Erratic Price, Smooth Dividend. Variance Bounds. Present Value. Ex Post Rational Price. Standard and Poor s Composite Stock-Price Index

Transcription

1 Erraic Price, Smooh Dividend Shiller [1] argues ha he sock marke is inefficien: sock prices flucuae oo much. According o economic heory, he sock price should equal he presen value of expeced dividends. However dividends are very sable; hey flucuae very lile abou an upward rend. Expeced dividends should herefore also flucuae lile, and consequenly sock prices should be sable. Variance Bound Shiller shows how he variabiliy of he dividend ses an upper bound o he variabiliy of he sock price. Le p denoe he sock price a ime, and le d denoe he dividend a ime. Le us assume ha p and d follow a saionary Iô sochasic process. In fac, sock prices flucuae wildly. 1 2 Presen Value According o economic heory, he price is he presen value of expeced dividends, p = The equivalen rae-of-reurn condiion is E (d τ )e r(τ ) dτ. (1) d d+ E (dp )=rp d, he dividend plus he capial gain equals he marke ineres rae imes he price. Ex Pos Raional Price Define he ex pos raional price p as he presen value of acual dividends, p := d τ e r(τ ) dτ. (2) Economic heory disinguishes beween ex pos meaning afer and ex ane meaning before. The ex ane (raional) price is jus he acual price, based on he presen value of expeced dividends. 3 4 Sandard and Poor s Composie Sock-Price Index Figure 1: Sandard and Poor s Composie In figure 1 [1, p. 422], he price is he Sandard and Poor s Composie Sock-Price Index for , derended by an exponenial growh facor. The ex pos raional price is calculaed from he dividend. Whereas he ex pos raional price is sable, he acual price is erraic. 5 6

2 Forecas By (1) and (2), he price is he forecas of he ex pos raional price, We wrie p = E(p ). p = p + e, (3) Variance Decomposiion If he forecas (3) is raional, hen p and e mus be uncorrelaed. Consequenly Var(p )=Var(p )+Var(e ). (4) in which e denoes he forecas error. 7 8 Forecas Variance The inequaliy Variance Bound Var(p ) Var(p ) is a variance bound. The variance of he ex pos raional price is an upper limi (upper bound) on he variance of he price. We calculae he difference beween he wo variances, by finding he variance of he forecas error. The price follows he sochasic differenial equaion dp =(rp d )d+ σ dz, in accord wih he rae-of-reurn condiion. The ex pos raional price saisfies dp =(rp d )d Since Moving Average de = dp dp = re d σ dz, herefore e = e r(τ ) σ dz τ. (5) The forecas error depends on he fuure unexpeced changes in he price. The forecas error is a moving average of hese unexpeced changes and exhibis considerable auocorrelaion. This auocorrelaion is eviden in figure 1. Forecas Error Variance For uncorrelaed random variables, he variance of he sum is he sum of he variances. Analogously, he moving average (5) expresses he forecas error as an inegral of uncorrelaed erms. Hence he variance of he forecas error formula is he inegral of he variance of hese erms, Var(e )= e 2r σ 2 d = σ 2 0 2r

3 Variance-Bound Tes Unforunaely he variance bound is violaed. Figure 1shows ha for derended daa he variance of he price is much larger han he variance of he ex pos raional price. Hence he sock marke is inefficien. Dividend Yield This inefficiency means ha one can forecas he rae-of-reurn on socks from he dividend yield (he dividend/price raio). A profiable rading rule is o buy when he dividend yield is high (because he price is hen oo low) and o sell when he dividend yield is low (because he price is hen oo high) Dividend Variance Bound The following proves he heorem: The variance of he dividend ses an upper bound o he insananeous variance of he price. Theorem 1 (Dividend Variance Bound) Var(d ) 2rσ 2. (6) Shiller finds ha his inequaliy is also violaed, by a large margin: alhough he dividend is sable, he shor-run flucuaion of he price is large. Var(d )=Var[rp E (dp )/d] 4Cov[rp,E (dp )/d] = 2rσ 2. The iniial equaliy is he rae-of-reurn condiion. The inequaliy is an insance of he general relaion Var(x y) 4Cov(x, y) (wih equaliy if and only if Var(x+y)=0). The final equaliy is lemma 2 below, a general propery of a saionary process The dividend variance bound (6) holds wih equaliy if and only if E (dp )= r[p E(p )]d. (7) he coninuous-ime analogue of a firs-order auoregression. Equivalenly, E (dd )= r[d E(d )]d, (8) for he dividend. We show his equivalence below. Subsiue E(p )=E(d )/r ino he rae-of-reurn condiion: E (dp )=(rp d )d ={rp d + r[e(d )/r E(p )]}d ={r[p E(p )] [d E(d )]}d. Combining he laer wih (7) yields p E(p )= 1 2r [d E(d )]

4 Taking he sochasic differenial obains E (dd )=2r E (dp ) = 2r{ r[p E(p )]d} { } 1 = 2r 2 2r [d E(d )] d = r[d E(d )]d, which proves (8). Undersaemen of he Uncerainy A criicism is ha he price and he dividend are no saionary, bu end o rise as ime passes. The uncerainy abou his rae of increase is an imporan par of he uncerainy. When Shiller derends he price and he dividend, implicily he assumes ha he long-run rae of increase is known. A change in he expeced long-run rae of increase can cause a grea change in he price. Consequenly he derending undersaes he uncerainy abou he price and he dividend Saionary Sochasic Process Suppose ha a saionary random variable x follows he sochasic differenial equaion dx = µ d+ σ dz, in which z is Wiener-Brownian moion. Equivalenly, x +d x = µ d+ σ dz. Taking he uncondiional expeced value gives 0=E(dx ) E(µ )d E(σ dz ) = 0 E(µ )d 0, so he mean E(µ )= Covariance of Level and Mean Change Taking he uncondiional variance gives Var(x +d )=Var(x + µ d+ σ dz ) = Var(x )+Var(µ )(d) 2 + Var(σ dz ) + 2Cov(x, µ ) d+ 2Cov(x,σ dz ) + 2Cov(µ d,σ dz ) = Var(x )+σ 2 d+ 2Cov(x, µ ) d. Lemma 2 (Covariance of Level and Mean Change) Cov(x, µ )= 1 2 σ 2. The covariance of he level and he mean change mus be negaive. By saionariy, Var(x +d )=Var(x ), which yields he following lemma

5 References [1] Rober J. Shiller. Do sock prices move oo much o be jusified by subsequen changes in dividends? American Economic Review, 71: , June , 5 25