8 Consumption-based asset pricing

Transcription

1 8 Consumpton-based asset pcng Pupose of lectue:. Exploe the asset-pcng mplcatons of the neoclasscal model 2. Undestand the pcng of nsuance and aggegate sk 3. Undestand the quanttatve lmtatons of the model 8. The fundamental asset pcng equaton Consde and economy whee people lve fo two peods... max t)+β u (c )} {c t,c,a } subject to y t = c t + p t a c = y +(p + d ) a, whee y t s ncome n peod t, p t s the (ex-dvdend) pce of the asset n peod t, andd t s the dvdend fom the asset n peod t. Substtute the two constants nto the utlty functon: max {u (y t p t a )+β u (y +(p + d ) a )} a and dffeentate w..t. how much of the asset to puchase, a : 0 = p t u (y t p t a )+β {u (y +(p + d ) a ) (p + d )} p t u = β {u (c ) (p + d )}. Intepetaton: the left-hand sde s the magnal cost of puchasng one addtonal unt of the asset, n utlty tems (so pce * magnal utlty), whle the ght-hand sde s the expected magnal gan n utlty tems (.e., next-peod magnal utlty tme pce+dvdend). Rewte to get the fundamental asset-pcng equaton: βu (c ) p t = u (p + d ), whee the tem βu (c ) u (c t) kenel ). s the (stochastc) dscount facto (o the pcng 46

2 Key nsght #: Suppose the household s not boowng constaned and suppose the household has the oppotunty to puchase an asset. Then ths equaton and the consumpton pocess fo ths household can be used to pce that asset. Ths apples to any asset and to the consumpton pocvess fo any household. What goes wong f the household s boowng constaned? The poblem s that f the houshold s constaned n peod t. Consde, fo example, a case when the household would lke to boow so as to ncease cuent consumpton, but s not allowed to do so. Then magnal utlty n peod t s vey hgh and the Eule equaton (fo a skless bond that pays one unt of consumpton fo sue next peod) becomes an nequalty: u > q t β {u (c )} Intuton: the household feels that the bond s vey expensve (.e., that the nteest ate s vey low), so t would lke to sell bonds (.e., boow fom the bank), but the bank does not allow the houshold to do so. Clealy, the consumpton steam of ths household cannot be used to pce the asset. 8.2 Whch assets ae expensve? Defne the etun of an asset as + = p + d p, t so the asset-pcng equaton can be ewtten as βu (c ) = u + Compae the pce of a sky asset wth that of a safe asset (.e., a bond wth a safe etun ): βu (c ) = u + βu (c ) βu (c ) = u + u βu (c ) βu (c ) = u ( + ) = u ( + ), whee the last equaton follows fom the fact that s skfee. Combne these equatons to obtan: βu (c ) u βu (c ) u βu (c ) = u = 0, (6) 47

3 whee the tem s the (stochastc) excess etun on the sky asset. Recall that the fomula fo covaance between two stochastc vaables X and Y s cov (X, Y ) = E {X Y } E (X) E (Y ) E {X Y } = E (X) E (Y )+cov (X, Y ) ultply equaton (6) by u /β on both sdes, and use the covaance fomula: 0 = u (c ) = cov u (c ), + Et {u (c )} = cov u (c ), + Et {u (c )} E t (7) The tem s the sk pemum,.e., the expected excess etun, o the expected etun on asset elatve to the etun on the skless bond. ake two key assumptons:. Suppose the economy ns a epesentatve-agent economy, so that each household s consumpton s a constant shae of the aggegate consumpton (c t = ) 2. Suppose (fo smplcty) that the utlty functon s quadatc,.e., that u (c) = c a c2 2 u (c) = ac Inset ths expesson fo u nto equaton (7): 0 = cov ( ac ), + Et u (C ) E t = a cov C, + Et u (C ) E t (8) Consde thee cases:. The maket potfolo: Suppose asset s the maket potfolo (.e., the whole stock maket). Note that both the stock maket and aggegate consumpton ncease when tmes ae good, so the etun on stocks s hghly coelated wth consumpton: cov C, > 0. 48

4 Equaton (8) then mples that u (C ) E t > 0, so the etun on the maket potfolo must be hghe than the safe etun (snce u (c) > 0): > Key message: thesk pemum on the maket potolo s postve because the aggegate stock maket s coelated wth aggegate consumpton. 2. An asset wth non-systematc sk: Suppose the asset s sky, but the etun s completely uncoelated wth aggegate consumpton: Equaton (8) then mples that cov C, =0. u (C ) E t =0, whch agan mples that ths asset has the same expected etun as the safe bond, even f t s sky: =, Key message: thee s zeo pemum fo holdng dosyncatc sk (.e., sk whch s uncoelated wth aggegate consumpton). 3. Insuance: Suppose the asset s sky, but the etun s negatvely coelated wth aggegate consumpton: cov C, < 0, so the etun s hgh peccely when consumpton s low. Ths s an example of an asset whch seves as nsuance. Equaton (8) then mples that u (C ) E t < 0 <, so ths assethas a lowe etun than the safe bond (.e., a negatve sk pemum). Key message: households ae wllng to pay apemum n ode to get nsuance. 49

5 8.3 Consumpton-based CAP Rewte equaton (8) as follows: = a cov C, [u (C )] so the expected pemum etun on the maket potfolo s = a cov C, [u. (C )] Rewte the expected etun on asset as = cov C, cov C, a cov C, [u + (C )] = cov C, cov C, + = co C, std (C ) std co C, std (C ) std + = std std co C, co C, + std BETA E t +, whee the consumpton BETA sdefned as BETA std co C, co C, Intepetaton: the tem BETA fo an asset eveals the sk pemum of addtonal sk of ths asset. 8.4 Equty pemum puzzle Go back to equaton (6), m =0, whee the stochastc dscount facto m s m = βu (c ) u. Usng the fomula fo the covaance, ewte ths as 0 = {m } + cov m, = {m } + co m, std (m ) std std (m ) = {m } Et std co m,, 50

6 Recall that fo the safe asset we have βu (c ) = + u = {m }, so we can ewte the equaton above as std (m )= co m, + std Note that co m, < 0 (snce aggegate consumpton and ae postvely coelated) and that by defnton, co m,. Theefoe, the smallest possble value of the tem s one: co(m,) co m,. Ths mples a (Hansen-Jaganathan) bound on the vaablty of m : std (m ) + std (9) The tem Et{ } s the Shape ato (afte the Nobel Laueate std() Wllam F. Shape),.e., the etun pe unt of sk. Data: The Shape ato s typcally aound 40% on an annual bass n developed countes, whle the annual safe nteest ate has been close to zeo on aveage. Thus, the standad devaton of the stochastc dscount facto should be aound 40% Suppose the utlty functon exhbts constant elatve sk aveson,.e., u (c) = c γ γ, so the dscount facto becomes γ C m = β, whch s close to on aveage (at least when the tme peod s shot). The standad devaton of m s theefoe appoxmately γ C C std log β = std log (β)+γlog C = γ std log 5

7 Usng U.S. data, the volatlty of consumpton gowth s about C std log 3%. Equaton (9) then mples that γ must be at least γ =40/3 3, whch s vey lage. Usng othe appoxmatons and bounds, t s staghtfowad to show that n ths model, the sk aveson must be at least 50 n ode to account fo a sk pemum of 6%, when the standad devaton of consumpton gowth s just 3% and the vaablty of the stock maket s std = 6%. Ths s the equty pemum puzzle. Note that estmates of the sk aveson (usng mco data) mples a sk aveson somewhee n the ange of γ [, 5]., whch s UCH lowe than 50. Fo example, wth a sk aveson of 25, a household who s offeed a 50/50 change of a gan o loss of 20% of lfetme consumpton, would pefe to athe take a 7% loss fo sue. 52