Consumption Based Asset Pricing

Transcription

1 Consumpton Based Asset Prcng Mchael Bar Aprl 25, 208 Contents Introducton 2 Model 2. Prcng rsk-free asset Prcng rsky assets Bubbles Introducton Tradtonal nance studes nancal markets and asset prces n solaton from other markets for goods and servces. Dynamc General Equlbrum theory used n macro economcs (Stochastc Neoclasscal Growth Model) however, s a framework that allows studyng all markets and all trades n goods, servces, and nancal assets jontly. These notes provde a short ntroducton nto the un ed theory of macroeconomcs and nance - macro nance. 2 Model The model we use s smlar to the stochastc NGM, wth nelastc labor supply. The representatve household earns ncome from wages, w t, and from = ; :::; n assets. Asset holdng at tme t s x t, where = ; 2; :::; n are ndces of assets. Asset prces are p t, and dvdends d t. The household s problem s: + s:t: p t x t+ = w t + = max f;x t+ g =;:::;n; ;;::: E 0 p t x t + = X t u ( ) d t x t = 8t San Francsco State Unversty, department of economcs.

2 The expectaton operator s there because we allow future ncome, prces and dvdends to be uncertan. The Lagrange functon s: ( #) X X L =E 0 t u ( ) t " + p t x t+ w t p t x t d t x t F.O.C. = [ ] : t u 0 ( ) t = 0 8t = ; 2; ::: [x t+ ] : t p t + E t t+ (p t+ + d t+ ) = 0 8 = ; :::; n; 8t = ; 2; ::: Combnng the condtons for ; + and x t+, gves the Euler Equaton: or = t u 0 ( ) p t + E t t+ u 0 (+ ) (p t+ + d t+ ) = 0 u 0 ( ) p t = E t [u 0 (+ ) (p t+ + d t+ )] The economc ntuton behnd the Euler Equaton should be famlar, as t s very smlar to the Euler Equaton n the Stochastc Neoclasscal Growth Model wth one asset - physcal captal. Investng one extra unt n asset entals gvng up p t unts of current consumpton. Thus, the left hand sde represents the utlty loss ("pan") from such nvestment. In the next perod, the return on the nvestment s p t+ + d t+ (prce of the asset + dvdends), whch s the gan n future consumpton. Multplcaton by margnal utlty, translates the gan n consumpton nto utlty gan. Therefore, the rght hand sde s the present value of expected future gan from nvestng one unt n asset. Rearrangng, gves the consumpton based asset prcng equaton: p t = E t u0 (+ ) u 0 ( ) (p t+ + d t+ ) () The term m t+ u0 (+ ) u 0 ( ) s called the stochastc dscount factor, and t s the margnal rate of substtuton between + and. The asset prcng formula () can be rewrtten n terms of (gross) return on asset : Pluggng R t+ nto (), gves: + r t+ = R t+ = p t+ + d t+ p t = E t (m t+ R t+ ) (2) where R t+ s the gross return (such as.05) on asset and r t s the net return (such as 0.05 or 5%) on asset. Equaton (2) s a very general asset prcng formula. Most theores of nancal asset prcng can be expressed n terms of ths formula. = 2

3 2. Prcng rsk-free asset Suppose that there exsts an asset wth net return r f t+ (real nterest rate), whch s guaranteed wth 00% certanty. Usng the asset prcng formula (2) gves: h = E t m t+ + r f t+ = E t (m t+ ) + r f t+ + r f t+ = We have the followng results: E t (m t+ ) = (3) E u 0 (+ ) t. Hgher means that people are more patent (put more value on future consumpton), and they are wllng to accept lower nterest rate when they trade current for future consumpton. 2. Real nterest rates are hgh when consumpton growth s hgh. To see ths, recall that margnal utlty s dmnshng, and wrte u 0 (+ ) u 0 ( ) u 0 () = u0 ( ( + g ct+ )) u 0 ( ) The hgher s the growth of consumpton, g ct+ = +, the lower s the above rato, leadng to hgher r f t+ n equaton (3). Intutvely, wth hgher nterest rates, current consumpton becomes more expensve relatve to future consumpton, and consumers want to lower consumpton today and nvest more (consume more n the future). 3. Usng rst order (lnear) Taylor expanson of u 0 (+ ) around gves Pluggng ths nto equaton (3) + r f t+ + r f t+ u 0 (+ ) u 0 ( ) + u 00 ( ) (+ ) h = h E u 0 ()+u 00 ()(+ ) t u 0 () E t + u00 () c u 0 () t E t [ RRA g ct+ ] = [ RRA E t (g ct+ )] ct+ where RRA s the Arrow-Pratt coe cent of relatve rsk averson. Once agan, we see that faster expected growth rate of consumpton s assocated wth hgher nterest rates. However, the real nterest rates are more senstve to consumpton growth when rsk averson, RRA, s hgher. Intutvely, wth hgher rsk averson, consumers want smoother consumpton path, and t takes larger nterest rate to compensate them for a gven consumpton change. In a specal case of rsk neutralty, RRA = 0, we have + r f t+ = + Thus, real nterest rate s equal to the utlty dscount factor, and do not depend on consumpton growth. 3 (4)

4 Example Suppse that u (c) = ln (c), the dscount factor s = 0:97, and expected future growth rate of consumpton s E t (g ct+ ) = 2%. Usng the approxmate rsk-free asset prcng formula (4), what should be the rsk-free real nterest rate? Soluton 2 Recall that the relatve rsk averson for logarthmc preferences s. Thus, + r f t+ r f t+ 5:2% 0:97 [ 2%] = 0:97 0:98 :052 Wthout rsk averson, we would have RRA = 0, and 2.2 Prcng rsky assets r f t+ 3:% Recall that from the de nton of covarance between two random varable X and Y, t follows: Cov (X; Y ) = E (XY ) E (X) E (Y ) Applyng ths to asset prcng formula (2) and manpulatng: = E t [m t+ ( + r t+ )] = E t (m t+ ) E t ( + r t+ ) + Cov [m t+ ; ( + r t+ )] = E t ( + r t+ ) + Cov [m t+; ( + r t+ )] E t (m t+ ) E t (m t+ ) + r f t+ = E t ( + r t+ ) + Cov [m t+; ( + r t+ )] E t (r t+ ) = r f t+ E t (r t+ ) = r f t+ E t (m t+ ) ; ( + r u 0 () t+) h E t u0 (+ ) h Cov u0 (+ ) u 0 () Cov [u 0 (+ ) ; ( + r t+ )] E t (u 0 (+ )) (5) The rst term n (5) s the rsk free return, and the second term n (5) s rsk adjustment or rsk premum. Thus, f an asset return s uncorrelated wth consumpton, ts expected return s equal to the rsk free return, and there s no premum. Snce u 0 (c) s dmnshng, asset returns that are postvely correlated wth consumpton are negatvely correlated wth margnal utlty. Thus, assets that are postvely correlated wth consumpton, and therefore Cov (u 0 (+ ) ( + r t+ )) < 0, must promse hgher expected return (because these assets make consumpton more volatle, or ncrease rsk). On the other hand, assets that are negatvely correlated wth consumpton, and therefore Cov (u 0 (+ ) ( + r t+ )) > 0, can o er expected returns that are lower than the rsk free return (because they help smooth consumpton, or reduce rsk - serve as nsurance). 4

5 2.3 Bubbles In ths secton we llustrate the possblty of speculatve prce bubbles. prcng equaton () and substtutng p t+ nto the formula for p t, gves Usng the asset p t = E t u0 (+ ) u 0 ( ) p t+ + u0 (+ ) u 0 ( ) d t+ 2 = E t 6 4 u0 (+ ) u 0 ( ) E t+ u0 (+2 ) u 0 (+ ) p t+2 + u0 (+2 ) u 0 (+ ) d t+2 + u0 (+ ) u 0 ( ) d t+7 5 p t+ = E t E t+ 2 u 0 (+2 ) u 0 ( ) p t u 0 (+2 ) u 0 ( ) d t+2 = E t 2 u 0 (+2 ) u 0 ( ) p t u 0 (+2 ) u 0 ( ) d t+2 + u0 (+ ) u 0 ( ) d t+ + u0 (+ ) u 0 ( ) d t+ The last step uses the law of terated expectatons E t [E t+ (X)] = E t (X). If we keep substtutng agan for p t+2 ; p t+3 ; ::: from the asset prcng equaton (), we obtan: " p t = lm E t s u 0 # (+s ) s! u 0 ( ) p X t+s + E t s t u 0 (c s ) u 0 (c s=t+ t ) d s b t f t The second term, f t, s the expected present dscounted value of future dvdends, where the dscount factors are ntertemporal margnal rates of substtuton. The second term therefore represents the fundamental part of the asset prce p t. The rst term, b t, s usually assumed to be zero, and ths assumpton s equvalent to absence of speculatve prce bubble. However, n realty, there are cases where asset prces do not appear to be valued accordng to ther fundamentals alone. Examples nclude the dot-com bubble n the 90s and the housng bubble of A bubble n asset exsts when the rst term b t 6= 0. Whle some researchers conclude that the presence of bubbles s evdence of rratonal behavoral on the part of nvestors, other researchers have developed theores n whch bubbles are consstent wth perfectly ratonal behavor. 3 References [] Lucas, Robert 978. "Asset Prces n an Exchange Economy," Econometrca 46,