Coordination, incomplete information and crises.

The contagion argument, currency crises, bubbles and

crashes.

The contagion argument.

Incomplete information provides the Graal ?

A normal form game :complete information.

The game Carlsson-Van Damme (1993)

If is known for sure. >1 : a1, a2, dominant, <0 : b1,b2, dominant. Intermediate :

2 Nash equilibria in pure strategies, 1 in mixed.

Note : >1/2 If 2 plays a2 with probability ½ at least, 1’s BR is a1 If <1/2 : replace a by b., 1 by 2

, , -1, 0-1, 0

0 , 0 , -1-1 0 , 0

a1

b1

a2 b2

Incomplete information : a reminder. Assume s(i)=+ev(i),

e scaling factor, v(i) noise e going to zero : almost complete information

However : Strategies have to be defined as functions of s(i), not

. The « eductive » anchor to the argument :

For s(i) >>1, (>1), then (a,a) is played For s(i)<<0, then (b,b) is played.

The contagion argument. If you believe that your opponent plays a for s ≥s’>

1/2.. You play a for s ≥s’- (because when you receive s’,

you believe that the probability of your opponent receiving more than s’ is greater than ½).

Currency crises

Predictable or unpredictable ?

The Model.

Currency crisis « fundamentals » :

[m-, M+], Optimal defense

strategy for the « Centre » :

If m <m- , devalue uniformly,

If m-≤m ≤M+, fight if the proportion of speculators attacking is smaller than a(m), a increasing.

The Govt’s optimal reaction.

m- M+

a(m)

Crisis and multiple self-fulfilling beliefs. Currency crisis

« fundamentals » : [m-, M+], Optimal defense strategy for the « Centre » : a<a(m) If attack, individual cost t If m ≥ M+, no attack. If attack and success : gain increasing with the

« overvaluation ». Consequence : perfect observability of the

fundamentals Outside [m-, M+], unique equilibrium Inside two equilibria : everybody attacks, nobody

attacks … Multiplicity, sunspot ?

Crisis and incomplete information.

Assumption : signal s(i) = m+e, (e uniform, support [–d,+d]) m is not CK.

Strategies Strategies are not actions 0-1, but actions/signal. Trigger strategies.

Consequence : There is a unique equilibrium in trigger strategy : s≤s*, attack, s >s*, do not attack. It is the unique SR, (ES, DS) equilibrium.

s*

Best response and equilibrium.

Induction argument: Evbd obs s < s’ attacks BR : trigger strategy S(s’)

Best response : 1, attacks, 2,3 Expected prop./

attack.people, expect. success and expect. gain decr. …when s increases.

S(s’) : BR trigger strategy Properties.

S(s’) increasing. dS/ds’<1

Signal realisation :x, What is the probability that

less than z of my opponents has received a higher signal =z

s’

1 2 3 4

m- -2d

Improving understanding

Uniqueness Clear predictions

Particular case : Dirac : m* / probabilty (not ½)

a(m*) of no devaluation makes speculators indifferent.**

(a(m*))G(m*)=t Policy relevant

conclusions.

Connection with CVD ? Similar ingredients,

signals, form of strategies, but also strategic complementarities

Remaining mysteries

Trigger strategy s

S’

Prop/Attack.

m

Next….

Modelling options : Simpler on the gain if devaluation, …1 On the defense a<m, defend More complex on the information on m

N(s,1/a). And more precise on the noise N(0,1/b).

Results : b 2 >a 2 /2 Uniqueness in monotone

strategies b large, a fixed, treshold 1-c

The Stock Market

Facts and theories.

Shares and values : Introduction.

Shares : Some words on history.

Risk Sharing. Liquidity.

Ownership rights / firms. Residual claimant, but limited liability. Formal control outside bankruptcy. Tradable.

The stock market and the firm. Financing (Modigliani Miller) Governance…

Questions……

The curious outsider’s view : dynamics /stock prices Random walk (Bachelier…) ? « Fat tail » (Mandelbrot) ?

The investors’ eyes : how to make money ? Can you predict future stock prices ? Can you beat the market ?

Bachelier no. Modern version : is the market efficient ?

The economist’s questions : The same i.e

What determines prices and their dynamics.. Plus financing the firm and governance..

Theoretical tools…

Fundamental value, Riskiness,

Market « efficiency ».

Stock valuation : the standard theory…

Two polar models for valuation : No dividend :

value = selling asset value. Fixed or random selling date, liquidation ?

Standard theory : Shares give rigth to get dividents. A share identified to an infinite sequence of dividends. :

d(t). Intermediate theories.

Choice : self finance, distribute dividends. Endogenous, logic of dividends, profitability of reinvested

funds. . Option :

Standard theory … And elementary …..

The fundamental value-1

Setting : Certain dividend, Common point expectations on next period price, Safe interest rate r

The basic connection. p(t) = {1/(1+r)}{pe(t+1/t) + d(t+1)} The asset price depends on its price to-morrow, etc… True with uncertainty : p(t) = {1/(1+r)}{E(pe(t+1) + E(d(t+1)}

The dynamics with common point expectations pe(t+1/t) ) = {1/(1+r)}{pe(t+2/t+1) + d(t+2)} …

p(t) = t+S

t+1 {1/(1+r}T-t {d(T)} + {1/(1+r)}t+S pe

(t+S/t+S+1)+d(t+S)}. Si for S large, pe (t+S) grows more slowly than (1+r)S, the 2d term tends

to zero p(t) = +∞

T=t+1 {1/(1+r}T-t {d(T)} , is the fundamental value. Partial equilibrium, common expectations…

Remark on the fundamental value : It is the perfect foresight equilibrium.

Assume p(t+1)=pe(t+1), Then, the above formula holds true. Also the rational expectations equilibrium..

But not the only one … Bubble solution : p(t)+Δ, p(t+1)+(1+r) Δ, …p(t+t’)+(1+r)t’Δ, is also a solution.

It is a locally SREE . (« eductively stable ») It is CK that p (t+S) and d(t+S) grow less quickly than (1+r)S :

p(t+S) I, I/ (1+r)S «small », for some S when d (t+S)<D, D/(1+r)S

Argument p(t+S-1) I/(1+r) +d(t+S)/(1+r)….. p(t+S-2) I/(1+r)2 +d(t+S-1)/(1+r) +d(t+S)/(1+r)2

It is almost CK that p(t) =valeur fondamentale.

Stochastic version. p (t+S) and d(t+S) grow < than (1+r)S , p(t+S) I

The fundamental value : other formulaS..

The kernel : p(t) = +∞

T=t+1 {1/(1+r}T-t {E(d(T))} Price equals the expectation of the fundamental

value. Illustrations. deterministic case.

Dividends grow at the rate g P(t) = d(0)(1+g)t/(r-g) =d(t)/(r-g). g=0

Comments. r increases, P decreases : intuition. If r=0,05, g=0,02, p =33 times the dividend, Si g=0,03, 50 times, si g=0,04, 100, si 0,01, 25 times.

Sensitivity to forecasts. Illustrations : stochastic dvidends iid

d(t)=d + ; zero mean, finite variance. p(t) = d/r, price constant.

The fundamental value : other formulaS..

The kernel : p(t) = +∞

T=t+1 {1/(1+r}T-t {E(d(T))} Price equals the expectation of the fundamental

value. Illustrations : stochastic case b :

d(t)=d +a (d(t-1) - d ) + p(t) = d/r + a(d(t)- d)/(1+r- a), a =1, 0

Markov Chain : d takes two values, h, b=0, Markov transition, c probab. Change. « ergodic » prob. : P

=(1/2, 1/2) P(t) two values Markov chain Stationary. support between 0 et (h/r), fluctuates like dividends.

Complexification.

More complex processes. ARMA, etc… May depend on the past.

Asymetric information ? « Smart et noisy traders », ..(Campbell, Shiller..) Smart infinitely lived Dividends sum of Brownian and AR1, P(t) = VF(t) –h/(r-g) + y(t).

Common property : Price fluctuate less than reconstituted fundamental

value !! v(t) = T=t+1 {1/(1+r}T {d(T))} P(t) = E(v(t)), v(t)=p(t)+e(t), Var(p)<Var(v), ergodicity

Illustrations.

prix

4

Prices.-

cas1

t

prix

Cas 2

Cas 3Cas 4

Risky assets

Valuation of a risky asset. q(j)/q(0) = [1/(1+r)][s A(j,s) P(s)], P(s) = (s)/ (s) is the « risk-neutral probability ».. Price = expectations of the discounted value of incomes

with corrected probabilities. (probability marginal utility of income).

Hence take corrected probability of the fundamental value…

Relative valuation of assets : (CAPM) [(ER(j) – (1+r)] =

[E(R(*)-(1+r)][Cov[(R(*),R(j)]/[Var (R(*)] * is the market portfolio. …

A stock return. q= E(A)/(1+r) – [Cov(A,c(m)]/ [(1+r)T(.)].

Market efficiency and asymetric information.

The theory of fundamental value and its predictions. The evolution of prices Les prix et leur

évolution: 1-A: 2 firms with the same flow of dividends have equal

value. 1-C : The risk premium is reasonable….. 1-D : Statistical evolution : prices vary less than the

reconstitued fundamental value.. 1-E : No bubble.

Prix, information et stratégies des acteurs. 2-A : No information in to-day prices ? 2-B : You cannot beat the market (ovm) 2-C : No (public) information to beat the market. 2-E : Crash : a lot of information. .

The stock market and the firm. 3-A : appropriate valuation. 3-B : good Signal for investment. 3-B « Discipline »