d’une certaine amplitude. La détermination de la loi de probabilité qu’admet le marché à un instant donné sera l’objet de cette étude,
6.2 Empirical evidence
6.2.1 Serial correlation
Exercise 6.1 For this exercise, use historical series of SP500 constituents available at datasetsSorbonne. 1. Compute returns from prices 1. Compute average autocorrelation of several orders across stocks 1. Test if significant 1. Test possible volume impact
6.2.2 Active asset management
Some references
- John Bogle & the Bogleheads
- The SPIVA index: SPIVA Index
- CTA investing: CTA Investing
- Mutual Fund Ratings and Performance Persistence
6.3 Toy models of markets with asymmetric information
Considering the importance of this topic, is there a way to have a model where those assumptions could be tested ? Surprisingly, even the simplest possible model become quickly complex.
6.3.1 Grosman and Stiglitz
Now that we know how an investor should behave in a simplified framework, a model can be build where different investors with different information will exchange together and what would be the resulting equilibrium. This is the objective of Grossman and Stiglitz (1980).
6.3.1.1 The model
As above, this is a one period model with one risky asset and one non risky asset. Interest rates are taken equal to zero which is equivalent to think in excess returns terms. The final value of the risky asset is the sum of two random components: \[u=\theta+\epsilon\] where \(\epsilon\) is a noise with variance \(\sigma_{\epsilon}\) and \(\theta\) is a noise with variance \(\sigma_{\theta}\) and average value \(\bar{\theta}.\) \((\theta,\epsilon)\) is assumed to be Gaussian vector with zero correlation.
The quantity of stock offered is equal to \(x\) with \(\bar x = 0.\) The price is a random variable \(p\) which will depend on market equilibrium. Both investors know the parameters of distribution as well as the price.
There exists a procedure “à la Walras” to fix the price. That procedure ensures that all investor know the parameters of the distribution as well as the price. The equilibrium is reached when the sum of demand is equal to the offer.
While both investors know the price and parameters, the Informed Investor (II) is supposed to have paid a cost \(c\) to know the outcome of \(\theta\) while the Uninformed Investor (UI) does not know anything else.
6.3.1.2 Equilibrium
The demand function of investor \(i\) is noted \(X_i\). Both investors are supposed to share the same risk aversion noted \(\lambda\) For the informed investor, an equivalent of formula (5.4) (maximisation in absolute terms, not relative) yields \[X_{I}=\frac{1}{\lambda}\frac{\theta-p}{\sigma_{\epsilon}^2}\]
while for the uninformed investor:
\[X_{U}=\frac{1}{\lambda}\frac{\mathbb E(u\mid p)-p}{var(u\mid p)}\] While for the informed investor, the demand function decreases with the price, it is less obvious for the uninformed investor. This will depend on the behavior of \(\mathbb E(u\mid p)\).
The price \(p\) which realises the equilibrium can only be a function of \(\theta\) and\(x\). We limit the analysis to linear relationships and make the assumption that the different random variables form a Gaussian vector. The following computations are direct application of the equations provided in section 11.1.
One has \(\mathbb E(u\mid p)=\bar \theta+\beta (p-\bar p)\) where \(\beta\) and \(\bar p\) are constants to be determined.
Let \(k\) be the proportion of informed investor. The equilibrium equation is \(kX_{I}+(1-k)X_{U}=x\) or, equivalently. This implies the following relationship between \(\theta\), \(p\) and \(x\):
\[\begin{align*} & p\times\left(-\frac{k}{\sigma_{\epsilon}^{2}}+\left(1-k\right)\frac{\beta-1}{v_{up}}\right)\\ + & \left(\theta\frac{k}{\sigma_{\epsilon}^{2}}-\lambda x\right)\\ = & \left(k-1\right)\frac{\bar{\theta}-\bar{p}}{v_{up}} \tag{6.1} \end{align*}\]where \(\beta\equiv\frac{<u,p>}{var()u|p},\;v_{up}\equiv var(u|p),\;\bar{p}\) are constants to be determined.
Equation (6.1) show that the price is an affine function of a random variable \(w=\theta-\frac{\lambda\sigma_{\epsilon}^{2}}{k}x\). As a result, conditioning by \(p\) is equivalent to condition by \(w\). Since is a linear combination of \((\theta,\epsilon, x)\) and w is also a linear combination of the same vector formula (eq:conditioningGaussian) can be used. It comes that
\[\begin{align*} v_{up} & = \sigma_{\theta}^{2} + \sigma_{\epsilon}^{2}-\frac{\sigma_{\theta}^{4}} {\sigma_{\theta}^{2}+\frac{\lambda^2\sigma_{\epsilon}^{4}}{k^2}\sigma_{x}^{2}}\\ \beta & = \frac{\sigma_{\theta}^{2}} {\sigma_{\theta}^{2}+\frac{\lambda^2\sigma_{\epsilon}^{4}}{k^2}\sigma_{x}^{2}} \\ \bar p &= \bar \theta \end{align*}\]Comments:
- \(\beta<1\)
- vup small, informative market
6.4 Capital Asset Pricing Model
6.4.1 Equilibrium & CAPM
At equilibrium the sum of demand for stocks equals the sum of offered stocks which is equal to the market cap weighted portfolio which we denote \(\alpha_m\). If the aggregated demand of agents can be modeled by a representative agent (references ?) then the market portfolio shall be in proportion of the representative agent portfolio: \(\pi=\lambda\Sigma\alpha\). Let \(\pi_m\) be the expected premium of the market portfolio and \(\sigma^2_m\) its variance. It is clear that \(\lambda=\frac{\pi_m}{\sigma^2_m}\). If we inject this value in the preceding equation and write it line by line we obtain:
\[\begin{equation} \pi=\frac{\pi_m}{\sigma^2_m}\Sigma \alpha_m \end{equation}\]or equivalently, line by line,
\[\begin{equation} \forall i: \pi_i=\beta i\pi_m \tag{6.2} \end{equation}\]since by definition \(\beta_i=<e_i,\alpha_m>_{\Sigma} / \sigma^2_m\).
Comments:
- Return only related to the “common part” with the market portfolio
- no correlation => no return
- opposite of financial analysis