j=0 s t t+1 + q t are vectors of length equal to the number of assets (c t+1 ) q t +1 + d i t+1 (1) (c t+1 ) R t+1 1= E t β u0 (c t+1 ) R u 0 (c t )
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1 1 Aet Prce: overvew Euler equaton C-CAPM equty premum puzzle and rk free rate puzzle Law of One Prce / No Arbtrage Hanen-Jagannathan bound reoluton of equty premum puzzle Euler equaton agent problem X X max β t u c t t Pr t j=0 t c t t + q a t a t+1 t W t t t W t+1 t+1 = y t+1 t+1 + q t a +1 t+1 + d t+1 t+1 a t+1 t a comment: a t and qt Euler equaton are vector of length equal to the number of aet u 0 a (c t ) q t = βe t u 0 a (c t+1 ) q t +1 + d t+1 (1) u 0 (c t )= βe t u 0 (c t+1 ) R t+1 1= E t β u0 (c t+1 ) R () u 0 t+1 (c t ) tranveralty condton a lm β j E 0 u 0 (c t+j ) q t+j a t+j =0 j prcng formula repeated ubttuton of (1) X u 0 a (c t+j ) q t = β j E t d t+j (3) u 0 (c t ) j=1 no bubble 1
2 tranveralty and t =1 complete market contency check revew A-D prce wth complete market (3) q t+j, = β c ( t, j ) Pr u 0 (c t ( t )) t t j u 0 t+1 j t 3 CCAPM (Conumpton Captal Aet Prcng Model) make ()and (3)operatonal: CCAPM ue aggregate conumpton n above equaton jutfcaton: equlbrum of repreentatve agent economy (Luca / Breeden) equlbrum wth complete market (Contantnde) complete market Pareto Optma repreentatve conumer (weghted utlty) back to Euler equaton u 0 (c t+1 ) R 1=E t β t+1 u 0 (c t ) Abence of arbtrage mple that there ext ome m t+1 uch that 1=E t mt+1 R t+1 THE emprcally tetable condton (agan) ntutve decompoton t the covarance that matter! 1=βE t µ u 0 (c t+1 ) E t Rt +1 + βcovt µ u 0 (c t+1 ),Rt +1 u 0 (c t ) u 0 (c t ) 4 Equty Premum Puzzle Euler equaton wth data on R tock market and R bond
3 mple log-normal calculaton preference and conumpton u 0 (c) = c γ ½ ¾ c t+1 1 = c exp ε c σ c c t ε c v N μc,σ µ c E c t+1 c t = μ c return ½ ¾ 1 R = 1+ r exp ε σ ε v N μ c,σ c E R = R =1+ r Euler takng log... tock and bond: " µ # γ R c t+1 1 = βe c t µ = β 1+ r ( c ) γ E t exp ε σ γε c + γ σ c µ 1 = β 1+ r ( c ) γ E t exp (1 + γ) γ 1 σ c γσ c 1 log 1+ r = log β + γ log c (1 + γ) γ σ c + γσ c 1 r f log 1+ r f = log β + γ log c (1 + γ) γ σ (4) c 1 r log (1 + r )= log β + γ log c (1 + γ) γ σ c + γσ c (5) premum: r r f log (1 + r ) log 1+ r f = γσ c (6) 3
4 Table removed due to copyrght retrcton. Kocherlakota, Narayana R. "The Equty Premum Puzzle: It' Stll a Puzzle." Journal of Economc Lterature 34, no. 1 (1996): 47 (Table 1). US data (from Mehra and Precott): r = 7% r f = 1% σ rc =.19% Kocherlakota need γ = 7 to match (6) equty premum puzzle to match (4) we need γ very hgh or very low rk free rate puzzle 4
5 Table removed due to copyrght retrcton. Kocherlakota, Narayana R. "The Equty Premum Puzzle: It' Stll a Puzzle." Journal of Economc Lterature 34, no. 1 (1996): (Table and 3). 5 Dcount Factor: LOP and NA I follow Cochrane and Hanen (199) cloely great paper to read two perod "now" and "then" (t and t +1f you prefer) J fundamental aet: x j payoff then q j now prce tack nto x and q (column) vector payoff pace for "then" P {p : p = c x for ome c R} prcng functon π (p) :P R then π (x) =q 5
6 defnton: Law of One Prce (LOP) hold f the prcng functon lnear c x = c 0 x then c q = c 0 q 1 π (c x) =c π (x) =c q defnton: dcount factor y P π (p) =E (yp) Rez repreentaton Theorem LOP (tochatc) dcount factor y P Let Y be the et of all dcount factor note: y may be negatve example: y = x 0 (Exx 0 ) 1 q note: f Exx 0 non-ngular then remove aet from x untl t! a non-ngular Exx 0 mean that (a) there a rk-free aet (b) there are two way of gettng the ame payoff Defnton: No Arbtrage (NA) hold p 0 π (p) 0 p > 0 (wth potve prob.) π (p) > 0 reult NA trctly potve dcount factor y > 0 Let Y ++ be the et of all dcount factor that are potve example 1 proof: m = β t u 0 (c then ) u 0 (c now ) π (c x) = π (c 0 x) cπ (x) = c 0 π (x) cq = c 0 q 6
7 6 Hanen-Jagannathan bound all theore: q = E (mp) m = f (data, parameter) (ee Cochrane book) note p /q rate of return H-J bound: dagnotc tool for model of m pecal cae: data on a ngle exce return relatve to ome baelne aet then π (r) =0o that ntuton: need volatle σ m r = p/q p 0 /q 0 0 = Emr = EmEr + cov (m, r) = EmEr + σ m σ r corr (m, r) EmEr 1 σm σ r = corr (m, r) 1 EmEr 1 σ m σ r σ r σ m Er Em note: Em = 1/R f f therea rk free rate R f let generalze: for any random vector x we can conder the populaton regreon: m = a + x 0 b + e whch jut defne e unquely a havng Ee = 0and cov(x, e) =0 by defnton cov (e, x) =0 var (m) var (x 0 b) 7
8 dea compute x 0 b and var (x 0 b) to get lower bound check whether theore for y pa th tet b = [cov (x, x)] 1 cov (x, y) a = Ey Ex 0 b How to compute b? dea: f x = p then theory help... aume x = p note that o: cov (x, y) = q E (y) E (x) b = [cov (x, x)] 1 [q E (y) E (x)] var x 0 [cov (x, x)] 1 [q E (y) E (x)] = var (x)[var (x)] E (y) E (x) f we knew E (y) we have a lower bound otherwe feable regon for par (E (y),var(y)) convenent no need to recompute lower bound for each theory help ee where the theory fal 3 cae: rk-le return E (y) pnned down and rky return one exce-return q = 0 Sharpe rato and market prce of rk (what we dd before!) general cae very flexble, ee CH paper fgure.1: exce.,.3,.4 from CH paper 8
9 7 Reoluton (?) 7.1 Exotc Preference Rk Averon v. IES (Wel / Epten-Zn) frt-order rk averon (Epten-Zn) habt pertence e.g. u(c t αc t t ) (Abel / Campbell-Cochrane) lo-averon 7. Heterogenou Agent Incomplete Market unnured doyncratc rk (Mankw / Contantnde-Duffe) borrowng contrant (Euler wth nequalty) (Luttmer / Heaton-Luca) contraned optma wth lmted commtment (Alvarez-Jermann) 7.3 Knghtan Uncertanty rk v. uncertanty fear of not undertandng return / uncertanty over probablty dtrbuton / dere for robut decon (Hanen and Sargent) 7.4 No rk premum! no rk premum to explan... htorcal returnon tockwereunexpected (McGratten-Precott) bond are money low return tock more rky than ample (low probablty of a crah) (ee Retz, Cochrane, Wetzman and Barro) 9
10 8 Concluon Rk premum puzzle great example of nterplay between theory and data no trong conenu on reoluton yet many new dea new model hould explore revt the welfare cot of BC (Alvarez and Jermann) 10
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