The Stock Market Facts and theories.

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..

Facts :aggregate evolution of stock prices

A century’s evolution..

Long period returns : US

Long period returns…

US : Volatility of returns.

US : Volatility of returns, next..

The predictability of returns…… ?

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 fundmental 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 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 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

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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 »

1-A : Price Difference and equal fundamental value.

1C: Unreasonable risk premium ? 1-C :The risk premium

is reasonable US (1889-1978) 3 variables Real return of stocks

(SP 500) : 7% (Rs) Real return of safe

bonds : 1% (Rs) Per capita

consumption growth : 1,8% / an (ct+1 / ct) Si alpha < 9, « il » faut acheter des actions : equity

premium puzzle. Mais si beta = 0,99, alpha > 1, « il » faut désépargner: « risk free rate puzzle ».

0

1

sstst cE

Cov(Cov(RsRs, , ctct+1 / +1 / ctct) > 0) > 0

1-C : (Equity Premium Puzzle, EPP)

US 1889-1978) 3 variables

Real return of stocks (SP 500) : 7% (Rs) Real return of safe bonds : 1% (Rs) Per capita consumption growth :

1,8% / year (ct+1 / ct) Puzzle (Mehra-Prescott 1985)

Incompatible with standard behaviour under risk.

Risk aversion and time preference. Cov(Rs, ct+1 / ct) > 0, but small, justifies a small

risk premium with iso-elastic utility and standard preferences.

1D, 2C – Other predictions

1-D : Prices vary less than reconstitued fundamental values.

2-B : You cannot beat the market (ovm)

2-C : No public information to beat the market.

1-C : No bubble.

Tulip mania Internet bubble.

2-E : Crash : a lot of information. Doubts

Crisis , Law… 1929, Crash 1987 Change Crises.

2-D : Objections to « efficiency ».

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 fundamental values.

No bubble…

The tulip mania ; Feb. 1637 :

Bulb = 20 times a yearly craftsman income.

5 ha land. Facts..

Explanations. Law/ possibility of

cancelllation of future contracts…

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12-11 03-02

Others

Internet Bubble… 2001

Crashes. 1929. 1987 2008….

Aller plus loin ou abandonner le paradigme …..

La première option Garder l’hypothèse d’anticipations rationnelles Tester encore version savante étendue: horizon court, multiplicité. Introduire une dose limitée de « rationalité limitée ».

Rationalité non standard :neu Myopie, …

La seconde option : Rationalité limitée plus radicale au niveau individuel. Ou au niveau collectif : test de la plausibilité de l’équilibre….

Divinatoire : conséquence de CK du modèle et de la rationalité), Mise en cause de la « rationnalisabilité » de l’équilibre.

Évolutive; Première option : une galerie de phénomènes compatibles

Fluctuations « erratiques » des cours. Comportement « moutonnier » d’imitation… Krachs de « multiplicité ». Bulles en information asymétrique. Bulles avec agents « irrationnels ».