Chapter 4-the value of information

Introduction

An important aspect of risk management is the flow of information linked to the distribution of future risk.

In a very large number of circumstances, there can be some signal or some events that give some information on the probability distibution of the future risk.

The value of information: the model

Information structure

We consider a risky situation that is characterizes by a set \(\Omega\) of “states of nature” : \(\Omega= \left\{\omega_{1},\omega_{2}, ...\omega_{N}\right\}\) The vector \(\Pi=(\Pi_{1},\Pi_{2},...,\Pi_{N})\) is the “prior” probability distibution on \(\Omega\).

The decision maker can observe a signal. Let \(\mathcal{M}\) the set of signals we can set (without los of generality) \(\mathcal{M}=\left\{1,2,...M\right\}\). The vector \(q\) denotes the vector of probabilities to observe the signals : \(q_{m}\) is the probability to observe the signal \(m\). The signal \(m\) carries some information that leads to revise the probability distribution on \(\Omega\). We note \(\pi_{mi}\) the probability of state \(i\) if the message \(m\) is observed.

We have hence : \[\Pi_{i}=\sum_{m=1}^{M}q_{m}\pi_{mi}\] that is \[^{t}\Pi=^{t}q \pi\] where \(\pi\) is the matrix \(\pi_{mi}\)

The decision problem

Posterior to the observation of the signal but prior the observation of \(\omega\), the decision maker must take a decison \(\alpha\) , that is pick an \(\alpha\) is a set \(B \subset\mathbb{R}^{n}\). He will obtain a final utility that will depend on \(\alpha\) and \(\omega\) : \(v(\alpha,\omega)\).

What is the maximal expected utility achieved by the decsion maker. The idea is the following, for any \(m\) he will take the good decision that is the decsion that maximizes : \[\sum_{i}\pi_{mi}v(\alpha,\omega_{i})\] So that the ex ante expected utility achieved is hence : \[V(\pi,q)=\sum_{m}q_{m}\max_{\alpha \in B}\sum_{i}\pi_{mi}v(\alpha,\omega_{i})\]

We have to compare it to the “no information case” where no signal is obeserved : \[W(\Pi)=\max_{\alpha \in B}\sum_{i}\Pi_{i}v(\alpha,\omega_{i})\]

In fact (and it is completely obvious) \(W\) is always smaller then \(V\).

The reason is simple. Take first \(\widehat{\alpha}\) the optimal ex ante decsion when no signal is observed (that is the solution of the maximization program defining \(W\). Then in the program defining \(V\) , replace \(\max_{\alpha \in B}\sum_{i}\pi_{mi}v(\alpha,\omega_{i})\) by \(\sum_{i}\pi_{mi}v(\widehat{\alpha},\omega_{i})\) which is by definition smaller! But doing so and using the fact that \(\Pi_{i}=\sum_{m=1}^{M}q_{m}\pi_{mi}\), we obtain exxactly \(W\).

The action chosen without information is also possible when information is available.Therefore the informed agent can at least duplicate his rigid strategy by chossing uniformly the same action no matter the information he received. “the decsion maker is free to ignore information”

Mathematically it is clear : let \(g\) the function \(\mathbb{R}^{N}\rightarrow \mathbb{R} : g(x)=\max_{\alpha \in B}\sum_{i=1}^{N}x_{i}v(\alpha,\omega_{i})\) \(g\) is the supremum of linear functions (linear forms). Each of the linear functions beeing obviously convex, \(g\) is convex as the upper enveloppe of convex functions.

As \(g\) is convex we have : \[g\left(\sum_{m=1}^{M}q_{m}\pi_{m}\right) \leq \sum_{m=1}^{M}q_{m}g(p_{m})\]

That is exactly : \[W\leq V\]

Refining the information structure

We want to compare two information structures \(\pi^{1},q^{1}\) and \(\pi^{2},q^{2}\) with the same “prior” probability : \(^{t}\Pi=^{t}q^{1} \pi{1}=^{t}q^{2} \pi{2}\)

We would like to compare the two information structure.

Definition

1 is a better information structure than 2 if and only if all agents prefer the first one to the second one : \(V( \pi{1},q^{1})\geq V( \pi{2},q^{2})\) for all decision problems.

For instance an agent faces two possible experiments and must select one from whose he will extract a signal before taking a decision. One faces this kind of problem when one has to evaluate medical diagnoses process or to improve whether forecasing and so on... Imagine that the 2nd information structure is obtained through a garbling machine that mixes the signal sent in the first information structure. If (in the second) the signal \(i\) is observed, the probability that the “original message (in the first) is \(j\) is \(k_{ij}\). So : \(k_{ij}=\Pr(m_{1}=j \mid m_{2}=i)\)

We obviously have : \(\forall i \sum_{j}k_{ij}=1\). Doing so one has the intution that one garbles the original message : the messages observed in experiment 2 don't give a perfect information on the signal observed in experiment 1. We have : \[\pi^{2}=K\pi^{1}\] But we must also have \[^{t}q^{2} \pi{2}=^{t}q^{1} \pi{1}\] So we must have : \[^{t}q^{1} =^{t}q^{2}K \] This means that \(\pi_{2}\) is a mean preserving spread of \(pi_{1}\). We have the following proposition.

Proposition

Given two information structures \(\pi^{1},q^{1}\) and \(\pi^{2},q^{2}\) with the same “prior” probability : \(^{t}\Pi=^{t}q^{1} \pi{1}=^{t}q^{2} \pi{2}\), then 1 is better than 2 if and only if there exists a positive matrix \(K\) with \(\forall i \sum_{j}k_{ij}=1\) such that \[\pi^{2}=K\pi^{1}\] and \[^{t}q^{1} =^{t}q^{2}K \] We say that \(\pi_{2}\) is a mean preserving spread of \(pi_{1}\).

Information and market : efficiency of financial markets

The concept of efficiency for financial market corresponds to the idea that the current prices of financial assets perfecly reflect all the information available to investors.

In other words, financial markets are efficient when an informed trader cannot make systematically more money than other who only observe current prices.

There are three different degrees of market efficiency.
To explain this, take the example of one asset (distributing no dividends) whose price is \(p_{t}\) at date t. At date t the price \(p_{t+1}\) is “random”. The weakest notion of market efficiency says that price is a a “martingale” for some probability distribution :
\[p_{t}=\mathbb{E}\left(p_{t+1}/p_{t}\right)\] The semi-strong efficiency says that the price at date t reflects all public information of date t: \[p_{t}=\mathbb{E}\left(p_{t+1}/\mathcal{I}_{t}\right)\]
The strong efficiency assumption says that the price reflects all public and private information : the equilibrium process makes public all private information. To precise these concepts we are going to study one very simple model proposed by Grosman and Stiglitz.

Supply and demand

There is a single, risky asset with random liquidation value \(\widetilde{\theta}\) and riskless asset (with unitary return). These are traded by risk averse agents and “noise traders.”

The utility derived by a trader i for the (random) profit \(w_{i}=(\widetilde{\theta}-p)x_{i}\) of buying \(x_{i}\) units of the asset at price \(p\) is of the CARA type: \[U(w_{i})=-\exp{-\rho_{i}w_{i}}\], where\(\rho_{i}>0\) is the CARA coefficient that measures risk aversion. We call risk tolerance the inverse of risk aversion \(t_{i}=\frac{1}{\rho_{i}}\).

Initial wealth of each trader i is normalized to 0 (wlog). Trader i is endowed with a piece of private information about \(\widetilde{\theta}\). Noise traders are assumed to trade for liquidity reasons submitting a random trade \(\widetilde{u}\). Suppose that a fraction of traders \(\mu\in[0,1]\) receives a private signal \(\widetilde{s}\) about \(\widetilde{\theta}\) , we call them Informed, subscript \(I\), while the complementary fraction does not, Uniniformed, subscript \(U\) .

Both classes of traders condition their orders on the price \(p\). Let \(\rho_{i}=\rho_{I}>0\) for Informed and \(\rho_{i}=\rho_{U}\ge0\), for Uninformed. \(\widetilde{s},\:\widetilde{\varepsilon},\:\widetilde{u}\) are (pairwise independent) normally distributed: \[\widetilde{s}\rightsquigarrow N(\bar{\theta},\sigma_{s}^{2})]\[\widetilde{\theta}=\widetilde{s}+\widetilde{\varepsilon}\quad,\widetilde{\varepsilon}\rightsquigarrow N(0,\sigma_{\varepsilon}^{2})\widetilde{u}\rightsquigarrow N(0,\sigma_{u}^{2})\] We call “precision” the inverse of the variance for \(j=s,\varepsilon\) or \(u\) : \(\tau_{j}=\frac{1}{\sigma_{j}^{2}}\)

If the price is \(p\) , what is the demand of a trader?

The quantity demanded maximizes the expected utility, the expectation being conditionnal to the information \(\mathcal{J}\) detained :

\[X_{i}(p/J)=\arg\max\left(\mathbb{E}\left[U_{i}\left(\left(\widetilde{\theta}-p\right)x_{i}\right)/\mathcal{J}\right]\right)\]

But we know that \(\widetilde{\theta}\) is normally distributed so that :

\[\mathbb{E}\left[U_{i}\left(\left(\widetilde{\theta}-p\right)x_{i}\right)/\mathcal{J}\right]=U_{i}\left(\mathbb{E}\left[\left(\widetilde{\theta}-p\right)x_{i}/\mathcal{J}\right]-\frac{1}{2}\rho_{i}\mathbb{\textrm{var}}\left(\left(\widetilde{\theta}-p\right)x_{i}/\mathcal{J}\right)\right)\]

So that i's maximization amounts to :

\[\max\left\{ \mathbb{E}\left[\left(\widetilde{\theta}-p\right)x_{i}/\mathcal{J}\right]-\frac{1}{2}\rho_{i}\mathbb{\textrm{var}}\left(\left(\widetilde{\theta}-p\right)x_{i}/\mathcal{J}\right)\right\}\] which gives : \[X_{i}(p/\mathcal{J})=\frac{\mathbb{E}\left[\widetilde{\theta}/\mathcal{J}\right]-p}{\rho_{i}\mathbb{\textrm{var}}\left(\widetilde{\theta}/\mathcal{J}\right)}\]

This expression is quite intuitive : the demand is proportionnal to the spread between expected value and price. The coefficient of proportionality is large when risk aversion is low and/or risk (measured through variance) is low. Before computing the equilibrium we must recall some simple formulas when random variables are normal. Let \(\widetilde{x},\widetilde{y}\) a gaussian vector (i.e the variables are such that every linear combination is gaussian). we have :

Lemma

\(\widetilde{x},\widetilde{y}\) a gaussian vector, we have :\[\mathbb{E}\left(\widetilde{x}/\widetilde{y}\right)-\mathbb{E}\left(\widetilde{x}\right)=\frac{\textrm{cov}\left(\widetilde{x},\widetilde{y}\right)}{\textrm{var }\left(\widetilde{y}\right)}\left[\widetilde{y}-\mathbb{E}\left(\widetilde{y}\right)\right]\] \[\textrm{var }\left(\widetilde{x}/\widetilde{y}\right)=\textrm{var}\left(\widetilde{x}-\mathbb{E}\left(\widetilde{x}/\widetilde{y}\right)\right)=\mathbb{\textrm{var}}\left(\widetilde{x}\right)-\frac{\left(\textrm{cov}\left(\widetilde{x},\widetilde{y}\right)\right)^{2}}{\mathbb{\textrm{var}}\left(\widetilde{y}\right)}=\left(1-\frac{\left(\textrm{cov}\left(\widetilde{x},\widetilde{y}\right)\right)^{2}}{\mathbb{\textrm{var}}\left(\widetilde{x}\right)\mathbb{\textrm{var}}\left(\widetilde{y}\right)}\right)\mathbb{\textrm{var}}\left(\widetilde{x}\right)\]

Naïve equilibrium

Now, consider a first “naive” equilibrium. Each trader optimize with his information. Uninformed has no information (noted \(\mathcal{J}_{U}^{0}\) ) so that \[\mathbb{E}\left[\widetilde{\theta}/\mathcal{J}_{U}^{0}\right]=\bar{s}\] and \[\mathbb{\textrm{var}}\left(\widetilde{\theta}/\mathcal{J}_{U}^{0}\right)=\sigma_{s}^{2}+\sigma_{\varepsilon}^{2}\] :
\[X_{U}(p/\mathcal{J}_{U}^{0})=\frac{\bar{\theta}-p}{\rho_{U}\left(\sigma_{s}^{2}+\sigma_{\varepsilon}^{2}\right)}=t_{U}\tau_{\theta}\left(\bar{\theta}-p\right)\] (with \(\frac{1}{\tau_{\theta}}==\frac{1}{\tau_{s}}+\frac{1}{\tau_{\varepsilon}})\)
Informed observe \(\widetilde{s}\) so that
\[ \mathbb{E}\left[\widetilde{\theta}/\mathcal{J}_{I}^{0}\right]=s \] and \[\mathbb{\textrm{var}}\left(\widetilde{\theta}/\mathcal{J}_{I}^{0}\right)=\sigma_{\varepsilon}^{2} \] .
So : \[ X_{I}(p/\mathcal{J}_{I}^{0})=\frac{s-p}{\rho_{I}\sigma_{\varepsilon}^{2}}=t_{I}\tau_{\varepsilon}\left(s-p\right)\]
The supply by noise trader is u. So that market clearing gives :
\[(1-\mu)t_{U}\tau_{\theta}\left(\bar{\theta}-p\right)+\mu t_{I}\tau_{\varepsilon}\left(s-p\right)=u\]
Which gives the price :\[p=\frac{(1-\mu)t_{U}\tau_{\theta}\bar{\theta}+\mu t_{I}\tau_{\varepsilon}s-u}{(1-\mu)t_{U}\tau_{\theta}+\mu t_{I}\tau_{\varepsilon}}\]

The first remark that can be done is that when \(\sigma_{\varepsilon}=0\), that is when the Informed traders are “perfectly” informed, then their demand curve is horizontal \(p=s\). The only possible equilibrium is hence \(p=s\) perfectly revealing their info. In the general case, remark that the equilibrium price depends (linearly) on \(s\) and \(u\).
For instance, when \(u=0\), \(p\) is larger than \(\bar{\theta}\) when \(s\) is larger than \(\bar{\theta}\), that is when Informed has “a good news about \(\theta\)”.
This dependence implies that the price conveys information about \(s\)! Indeed we have at equilibrium:
\[ \widetilde{s}=\frac{\left[(1-\mu)t_{U}\tau_{\theta}+\mu t_{I}\tau_{\varepsilon}\right]\widetilde{p}-\left(\left(1-\mu\right)\tau_{\theta}t_{U}\right)\bar{\theta}+\widetilde{u}}{\mu t_{I}\tau_{\varepsilon}}\]
So that the best prediction of \(s\) varies with \(p\) :
\[ \mathbb{E}\left(\widetilde{s}/p\right)=\frac{\left[(1-\mu)t_{U}\tau_{\theta}+\mu t_{I}\tau_{\varepsilon}\right]p-\left(\left(1-\mu\right)\tau_{\theta}t_{U}\right)\bar{\theta}}{\mu t_{I}\tau_{\varepsilon}}\]

In fact, Uninformed (but sophiticated) traders should have used this information to set their demand (which they have not at this first naïve stage).
But if they modify their demand function accordingly to take into account this information, this will modify the formula of the price equilibrium, function of s and u! this will in turn modify the information inferred by Uninformed...and so on! The idea of Rational Expectation Equilibrium consists in finding a price formula which is “self fulfilling”. If the price formula is \(p=f(s,u)\) and if the traders use this information then the equilibrium price will be precisely \(f(s,u)\) !

Definition

A REE is a price function \(p=f\left(s,u\right)\) such that, if traders know this price function, they infer information from price and set their demand accordingly. Doing this it turns out that equilibrium price will be precisely \(f(s,u)\)

. In some sense, this type of equilibrium is the limit of the sequence of inference mentioned above.

Rational Expectation Equilibrium (Grossman and Stiglitz)

The idea is to find a linear price formula \(p=a+bs-\lambda u\) which is self fulfilling.

Let us be more precise. Uninformed only observe \(p\) : \[\mathbb{E}\left(\widetilde{\theta}/\mathcal{J}_{U}\right)=\mathbb{E}\left(\widetilde{\theta}/a+b\widetilde{s}-\lambda\widetilde{u}=p\right)=\mathbb{E}\left(\widetilde{s}+\widetilde{\varepsilon}/b\widetilde{s}-\lambda\widetilde{u}=p-a\right)\]
which gives, using the fact that all variables are normal :
\[ \mathbb{E}\left(\widetilde{\theta}/\mathcal{J}_{U}\right)=\bar{\theta}+\frac{b\sigma_{s}^{2}}{b^{2}\sigma_{s}^{2}+\lambda^{2}\sigma_{u}^{2}}\left(p-a-b\bar{\theta}\right)\]
or, equivalently
\[\mathbb{E}\left(\widetilde{\theta}/\mathcal{J}_{U}\right)=\bar{\theta}+\frac{b\tau{}_{u}}{b^{2}\tau{}_{u}+\lambda^{2}\tau{}_{s}}\left(p-a-b\bar{\theta}\right)=\bar{\theta}+\frac{b\tau{}_{u}}{k}\left(p-a-b\bar{\theta}\right)\]

In the above formula we have set \( k=b^{2}\tau{}_{u}+\lambda^{2}\tau{}_{s}\) , (which depends on \(b\) and \( \lambda)\). In the same way :
\[ \textrm{var}\left(\widetilde{\theta}/\mathcal{J}_{U}\right)=\mathbb{\textrm{var}}\left(\widetilde{s}+\widetilde{\varepsilon}/b\widetilde{s}-\lambda\widetilde{u}=p-a\right)=\sigma_{s}^{2}+\sigma_{\varepsilon}^{2}-\frac{b^{2}\sigma_{s}^{4}}{b^{2}\sigma_{s}^{2}+\lambda^{2}\sigma_{u}^{2}}=\sigma_{\varepsilon}^{2}+\frac{\lambda^{2}\sigma_{u}^{2}\sigma_{s}^{2}}{\left(b^{2}\sigma_{s}^{2}+\lambda^{2}\sigma_{u}^{2}\right)}=\sigma_{\varepsilon}^{2}+\frac{\lambda^{2}}{k}\]
The Uninformed demand is hence :
\[ X_{U}(p/\mathcal{J}_{U})=t_{U}\frac{1}{\sigma_{\varepsilon}^{2}+\frac{\lambda^{2}}{k}}\left[\bar{\theta}+\frac{b\tau{}_{u}}{k}\left(p-a-b\bar{\theta}\right)-p\right]\]
The Informed has the same demand function :
\[ X_{I}(p/\mathcal{J}_{I})=t_{I}\tau_{\varepsilon}\left(s-p\right) \]
Market clearing gives :
\[ \mu t_{I}\tau_{\varepsilon}\left(s-p\right)+(1-\mu)t_{U}\frac{1}{\sigma_{\varepsilon}^{2}+\frac{\lambda^{2}}{k}}\left[\bar{\theta}+\frac{b\tau{}_{u}}{k}\left(p-a-b\bar{\theta}\right)-p\right]=u\]

Identifying with\( p=a+bs-\lambda u\) gives for the coefficients of \(u\) and \(s\) , and for the constant \(a\) :
\[\mu t_{I}\tau_{\varepsilon}\lambda+(1-\mu)t_{U}\frac{1}{\sigma_{\varepsilon}^{2}+\frac{\lambda^{2}}{k}}\left[\frac{-b\lambda\tau{}_{u}}{k}+\lambda\right]=1\]\[ -\mu t_{I}\tau_{\varepsilon}b+(1-\mu)t_{U}\frac{1}{\sigma_{\varepsilon}^{2}+\frac{\lambda^{2}}{k}}\left[\frac{b\tau{}_{u}}{k}b-b\right]=-\mu t_{I}\tau_{\varepsilon} \] \[\left(-\mu t_{I}\tau_{\varepsilon}-(1-\mu)t_{U}\frac{1}{\sigma_{\varepsilon}^{2}+\frac{\lambda^{2}}{k}}\right)a+(1-\mu)t_{U}\frac{\bar{\theta}\tau{}_{s}}{\sigma_{\varepsilon}^{2}+\frac{\lambda^{2}}{k}}\left(\frac{\lambda^{2}}{k}\right)=0\]
That gives :
\[\lambda\mu t_{I}\tau_{\varepsilon}=b\]
Such that :
\[ p=a+\lambda\left(\mu t_{I}\tau_{\varepsilon}s-u\right)\]
reinjecting in \( k=b^{2}\tau{}_{u}+\lambda^{2}\tau{}_{s}\) gives :
\[ k=\lambda^{2}\left(\left(\mu t_{I}\tau_{\varepsilon}\right)^{2}\tau_{u}+\tau_{s}\right) \]
We have hence :
\[ \textrm{var}\left(\widetilde{\theta}/J_{U}\right)=\sigma_{\varepsilon}^{2}+\frac{1}{\left(\mu t_{I}\tau_{\varepsilon}\right)^{2}\tau_{u}+\tau_{s}}=\frac{1}{\tau} \]
which allows to find equations giving \(\lambda\) and \(a\) :
\[\mu t_{I}\tau_{\varepsilon}\lambda+(1-\mu)t_{U}\tau\left[-\frac{\mu t_{I}\tau_{\varepsilon}\tau{}_{u}}{\frac{1}{\tau}-\frac{1}{\tau_{\varepsilon}}}+\lambda\right]=1\]\[ \left(-\mu t_{I}\tau_{\varepsilon}-(1-\mu)t_{U}\tau\right)a+(1-\mu)t_{U}\tau\bar{\theta}\left(\frac{\tau{}_{s}}{\frac{1}{\tau}-\frac{1}{\tau_{\varepsilon}}}\right)=0\]
It is interesting that \(w=s-\frac{1}{\mu t_{I}\tau_{\varepsilon}}u\) is the “noisy” information conveyed by \(p\) on \(s\). Indeed :
\[\mathbb{E}\left(w/s\right)=s \]and \[\textrm{var}\left(w/s\right)=\frac{1}{\mu t_{I}\tau_{\varepsilon}\tau_{u}}\]

Informed perfectly informed .Obviously, as in the naïve equilibrium, when \(\tau_{\varepsilon}\) is infinite (Informed are perfectly informed) then the price gives perfect information on \(s\) (and \(p=s\)). •

No noisy traders. More interestingly, if there are no “noisy” traders \(\sigma_{u}=0\), then the price gives also perfect information on s. then \[ \textrm{var}\left(\widetilde{\theta}/J_{U}\right)=\sigma_{\varepsilon}^{2}=\textrm{var}\left(\widetilde{\theta}/J_{I}\right)\]. This gives also \(p=s\)! Indeed market clearing gives : \[\mu t_{I}\tau{}_{\varepsilon}(s-p)+\left(1-\mu\right)t_{U}\tau{}_{\varepsilon}\left[\bar{\theta}+\frac{\left(p-a-b\bar{\theta}\right)}{b}-p\right]=0\]which implies \(b=1\) and \(a=0 \) as soon as \(\mu>0\). •

No insider . What happens when \(\mu=0\] (no insider)? In that case we have : \[\lambda=\frac{\sigma_{\theta}^{2}}{t_{U}} a=\bar{\theta}\] Which means : \[p=\bar{\theta}-\frac{\sigma_{\theta}^{2}}{t_{U}}u\]

To sum up :

Proposition

In the Rational Expectation Equilibrium, if \(\sigma_{u}=0\) (no noisy traders), and \(\mu>0\) then the equilibrium price is \(p=s\). That means that the market is strongly efficient : the price reflects all public and private information. When \(\mu=0\) the equilibrium price is \(\bar{\theta}-\frac{\sigma_{\theta}^{2}}{t_{U}}u\), which gives obviously \(p=\bar{\theta}\) when there are no noisy traders.

We can formulate a final remark :

Remark

Suppose that prior to trading people decide weather to acquire or not the signal \(s\) at a fix cost \(k\), and that there are no noisy traders. If \(\mu=0\) and if \(k\) is not too large, it is interesting to buy information : it can be easily shown indeed that the expected utility achieved when informed is larger than the one when everybody is non informed. But as soon as \(\mu>0\) , the price becomes fully informative and it is not worth while to buy information since this information becomes free through price!