Technical Appendix. Political Uncertainty and Risk Premia

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1 echnical Appendix o accompany Poliical Uncerainy and Risk Premia Ľuboš Pásor Universiy of Chicago, CEPR, andber Piero Veronesi Universiy of Chicago, CEPR, andber Sepember 4, 011

2 Proof of Lemma 1: he same argumen leading o equaion IA.8 in he Inerne Appendix of Pasor and Veronesi 011 implies ha condiional on policy n, n 0, 1,..,, being chosen a ime τ, aggregae capial is given by B B τ e μ+gn 1 σ τ +σz Z τ hus, exploiing W B we have W 1 E τ 1 γ policy n B1 τ 1 γ e1 τ μn g τ σg,n +μ γ σ τ 1 I follows immediaely ha W 1 W 1 E τ 1 γ policy n >E τ 1 γ policy m if and only if μ n μ n g γ τσ g,n >μm g γ τσ g,m μm Q.E.D. Proof of Proposiion 1. he governmen chooses policy n {0, 1,..., } if and only if for all m n, m 0, 1,...,, C n W 1 E τ 1 γ policy n C m W 1 > E τ 1 γ policy m where recall ha C 0 1. he same calculaions as in Lemma 1 lead o he inequaliy μ n g σ g,n τγ 1 c n γ 1 τ >μm g σ g,m τγ 1 c m γ 1 τ B1 he claim follows from he definiions of μ n and c n in equaions 15 and 3. Q.E.D. ProofofCorollary1. Immediae from Proposiion 1 and equaions 16 and 17. ProofofCorollary.Asofime, wehaveforeachn 1,..., c n ĉ n, σ c, B Given he condiion in Proposiion 1 ha policy n {1,..., } is chosen if and only if μ n c n > μ m c m m n, m 1,..., B3 μ n c n > x τ B4 1

3 he condiional probabiliy a ha policy n is chosen a τ is given by μ p n Pr n c n > μ m c m for m n μ n c n >x τ c Pr n μ n + μ m < c m for m n μ n c n c n φ >x ec n c n d c n τ Π m n Pr c n μ n + μ m < c m c n Pr μ n c n >x τ c n φ ec n c n d c n Π m n 1 Φ ec m c n μ n + μ m Φ x μ n c n ĝ φ ec n c n d c n where we used he fac ha c m s are independen of each oher as well as of x τ.moreover, from he definiion of x τ ĝ τ bσ τ τγ 1 see equaion 16 we have x τ ĝ ĝ bσ τ τγ 1, σ σ τ. We noe wo properies: 1. As ĝ,henp n 0 for all n {1,...,}, asφ x μ n c n ĝ 0.. As τ we have Φ x μ n c n ĝ eμ n ec n φ x x ĝ dx 1 {xτ <eμ n ec n} B5 so ha p n p n τ eμ n x τ Π m n 1 Φ ec m c n + μ m μ n Φ x μ n c n ĝ φ ec n c n d c n Π m n 1 Φ ec m c n + μ m μ n φ ec n c n d c n Q.E.D. Proof of Lemma A1. Using he same argumens as o obain equaion IA.0 in he Inerne Appendix of Pasor and Veronesi 011, afer he announcemen of policy n a ime τ+, he sae price densiy is given by E τ + π policy n πτ n + λ 1 B τ +e μn g τ e μ+ 1 γγ+1σ τ + γ τ σg,n B6 herefore, using also B τ B τ +, he sae price densiy a τ is π τ p n τ πτ n + n0

4 λ 1 n0 p n τ B τ λ 1 B τ e μ+ 1 γγ+1σ τ e μn g τ e μ+ 1 γγ+1σ τ + γ τ σg,n Using he definiion μ 0 g ĝ τ in equaion 18 and he condiion p n τ e μn g τ + γ τ σg,n n0 p 0 τ 1 n1 p n τ B7 we can rewrie he sae price densiy a τ as π τ λ 1 Bτ 1+ e μ+ 1 γγ+1σ τ bg τ τ + γ p n τ n1 τ bσ τ e μn g bg τ τ + γ τ σ g,n bσ τ 1 Similarly, afer he announcemen of policy n, n 0, 1,...,, aimeτ+, we have E τ + B Bi policy n i,n τ + B τ +Bτ i + e1μn g τ e 1μ+ 1 γγ 1σ τ + 1 herefore, we have E τ B Bi n0 p n τ i,n τ + B τ B i τ e 1μ+ 1 γγ 1σ τ Bτ 1+ p n τ e 1μn g n0 Bτ i e 1μ+ 1 γγ 1σ τ +1bg τ τ + 1 n1 p n τ τ σ g,n 1 τ + τ σ g,n τ bσ τ e 1μn g bg τ τ + 1 τ σ g,n bσ τ 1 B8 he claim follows from aking he raio M i τ Eτπ B i π τ Eτλ 1 B Bi π τ.q.e.d. ProofofLemmaA. From B6 and B8 we obain ha if policy n, n 0, 1,...,, is seleced a τ+, hen Mτ i + E τ + B Bi policy n E τ + B policy n Bi τ +e μσ+μn g τ + 1 τ σ g,n B9 Q.E.D. 3

5 ProofofProposiion. From Lemmas A1 and A, he gross announcemen reurn from announcing policy n is 1+R n ĝ τ e μn bg τ τ + 1 τ σg,n bσ τ 1+ n1 pn τ 1+R 0 ĝ τ 1+ n1 pn τ e μn g bg τ τ + γ τ σg,n bσ τ 1 e 1 μ n 1 g bgτ τ + τ σg,n bσ τ 1 Similarly, recalling he noaion μ 0 g ĝ τ and σ g,0 σ τ, from Lemma A1 and A he gross announcemen reurn from announcing no policy change is 1+ n1 pn τ e μng bgτ τ + γ τ σ g,n bστ 1 1+ n n1 pn τ e 1μn g bg τ τ + 1 τ σ g,n bσ τ 1 B10 herefore, we can wrie more compacly, for n 1,...,, 1+R n ĝ τ e μn g bg τ τ + 1 τ σ g,n bσ τ 1+R 0 ĝ B11 We can express all he formulas in erms of μ n and x τ. Using he definiions μ n + σ g,n τγ 1 μn g B1 x τ + σ τ τγ 1 ĝ τ B13 we have σ μ n ĝ τ μ n g,n σ τ x τ + τγ 1 B14 he claim of Proposiion hen follows quickly. Q.E.D. Proof of Corollary 3. Immediae from Proposiion. Q.E.D. Proof of Corollary 4. Immediae from Corollary 3: for any wo policies n and m wih μ n μ m, he resul follows from equaion 31. Q.E.D. ProofofProposiion3: o prove his proposiion, we need hree lemmas: Lemma B1.Δb τ and ĝ τ are perfecly correlaed, and we can wrie Δb τ E Δb τ +ĝ τ E ĝ τ σ / σ +τ E Δb τ +x τ E x τ σ / σ +τ Proof of Lemma B1: From Lemma A5 in Pasor and Veronesi 011, we have ha b τ logb τ andĝ τ have he condiional join disribuion bτ b E Δb τ Vb,C ; g,b B15 ĝ τ E ĝ τ C g,b,v bg 4

6 where E Δb τ μ + ĝ 1 σ τ E ĝ τ ĝ V b τ σ + σ τ V bg σ σ τ C bg,b σ τ We now see ha b τ b and ĝ τ are perfecly correlaed. In fac, Using he fac ha we find Corr Corr C bg,b Vb V bg σ τ 1 bσ σ τ τ σ + σ τ σ σ τ τ σ σ σ σ + σ τ σ τ τ σ + σ τ σ σ τ τ σ σ + σ τ σ σ bσ σ σ +bσ τ B16 B17 σ τ 1 τ σ τ I follows ha we can wrie Δb τ E Δb τ +{ĝ τ E ĝ τ } C b,bg V b E Δb τ +{ĝ τ E ĝ τ } V bg V bg E Δb τ +{ĝ τ E ĝ τ } σ τ E σ σ τ Δb τ +{ĝ τ E ĝ τ } σ / σ +τ where we also used he equaliy σ σ τ σ σ σ σ + σ τ σ τ σ + σ τ From he definiion of x τ, i also follows ha x τ E x τ ĝ τ E ĝ τ. Q.E.D. 5

7 Lemma B: he condiional disribuion of Δb τ b τ b logb τ /B condiional on ime- informaion and policy n being chosen a ime τ is f Δb τ S,n a τ φ Δb τ Δb τ p n eμ n bσ E x τ Δb τ E Δb τ τ bσ +σ B18 Π m n 1 Φ ec m c n μ n + μ m φ ec n c n d c n B19 where φ Δbτ Δb τ is he normal densiy wih mean E Δb τ μ + ĝ 1 σ τ and variance V b τ σ + σ τ. In addiion, E x τ ĝ bσ τ τγ 1. oe ha f Δb τ S,κ a τ does no depend on he curren value of log capial, b, hence he condiional dependence only on S and ime. ProofofLemmaB. he condiional CDF is Pr Δb x F Δbτ Δb S, τ < μ n c n τ < Δb, c n μ n + μ m < c m for m n Pr x τ < μ n c n c n μ n + μ m < c m for m n S x τ < μ n c n c n μ n + μ m < c m for m n S B0 he denominaor is jus p n from Corollary. Consider he numeraor. From Lemma B1: Δb τ E Δb τ {x τ E x τ } σ / σ +τ which implies σ x τ E x τ +{Δb τ E Δb τ } σ + σ τ hus, he join disribuion can be wrien as x Pr Δb τ < Δb, τ < μ n c n c n μ n + μ m < c m S bσ E x τ +{Δb τ E Δb τ } Pr Δb τ < Δb, σ +bσ τ < μn c n S c n μ n + μ m < c m c n < μ n bσ E x τ {Δb τ E Δb τ } Pr Δb τ < Δb, σ +bσ τ c n,s φ ec c n μ n + μ m < c m n c n d c n Δb eμ n bσ E x τ {Δb τ E Δb τ } σ +bσ τ Π m n 1 Φ ec m c n μ n + μ m φ ec n c n d c n φ Δbτ Δb τ dδb τ where we exploied he independence across c m and wih respec o Δb τ. Subsiuing ino B0 and aking he firs derivaive wih respec o Δb, we obain he densiy B19. Q.E.D. 6

8 Lemma B3: he disribuion of ĝ τ condiional on ime- informaion and no new policy being chosen a ime τ is f ĝ τ no policy change a τ φ bg τ ĝ τ ĝ Π p 0 n1 1 Φ ec n μ n ĝ τ + σ τ τγ 1 B1 where φ bgτ ĝ τ ĝ is he condiional normal densiy of ĝ τ,namely,ĝ, σ σ τ. ProofofLemmaB3:he condiional CDF is given by F bgτ g no policy change a τ F bgτ g x τ > μ n c n for all n Pr ĝ τ <g& ĝ τ > μ n c n + bσ τ τγ 1 for all n Pr x τ > μ n c n for all n Pr ĝ τ <g& ĝ τ > μ n c n + bσ τ τγ 1 for all n ĝ τ φ bgτ ĝ τ ĝ dĝ τ p 0 1 {bg τ <g}π n1 1 Φ ec n μ n ĝ τ + bσ τ τγ 1 φ bgτ ĝ τ ĝ dĝ τ p 0 g Πn n1 1 Φ ec n μ n ĝ τ + bσ τ τγ 1 φ bgτ ĝ τ ĝ dĝ τ p 0 aking he firs derivaive wih respec o g, we obain he densiy B1. Q.E.D. ProofofProposiion3:We know ha π E π τ + E π τ + n a τ p n oe ha for n 1,..., E π τ + n a τ E λ 1 B τ +e μn g τ e μ+ 1 γγ+1σ τ + γ τ σg,n n a τ n0 B λ 1 e μ+ 1 γγ+1σ τ μ n g τ + γ τ σg,n e b E e b τ b n a τ λ 1 B e μ+ 1 γγ+1σ τ e μn g τ + γ τ σg,n e Δbτ f Δb τ S,n a τ dδb τ Similarly, for n 0wehave E π τ + 0aτ E λ 1 B τ +e bgτ τ e μ+ 1 γγ+1σ τ + γ τ bσ τ 0aτ 7

9 λ 1 e μ+ 1 γγ+1σ τ + γ τ bσ τ e b E e Δb τ bg τ τ 0aτ λ 1 e μ+ 1 γγ+1σ τ + γ τ bσ τ e b r «V E Δb τ +{bg τ E bg τ } b bg V τ τ E bg 0aτ e λ 1 B e μ+ 1 γγ+1σ τ e e γ bg τ + τ bσ τ E Δb τ +{bg τ E bg τ }r Vbτ V bgτ «τ bg τ bg f ĝτ 0aτ dĝ τ he resul follows from comparing he erms in Equaions 36 and A3 wih he ones above, and defining in his proposiion μ 0 g ĝ and σ g,0 σ τ. Q.E.D. ProofofProposiion4:he resul follows from an applicaion of Io s Lemma o equaion 36, and recalling ha π is a maringale, and hus E dπ /π 0. Q.E.D. Proof of Corollary 5: From propery 1 in he proof of Corollary, for a given disribuion of c n,wehavep 0 1asĝ. I follows ha he sae price densiy converges o one ha assigns zero probabiliy o a policy change: π ΩS E π τ + 0an λ 1 E B λ 1 B 0an e bg e μ+ 1 γγ+1σ + γ bσ Since his sae price densiy does no depend on any ĉ n 1 ΩS,wehave ΩS 0. Q.E.D. bc n Proof of Proposiion 5: he proof is idenical o he proof of Proposiion 3, excep ha we have o calculae E πτ + Mτ i + p n E πτ + Mτ i + n a τ n0 From B8, for n 1,.., : E πτ + Mτ i + n a τ λ 1 E i τ + n a τ λ 1 E B τ +Bτ i + e 1μn g τ e 1μ+ 1 γγ 1σ τ + 1 τ σ g,n n a τ λ 1 e 1μn g τ e 1μ+ 1 γγ 1σ τ + 1 τ σ g,n E e bτ +bi τ n a τ ow, recall which implies B i τ B i B τ e 1 σ 1 τ +σ 1Zτ i Z i B3 B e bi τ e b i +bτ b 1 σ 1 τ +σ 1Z i τ Zi B4 8

10 For n 1,...,, wehenhave: E πτ + M i τ + n a τ λ 1 B Be i 1μn g τ e 1μ+ 1 γγ 1σ τ + 1 τ σ g,n E e 1Δb τ n a τ λ 1 B B i e1μn g τ e 1μ+ 1 γγ 1σ τ + 1 τ σ g,n e 1Δbτ f Δb τ S,n a τ dδb τ Similarly, for n 0,wehave: E πτ + Mτ i + 0 aτ λ 1 E i τ + 0aτ λ 1 E B Bτ i e 1bgτ τ e 1μ+ 1 γγ 1σ τ + 1 τ bσ τ 0aτ τ λ 1 e 1μ+ 1 γγ 1σ τ + 1 τ bσ τ E e bτ +bi τ +1bg τ τ 0aτ λ 1 B λ 1 B E λ 1 B B i 1 e1μ+ γγ 1σ τ + 1 τ bσ τ E e 1Δb τ +1bg τ τ 0aτ Be i 1μ+ 1 γγ 1σ τ + 1 τ bσ τ r e 1 E Δb τ +{bg τ E bg τ } «V b V bg +1bg τ τ n a τ Be i 1bg τ +1μ+ 1 γγ 1σ τ + 1 τ bσ τ r e 1 E Δb τ +{bg τ E bg τ } «V b V bg +1bg τ bg τ f ĝτ S,n a τ dĝ τ he resul follows from comparing he erms in Equaions 41 and A4 wih he ones above, and defining in his proposiion μ 0 g ĝ and σ g,0 σ τ. Q.E.D. ProofofProposiion6. he claim follows from an applicaion of Io s Lemma o he price M i in Proposiion 5, and he equilibrium resricion μ i dm i M Cov, dπ M i π. Q.E.D. ProofofProposiion7. he expression for he jump risk premium follows immediaely from JS τ p n τ R n x τ n0 where R n x τ are given in Proposiion. We now see ha M i J S τ Cov τ + τ 1, π τ + 1 {E Mτ i τ J M J π E τ J M E τ J π } π τ where, if policy n is chosen, we denoe JM n M τ+ n M τ and Jπ n πn τ+ π τ. Recall from Proposiion ha J n M 1+Rn x τ 9

11 1+ e eμn x τ τ γ τ σg,n bσ κ1 τ pκ τ e τ eμκ x τ + γ τ σg,κ bσ τ 1 1+ κ1 pκ τ e 1 τ eμκ x τ 1 We can compue a similar expression now for he sochasic discoun facor. From he expressions for πτ n + and π τ in he proof of Lemma A1, i follows ha for n 1,.., where Jπ n x τ πn τ + e eμn γ x τ τ + σg,n bσ τ τ Jπ 0 x τ π τ J 0 π x τ π0 τ + π τ 1 1+ κ1 pκ τ e eμk x τ τ + γ τ σg,k bσ τ 1 his implies ha J n π x τ J n M x τ J 0 π x τ J 0 M x τ e 1eμn x τ τ 1+ for n 1,..., κ1 pκ τ e1 τ eμκ x τ κ1 pκ τ e 1 τ eμκ x τ 1 I follows ha E τ J π x τ J M x τ p κ e 1eμκ x τ τ τ κ1 1+ κ1 pκ τ e 1 τ eμκ x τ p κ 1 τ κ1 1+ κ1 pκ τ e1 τ eμκ x τ p κ x τ τ κ1 pκ τ e 1 τ τ e1eμκ +1 eμκ x τ 1 κ1 1 κ1 p κ τ Similarly, E τ J π x τ 1 } {e p κ eμn τ e γ σ g,n τ τ J 0 κ1 e xτ τ + γ bσ τ τ π x τ + 1 p κ τ Jπ 0 x τ κ1 p κ τ e eμn x τ τ e γ σg,n bσ τ τ +1 Jπ 0 x τ κ1 κ1 p κ τ 10

12 hus, we finally obain Jx τ Cov τ J M,J π {E τ J M J π E τ J M E τ J π } E τ J M 1 Q.E.D. REFERECES Pasor, Lubos, and Piero Veronesi, 011, Uncerainy abou Governmen Policy and Sock Prices, Journal of Finance, forhcoming. 11