11.3 Expectations and Efficient Markets

Market efficiency is a very general concept in economic theory. A market is called efficient if at every point in time all relevant information is completely reflected in the price of the traded object. This general definition must be defined more concretely, in order to say what "completely reflected" means. To this end we require the concept of rational expectations. In general one speaks of rational expectations when by the forecast of a stochastic process $ P_t$ all relative and available information $ {\cal F}_{t-1}$ (see Definition 5.1) is `optimally' used. Optimal means that the mean squared error of the forecast is minimized. This is the case when the conditional expectation (see Section 3.5) $ {\mathop{\text{\rm\sf E}}}[P_t \vert {\cal
F}_{t-1}]$ is used as the forecast.

Theorem 11.1  
For every $ h > 0$ using the conditional expectation $ {\mathop{\text{\rm\sf E}}}[P_{t+h}
\mid {\cal F}_t]$ as a forecast, $ P_{t+h\vert t}^*$ minimizes the mean squared error $ {\mathop{\text{\rm\sf E}}}[(P_{t+h}-P_{t+h\vert t}^*)^2]$ given all relevant information $ {\cal F}_t$ at time $ t.$

Proof:
Given any forecast $ P_{t+h\vert t}^*$, that can be written as a (in general nonlinear) function of the random variables at time $ t$, which determines the information set $ {\cal F}_t$, then the mean squared error can be written as


$\displaystyle {\mathop{\text{\rm\sf E}}}[(P_{t+h}-P_{t+h\vert t}^*)^2]$ $\displaystyle =$ $\displaystyle {\mathop{\text{\rm\sf E}}}[(P_{t+h}-{\mathop{\text{\rm\sf E}}}[P_...
...] + {\mathop{\text{\rm\sf E}}}[P_{t+h}\vert{\cal F}_{t}]
- P_{t+h\vert t}^*)^2]$  
  $\displaystyle =$ $\displaystyle {\mathop{\text{\rm\sf E}}}[(P_{t+h}-{\mathop{\text{\rm\sf E}}}[P_{t+h}\vert{\cal F}_{t}])^2]$  
    $\displaystyle + {\mathop{\text{\rm\sf E}}}[( {\mathop{\text{\rm\sf E}}}[P_{t+h}\vert{\cal F}_{t}] - P_{t+h\vert t}^*)^2],$ (11.5)

since the cross product is equal to zero:
$\displaystyle 2 {\mathop{\text{\rm\sf E}}} [ \left( P_{t+h}- {\mathop{\text{\rm...
... {\mathop{\text{\rm\sf E}}}
[P_{t+h} \vert {\cal F}_{t}] - P_{t+h\vert t}^* ) ]$ $\displaystyle =$    
$\displaystyle 2{\mathop{\text{\rm\sf E}}} [
{\mathop{\text{\rm\sf E}}} [ P_{t+h...
...]
( {\mathop{\text{\rm\sf E}}}[P_{t+h}\vert{\cal F}_{t}] - P_{t+h\vert t}^* ) ]$ $\displaystyle =$    
$\displaystyle 2{\mathop{\text{\rm\sf E}}} [ 0 \cdot ( {\mathop{\text{\rm\sf E}}}[P_{t+h} \vert{\cal F}_t] -P_{t+h\vert t}^*) ]$ $\displaystyle =$ $\displaystyle 0.$  

The second term on the right hand side of (10.5) is nonnegative and is equal to zero when $ {\mathop{\text{\rm\sf E}}}[P_{t+h}\vert{\cal F}_{t}]
= P_{t+h\vert t}^*$.

$ {\Box}$

Not all economic variables have sufficient information available to estimate $ {\mathop{\text{\rm\sf E}}}[P_t \mid {\cal F}_{t-1}].$ This has to do with the type of underlying process that determines $ P_t$ and the relative level of the necessary information for the forecast. In order to shed light upon this conceptual problem, hypotheses have been developed in the macro-economic theory, which do not require the use of mathematical expectations $ {\mathop{\text{\rm\sf E}}}[P_t \mid {\cal
F}_{t-1}]$. The hypothesis on adaptive expectations assumes for instance that the forecast at time $ t-1$ of $ P_t$, $ {\mathop{\text{\rm\sf E}}}^a_{t-1}[P_t]$, is generated by the following mechanism:

$\displaystyle {\mathop{\text{\rm\sf E}}}^a_{t-1}[P_t]-{\mathop{\text{\rm\sf E}}}^a_{t-2}[P_{t-1}] = \theta(P_{t-1} - {\mathop{\text{\rm\sf E}}}^a_{t-2}[P_{t-1}])$ (11.6)

with a constant parameter $ \theta, \:\: 0<\theta<1$. Changes in the forecast result from the last forecast error weighted by $ \theta$.

Theorem 11.2  
The adaptive expectation in (10.6) is optimal in the sense of the mean squared error exactly when $ P_t$ follows the process

$\displaystyle P_t = P_{t-1} + \varepsilon_t - (1-\theta)\varepsilon_{t-1}$ (11.7)

where $ \varepsilon_t$ is white noise.

Proof:
With the Lag-Operator $ L$ (see Definition 10.13), (10.6) can be represented as

$\displaystyle \left\{1 - (1-\theta)L\right\}{\mathop{\text{\rm\sf E}}}^a_{t-1}[P_t] = \theta P_{t-1}.
$

Since $ 0<\theta<1$ and $ \left\{1 - (1-\theta)z\right\}^{-1} =
\sum_{i=0}^\infty (1-\theta)^i z^i$ this can be written as

$\displaystyle {\mathop{\text{\rm\sf E}}}^a_{t-1}[P_t] = \theta \sum_{i=0}^\infty (1-\theta)^i
P_{t-i-1}.
$

The process (10.7) can be rewritten as

$\displaystyle \left\{1-(1-\theta)L\right\}\varepsilon_t = P_t - P_{t-1}
$

and
$\displaystyle \varepsilon_t$ $\displaystyle =$ $\displaystyle \sum_{j=0}^\infty (1-\theta)^j (P_{t-j} - P_{t-j-1})$  
  $\displaystyle =$ $\displaystyle P_{t} - \theta \sum_{j=0}^\infty (1-\theta)^j P_{t-j-1},$  

so that $ P_t - {\mathop{\text{\rm\sf E}}}^a_{t-1}[P_t]$ is white noise. Thus $ {\mathop{\text{\rm\sf E}}}^a_{t-1}[P_t]$ is the best forecast for $ P_t$ in the sense of the mean squared error.

$ {\Box}$

The process (10.7) is also referred to as the integrated autoregressive moving average process (ARIMA) of order (0,1,1). The family of ARIMA models will be discussed in more detail in Chapter 11. In general exogenous factors, for example, supply shocks, could also be involved in determining the equilibrium prices. In this case adaptive expectations would be suboptimal. If $ X_t$ is the stochastic exogenous factor and $ {\cal F}_t$ is a family of results which are determined from the observations $ \{p_t,p_{t-1},\ldots, x_t,x_{t-1},\ldots\}$ available at time $ t$, then the optimal process $ {\mathop{\text{\rm\sf E}}}[P_t \mid {\cal
F}_{t-1}]$ is in general a function of $ \{p_t,p_{t-1},\ldots\}$ and of $ \{x_t,x_{t-1},\ldots\}$. Special cases do exist in which adaptive expectations coincide with rational expectations, for example, in a linear supply/demand system with $ X_t$ as an exogenous shock that follows a random walk. If $ X_t$ is instead an AR(1) process, then forecasts with adaptive expectations have a larger mean squared error than forecasts with rational expectations. If the factor $ X_t$ is common knowledge, i.e., available to the public, then rational expectations in this example would mean that the price would be optimally forecasted by using this information.

However, when the factor $ X_t$ is not observable for everyone, in principle the uninformed agent could learn from the prices offered by the informed agent. This means that through observation of prices they could obtain information on the status of $ \omega$, above and beyond what is in their private information set $ F_i$. Here it is assumed that the information function of prices is correctly interpreted.

In order to illustrate what role the price plays in forming expectations, imagine purchasing a bottle of wine. In the store there are three bottles to choose with the prices EUR 300, EUR 30 and EUR 3. Since the bottle for EUR 300 exceeds the budget, only two bottles for EUR 3 and EUR 30 are considered. Now assume that someone who is not a wine expert could not evaluate the quality of the wine from instructions written on the label. Since one is pressed by time, collecting information from other people is time consuming. What remains is the information included in the price. Assume further that one has learned through previous shopping experiences that the more expensive wine tends to be better than the cheaper wine. Thus one constructs a function of the price with respect to the quality, i.e., how good the wine is. One would choose the wine for EUR 30 if the better quality and more expensive wine was valued more in the utility function than the price advantage of the cheaper wine. The buyer behaved rationally, since he optimized his decision (here maximizing his utility function) with the help of the available information and the price function, assuming that the function was right.

In addition let's take a look at another example of an experimental market which is taken from the literature. We have a security that is traded in two periods P1 and P2 and in each period it pays various dividends according to the type of investor. The trading system is an auction in which at the time of an offer both bid and ask prices are verbally given. There are three types of investors and from each type there are three investors, i.e., a total of nine investors can trade the security, among other instruments. Each investor has an initial capital of 10000 Franks (1 `Frank' = 0.002 USD) and two securities. The initial capital of 10000 Franks must be paid back at the end of the second period. Every profit which results from trading the security may be kept. When the investor is in possession of the security at the end of P1 or P2, he will receive the dividend with respect to what type of investor he is. Table 10.2 displays information on the dividend payments.


Table 10.2: Payments in periods P1 and P2 according to type of investor
Type P1 P2
I 300 50
II 50 300
III 150 250


Every investor knows only his own dividend payment, no one else. The question is, whether and if so how fast the investors `learn' about the pricing structure, i.e., gain information on the value of the security to the other investors. There are two underlying hypotheses:

  1. Investors tell each other through their bids about their individual dividends only in P1 and P2 (`naive behavior').
  2. The investors draw conclusions through the observed price on the value of the security for the other investors and use this information in their own bids (`rational behavior').
Since the experiment is over after the period P2, only the individual dividend payments of each investor are of interest, so that in P2 both hypotheses coincide: The equilibrium price is 300 Franks, since type II is just willing to buy at this price and there is competition among the type II investors. At the beginning of P1, before any trading begins, each investor has information only on his own dividends, so that at first one applies naive behavior: type I and type II would offer a maximum of 350, type III would offer a maximum of 400, thus the equilibrium price according to the hypothesis of naive behavior is 400 Franks. This hypothesis performed well in empirical experiments. When the experiment is repeated with the same dividend matrix, the investors can learn through the prices of the previous experiment, which value the security has for the other types of investors in P2. In particular under the hypothesis of rational behavior, type I could learn that the equilibrium price in P2 is higher than what his own dividend would be, thus he could sell the security at a higher price. The equilibrium price in P1 is under the rational hypothesis 600 Franks. Type I buys at the price in P1 and sells in P2 to type II at a price of 300.

In repeated experiments it was discovered that the participants actually tended from naive behavior to rational behavior, although the transition did not occur immediately after the first experiment, it was gradual and took about 8 repetitions. Other experiments were run, including a forward and futures market in which in the first period P1 the price of the security in P2 could already be determined. Here it was shown that through the immediate transparency of the security's value in future periods the transition to rational expectations equilibrium was much quicker.

The observed market price is created through the interaction of various supplies and demands an aggregation of the individual heterogenous information sets. Assume that the price at time $ t$ is a function of the state of the economy, the price function $ p_t(\omega)$, $ \omega \in \Omega$. We define in the following an equilibrium with rational expectations.

Definition 11.16 (RE-Equilibrium)  
An equilibrium at $ t$ with rational expectations (RE-equilibrium) is an equilibrium in which every agent $ i$ optimizes his objective function given the information set $ {\cal F}_{i,t}$ and the price function $ p_t(\omega)$.

Definition 10.16 assumes in particular that every agent includes the information function of the prices correctly in his objective function.

The concept of efficient markets is closely related to the concept of rational expectations. According to the original and general definition, a market is efficient if at every point in time all relevant information is reflected in the price. This means, for example, that new information is immediately incorporated into the price. In the following we define efficient markets with respect to an information set $ {\cal G}$.

Definition 11.17 (Efficient Markets)  
A market is efficient with respect to $ {\cal G} = (G_t)$, $ t \in \mathbb{N}$, $ G_t \subset {\cal F}_t$, if at every time $ t$ the market is in RE-equilibrium with the price function $ p_t(\omega)$ and if for every agent $ i$ and every time $ t$ the following holds

$\displaystyle G_t \subset \{ {\cal F}_{i,t} \cup p_t(\omega) \}. $

Typically three cases are identified as weak, semi-strong and strong efficiency.

  1. The market is weak efficient, when efficiency refers only to historical prices, i.e., the set $ {\cal G} = (G_t)$,
    $ G_t = \{p_t, p_{t-1}, p_{t-2},\ldots \}$. This is, for example, achieved when for all $ i$ it holds that $ \{p_t, p_{t-1},
p_{t-2},\ldots \} \subset F_{i,t}$, that is when the historical prices are contained in every private information set.

  2. The market is semi-strong efficient, when efficiency refers to the set $ {\cal G} = (G_t)$, $ (\cap_i F_{i,t}) \subset G_t \subset (\cup_i
F_{i,t})$, which includes all publicly available information.

  3. The market is strong efficient, when efficiency refers to the set $ {\cal G} = (G_t), G_t = \cup_i F_{i,t}$, i.e., when all information (public and private) is reflected in the price function. In this case one speaks of a fully revealing RE-equilibrium.

An equivalent definition says that under efficient markets no abnormal returns can be achieved. In order to test it one must first determine what a `normal' return is, i.e., one must define an econometric model. Efficient markets can then be tested only with respect to this model. If this combined hypothesis is rejected, it could be that markets are inefficient or that the econometric model is inadequate.

The following is a brief summary of the typical econometric models that have been proposed for financial data. For each of the most interesting financial instruments, stocks, exchange rates, interest rates and options, a corresponding theory will be presented, which are considered to be classic theory in the respective areas. In later chapters we will refer back to these theories when we discuss empirically motivated expansions.