Financial Innovation and Asset Price Volatility. Online Technical Appendix

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1 Fnancal Innovaon and Asse Prce Volaly Onlne Techncal Aendx Felx Kubler and Karl Schmedders In hs echncal aendx we derve all numbered equaons dslayed n he aer Equaons For he wo models n he aer, he frs se of numbered equaons, = eh, = κm,h, c o,h = κ o,h wh h =,, follows already from Huffman 987 and Kubler and Schmedders 0 Here we derve hs resul agan n some more generaly for economes wh I long-lved secures, J shor-lved secures and S saes For he urose of hs dervaon only, we need o nroduce some addonal noaon I suffces o examne a sngle ndvdual s uly omzaon roblem n he OLG economy and so we om he ye suerscr h and he me subscr We need o exlcly refer o he deendence of he decson varables on he exogenous shocks and denoe consumon a mddle age by c m and consumon a old age by c o,s for, s S S The agen can nves n I dfferen socks, I = {,,, I}, and J dfferen one-erod secures, j J = {,,, J} We denoe he sock dvdends by d s and he ayoffs of he one-erod secures by fs j for s S The corresondng rces are denoed by s and qs, j resecvely We consder an ndvdual s uly omzaon roblem n he OLG economy The decson varables are he consumon choces, c y, c m for S, c o,s for, s S S, he orfolo choces of long-lved secures, φ y, for I and for I and S, and he osons of shor-lved secures θ y,j for j J and θs m,j for j J and S max lnc y π s lnc m = = s = π s,s lnc o,s c m q j c y φ y, s d φ y, c o,s s,s d m, q j θ y,j e = 0 a fs j θ y,j = 0 S b = 0, s S S c Kubler: De of Bankng and Fnance, Unversy of Zurch and Swss Fnance Insue, fkubler@gmalcom Schmedders: De of Busness Admnsraon, Unversy of Zurch and Swss Fnance Insue, karlschmedders@busnessuzhch We hank Johannes Brumm and Smon Scheurng for dealed commens on an earler draf

2 We denoe he Lagrange mullers for he budge consrans a, b, and c by λ, µ s for all S, and ν s,s for all, s S S, resecvely Then he necessary and suffcen frs-order condons for hs omzaon roblem are as follows λ S c y λ = 0 d π s c m µ s = 0 S e π s,s c o ν s,s,s = 0, s S S f µ s s d = 0 I g λq j µ s fs j = 0 j J h S µ s ν s,s s,s d s Mullyng g by φ y, and by λ φ y, µ s s d φ y, S = 0 I, S µ s qs j ν s,s fs j = 0 j J, S j for all I and hen summng over all I and S leads o S Smlarly, mullyng h by θ y,j and j by θs m,j leads o λ q j θ y,j µ s fs j θ y,j S µ s S ν s,s s,s d s φ m, = 0 for all j J and hen summng over all j J and S S µ s q j S ν s,s = 0 Addng he las wo equaons and relacng he Lagrange mullers λ, µ s, and ν s,s by he exressons from d, e, and f, resecvely, leads o c y φ y, q j θ y,j = π s c m s d φ y, fs j θ y,j qs j s S π s,s c o s,s d m, fs j,s S Combnng he las equaon wh he budge equaons b and c leads o c y φ y, q j θ y,j = π s π s,s = S S The budge equaons a for he frs erod now mly c y = e whch s he frs equaon n condon Mullyng by for all I and hen summng over all I leads o µ s ν s,s s,s d s φ m, = 0 S Smlarly for j, µ s q j ν s,s = 0 S

3 Addng he las wo equaons and relacng he Lagrange mullers µ s and ν s,s by he exressons from e and f, resecvely, leads o π s c m qs j = π s,s c o s,s d m, fs j s,s = and so, snce π s = s π S,s, follows ha c m = π s,s q j = s d φ y, fs j θ y,j whch s he second se of equaons n condon The budge equaons c are he hrd and las equaons n condon Ths comlees he roof of equaons In he remander of hs echncal aendx we emloy he same noaon as n he aer Equaons In a comeve equlbrum he marke-clearng condon for he consumon good requres c o,h = e d = e κm κ o h= Combnng hs condon wh he marke-clearng condon for he sock marke, κ m oban e d κm = e d Solvng for he sock rce hen yelds he exressons κ o = d, we = e κm = 4 e d κ o Equaon In he economy wh ncomlee markes, he sock s he only secury avalable for rade The sock holdngs of he young agens sasfy φ y, φ y, = e Combnng hs equaon wh exresson for he sock rce yelds he aggregae sock holdngs of he young agens, φ y, φ y, e = e κm The aggregaed cash-a-hand osons of he mddle-aged agens sasfy κ m = φ y, φ y, d Combnng he las wo equaons and relacng accordng o leads o κ m e = e κm e κm d Solvng for κ m yelds he rule for he evoluon of he aggregaed cash-a-hand osons of he mddle-aged agens, κ m = e d 4 κ m e

4 Equaon 4 In he economy wh comlee markes, he rces q, and q, of he wo Arrow secures sasfy he frs-order condons of all agens radng n erod The Euler equaons of he young agens sae π h s = q s, for h =, and s {, } Smlarly, he Euler equaons of he mddle-aged agens sae π h s = q s,c o,h Combnng he wo ses of Euler equaons leads o h= πs h πs h q s, = h= co,h Equaon mles c y = e and so aggregae consumon of he mddle-aged and old agens sasfes Wh κ m,h h= c o,h = e d = from equaon, we can exress he rces of he Arrow secures by h= π h s eh π h s κm,h q s, = e d Once agan usng he Euler equaons of he young agens we oban κ m,h = cm,h = πh s q s, Combnng he las wo equaons yelds he rule for he evoluon of he cash-a-hand osons of he mddle-aged agens of ye h n he comlee-markes economy, κ m,h = πs h e h 4 e d 4 πs e π e πs κ m, π κ m, Equaon 5 Subsung d = e no equaon leads o κ m = 4 κ m e e The defnon of he normalzed aggregaed cash-a-hand oson ˆκ m = κm e of he mddle-aged agens hen yelds ˆκ m = 5 4 ˆκm Equaon 6 Indvdual endowmens of he young agens sasfy e s = e s = es Agens of ye have belefs η, η and agens of ye hold he belefs η, η for he wo ossble exogenous saes wh η The Euler equaons of he young agens mly = η 6 e, c m, q = η 6 e and c m,, q = η 6 e, c m,, q = η 6 e, q, c m, 4

5 These equaons mly and hus c m, = κ κ m, = η η c m, m, and c m, = κ κ m, = η η c m, m, κ m, = ηκ m, κ m, = ηκ m and κ m, = ηκ m, κ m, = ηκ m Snce πs π = and e s = e s = es equaon 4 leads o κ m = κ m, κm, For s = s he second erm n he denomnaor s whle for s s hs erm s = e e e πs κ m, π κ m, πs κ m, π κ m, = η η κ m πs κ m, π κ m, = η η κm And so he evoluon of he normalzed aggregaed cash-a-hand oson of he mddle-aged agens n he comlee-markes economy sasfes ˆκ m = 4 Π ˆκ m 6 wh Π = { η η f s = s, 4η η f s s 5