Negaive Swap Spreads and Limied Arbirage Urban Jermann Wharon School, Universiy of Pennsylvania
Ineres rae swap Pay LIBOR, ge fixed rae periodically Swap spread Fixed swap rae minus Treasury of same mauriy Why a spread? LIBOR vs TBill rae Defaul/Liquidiy Limis o arbirage
Swap spreads 200 150 2 year 5 year 10 year 20 year 30 year 100 50 0 50 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan04 Jan05 Jan06 Jan07 Jan08 Jan09 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 Jan18
This paper Develops model wih limied arbirage for swaps Analyically, wih srong fricions negaive swap spread Quaniaive model wih negaive swap spreads Addiional empirical suppor for model
Lieraure Swap spreads: Liu, Longsaff & Mandell (2006), Feldhueer & Lando (2008), Hanson (2014), Gupa & Subrahmanyam (2000), Collin-Dufresne and Solnik (2001), Eom, Subrahmanyam & Uno (2002), Johannes & Sundaresan (2007), Smih (2015), Klinger & Sundaresan (2016) Limied arbirage: Shleifer & Vishny (1997), Dow & Goron (1994), Garleanu, Pedersen & Poeshman (2009), Gabaix, Krishnamurhy & Vigneron (2007), Vayanos & Vila (2009), Liu & Longsaff (2004), Tuckman & Vila (1992), Gromb & Vayanos (2010) Effecs of recen financial regulaion: Duffi e (2016), Du, Tepper & Verdelhan (2016), Boyarchenko e al. (2018) Firs limied arbirage model for swaps
Model ouline Dealer selecs bonds and swaps Bond prices exogenous, swap price endogenous Holding coss for bonds Laer exensions: Capial requiremens, demand effecs, swap holding coss
Model Shor-erm riskless deb q ST (z) = exp ( y ST (z)) LIBOR deb q LIB (z) = exp ( (y ST (z) + θ (z))) wih θ he TED spread Long-erm deb pays c LT + λ + (1 λ) q LT ( z ) wih price and yield q LT (z) = wih erm spread τ (z) c LT + λ exp (y LT (z)) 1 + λ y LT (z) = y ST (z) + τ (z)
Swap Swap pays (o fixed receiver) 1 c Sw ( 1) + (1 λ) m q LIB (z) wih m value nex period For new swap, find c Sw so ha and is he swap rae m (z, ω) = 0, y Sw (z, ω) = c Sw If zero ne demand for swaps, can creae a-marke swap and y Sw for each (z, ω). Oherwise assume here is only one swap.
Fricions The cos for holding long-erm deb is given by j ( α LT ) κ = LT 2 wih α LT amoun of long-erm deb Similarly for shor-erm deb h ( α ST ) κ = ST 2 ( α LT ) 2 ( α ST ) 2
Dealer s problem V (ω, z) = max u (c) + β (ω, z) E ( V ( ω, z )) c,α ST,α LT,s subjec o c = ω α ST q ST (z) α LT q LT (z) s m h ( α ST ) ( ) j α LT and ω = α ST e µ(z ) + α [ ( LT clt + e µ(z λ + (1 λ) q ) LT z )] [ ( ) ] + s 1 e µ(z c ) Sw q LIB (z) 1 + (1 λ) m +π ( z )
Equilibrium Swap marke clears s = d (z) Firs-order condiions h 1 (α ST ) = βe u 1(c ) u 1 (c)e µ(z ) q ST ( ) j 1 (α LT ) = βe u 1 (c ) u 1 (c)e µ(z ) [c LT + λ + (1 λ) q LT (z )] ( m = βe u 1 (c ) u 1 (c)e µ(z ) [ c Sw ( 1 q LIB (z) 1 ) + (1 λ) m ]) q LT
No-arbirage case m = E ( Λ+1 Λ [ ( 1 c Sw q ST Assume Λ prices all bonds ) ]) 1 θ + (1 λ) m +1 Then m = ( ) c Sw + λ Ω ({1}) 1 Ω ({θ }) wih Ω ({x }) = (1 λ) j Λ +1+j E x +j. Λ j=0
(A-marke) swap rae y Sw 0 =, ( ) y Sw + λ y Sw = 1 + Ω ({θ }) λ Ω (1) Ω (1) 1 Ω ({θ }) Price of long-erm bond ( ) q LT = c LT + λ Ω (1) Yield of long-erm bond y LT 1 = y LT =, ( ) y LT + λ Ω (1) 1 Ω (1) λ
No-arbirage case If θ = θ hen If θ 0, y Sw y LT = Ω ({θ }) Ω (1) y Sw y Sw y LT = θ y LT 0
Swap pricing wih very srong fricions Assume holding cos parameer, κ LT and κ ST, very large Consan endowmen of oher profis, π (z) = π Zero-ne demand for swaps, d (z) = 0 No-inflaion uncerainy, µ (z) = µ Then, swap price equals m = βe ([ c Sw y LIB + (1 λ) m +1 ]) Seing m = 0 defines he swap rae as y Sw ( ) = w j E y+j 1 LIB j=1
Taking uncondiional expecaion ( ) ( ) ( E = E = E y Sw y LIB y ST ) + E (θ ) Bond yield ( E y LT ) ( = E y ST ) + E (τ ) Swap spread ( E y Sw ) ( E y Sw y LT ) = E (θ ) E (τ ) Hisorically, E (θ ) = 0.6% and E (τ ) = 1.7%, so ha ( ) ( ) E E = 1.1% y LT
Swap pricing wih fricions from model Dealer s firs-order condiions q ST = E ( Λ+1 Λ ) κ ST α ST +1 q LT ( Λ+1 [ ] ) = E c LT + λ + (1 λ) q+1 LT κ LT α LT +1 Λ Implied swap pricing y Sw y LT = Ω ({θ }) Ω + Ω (1) Term spread, q LT ( κ ST {α ST +1} κ LT q LT Ω (1), { α LT +1}, Swap spread { } ) Λ Λ α LT +1 +1
Calibraion VAR(1) for four dimensional sae vecor Transiion: Var-Cov: 10 6 [y ST (z), τ (z), µ (z), θ (z)].91 0.07 0 0.87 0 0 0 0.76 0 0 0.06.72 5.1 3.4 4.3 0 3.2 2.6 0 24.8 1.1 0.5
Calibraion Symbol Parameer Value ȳ ST Shor rae level 0.01156 τ Term spread level 0.00429 µ Inflaion level 0.00938 θ TED spread level 0.00158 γ Risk aversion 2 ν Discoun elasiciy 1 1/λ Mauriy of long-erm deb and swap 120
Quaniaive resuls 30Y SS κ LT TED E() Sd() E(α LT ) E() Daa 7/1997 9/2008 57 27 58 10/2008 10/2015 18 12 35 Model κ LT = 0.0001 62 8 21 63 κ LT = 0.0025 9 55 96 63 κ LT = 0.01 54 89 119 63 Pos 10/2008 TED 18 41 80 35 and κ LT = 0.0014 High ri. av., γ = 4 27 60 113 63 Low disc.elas., ν = 0.8 17 61 101 63 ST deb cos, κ ST = 0.0025 9 75 97 63 Consan TED 16 62 96 63 Consan Inflaion 17 57 107 63
Holding cos esimae Enhanced Supplemenary Leverage Raio (US G-SIB) of 5% or 6% Wih κ ST = 0, ( E y Sw 5% 10% = 50, or 60 basis poins. +E wih χ 1 ) y LT = Eθ κ LT Eα LT +1 χ Ω ({θ E θ } κ LT q LT { Λ Λ (α LT +1 +1 E αlt +1) }) Ω (1)
Demand effecs and Swap holding coss Swap Demand LT bond Swap cos rae sensiiviy posiion impac Benchmark (κ LT = 0.0025) d = 0 6.17 1.23 d =.2 6.04.88 1.33 d =.2, κ Sw =.005 5.64.4056 Low fricion (κ LT = 0.0001) d = 0 7.00 3.52 d =.2 6.99.14 3.72 d =.2, κ Sw =.005 6.58.4056 High fricion (κ LT = 0.01) d = 0 5.67.46 d =.2 5.40 1.28.51 d =.2, κ Sw =.005 5.00.4028
Leverage consrain max (α ST q ST (z), 0) + max (α LT q LT (z), 0) ξ ω E(30Y SS) freq(ss<0) freq(η > 0) ξ = 62 0 0 ξ = 20 48.001.05 ξ = 10 19.20.14 ξ = 5 25.66.29
10/2008-2018: Swap Mauriy 2 5 10 20 30 TERM 0.045 0.008 0.116 0.115 0.187 TED 0.337 0.179 0.102 0.285 0.253 MBSD 0.026 0.015 0.067 0.065 0.100 3MTB 0.151 0.154 0.011 0.022 0.036 VIX 0.003 0.003 0.002 0.000 0.001 R2adj 0.56 0.38 0.28 0.32 0.31
1997-2018: Swap Mauriy 2 5 10 20 30 TERM 0.036 0.044 0.157 0.128 0.205 TED 0.220 0.089 0.025 0.012 0.007 MBSD 0.030 0.043 0.088 0.058 0.104 3MTB 0.012 0.000 0.083 0.096 0.100 VIX 0.004 0.006 0.002 0.000 0.000 R2adj 0.50 0.28 0.22 0.04 0.18
Conclusion Model for swaps wih limied arbirage Wih reasonable fricions on bond holdings ge negaive swap spreads Negaive swap spreads even wihou explici demand effecs