FINM6900 Fiace Theory How Is Asymmetric Iformatio Reflected i Asset Prices? February 3, 2012 Referece S. Grossma, O the Efficiecy of Competitive Stock Markets where Traders Have Diverse iformatio, Joural of Fiace, 31, 1976), 573-85. Accordig to the ratioal expectatios equilibrium REE) models, prices ifluece demad also through iformatio. A example of a REE model is a fully-revealig ratioal expectatios equilibrium model i which the price is a sufficiet statistic for all private iformatio. As log as the price is give, the idividuals are ot iterested i the private sigals of the expected payoff strog efficiet market hypothesis). I the Grossma paper a model with idepedetly ad idetically distributed private sigals is used to show that there exists a fully revealig ratioal expectatios equilibrium. I this model the price fuctio, coditioal o all private sigals, is a liear fuctio of the average sigal. This implies that the equilibrium is ot oly ratioal but also fully revealig because the price a sufficiet statistic for the set of private sigals. Thus, the price aggregates ad reveals all iformatio. This model does ot take ito accout costly iformatio. If iformatio is costly, there will be o icetive to collect it. Therefore, if o oe collects the iformatio, it will ot be revealed i the price. The fully revealig ature of the model oly depeds o oe piece of iformatio, amely the average sigal. Because of this ivestors are capable of ivertig the iformatio from the prices. O the cotrary, i the oisy ratioal expectatios model this is o loger the case. 1
Extesios: Aat R. Admati, A Noisy Ratioal Expectatios Equilibrium for Multi-Asset Securities Markets, Ecoometrica, Vol. 53, No. 3. May, 1985), pp. 629-658. supply is oisy, the price is o loger fully revealig, oisy ratioal expectatios, Albert S. Kyle, Cotiuous Auctios ad Isider Tradig, Ecoometrica, Vol. 53, No. 6. Nov., 1985), pp. 1315-1336. partially-revealig equilibrium, market maker, oise traders, isider, market maker s price schedule fuctio of aggregate demad), isider s optimal demad, Kyle lambda market depth), secod source of ucertaity amout of oise tradig), Set-up: 1. W 0i - iitial wealth of trader i, 2. X F i - value of risk-free asset purchased i period 0, 3. X i - umber of uits of risky assets purchased i period 0, 4. P 0 - curret price of the risky asset, 5. r - rate of retur o the risk-free asset, 6. P1 - payoff per uit o the risky asset, 7. wealth i period 1 is give by W 1i = 1 + r)x F i + P 1 X i, 1) 8. budget costrait W 0i = X F i + P 0 X i, 2) 2
9. substitutig 2) ito 1) gives W 1i = 1 + r)w 0i P 0 X i ) + P 1 X [ i ] W 1i = 1 + r)w 0i + P1 P 0 1 + r) X i 3) At time 0, trader i observes sigal y i, where y i = P 1 + ɛ i 4) ad P 1 is a realisatio of the radom variable P 1. Thus, a fixed, but ukow, realisatio of P 1 mixes with oise, ɛ i, to produce the observed y. Let I i be the iformatio available to the ith trader. Trader i has a egative expoetial utility fuctio give by U i W 1i ) = e a i W 1i, a i > 0 5) where a i is the coefficiet of absolute risk aversio ad W 1i is assumed to be ormally distributed coditioal o I i. Each trader is assumed to maximise the expected value of U i W 1i ) coditioal o I i. The by the momet geeratig fuctio 1 of the ormal distributio we ca write E[U i W [ 1i ) I i ] = exp a i E[ W 1i I i ] a i 2 V ar[ W ]) 1i I i ] It follows that to maximise E[U i W 1i ) I i ] is equivalet to maximisig the followig problem: is 6) a i E[ W 1i I i ] a2 i 2 V ar[ W 1i I i ] 7) 1 I probability theory ad statistics, the momet-geeratig fuctio of a radom variable X M X t) = Ee tx ), t R wherever this expectatio exists. The momet-geeratig fuctio geerates the momets of the probability distributio, ad thus uiquely defies the distributio of the radom variable. For a ormal distributio, it ca be show that the momet geeratig fuctio is M X t) = exp µt + σ2 t 2 ). 2 Source: Wikipedia, The Free Ecyclopedia) 3
Note by equatio 3): E[ W 1i I i ] = 1 + r)w 0i + E[ P ) 1 I i ] 1 + r)p 0 X i 8) [{ V ar[ W } ] 1i I i ] = V ar 1 + r)w 0i + P1 P 0 1 + r) X i I i Hece trader i maximises = X 2 i V ar[ P 1 I i ] 9) a i 1 + r)w 0i + a i E[ P 1 I i ] 1 + r)p 0 ) X i a2 i 2 X2 i V ar[ P 1 I i ] 10) with respect to X i. The F.O.C. is: a i E[ P 1 I i ] 1 + r)p 0 ) 2 a2 i 2 X iv ar[ P 1 I i ] = 0 X d i = E[ P 1 I i ] 1 + r)p 0 a i V ar[ P 1 I i ] 11) where X d i deotes optimal X i. Thus, the demad for the risky asset depeds o its expected price, variace ad the iformatio the trader receives, is icreasig i the expected pay-off, is decreasig i the variace, is decreasig i the risk aversio a i.) Let X be the total stock of the risky asset. A equilibrium price i period 0 must have demad equal supply, i.e. N i=1 X d i = X. From 11), the ith trader s demad for the risky asset depeds o the iformatio s)he receives. This depeds o the observatio s)he gets, y i. Thus, sice the total demad for the risky asset depeds o y 1, y 2,..., y, it is atural to thik of market clearig price as depedig o the y i, i = 1, 2,...,. Let y y 1, y 2,..., y ), the the 4
equilibrium price is some fuctio of y, P 0 y). That is, differet iformatio about the retur o a asset leads to a differet equilibrium price of the asset. There are may fuctios of y. For a particular fuctio, P 0 y) to be a equilibrium we require that: for all y, i=1 [ ] E P1 yi, P0 y) a i V ar [ Pi yi, P 0 y) 1 + r)p0 y) ] = X 12) expected value of P 1 is coditioal o the iformatio set ad price i time 0 - everyoe observes the price, the total demad for the risky asset must equal the total supply, the ith trader s iformatio I i, is y i ad P 0 y) he is able to observe his ow sample y i ad some iformatio about the sample of other traders P 0 y), the ith trader s demad fuctio uder price system P 0 y) is X d i P 0, y i ) = [ ] E P1 yi, P0 y) a i V ar [ Pi yi, P 0 y) 1 + r)p0 ] 13) Sufficiet statistic Assume that ɛ i i 4) i N0, 1) i.e. all traders sigals are assumed to have equal precisio - same variace). The ɛ 1 i=1 ɛ i N 0, ) 1, fy i P 1 ) NP 1, 1). Further assume that ɛ 1, ɛ 2,..., ɛ is joitly ormally distributed with Covɛ i, ɛ j ) = 0 if i j. It follows that the joit desity of y give P 1, say fy P 1 ) is multivariate ormal, its mea vector is E[y P 1 ] = P 1, P 1,..., P 1 ), its covariace matrix Cov[y P 1 ] is a idetity matrix, fy P 1 ) NP 1, I), 5
traders believe that P 1 is distributed idepedetly of ɛ 1, ɛ 2,..., ɛ ad P 1 N P 1, σ 2 ). Fially, we also have P 1 N P1, σ 2). If we have the average iformatio, we do t eed ɛ i, because if the distributio is ormal, the mea ad variace are sufficiet statistics. Theorem 1. Uder the above assumptio about the joit distributio of y ad P 1, there exists a ratioal expectatios equilibrium that is a solutio to 12) with P0 y) give by P0 y) = α 0 + α 1 ȳ, 14) where ȳ i=1 y i 15) ad α 0 α 1 P 1 1 i=1 a i σ 2 X 1 + σ 2 )1 + r) 1 i=1 a i 16) σ 2 1 + σ 2 )1 + r) 17) 6
Commets ȳ is the sample mea of the y i. The equilibrium price depeds o the iformatio y oly through ȳ. The average sigal ȳ is a sufficiet statistic for the iformatio cotaied i all the other sigals. Ay trader by observig the value of P0 y) ca lear ȳ from 14). ȳ is a more precise estimate of P 1 tha is y i. The market price aggregates all the iformatio collected by the traders i a optimal way, to the extet that ȳ is a sufficiet statistic for the family of desities fy P 1 ). The above equilibrium is a fully-revealig ratioal expectatios equilibrium, i.e. the equilibrium price fully reveals all private iformatio. Lemma 1. If h i y i, ȳ P 1 ) is the joit desity of ȳ ad y i coditioal o P 1, the ȳ is a sufficiet statistic 2 for h i y i, ȳ P 1 ). Lemma 2. Let m P 1 ȳ) be the desity of P 1 coditioal o ȳ. Let ˆm P 1 ȳ, y i ) be the[ desity ] of [ P 1 coditioal ] o [ ȳ ad] y i. The[ m P 1 ȳ = ˆm P 1 ȳ, y i ) ad hece E P1 ȳ = E P1 ȳ, y i ad V ar P1 ȳ = V ar P1 ȳ, y i. Commets if trader is give iformatio about ȳ ad y i the ifereces about P 1 will be made idepedetly of y i, y i is extraeous iformatio if ȳ is kow 2 I statistics, oe ofte cosiders a family of probability distributios for a radom variable X, parameterized by some parameter θ. A quatity T X) that depeds o the observable) radom variable X but ot o the uobservable) parameter θ is called a statistic. A statistic that captures all of the iformatio i X that is relevat to the estimatio of θ is called a sufficiet statistic. Source: Wikipedia, The Free Ecyclopedia) 7
Proof of Theorem 1 Note that sice y i = P 1 + ɛ i ad P 1 N P1, σ 2), ɛ i N 0, 1), P 1 ad ɛ i are ucorrelated the E [y i ] = E P1 + ɛ i = P 1, V ar [y i ] = V ar P1 + ɛ i = σ 2 + 1, y i N P1, σ 2 + 1 ) ȳ N P1, σ 2 + 1 ), sice [ 1 E [ȳ] = E P1 + ɛ 1 +... + P 1 + ɛ [ 1 = E P 1 + ɛ 1 +... + ɛ = E P1 + E [ ɛ] = P 1 ad [ 1 V ar [ȳ] = V ar P1 + ɛ 1 +... + P 1 + ɛ [ 1 = V ar P 1 + ɛ 1 +... + ɛ = V ar P1 + V ar [ ɛ] = σ 2 + 1 ) P1 f P1, ȳ N P 1 ), σ 2 σ 2 σ 2 σ 2 + 1 )), 8
because Cov P1, ȳ = E[ P1 E[ P 1 ] ȳ E[ȳ]) = E[ P1 P ) ȳ 1 ] P1 = E[ P1 P ) i 1 ] P 1 + ɛ i ) = E[ P1 P ) 1 ] P 1 + i ɛ i P 1 P 1 = E[ P1 P ) 1 ] P1 + ɛ P 1 = E[ P1 P ) 1 ] P1 P 1 = V ar[ P 1 ] = σ 2 thus, by the well-kow formula for coditioal ormal, 3 ad E[ P 1 ȳ] = P 1 + σ 2 σ 2 + 1 ) 1 ȳ ) P1 [ = P 1 1 σ2 = σ 2 + 1 P 1 σ 2 + 1 + σ2 ȳ σ 2 + 1 ] + σ2 σ 2 + 1ȳ V ar[ P 1 ȳ] = σ 2 σ 2 σ 2 + 1 ) 1 σ 2 [ ] = σ 2 1 σ2 σ 2 + 1 σ 2 = σ 2 + 1 We assume P0 ȳ) = α 0 + α 1 ȳ. By Lemma 2, E P yi, α 0 + α 1 ȳ = E P yi, ȳ ) ) )) x 3 µx Σxx Σ If N, xy, the the distributio of x give y = y y µ y Σ yx Σ is ormal yy with mea µ x + Σ xy Σ 1 yy y µ y ) ad variace Σ xx Σ xy Σ 1 yy Σ yx. = 9
E P ȳ ad V ar P yi, α 0 + α 1 ȳ = V ar P yi, ȳ = V ar P ȳ From 12) we have X = = = i=1 E[ P 1 y i, P 0 ] 1 + r)p 0 a i V ar[ P i y i, P 0 ] P 1 σ 2 +1 + i=1 P 1 + σ 2 +1 σ2 σ 2 +1ȳ 1 + r)p 0 σ a 2 i σ 2 +1 σ2 σ 2 +1ȳ 1 + r)p 0 σ 2 σ 2 +1 P0 = 1 P 1 i a i Xσ 2 1 i a i 1 + r)1 + σ 2 ) + σ2 1 + σ 2 )1 + r)ȳ P0 = α 0 + α 1 ȳ Thus, if P 0 y) = α 0 + α 1 ȳ ad α 0 ad α 1 are as above give by equatios 16) ad 17)), the P 0 y) will be a equilibrium, i.e. it will be the price that clears the market. i=1 1 a i Discussio 1. I equilibrium the curret price summarises all the iformatio i the market. Each trader fids his ow y i redudat. This creates strog disicetive for ivestmet i iformatio, sice each trader could do as well by observig oly the spot price as he could if he also purchased y i. 2. Perfect competitio is assumed amog the traders, so each idividual trader of type i assumes that his tradig activity has o effect o P 0. Thus, whe oe type i trader stops gettig iformatio via y i, P 0 is ot affected ad ȳ ca still be deduced from P 0. 3. If it costs C > 0 dollars to become iformed the equilibrium will ot exist. Each trader of type i stops collectig iformatio because the iformatio i P 0 is superior to y i ad free. 4. Suppose there are oly a few iformed traders. How does it affect thigs? It does t. Let there be m types of iformed traders, the the price will be a liear fuctio of m j=1 y j ad thus trasmits all iformatio to uiformed 10
traders. Cosider ay give iformed trader of type m; he feels that he could stop payig C dollars ad though he would o loger get the iformatio y m, the price system reveals the superior iformatio, m j=1 y j. Hece, it is ot a equilibrium. 5. If o traders are iformed, the for sufficietly small cost of beig iformed) each trader would wat to become iformed because he gets o iformatio for free via the price system. Hece, with iformatio cost positive equilibrium does ot exist. No matter how may types of iformed traders there are, the price system perfectly aggregates their iformatio ad removes icetive from a trader of a particular type to become iformed. This is because traders are price takers ad assume the price system is ot affected by their actios. 6. The other paradox of the markets where prices are sufficiet statistics is that each trader s demad fuctio is a fuctio oly of the price ad ot his ow iformatio. If all traders igore their iformatio how does iformatio get ito the price? This poit is strogly related to the fact that the demad fuctio 13) is ot a ordiary demad fuctio. It gives the demads of traders i equilibrium. I models where the price coveys iformatio, there is o loger the classical separatio betwee demad fuctios ad equilibrium prices i.e. classically, demad fuctios ca be derived idepedetly of the distributio of equilibrium prices, ot the case here). 7. With ay fiite cost of obtaiig iformatio, the equilibrium would ot exist because each idividual receives o additioal beefit from kowig y i give they ca observe ȳ from the price. I other words, a give idividual does ot persoally beefit from havig private iside) iformatio i a fully-revealig equilibrium. I order for idividuals to beefit from obtaiig costly) iformatio, we eed a equilibrium where the price is oly partially revealig. For this to happe, there eeds to be oe or more additioal sources of ucertaity that add oise to idividuals sigals, so that agets caot ifer it perfectly - oisy ratioal expectatios. 11
Theorem 2. If P 0 y) is a equilibrium, ad P 0 y) = H P 0 y)), where H ) is a strictly mootoe fuctio which is ot the idetity mappig, the P 0 y) is ot a equilibrium. Proof is based o the fact that E P yi, P0 y) = E P yi, P0 y) ad V ar P yi, P0 y) = V ar P yi, P0 y) sice P 0 ad P 0 cotai the same iformatio. There caot be two equilibria with the same iformatio cotet, so if P 0 y) is a equilibrium, the P 0 y) is ot a equilibrium. 12