Financial markets What goes wrong with existing theory ?

Financial markets The subject. Stock market, Other financial markets, Speculative markets Three anchors to existing theory. The fondamental value reference : prices tend to be attracted to fundamental values. stock markets. The efficient market hypothesis. Different forms : Markets transmit all (publicly) available information. You cannot beat the market. stock and financial markets. The social role of speculation. Speculation is stabilizing (Friedman..) More markets, more participants is better. To-day : emphasis on point 1.

The stock market What goes wrong with the fundamental value ?

The fundamental value-1 Setting : Common point expectations on next period price, Safe interest rate r The basic connection. p(t) = {1/(1+r)}{p e (t+1/t) + d(t+1)} The asset price depends on its price to-morrow, etc With uncertainty : p(t) = {1/(1+r)}{E(p e (t+1) + E(d(t+1)} The dynamics with common point expectations p e (t+1/t) ) = {1/(1+r)}{p e (t+2/t+1) + d(t+2)} Si for S large, p e (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

The theory of fundamental value and its predictions. The evolution of prices : 1-A: 2 firms with the same flow of dividends have equal value. 1-B : Statistical evolution : prices vary less than the reconstitued fundamental value.. 1-C : No bubble. The connections of prices and information: 1-D : Crash : a lot of information. 1-E : The risk premium is reasonable.. Connected problems, but treated separately.

1-A : Price Difference and equal fundamental value.

US : Volatility of returns.

Long period returns

1-C : (Equity Premium Puzzle, EPP) US (1889-1978) 3 variables Real return of stocks (SP 500) : 7% (R s ) Real return of safe bonds : 1% (R) Per capita consumption growth : 1,8% / year (c t+1 / c t ) Puzzle (Mehra-Prescott 1985) Incompatible with standard behaviour under risk. Risk aversion and time preference. Cov( R s, c t+1 / c t ) > 0, but small, justifies a small risk premium with iso-elastic utility and standard preferences.

1E: Unreasonable risk premium ? 1-C :The risk premium is reasonable US (1889-1978) 3 variables Real return of stocks (SP 500) : 7% (R s ) Real return of safe bonds : 1% (R s ) Per capita consumption growth : 1,8% / an (c t+1 / c t ) Cov(Rs, ct+1 / ct) > 0 Cov(Rs, ct+1 / ct) > 0

1-B : Prices and reconstituted fundamental values : Illustrations. prix 4 Prices.- cas1 t prix Cas 2 Cas 3 Cas 4

Excess volatility puzzle. The diagram observed prices : 1860 to-day. Fundamental values reconstituted. With several assumptions on the discount rate Or on future dividends. Prices vary more than reconstituted f undamental values.

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 fois, 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 predictability of returns ?

2-D : Objections to efficiency .

1-C No bubble. But not the only one Bubble solution : p(t)+, p(t+1)+(1+r) , p(t+t)+(1+r) t , 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)

History. Real Sphere. Real Sphere. , r.v zero mean. Period 0: fundamental=0 Period 1: Period 2: shock => + . Period 3: Random shock => + Information Information Positive feedback traders : know past p t-i. Passive Investors: know in period 2, in 3. Speculators: receive (a signal of ) at period 1 Agents actions. Agents actions. Passive Invest., Speculators. Mean-Var

A summary of history. Demands DateEvent Positive feedback.. Passive Speculators 0 Reference 00Optimal(0) 1 Speculators learn 0 - p 1 Optimal : 2 Passive learn (p 1 - p 0 ) -(p2-)-(p2-)-(p2-)-(p2-) -(p2-)-(p2-)-(p2-)-(p2-) 3 Liquidation : +

Equilibrium mechanics. In the absence of speculators (u = 0) In the absence of speculators (u = 0) At period 1, no news => p 1 = p 0 = 0 At period 2, D f =0, Passive investors : expect no trade from positive feedback, hence equilibrium is no trade and D e = 0, p 2 = Destabilisation (u > 0) : Destabilisation (u > 0) : Perfect forecast of p 2 by specul. p 1 = p 2 If p 1 > 0, D f 2 > 0 => p 2 > . Equilibrium. Equilibrium. Period 2 equilibrium : 0 = p 1 + ( - p 2 ) p 1 = p 2 : p 2 * = /( - ) p 1 = p 2 : p 2 * = /( - ) Then p 2 * > Then p 2 * > Destabilisation by rational agents.

Equilibrium : destabilisation

Herd behaviour.. Herd behaviour and 1- Followers 2- Bubbles.

Herd behaviour The logic : reminder.. The logic : reminder.. Two choices, A and B. Two observations, R (A), V(B), N (No observation. Sequential decisions : Agents, 1,2, n have to decide sequentially. Cascade : Cascade : 1 observe R, go to A. 2 observes 1.R, go to A. 2.N, go to A. 3.V, go to B. 3 observes. 1.Go to A, if 1 or 2, even if B 2.Go Then, Characteristics. Characteristics. Rational expectations equilibrium : it is rational to follow the crowd. Fragile : everybody understands that a long queue at A carries little information.

Herd behaviour and financial markets. A simple model : A simple model : (from Avery-Zemsky AER, (98)) Each informed agent receives a signal on the value of fundamental V. He buys, (sells) if his expected value greater,(smaller) than the price The price of the asset equals its expected value given the history of trades (market maker). Noise traders come into the sequence with probability 1-u and act randomly.. With standard signals, With standard signals, no herd behaviour. The price reflects all previous public information Wtih fixed price, back to previous story.. The price converges towards the fundamental value..

Herd behaviour and financial markets. A simple model with event uncertainty. A simple model with event uncertainty. V=[0, , 1] A single type of informed tradersignal x x= , iif V= P(x=1/1)= P(x=0/0)=p> . If p

Herd behaviour and financial markets. A simple model with event uncertainty. A simple model with event uncertainty. V=[0, , 1] Informed tradersignal x x= , iif V= P(x=1/1)= P(x=0/0)=p> . But two types of informed traders : H, L. p(H)=1, p(L) > . Proportion of H,L unknown : well W or poorly informed P market A price bubble : A price bubble : () close to 1, strong a priori for W. p(L) close to , no H trader in the poorly informed market 50/50 in the W market. The truth is (0,P)

Herd behaviour and financial markets. Period 1 : (0-50) Period 1 : (0-50) Price almost fixed. Herding (buy..). Period 2. Period 2. the market-maker believes that activity reflects good news (since the market is a priori well- informed) and that a poorly informed market generates herd behaviour that mimicks activity of a well-informed market. price rises. Period 3. Period 3. Fall in activity due to herding, The market-maker learns that the market is poorly informed. Hence drop in prices. Remarks on the limit of the model : no forward-looking behaviour Remarks on the limit of the model : no forward-looking behaviour

Bubbles Without followers and/or herd behaviour

About bubbles. The bubble problem : The bubble problem : Price greater than fundamental value. Definition of FV Ruled out in general equilibrium ? Sunspots ? The internet bubble: The internet bubble: Reminder : March 2000, Tulip mania (1630), South Sea bubble (1720.Newton !) funds managers between Charybde and Scylla. Irrational, to play or not. Error 1: JR, Tiger Hedge Fund : dissolved end 1999. Error 2 : SD, Quantum Fund : resignation 04-2000.

The model. The Background. The Background. Vague : Price p(t)=exp(gt). The bubble. The bubble. Price >FV from t(0), (1-b(t-t(0)))p(t), b increasing. t(0) random, Poisson (t(0))=1-exp(-t(0)) Bursts. Bursts. for sure at t(0) + , Bursts if cumulative selling pressure. t(0)+. M">

Individual decision to ride the bubble ? Calculations. Calculations. For fixed strategies of others : Endogenous bursting : t(0)+T* > t(0)+. Min (T*,) bursting bubble. Loss / s, One unit. b(s-T*)p(s) or b()p() Criteria : compare Criteria : compare h( /t i )(b(t-T * ) and (g-r)., t=t i + h instantaneous probability of crash or h(t/t i ) and (g-r)/(b()). Hint: h ( / t i )=( /(1-exp(-(- ))), = T*

Equilibrium. Equilibrium with trigger strategies Equilibrium with trigger strategies Trigger strategy : waiting time /t i. Type 1 Equilibrium : bubble bursts exogenously. Type 2 endogenously. Type 1 Equilibrium. Type 1 Equilibrium. Each informed agent sells possibly / a waiting period of t = (1/)Log((g-r)/g-r-b()) Proof and conditions. Proof and conditions. /(1-exp(-( - ))) =(g-r)/(b()). If for = - , lhs < rhs : /(1-exp(- ) )< (g-r)/(b()). t> - , the bubble does not burst. Comments Comments Opinion dispersion + intensity prevent bursting.

Equilibrium. Type 2 Equilibrium : bubble bursts under attack. Type 2 Equilibrium : bubble bursts under attack. Each informed agent sells (if possible) after waiting * *= b -1 {(g-r)(1-exp(-( ))/)}- Proof and conditions. Proof and conditions. If all have the trigger strategy , The bubble will burst at t(0)+ +, (t(0) + ) = + Equilibrium Condition : = = * /(1-exp(-( ))=(g-r)/ (b( +*) ). Comments. Comments.

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Financial markets What goes wrong with existing theory ?