Internet Appendix for: Mortgage Convexity

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1 Inerne Appendix for: Morgage Convexiy Samuel G. Hanson Harvard Business School January 6, 013 A Exended Model wih Muliple Mauriies We now allow for muliple bond mauriies. This is simply a mechanical exension of he model wih -period bonds. I am rying o do somehing along he lines of Gabaix, Krishnamurhy, and Vigneron 007 here. This is also in he spiri of a more racable, discree-ime version of Vayanos and Vila 009 and Greenwood and Vayanos 011. A.1 Se-up Le r +1 = r 1 +1 = y 1 denoe he log reurn on 1 period riskless bonds beween ime and + 1. The log excess reurn on n-period bonds relaive o 1-period bonds is rx n ny n b =b +1 = n 1 y n 1 +1 r +1. Suppose ha here bonds wih mauriiies n =,..., N, le, b 3,...b N denoe he N 1 1 vecor of holdings in long-erm bonds, and le rx N 1 +1 =rx +1, rx 3 +1,...rx N +1 denoe he N 1 1 vecor of excess reurns over 1-period bonds. Assume ha he shor-erm rae is exogenously pinned down by moneary policy are ha here is a riskfree sorage echnology in perfecly elasic supply. Arbirageurs solve so heir demand for bonds is max {b E b rx N τ b V ar rx N 1 +1 b = τ V ar rx +1 1 E rx +1. b }, Suppose ha he ne supply of hese bonds is q N 1 y = q y, q 3 imposing for simpliciy ha D y q y = diagq y, q 3 clearing we require y 3,...q N y 3 y N E rx N 1 +1 = τ 1 V ar rx N 1 +1 q N 1 y = τ 1 Cov rx N 1 +1, rx P +1 1,...q N y N,. By marke

2 where rx P +1 = rx N 1 +1 q N 1 y is he reurn on he porfolio held by arbirageurs. Thus, since E rx P +1 = E rx N 1 +1 q N 1 y = τ 1 V ar rx P +1, we have a CAPM for bonds E rx N 1 +1 = Cov rx N 1 +1, rx P +1 V ar rx P +1 τ 1 V ar rx P Cov rx N 1 +1 = +1, rx P +1 E V ar rx P rx P +1, +1 wih he porfolio of bonds held by arbirageurs rx P +1 funcioning as he marke porfolio from he sandpoin of he pricing kernel. A. Equilibrium yields Le y = y 1, y 0, rx N 1 0, q y, q 3, y 3,...y N denoe he N 1 vecor of yields and rx +1 = rx 1 +1, rx +1, rx 3 +1,...rx N +1 he excess reurn on 1-period bonds is zero by definiion. Also, le q y = y 3,...q N y N = 0, q N 1 y. Le N 0 = diag1,, 3,..., N, N 1, N and N 1 = subdiag1,, 3,..., N, N 1 denoe he N N marices so ha Noe ha rx +1 = N 0 y N 1 y +1 r +1 1 N. V ar rx +1 = N 1 V ar y +1 N 1, so ha he N N marix V ar rx +1 is jus he N 1 N 1 V ar rx N 1 +1 plus a op border row of zeros and a lef border column of zero. Thus, a general version of he E rx +1 = τ 1 V ar rx +1 q y marke clearing condiion is N 0 y N 1 E y +1 r 1 N = τ 1 N 1 V ar y +1 N 1q y +1 = or y = N 1 0 r 1 N + N 1 E y +1 + τ 1 N 1 V ar y +1 N 1q y A.3 Adding facor srucure Suppose ha V ar y +1 has he following principal componens represenaion Tha, is we have V ar y +1 = k σ kβ k β k. y n +1 = µ n + K k=1 βn k x k,+1 where V ar x k,+1 = σ k and Cov x k,+1, x j,+1 = 0 for k j. For insance, suppose ha β n 1 = 1 for all n, so we can hink of he firs componen as he level facor in yields. We can wrie of N 1 marke clearing condiions plus he requiremen ha rx 1 +1 = 0 as E rx +1 = τ 1 V ar rx +1 q y = k N 1β k τ 1 σ kn 1 β k q y = k b kλ k y

3 where b k = N 1 β k and λ k y = τ 1 σ k N 1β k q y. Furhermore, yields mus saisfy he following fixed-poin condiion Thus, we have y = N 1 0 N 1 E y +1 + r +1 1 N + N 1 0 y n Forward raes ake he form = r +1 + n 1 E y n 1 +1 n k N 1β k τ 1 σ kn 1 β k q y. + n 1 k βn 1 k λ k y. n f n = ny n n 1 y n 1 = E n 1 y n 1 +1 n y n +1 + k n 1 βn 1 k n β n k λ k y = E f n k n 1 βn 1 k n β n k λ k y. Naurally, if all of he λ k = 0, he expecaions hypohesis holds and we have f n E r +n. A.4 Excess sensiiviy wih muliple mauriies I is easy o compue he derivaive of all yields wih respec o r +1. We have D r+1 y = so ha we have D r+1 y = Expecaion hypohesis effec {}}{ N 1 0 N 1 D r E y N + I N 1 0 k N 1β k τ 1 σ kn 1 β k Dy q y Endogenous change in risk premia = E f n 1 +1 = { }}{ N 1 0 k N 1β k τ 1 σ kn 1 β k Dy q y D r+1 y 1 N 1 N1 D r+1e y N. Clearly, we have D r+1 y = N 1 0 N1 D r+1e y N if Dy q y = 0. Muliplying his vecor by he marix I N 1 0 k N 1β k τ 1 σ k N 1β k D y q y 1 is jus a mulivariae represenaion of he excess sensiiviy resul shown abou wih wo mauriies. Anoher hing his ells you is ha risk premia are very responsve o changes in supply for mauriies ha are i longer and ii have greaer loadings on he common facor. A.5 Saionary Equilibria We can define a saionary equilibrium of he form y s = N 1 0 N 1 E y +1 s +1 s + r +1 1 N + N 1 0 k N 1β k, τ 1 σ k,n 1 β k, q y s 0 3

4 for some sae process s.since N 1 0 N 1 j 0 for j < n and N 1 0 N 1 n = 0, ineraing forward yields are y s = n 1 j=0 N 1 + n j=0 For insance, he n-year yield is y n = n 1 E n 1 j=0 r +j+1 s j 0 N 1 N N E r +j+1 s N 1 j 0 N 1 N 1 0 E k N 1β k,+j τ 1 +j σ k,+jn 1 β k,+j q +j y +j s +j s +n 1 n j=0 n 1 j E k βn 1 j k,+j τ 1 +j σ k,+jn 1 β k,+j q +j y +j s +j s, and he he n-year forward rae is f n = E r +n + n j=0 k E n 1 j β n 1 j k,+j n j β n j k,+j τ 1 +j σ k,+j N 1β k,+j q +j y +j s +j s. So he expecaions hypohesis holds if: i q +j y +j = 0 for all + j almos surely; or iii τ +j for all + j almos surely; or iii σ k,+j = 0 for all k and + j almos surely A.6 Index duraion and he duraion risk premium To clearly undersand he raionale for using DUR as a forecasing variable, I suppose ha he firs facor is a level facor in yields. Empirically, a level facor explains he vas majoriy of movemens in nominal yields Lierman and Scheinkman 1991 as well as real yields Gurkaynak, Sack, and Wrigh As in he erm srucure lieraure, one would like o derive he exisence of such a level facor in equilibrium based on a small number of exogenous sochasic primiives e.g., he shor rae, a supply variable, arbirageur risk olerance, ec. as opposed o assuming is exisence. As shown below, following he approach in Vayanos and Vila 009, one can close he model in his fashion and a similar inuiion emerges. Concreely, I assume ha β 1 = 1 so he firs facor shifs he yield curve in a parallel fashion. Thus, we have λ 1 y = τ 1 σ 1N 1 β 1 q y = N σ n= n 1 qn y n 1 τ = σ 1 N n= qn y n τ DUR { }}{ N n= n 1 qn y n N. n= qn y n 1 The laer fac is somewha of a puzzle from he sandpoin of heories which aribue he level facor in nominal yields o shifs long-run inflaion expecaions. 4

5 Resric aenion o duraion profiles such ha N ha λ 1 y = τ 1 σ 1Q DUR. n= qn y n = Q and ha σ 1, = σ 1, so In oher words, he duraion risk premium is he aggregae index duraion, imes he oal number of bonds divided by risk olerance, imes he volailiy of he level facor in yields. As a resul, we have E rx n +1 = n 1 λ 1 y + K k= bn 1 k λ k y = n 1 τ 1 σ 1Q DUR + K k= bn 1 k λ k y. Proposiion 1 The coeffi ciens on duraion in a mulivariae forecasing specificaion i.e., when we add a bunch of forecasers so we are soaking up all of K k= bn 1 k λ k y esimaed mauriy-by-mauriy should be b n = n 1 τ 1 σ 1Q. In oher words, he coeffi ciens on index duraion should be approximaely linear in n. Furhermore, his linear relaionship should be seeper when i σ 1 is higher, ii Q is larger, or iii τ is smaller. Thus, his simple heory provide a very direc moivaion for somehing exacly like a forward-looking measure of MBS index duraion as a forecasing variable when predicing bond excess reurns. This predicion emerges incredibly srongly in he daa. Imporanly, his linear paern by mauriy does no hold generally for oher predicive variables. This appears o be somehing unique and special abou duraion per se. A.7 Wha does his look like in forward space? We have f n = E r +n + n j=0 E τ 1 +j σ 1,+jQ +j DUR +j + n j=0 E K k= n 1 j βn 1 j k,+j Le s assume ha Q = Q, τ = τ, σ 1,+j = σ 1 and ha n j β n j k,+j λ k,+j y+j DUR +1 = DUR + ρ DUR DUR + ε D,+1 so ha E DUR +j = DUR + ρ j DUR DUR, 5

6 so we have = E r +n + τ 1 σ 1Q n j=0 DUR + ρ j DUR DUR + K k= E n j=0 n 1 j β n 1 j k,+j n j β n j k,+j λ k,+j y +j = E r +n + τ 1 σ 1Q n 1 DUR + 1 ρn 1 DUR DUR 1 ρ + K k= E n j=0 n 1 j β n 1 j k,+j n j β n j k,+j λ k,+j y +j. f n Obviously, we have and n ρ 1 ρ n 1 = 1 ρ 1 ρ n 1 > 0 1 ρ ln ρ ρn 1 1 ρ so i here is a large impac on more disan forwards and ii he impac is larger when shocks are more persisen. A.8 Wha does his look like in yield space? Under hese same assumpions we have y n = n 1 E n 1 j=0 r +j + n 1 τ 1 σ 1Q n j=0 n 1 j DUR + ρj DUR DUR Clearly +n 1 n j=0 n 1 j E K k= βn 1 j k,+j λ k,+j y+j = n 1 E n 1 n 1 1 j=0 r +j + τ 1 σ 1Q DUR + 1 ρ 1 ρn DUR n 1 ρ DUR +n 1 n j=0 n 1 j E K is increasing in ρ. Noe ha k= βn 1 j k,+j n j=0 n 1 j ρj n λ k,+j y +j > 0, = 1 1 ρ 1 ρn n 1 ρ n 1 j=0 n 1 j ρj = 1 ρ 1 n j=1 1 ρ j 1, so he formulaion in yields is consisen wih he formulaion in forwards. We also have 1 n 1 ρ 1 ρn n 1 ρ = 1 ρ n n 1 n n 1 ρ = n ρ n + ln ρ ρ n 1 ρ. This is also increasing in n. So larger effec when shocks are more persisen and large impac on disan 1-year forwards and disan yields. Obviously, he impac on disan insananeous 6

7 forwards would be zero a some horizon. A.9 Asymmeries beween duraion exensions and conracions As discussed in he main ex, i is naural o conjecure ha ρ is relaively low when DUR > DUR, bu ρ can be much higher when DUR < DUR. This emerges very srongly in he daa. Thus, one would like o esimae y n = a + b 1 DUR DUR + + b0 DUR DUR + c x + ε n where x + = max {x, 0} and x = min {x, 0} are he posiive and negaive pars of x. One wans o find ha b 1 > b 0. Alernaely, one could esimae y n = a + b 1 DUR DUR + + b0 DUR DUR + c x + ε n. One wans o find ha b 1 > b 0 and his appears o srongly be he case in he daa. However, heir is a huge simulaeneiy problem wih his approach. Specifically, we know ha DUR iself is a complicaed, nonlinear funcion of y n. Thus, one worries ha his regression will simply recover mechanical facs abou MBS duraion as opposed o he equilibrium effec of ineres. The problem here is ha DUR DUR + and DUR DUR are funcions of y n. Specifically, MBS become more negaively convex a lower yields han higher yields. So when yields are low, a given change in DUR DUR is associaed wih a small change in y n. And when yields are high, a given change in DUR DUR + is associaed wih a large change in y n. This is no saying anyhing ineresing abou he equilibrium deerminaion of ineres raes. Firs, o undersand he problem, suppose everyhing is symmeric and we would like o esimae β in = α + β DUR DUR + ε n. y n The underlying srucural parameer of ineres is 1 β = τ 1 σ 1Q 1 ρ 1 ρn n 1 ρ. The simulaneiy problem here is ha MBS duraion iself is a funcion of bond yields. For simpliciy, assume ha DUR DUR = θ y n + δ n. Thus, we have or y n y n = α + βθ y n + β δ n = 1 βθ 1 α + 1 βθ 1 β δ n + ε n + 1 βθ 1 ε n 7

8 and DUR DUR = 1 βθ 1 θα + 1 βθ 1 δ n + 1 βθ 1 θ ε n. Assume ha 1 > βθ so he sysem is sable. Thus, supposing ha Cov δ n have b OLS = Cov DUR DUR, y n V ar DUR DUR = σ δ σ δ + β + θ σ ε θ σ ε So b OLS is a weighed average of β and 1/θ. Noe ha σ δ + θ σ ε, ε n = 0, we 1 θ. b OLS θ σ ε = σ δ + θ σε θ σ ε + βθσ δ σ δ which is negaive for θ σ ε/σ δ + β > 1. Obviously, o consisenly esimae β, we need an insrumen z n for δ n i.e., shifs in duraion ha are orhogonal o any unmodeled deerminans of ineres raes so Cov z n 0 while Cov z n, δ n 0. Then we have, ε n = b IV = Cov z n, y n Cov z n, DUR DUR 1 βθ 1 β Cov z n = 1 βθ 1 Cov z n, δ n, δ n = β. In oher words, one needs an exogenous shif in MBS duraion or MBS supply ha is unrelaed o any unmodeled deerminans of ineres raes. Good luck! Specifically, le s say we have y n = a + β 1 DUR y n DUR + + β 0 DUR y n DUR + ε n where 1 β 1 = τ 1 σ 1Q 1 ρn ρ 1 n 1 ρ 1 > τ 1 σ 1Q 1 ρn 0 1 ρ 0 n 1 ρ 0 = β 0. Now suppose ha + DUR y n DUR = 1+ DUR y n DUR = 1 θ 1 y n θ 0 y n + δ n + δ n 1 + = 1 DUR > DUR and 1 = 1 DUR < DUR. The idea here is ha θ 0 > θ 1, so MBS become less negaively convex when hey are rading a a premium. Thus, we have y n = β 1 θ 1 1 α + 1 β 1 θ 1 1 β 1 δ n 1 β 0 θ 0 1 α + 1 β 0 θ 0 1 β 0 δ n + 1 β 1 θ 1 1 ε n + 1 β 0 θ 0 1 ε n. 8

9 I am going o sop for now wih he algebra, bu he generalized version of he simulaneiy problem suggess ha should approximaely his isn 100% correc have b 1 = σ δ σ δ + β θ 1σ 1 + θ 1σ ε 1 ε σ δ + θ +σ ε θ 1 and b 0 = σ δ σ δ + β θ 0σ 0 + θ 0σ ε 1 ε σ δ +. θ 0σ ε θ 0 Thus, wihou an insrumen i is impossible o know wheher he fac ha we find b 1 > b 0 is due o β 1 > β 0 or θ 1 < θ 0. Ideally, one would like o insrumen for DUR y n DUR + and DUR y n DUR using 1 + δ n and 1 δ n. A.10 Simulaneiy Suppose ha Q y = α + β y so ha y = r +1 + E r + / + E rx +1 / and or Thus, we have E rx +1 = τ 1 σ Q y = τ 1 σ α + τ 1 σ β y y = r +1 + E r + / + τ 1 σ α + τ 1 σ β y /. and y E rx +1 = τ 1 σ and = 1 τ 1 σ β 1 r+1 + E r + / + 1 τ 1 σ β 1 τ 1 σ α / = τ 1 σ = τ 1 σ α + β y α + β 1 τ 1 σ 1 β r+1 + E r + / + β 1 τ 1 σ 1 β τ 1 σ α / β r +1 + E r + / + α. 1 τ 1 σ β Suppose ha σ = σ and β = β, so his becomes y = 1 τ 1 σ β 1 r+1 + E r + / + 1 τ 1 σ β 1 τ 1 σ α / Thus, an increase in τ 1 β E rx +1 = τ 1 σ r+1 + E r + / + α 1 τ 1. σ β increase erm premia for wo reasons: here is a direc effec and 9

10 posiive feedback effec ha works hrough MBS duraion exension when β > 0. B Closing he Model a la Vayanos and Vila 009 Assume ha r +1 = r + ρ r r r + ε r,+1 so ha E r +j = r + ρ j r r r. Assume ha he ne supply ha arbirageurs mus hold is where q N 1 y = q 0 + q 1 s + Qy s +1 = ρ s s + ε s,+1 so ha E s +j = ρ j ss. Furhermore, I assume ha Q =diag0, Q,..., Q N. In he case of a pure mauriy shock, we have 1 q 1 = 0 whereas a supply shock is characerized by 1 q 1 > 0. One can hink of various ypes of supply or mauriy shocks. The shocks o governmen deb mauriy or supply considered by Krishnamurhy and Vissing-Jorgensen 01 and Greenwood and Vayanos 011 are quie persisen, e.g., ρ s 1 a an annual horizon. However, V ar ε s,+1 is probably quie small in his conex. By conras, he MBS mauriy shocks sudied here are characerized large values of V ar ε s,+1 and small values of ρ s e.g., perhaps near 0 a an annual horizon. Obviously, i is rivial o allow for muliple ypes of mauriy shocks. And, in principal, one could also exend he model o allow for differen shocks o he expeced physical pah of shor-raes. This is clear in he more general expressions given above. Of course, he model is ricky enough o solve as is, so I sick o he simples possible parameric assumpions following Vayanos and Vila 009. Vayanos and Vila 009 as well as Greenwood and Vayanos 011 boh assume ha Q n < 0 for all n. This is naural assumpion. The idea is eiher han i borrowers issue more long-erm deb when yields are low or ii preferred-habi invesors buy more long-erm deb when yields are high. As a resul, he ne supply duraion ineres rae risk han arbs need o bear is decreasing in he general level of ineres raes. Thus, long-erm yields will appear o underreac o movemens in shor-erm raes. However, wheher Q n < 0 or Q n > 0 is ulimaely an empirical issue. Moivaed by he fac ha i a wide variey of sudies argue ha long-erm raes appear o be excessively sensiive o movemens in shor-erm raes and ii han aggregae bond marke duraion is srongly increasing in he level of yields in he US over he pas 5 years due o negaive MBS convexiy because households borrow longer erm when raes are high, I will assume ha Q n > 0. This is a major poin of deparure from Vayanos and Vila 009. As a resul of his deparure, a single facor model in which only he shor-erm rae is exogenous will be inconsisen wih sandard erm srucure forecasing resuls. Specifically, The following analysis also highlighs ha, formally speaking, he Vayanos and Vila 009 model does no rely on he assumpion ha Q n < 0. However, wheher Q n < 0 or Q n > 0, may impac some of he uniqueness or regulariy argumens one would like o make. 10

11 such a model would counerfacually imply ha a high erm spread forecass low excess reurns. Thus, o reconsile he excess sensiiviy resuls and ha fac ha Q n > 0 in he daa wih he reurn predicabiliy findings, I will need o consider a model when i he shor-rae is exogenous and Q n > 0, ii here are shocks o duraion supply, and iii perhaps here are shocks o arbirageur risk olerance. B.1 Equilibrium yields Le y = y 1, y, y 3,...y N denoe he N 1 vecor of yields. Conjecure ha y = a 0 + a r r + a s s. We know ha a 0 1 = 0, a r 1 = 1, and a s 1 = 0. Excess reurns are Noe ha rx +1 = N 0 a 0 + a r r + a s s N 1 a 0 + a r r +1 + a s s +1 r 1 N = N 0 N 1 a 0 + r N 0 r +1 N 1 a r + s N 0 s +1 N 1 a s r 1 N E rx +1 = N 0 N 1 a 0 + r N 0 E r +1 N 1 a r + s N 0 E s +1 N 1 a s r 1 N = N 0 N 1 a 0 + r N 0 ρ r N 1 a r 1 ρ r rn 1 a r + s N 0 ρ s N 1 a s r 1 N = N 1 a 0 1 ρ r rn 1 a r + r N ρ r a r 1 N + s N ρ s a s where I use he noaion N ρ N 0 ρn 1. We also have V ar rx +1 = N 1 σ r a r a r + σ sa s a s + σ r,s a r a s + a s a r N 1. One can solve for he unknown coeffi ciens using he equilibrium condiion E rx +1 = τ 1 V ar rx +1 q y or N 1 a 0 1 ρ r rn 1 a r + r N ρ r a r 1 N + s N ρ s a s = τ 1 N 1 σ r a r a r + σ sa s a s + σ r,s a r a s + a s a r N 1 q0 + q 1 s + Qy = τ 1 N 1 σ r a r a r + σ sa s a s + σ r,s a r a s + a s a r N 1 q0 + q 1 s + Qa 0 + Qa r r + Qa s s. 11

12 Maching erms in he above, we require Consan : N 1 a 0 1 ρ r rn 1 a r = A q 0 + Qa 0 r : N ρ r a r 1 N = A Qa r s : N ρ s a s = A q 1 + Qa s. where A = τ 1 N 1 σ r a r a r + σ sa s a s + σ r,s a r a s + a s a r N 1. All of hese equaions are of he form expeced reurn drif equals price of risk A imes quaniy of risk erms in brackes afer A. Of course, his is fixed-poin ime probelm he a vecors deermine boh drif and risk i.e., sensiiviy o he underlying shocks. This is jus as in Vayanos and Vila 009 and Greenwood and Vayanos 011. We can rewrie his as Consan : N 1 a 0 1 ρ r rn 1 a r = τ 1 N 1 σ r a r a r + σ sa s a s + σ r,s a r a s + a s a r N 1 q0 + Qa 0 Now solve r : N ρ r a r 1 N = τ 1 N 1 σ r a r a r + σ sa s a s + σ r,s a r a s + a s a r N 1 Qar s : N ρ s a s = τ 1 N 1 σ r a r a r + σ sa s a s + σ r,s a r a s + a s a r N 1 q1 + Qa s. Consan : a 0 = N 1 AQ 1 Aq 0 + N 1 AQ 1 N 1 1 ρ r ra r = N 1 AQ 1 Aq ρ r r N 1 AQ 1 N 1 Nρ r AQ 1 1 N r : a r = Nρ r AQ 1 1 N s : a s = N ρ s AQ 1 Aq 1 So we need a marix ha solves he following marix fixed poin equaion σ r N ρ r AQ 1 1 N 1 N N ρ r AQ 1 A = τ 1 N +σ s N ρ 1 s AQ 1 A q 1 q 1A N ρ s AQ 1 +σ r,s N ρ r AQ 1 1 N q 1A N ρ s AQ 1 +σ r,s N ρ s AQ 1 Aq 1 1 N N ρ r AQ 1 N 1 Noe ha if Q is a diagonal marix wih posiive elemens i.e., if he MBS convexiy chanell is presen his will make a r and a s larger in magniude. In general, he above yields a sysem of 3 N 1 nonlinear equaions in 3 N 1 unknowns {a 0,n } N n=, {a r,n} N n=, and {a s,n} N n=.. B. Full Model for N = 3 I now solve he model in closed form for N = 3 which is suffi cien o explore he main economic inuiions. This model feaures an exogenous shor-erm rae and endogenous inermediae and long-erm yields. 1

13 B..1 General soluion The exogenous parameers are q 0 = q 0,, q 1 = q 1,, and Q = 0 Q 0. q 0,3 q 1,3 0 0 Q 3 We know ha a 0 1 = 0, a r 1 = 1, and a s 1 = 0, so we are looking for such ha where a 0 = a 0,, a r = a r,, a s = a s, a 0,3 a r,3 a s,3 Consan : N 1 a 0 = 1 ρ r rn 1 a r + A q 0 + Qa 0 r : N ρ r a r 1 N = AQa r s : N ρ s a s = A q 1 + Qa s. A = τ 1 N 1 σ r a r a r + σ sa s a s + σ r,s a r a s + a s a r N = τ 1 0 σ r a r, σ r + σ r,s a s, 0 a r, σ r + σ r,s a s, σ r a r, + σ s a s, + σ r,s a r, a s, Noe ha N ρ = 0 0 ρ and N ρ 1 = ρ/ 1/ ρ /3 ρ/3 1/3 So we are lef wih he following equaions: Consan : 1 0 a 0, 1 ρ r r a r, 0 3 a 0,3 0 0 a r, = τ 1 0 σ r a r, σ r + σ r,s a s, q 0, + Q a o, 0 a r, σ r + σ r,s a s, 4σ ra r, + 4σ sa s, + 8σ r,s a r, a s, q 0,3 + Q 3 a o,3 13

14 r : ρ r 0 a r, 1 0 ρ r 3 a r, = τ 1 0 σ r a r, σ r + σ r,s a s, Q a r, 0 a r, σ r + σ r,s a s, 4σ ra r, + 4σ sa s, + 8σ r,s a r, a s, Q 3 a r, s : ρ s 0 a s, 0 ρ s 3 a s, = τ 1 0 σ r a r, σ r + σ r,s a s, q 1, + Q a s, 0 a r, σ r + σ r,s a s, 4σ ra r, + 4σ sa s, + 8σ r,s a r, a s, q 1,3 + Q 3 a s,3 Simplifying, we are lef wih 6 non-linear equaions in 6 unknowns a 0, r 1 ρ r = 3a 0,3 a 0, r 1 ρ r a r, τ 1 σ r a r, σ r + σ r,s a s, q0, + Q a 0, a r, σ r + σ r,s a s, 4σ ra r, + 4σ sa s, + 8σ r,s a r, a s, q 0,3 + Q 3 a 0,3 ar, 1 + ρ r = 3a r,3 ρ r a r, 1 τ 1 σ r a r, σ r + σ r,s a s, Q a r, a r, σ r + σ r,s a s, 4σ ra r, + 4σ sa s, + 8σ r,s a r, a s, Q 3 a r,3 a s, 3a s,3 ρ s a s, = τ 1 σ r a r, σ r + σ r,s a s, q1, + Q a s, a r, σ r + σ r,s a s, 4σ ra r, + 4σ sa s, + 8σ r,s a r, a s, q 1,3 + Q 3 a s,3 The sysem decouples in he sense ha soluion for a r,, a r,3, a s,, and a s,3 involves solving four nonlinear equaions. The soluion for a 0, and a 0,3 is hen deermined linearly via a 0 = N 1 AQ 1 Aq ρ r rn 1 a r. Thus, he rick is solving he four nonlinear equaions for a r,, a r,3, a s,, and a s,3. For simpliciy, assume ha σ r,s = 0 i.e, shocks o he shor rae 14

15 and supply are orhogonal, so we have a r, 1 + ρ r = τ 1 σ r Q + Q 3 a r,3 a r, a s, = τ 1 σ r q 1, + q 1,3 a r, + Q a s, + Q 3 a s,3 a r, 3a r,3 ρ r a r, 1 = τ 1 σ r Q a r, + Q 3 a r, a r,3 + τ 1 σ sq 3 a s, a r,3 3a s,3 ρ s a s, = τ 1 σ r q1, a r, + Q a s, a r, + q 1,3 a r, + Q 3 a s,3 a r, +τ 1 σ s q1,3 a s, + Q 3 a s,3 a s, Rescale A r, = a r,, A r,3 = 3a r,3, A s, = a s,, A s,3 = 3a s,3, C = Q /, C 3 = Q 3 /3 and rewrie he sysem as A r, 1 + ρ r = τ 1 σ r C + C 3 A r,3 A r, A s, = τ 1 σ r q 1, + q 1,3 A r, + C A s, + C 3 A s,3 A r, A r,3 ρ r A r, 1 = τ 1 σ r C + C 3 A r,3 A r, + τ 1 σ sc 3 A s, A r,3 A s,3 ρ s A s, = τ 1 σ ra r, q 1, + C A s, + q 1,3 A r, + C 3 A s,3 A r, +τ 1 σ s q 1,3 + C 3 A s,3 A s,. In general, we probably need τ suffi cienly large o guaranee uniqueness as in VV 009 and GV 011. The firs equaion implies and hird equaion implies A r,3 = A r, = 1 + ρ r 1 τ 1 σ r C + C 3 A r,3 1 + ρ r A r, + τ 1 σ rc A r, 1 τ 1 σ rc 3 A r, τ 1 σ sc 3 A s,. This clearly shows how he MBS convexiy channel C, C 3 > 0 generaes excess sensiiviy of long-erm yields o shor rae shocks. The second equaion implies A s, = 1 τ 1 σ rc 1 τ 1 σ r q 1, + q 1,3 A r, + 1 τ 1 σ rc 1 τ 1 σ rc 3 A r, A s,3. Fourh equaion shows ha A s,3 = τ 1 σ ra r, q 1, + C A s, + q 1,3 A r, + ρ s A s, + τ 1 σ sq 1,3 A s, 1 τ 1 σ rc 3 A r, τ 1 σ sc 3 A s,, so he MBS convexiy channel also implies excessive sensiiviy o supply shocks. I now work hrough some special case where i is possible o obain closed-form soluions. 15

16 B.. Special case when σ s = 0 This is he case where he supply shifer s is assumed o be deerminisic. becomes The sysem A r, 1 + ρ r = τ 1 σ r C + C 3 A r,3 A r, A s, = τ 1 σ r q 1, + q 1,3 A r, + C A s, + C 3 A s,3 A r, A r,3 ρ r A r, 1 = τ 1 σ r C + C 3 A r,3 A r, A s,3 ρ s A s, = τ 1 σ ra r, q 1, + C A s, + q 1,3 A r, + C 3 A s,3 A r,, so ha we can firs solve for A r, and A r,3. Subsiue ino he hird equaion o obain A r,3 A r, = 1 + ρ r 1 τ 1 σ r C + C 3 A r,3 ρ r 1 + ρ r 1 τ 1 σ r C + C 3 A r,3 1 = τ 1 σ 1 + ρ r C + C 3 A r,3 r 1 τ 1 σ r C + C 3 A r,3 which yields he following cubic in A r,3 0 = A r,3 1 τ 1 σ r C + C 3 A r,3 ρr 1 + ρ r 1 τ 1 σ r C + C 3 A r,3 1 τ 1 σ r C + C 3 A r,3 τ 1 σ r C + C 3 A r,3 1 + ρ r = A r,3 3 τ 1 σ rc 3 A r,3 τ 1 σ rc 3 + τ 1 σ rc 3 1 τ 1 σ rc +A r,3 1 τ 1 σ rc 1 τ 1 σ rc + τ 1 σ rc 3 τ 1 σ rc 3 ρ r + 1 ρ r ρ r τ 1 σ rc + 1 τ 1 σ rc + τ 1 σ r ρ r + 1 C One you solve his cubic, he soluions for A s, and A s,3 solve he following sysem A s, = τ 1 σ r q 1, + q 1,3 A r, + τ 1 σ rc 3 A r, A s,3 1 τ 1 σ rc A s,3 = τ 1 σ ra r, q 1, + C A s, + q 1,3 A r, + ρ s A s, 1 τ 1 σ rc 3 A r,. B..3 Special case when Q = Q 3 = 0 i.e., no MBS convexiy channel, bu σ s > 0 In his case, we ge furher decompling. Now we have a r, = 1 + ρ r and a r,3 = 1 + ρ r + ρ r. 3 16

17 The equaions deermining a s ake he form which implies ha a s, = τ 1 σ r q 1, + q 1,3 a r, 3a s,3 ρ s a s, = τ 1 σ r q1, a r, + q 1,3 a r, + τ 1 σ sq 1,3 a s, a s, = τ 1 σ r q 1, + q 1,3 a r, = τ 1 σ r q 1, + q 1,3 1 + ρ r. Thus, a s, work jus like a ime-varying adjusmen o he duraion risk-premium on -period bonds. Finally, we have 3a s,3 = ρ s a s, + τ 1 σ r q1, a r, + q 1,3 a r, + τ 1 σ sq 1,3 a s, = 1 + ρ r + ρ s τ 1 σ r q 1, ρ r q 1,3 +τ 1 σ sq 1,3 τ 1 σ r q 1, + q 1,3 1 + ρ r. The firs erm in a s,3 is jus he naural loading on he ime-varying adjusmen o he duraion risk premium i.e., τ 1 σ r q 1, ρ r q 1,3 imes s. Where does he loading of 1 + ρ r + ρ s comes from? 3-period bonds have a duraion of 1 + ρ r wih respec o he shor rae omorrow ha is he firs erm. Tomorrow ime + 1 he duraion wih respec o shor raes a + will be 1, bu he ransiory componen of he duraion risk premium has persisence ρ s. Thus, 3-period yields rise oday boh because of he higher duraion risk premium oday, bu also because of he higher duraion risk premium omorrow on -period bonds. The final erm says ha supply risk from holding 3-period bonds depends on he level of duraion supply. In oher words, here is a duraion supply risk premium as disinguished from a duraion risk premium and i is increasing in he curren amoun of duraion supply. Suppose ha q 1,3 > 0 > q 1,, so a posiive value of s means ha curren duraion is high. Inuiively, here is a ime-varying amoun of noise-rader risk in bond marke. When s is large, arbs have o los of 3-year bonds oday which means ha here are bearing more supply risk beween oday and omorrow he risk ha he duraion risk premium will move omorrow due o a supply shock. Thus, he yield on 3-year bonds mus rise o induce me o bear his addiional supply risk. Conversely, when s is low, I am bearing less of his duraion supply risk, so he yield mus fall o induce me o hold less. Wihou loss of generaliy, le s assume ha q 1,3 = 1 and q 1, = 1. Thus, we have a r, = 1 + ρ r 3a r,3 = 1 + ρ r + ρ r a s, = τ 1 σ rρ r 3a s,3 = 1 + ρ r + ρ s τ 1 σ rρ r + τ 1 σ s τ 1 σ rρ r. 17

18 Finally, we jus need o deermine he inercep erms. The soluion saisfies a 0, 1 ρ r r = τ 1 σ r a r, σ r q0, 3a o,3 a 0, r 1 ρ r a r, a r, σ r σ r a r, + σ s a s, or, using he fac ha a r, = 1 + ρ r, we have a o, 1 ρ r r 3a o,3 a 0, 1 ρ r r So we have a 0, = 1 ρ r r + σ rτ 1 q 0, ρ r q 0, q0, = τ 1 σ r 1 + ρ r σ r 1 + ρ r σ r σ r 1 + ρ r + σ s τ 1 σ rρ r q 0,3 1 = σ rτ 1 q 0, ρ r q 0, + σ sτ ρ r τ 1 σ rρ r. q 0,3 3a o,3 = ρ r ρ r r ρr σ rτ 1 q 0, ρ r q 0,3 + σ sτ 1 τ 1 σ rρ r q0,3. Thus, yields ake he form Duraion Risk Premium {}}{ Expecaions componen {}}{ Time-varying piece y = r + r + ρ r r r / + σ rτ q 1 {}}{ 0, ρ r q 0,3 + ρ r s / q 0,3 and y 3 = Expecaions componen {}}{ r + r + ρ r r r + r + ρ r r r /3 Duraion Risk Premium { }} { Time-varying piece {}}{ + σ rτ ρ r q 0, ρ r q 0, ρ r + ρ s ρ r s /3 Duraion supply RP {}}{ + σ sτ Time-varying piece 1 τ 1 σ {}}{ rρ r q 0,3 + s /3 The duraion supply risk premium is he componen ha is more closely analogous wih MBS prepaymen risk. However, unlike Gabaix, Krishnamurhy, and Vigneron 007, he idea here is ha he duraion supply risk premium here ends up impacing he prices of all fixed income securiies, no simply MBS. The idea is ha random household refinancing behavior is major source of marke-level noise rader risk wihin US bond markes. In he Greenwood and Vayanos 011 conex sudying shocks o he mauriy srucure of US governmen deb, i is naural o assume ha σ s is very small and ρ s 1. In he conex of MBS prepaymen risk, 18

19 σ s is plausible much larger bu ρ s is small. This is like a form of household, noise-rader risk in bond markes. Noe ha he duraion supply risk premium is of order τ, will be quie small unless risk olerance is very low. B..4 Special case when Q 0 and σ s > 0, bu Q 3 = 0 Anoher special case is Q 0 and σ s > 0, bu Q 3 = 0. The inerpreaion is ha he supply of -period bonds rises and he supply of 1-period bonds falls when raes rise. However, here is no direc effec on he supply of 3-period bonds. This is jus a simplifying assumpion ha makes closed form expressions paricularly simple. Also, wihou loss of generaliy, assume ha q 1,3 = 1 and q 1, = 1. The sysem becomes a r, 1 + ρ r = τ 1 σ rq a r, a s, = τ 1 σ r 1 + a r, + Q a s, 3a r,3 ρ r a r, 1 = τ 1 σ rq a r, 3a s,3 ρ s a s, = τ 1 σ r a r, a r, 1 + Q a s, + τ 1 σ s a s, and Then we have a r, = 1 + ρ r τ 1 σ rq a r,3 = 1 + ρ ra r, + τ 1 σ rq a r, 3 = 1+ρ 1 + ρ r r τ 1 σ rq + τ 1 σ 1+ρr rq τ 1 σ rq 3. Thus, we have a r,/ Q > 0 and a r,3/ Q > 0: excess sensiiviy of long raes o shor raes is greaer when hen marke is negaively convex. This corresponds o Proposiion and Proposiion 3 in he main ex. Noe ha he soluion only makes sense if 1 > τ 1 σ rq which is he specific analog here o he general sabiliy condiion discussed in he main ex. And, again, all four coeffi ciens are increasing in Q. Also, since a r,/ Q > 0 and a r,3/ Q > 0, holding fixed σ r he volailiy of long-erm yields is increasing in Q which corresponds o Proposiion 4. We also have a a s, = τ 1 σ r, 1 r = τ 1 σ ρ r + τ 1 σ rq τ 1 σ r rq τ 1 σ rq 19

20 and a s,3 = ρ s a s, + τ 1 σ r = ρ r +τ 1 σ r Q a r, a r, 1 + Q a s, + τ 1 σ s ρ s τ 1 σ r + τ 1 σ τ 1 σ r Q r +τ 1 σ s τ 1 σ r = τ 1 σ r ρ r + τ 1 σ rq 3 τ 1 σ rq = a s, 3 ρ s ρ r 3 1+ρ r ρr +τ 1 σ τ 1 σ r Q rq τ 1 σ rq ρ r +τ 1 σ r Q τ 1 σ r Q a s, + Q τ 1 σ r ρ r +τ 1 σ r Q τ 1 σ r Q 3 ρ s ρ r + τ σ τ 1 σ rσ ρ r + τ 1 σ rq s rq τ 1 σ rq + τ σ τ 1 σ rσ ρ r + τ 1 σ rq s rq τ 1 σ rq Thus, we have a s,/ Q > 0 and a s,3/ Q > 0, so ha MBS convexiy also amplifies he response of yields o duraion supply shocks. 3 Now jus need o redo his wih Q AND Q 3 non-zero. However, I worry ha his will lose all racabiliy. B.3 Some Special Cases for General N B.3.1 Log expecaions hypohesis Wih i q 0 = q 1 = 0 and Q = 0 or ii τ, so A 0, we have Consan : a 0 = 1 ρ r r N 1 1 N 1 Nρ r 1 1 N = 1 ρ r r N 1 1 dur 1 ρ r. r : a r = N ρ r 1 1 N = N 1 0 dur ρ r. s : a s = 0 where dur ρ r = p 1 / r, p he duraions and dur 1 ρ r = 0, p 1 / r,..., p N / r is he vecor of negaive one imes / r,..., p N 1 / r. Using he fac ha N ρ r 1 1 N = 3 Consisency check. This reduces o he previous case when Q = 0. Specifically, we hen have a s, = τ 1 σ rρ r, so ha 3a s,3 = a s, ρ s ρ r τ 1 σ + τ σ rσ ρ r + τ 1 σ rq s rq τ 1 σ rq = ρ s ρ r τ 1 σ rρ r + τ 1 σ s τ 1 σ rρ r 0

21 N 1 0 dur ρ r and N 1 N 1 0 is a marix wih 1s on he subdiagonal, 4 we have Thus, he log expecaions hypohesis holds Since I follows ha y n = p n a 0 = 1 ρ r r N 1 1 N 1 Nρ r 1 1 N n 1 E j=0 r +j n = ny n = 1 ρ r r N 1 1 N 1 N 1 0 dur ρ r = 1 ρ r r N 1 1 dur 1 ρ r. = r + n 1 j=0 ρj r r r = r ρ n r r r n n 1 ρ r = nr n 1 j=0 ρj r r r = nr 1 ρn r 1 ρ r r r dur n ρ r = pn r = n 1 j=0 ρj r = 1 ρn r 1 ρ r is he relevan measure of duraion in a discree Vasicek syle model his is really closer o a key rae duraion because we are measuring he sensiiviy of bonds of all mauriies o he exogenous shor rae. B.3. Consan duraion risk premium Now consider he case wih i no supply shocks q 1 = 0 and ii inelasic ne duraion supply Q = 0, bu q 0 0. Then we have Consan : a 0 = N ρ r rn 1 a r + Aq 0 r : a r = N ρ r 1 1 N = N 1 0 dur ρ r s : a s = 0. where A = τ 1 N 1 σ r a r a r N 1 = τ 1 σ r N1 N ρ r 1 1 N 1 NN ρ r 1 N 1 N1 = τ 1 σ r N 1 0 dur ρ r N 1 0 dur ρ r = τ 1 σ rdur 1 ρ r dur 1 ρ r 4 Easy o see ha N 1 1 is he lower-riangular averaging marix wih N = 1. 1

22 Thus, we have a 0 = 1 ρ r r N 1 1 N 1 N ρ r 1 1 N + N 1 1 τ 1 σ r N1 N ρ r 1 1 N 1 NN ρ r 1 N 1 q0 = N 1 1 dur 1 ρ r 1 ρ r r + N 1 1 dur 1 ρ r τ 1 σ rdur 1 ρ r q 0 The firs erm is he expecaions hypohesis erm and he second erm is he ime-invarian duraion risk premium. Recalling ha we have a r = N ρ r 1 1 N = N 1 0 dur ρ r y = a 0 + a r r = N ρ r r dur 1 ρ r + dur 1 ρ r τ 1 σ rdur 1 ρ r q 0 + N 1 0 dur ρ r r. Expeced excess reurns are E rx +1 = N 1 a 0 1 ρ r rn 1 a r + r N ρ r a r 1 N + s N ρ s a s = N 1 a 0 1 ρ r rn 1 a r = N 1 N 1 1 A q ρ r rn 1 a r 1 ρ r rn 1 a r = Aq 0 = dur 1 ρ r τ 1 σ r dur 1 ρ r q 0 In oher words, expeced excess reurns on bonds ake he form E rx n +1 = dur n 1 τ 1 σ N r n= q ndur n 1 where dur n 1 = n 1 j=0 ρj r = 1 ρ n r / 1 ρ r. And yields ake he form 5 y n = r + n 1 dur n r r + n 1 n 1 j=1 dur j τ 1 σ N r n= q ndur n 1. 5 Consisency check. We have rx n +1 = nyn n 1 y n 1 r = r + dur n r r dur n 1 r +1 r + dur n 1 τ 1 σ N r n= q ndur n 1 so ha E rx n +1 = dur n 1 τ 1 σ r = dur n 1 N n= q ndur n 1 + r r dur n ρ r dur n 1 1 τ 1 σ N r n= q ndur n 1 since dur n = 1 + ρ r dur n 1.

23 B.3.3 No supply shocks, bu MBS convexiy channel Noe ha if q 1 = 0, bu Q 0, we have Consan : a 0 = N 1 AQ 1 Aq 0 + N 1 AQ 1 N 1 1 ρ r ra r = N 1 A Q 1 Aq ρ r r N 1 AQ 1 N 1 Nρ r AQ 1 1 N r : a r = Nρ r AQ 1 1 N s : a s = 0 Risk : A = σ rτ 1 N 1 a r a rn 1, so we need a marix ha solves he following marix fixed poin equaion:, A = τ 1 σ rn 1 N ρ r AQ 1 1 N 1 N N ρ r AQ 1 N 1. B.3.4 Adding Supply Shocks, bu No MBS Convexiy Channel Consider he case where Q = 0, bu q 0 0 and q 1 0. Now expeced reurns will vary over ime. Assuming ha σ r,s = 0, we have Consan : N 1 a 0 1 ρ r rn 1 a r = A q 0 r : N ρ r a r 1 N = 0 s : N ρ s a s = Aq 1. where As above, we have A = τ 1 N 1 σ r a r a r + σ sa s a s N 1. a r = N ρ r 1 1 N. Thus, a s solves he following non-linear equaions i.e., fixed-poin condiions N ρ s a s = τ 1 N 1 σ r N ρ r 1 1 N 1 N N ρ r 1 + σ sa s a s N 1 q 1 = σ rτ 1 N 1 N ρ r 1 1 N 1 N N ρ r 1 N 1q 1 + σ sτ 1 N 1 a s a sn 1q 1 = A con q 1 + σ sτ 1 N 1 a s a sn 1q 1 = τ 1 σ rdur 1 ρ r dur 1 ρ r q 1 + σ sτ 1 N 1 a s a sn 1q 1 where A con = τ 1 σ rdur 1 ρ r dur 1 ρ r is he reurn convariance marix above when q 1 = 0. The a s,n will be deermined by he resuling sysem of quadraic equaions. Then, finally we can solve for he a 0. 3