A Model of the Reserve Asset - PDF Free Download

A Model of the Reserve Asset Zhiguo He (Chicago Booth and NBER) Arvind Krishnamurthy (Stanford GSB and NBER) Konstantin Milbradt (Northwestern Kellogg and NBER) July 2015 ECB 1 / 40

Motivation US Treasury bonds have been the world reserve asset for a long time Safe asset portfolios tilted towards US Treasury bonds Convenience yield on US Treasury bonds; Higher premium in bad states ( negative β ) Despite increasing size of US debt, reserve asset status has persisted Despite deteriorating US fiscal position, US interest rates remain low German bund occupies a similar position in the Euro area History Gold; UK consol bond pre-wwi; Joint reserve asset: UK and US in interwar period Pre 2009 in Europe: German, French, Italian bonds are all negative β Since crisis, many proposals to create a Euro-wide government bond, to serve as a Euro reserve asset But no models of a reserve asset, so no framework to analyze different Eurobond designs 2 / 40

Literature International finance, economic history literature on reserve currency Eichengreen (many), Krugman (1984), Frankel (1992) Store of value, medium of exchange, unity of account, multiple equilibria Shortages of store of value Safe assets literature (Holmstrom and Tirole, 1998, Caballero, Farhi, Gourinchas 2008, Caballero and Krishnamurthy 2009, Krishnamurthy and Vissing-Jorgensen 2012) Multiplicity: Samuelson (1958) on money, Diamond (1965) on govt debt No formal models of endogenous determination of which asset is chosen as store of value Sovereign debt rollover risk and global games Calvo (1988), Cole and Kehoe (2000); Morris and Shin (1998), He and Xiong (2012) Highlight strategic substitution in asset market 3 / 40

Model Setup Investors (j): Measure 1 + f of investors with one unit of funds each Risk neutral, each investor must invest his funds in sovereign debt Countries/debt (i): Two countries, debt size s 1 = 1 and size s 2 = s < 1 Fundamental ( surplus ) s i θ i Foreign denominated debt: true surplus plus foreign reserves Domestic denominated debt: true surplus plus any resources CB is willing to provide to forestall a rollover crisis Debt of face value of s i (exogenous) issued at endogenous price p i Default if surplus plus bond proceeds insufficient for obligations s i θ i + s i p }{{} i < s }{{} i total funds available debt obligations Given price p i, default decision depends on θ i Recovery in default = 0 4 / 40

Multiple equilibria in a special case No default if, s i p i s i (1 θ i ) Suppose sufficient funding for both countries: 1 + f (1 θ }{{} 1 ) + s 2 (1 θ 2 ) }{{} total funds available funding needs (1) Possible equilibria 1. Country 1 is safe (=reserve asset), country 2 defaults: p 1 = 1 + f, p 2 = 0 Investor return = 1 1 + f 2. Country 2 is safe (=reserve asset), country 1 defaults: p 1 = 0, s p 2 = 1 + f Investor return = s 1 + f 3. Both countries safe, two reserve assets 5 / 40

Coordination and multiple equilibria No default if, s i p i s i (1 θ i ) Suppose sufficient funding for both countries: 1 + f (1 θ }{{} 1 ) + s 2 (1 θ 2 ) }{{} total funds available funding needs Possible equilibria 1. Country 1 is safe (=reserve asset), country 2 defaults: p 1 = 1 + f, p 2 = 0 Investors return = 1 1 + f 2. Country 2 is safe (=reserve asset), country 1 defaults: p 1 = 0, s p 2 = 1 + f Investors return = s 1 + f 3. Both countries safe, two reserve assets 4. If s 2 = 0, country 1 is safe (Japan?) 6 / 40

Full characterization as function of s, f and θ Global games technique: Unobserved relative fundamentals (higher δ means country 1 is stronger): 1 θ 1 = (1 θ) exp ( δ ) ; 1 θ 2 = (1 θ) exp ( + δ ). Each investor receives a noisy private signal before investing δ j = δ + ɛ j Take δ [ δ, δ ] (any cdf, but wide support) and noise ɛ j U [ σ, σ] We will (mostly) look at σ 0: fundamental uncertainty vanishes, but strategic uncertainty remains Timing assumptions: Investors place market orders to buy debt Country default decision after orders are submitted 7 / 40

Returns for δ = 0 (Rollover risk) Return to investing in bonds 1 1 θ Country 1 return Country 2 return 1 1 + f s 1 + f 0 s 1 θ 1 + f 1 θ 1 + f measure of agents investing in country Given proportion x investing in country 1, no default if: x 1 θ ( ) 1 δ (country 1) 1 x s ( ( 1 θ )) 2 δ 1 + f 1 + f (country 2) 8 / 40

Returns for δ = 0 (Liquidity) Return to investing in bonds 1 1 θ Country 1 return Country 2 return 1 1 + f s 1 + f 0 s 1 θ 1 + f 1 θ 1 + f measure of agents investing in country Liquidity/market depth: country 2 price-rises/return-falls faster Given proportion x investing in country 1, conditional returns are 1 p 1 = 1 (1 + f ) x and 1 p 2 = s (1 + f ) (1 x) 9 / 40

Strategy space Threshold Equilibrium: Let φ (δ j ) be investment in country 1 of agent with signal δ j Consider threshold strategies If δ j > δ invest in country 1 i.e. φ=1; otherwise country 2 i.e. φ=0 The equilibrium cutoff δ is determined by indifference of marginal investor with signal δ j = δ Must be indifferent between investing in country 1 versus 2 How restrictive are threshold strategies? We can prove that the threshold equilibrium is the unique equilibrium among monotone strategies φ ( ), given some parameter restrictions For non-monotone strategies, other equilibria might exist (see later) 10 / 40

Expected returns Marginal investor δ j = δ does not know other investors signals He asks, suppose fraction x [0, 1] of investors have signals> δj Marginal agent backs out true δ for given x as follows δ = δ + (2x 1)σ Take σ 0 so only strategic uncertainty remains... Global games result: x U [0, 1] from the view of marginal investor, for any prior of δ Integrating over possible values of x U [0, 1] gives expected profits 1 1 Π 1 = (1 + f ) x dx = 1 ( ln 1 + f ) 1 + f 1 θ + δ and Π 2 = (1 θ)e δ 1+f 1+f s(1 θ)e δ 1+f 0 s (1 + f ) (1 x) dx = s 1 + f ( ln s + ln 1 + f ) 1 θ δ 11 / 40

Expected returns Return to investing in bonds 1 1 θ Country 1 return Country 2 return 1 1 + f s 1 + f 0 s 1 θ 1 + f 1 θ 1 + f measure of agents investing in country For marginal agent, proportion of investors in country 1 is x: Π1 = Integral under green curve Π2 = Integral under red curve 12 / 40

Threshold Threshold δ is determined by the indifference condition Π 1 (δ ) = Π 2 (δ ) Solving for δ (recall that s (0, 1]) δ = 1 s 1 + s z s ln s + 1 + s }{{}}{{} negative, liquidity positive, rollover (2) where we define aggregate funding conditions z ln 1 + f 1 θ > 0 (3) High z means high savings ( savings glut ), good average fundamentals. Liquidity effect dominates Low z is opposite δ : lowest value of δ so that country 1 s bonds are reserve asset 13 / 40

Graphically δ as function of country 2 size Country 1 is reserve asset if fundamental δ > δ δ < 0 implies that the large country enjoys premium 14 / 40

Graphically δ as function of country 2 size Country 1 is reserve asset if fundamental δ > δ δ < 0 implies that the large country enjoys premium 15 / 40

When will world switch? In high z world (savings glut) US Treasury size: Debt = $12.7tn, (CB money $4.6tn) : maximum liquidity for the world Even if US fiscal position is worse than others (i.e. δ < 0)... Switch not on the horizon Unless macro moves to low z world US Treasury size becomes a concern can the country rollover such a large debt? Investors may start coordinating on countries with (a bit) smaller debt size Germany? Debt = $1.5tn 16 / 40

Era of UK consol bond: Size helps 6 200.0 180.0 5 160.0 4 140.0 120.0 3 100.0 80.0 2 60.0 1 40.0 20.0 0 0.0 1850 1860 1870 1880 1890 1900 1910 1920 UK Consol Yield UK Debt/UK GDP US Debt/UK GDP UK government debt was reserve asset until sometime after WWI US GDP exceeds UK GDP by 1870 In 1890, UK Govt Debt 3 US Govt Debt UK Debt/GDP = 0.43 vs. US Debt/GDP = 0.10, so US is safer 17 / 40

Relative fundamentals Relative fundamentals/ge in safe assets is central to our model Take model with no coordination, where repayment is equal to surplus (θ) and world interest rate is normalized to one. Our model (for δ = 0) p 1 = E [min (θ 1, 1)], p 2 = E [min (θ 2, 1)] θ 1 > θ 2 p 1 = 1 + f, p 2 = 0 θ 1 < θ 2 p 2 = 1 + f, p 1 = 0 US fiscal position is weaker now than before, but still better than everyone else Same for Germany within Eurozone 18 / 40

Negative β Take an extreme case where country 1 is a.s. reserve asset, δ >> δ Say s = 1 and suppose there is some recovery in default l i Country 1 bond price and return (R) p 1 = 1 + f p 2 R = 1 p 1 = 1 1+f p 2 Country 2 bond price p 2 = l 2 R (no arbitrage, offering the same return) Solving: p 1 = 1 + f and p 2 = 1 + l 1+l 1+f 2 2 l 2 Shocks to l 1 has no effect on anything, but l 2 p 1 and p 2 Say shocks to average fundamentals hurt l 1, l 2 equally: Reduces p2, increases p 1 Reserve asset has negative β 19 / 40

Negative β β 1 θ U[0.1,0.6], s=0.9, f=0.1, l=0.7 2 1 0.015 0.020 0.025 0.030 0.035 0.040 0.045 δ -1-2 -3 Country 1 β 1 = Cov(p 1,θ) Var(θ), as function of relative fundamental δ. 20 / 40

Switzerland? What if there were full-commitment reserve assets available to investors? Switzerland: Debt = $127bn, (CB money $500bn) Denmark: Debt = $155bn US: Debt = $12.7tn, (CB money $4.6tn) Implicit assumption in our analysis is that substantially all of reserve asset demand is satisfied by debt subject to rollover risk Say s is quantity of full-commitment assets and define ˆf f p s s In equilibrium ps is set using the expected return from investing in country 1/country 2. Otherwise, model is as analyzed based on total demand of ˆf 21 / 40

Two reserve assets ( joint safety ) Monotone threshold strategies, only one reserve asset φ(δj ) = 1 if δ j > δ, otherwise 0 (recall φ investment in country 1) 22 / 40

Two reserve assets ( joint safety ) Monotone threshold strategies, only one reserve asset φ(δj ) = 1 if δ j > δ, otherwise 0 (recall φ investment in country 1) If we allow for non-monotone oscillating strategies: φ(δj ): 0,1,0,1,0,1,... in a non-monotone fashion (not quite mixing, but very similar) Then, for high z > zhl, joint safety for values of δ in GRAY s=0.25 1.0 0.5 z HL δ * 0.0-0.5-1.0 0.0 0.5 1.0 1.5 2.0 z 22 / 40

Sovereign choices Debt size (s), fundamentals (θ), are choice variables Externalities in model Role for coordination Security design as coordination 23 / 40

Eurobonds and coordination Policy proposals to create a Euro-area reserve asset Proceeds to all countries, so all countries get some seignorage Flight to quality is a flight to all, rather than just German Bund We study: Countries issue two bonds: A common bond in α share An individual bond in (1 α) share Common bond is pooled bond (essentially a bundle ), for which each country is responsible for paying its respective share of the obligation No cross-default provisions (structure is closest to Euro-Safe-Bonds, or ESBies) We set aside moral hazard considerations which are likely first-order 24 / 40

Common bond and individual bonds Two-stage game Stage 1: issuance of common bonds Countries issue common bonds: α (large) and αs (small) Investors pay f ˆf, so common bond price Split proceeds αp c s i s i +s i p c = f ˆf α (1 + s) Stage 2: issuance of individual bonds Investor gets signal δ j Individual country bonds issued at prices p 1 and p 2 Investors invest remainder of funds 1 + ˆf into individual (non-bundled) bonds 25 / 40

Common bond and individual bonds If country i defaults, it does so on both individual and portion of common bond New no-default condition: (1 α)p 1 + θ 1 + Common bond proceeds {}}{ αp c 1 (1 α)p 2 + θ 2 + αp c }{{} Common bond proceeds 1 Importantly, common bond proceeds are allocated in a state-independent way across the two countries Contrast this with the winner takes all funding provided by the individual bonds; this is a state-dependent allocation 26 / 40

Why might this work? In basic model (α = 0) no default if, s i p i s i (1 θ i ) Suppose global funds exceeds funding need: 1 + f (1 θ }{{} 1 ) + s 2 (1 θ 2 ) }{{} total funds available sum of individual funding needs Multiple equilibria... When α = 1, neither country defaults if, 1 + f (1 θ }{{} 1 ) + s 2 (1 θ 2 ) }{{} total funds available funding need of common bond Security design coordinates investor actions Flight to the reserve asset generates stable funding for both countries 27 / 40

Common bond equilibrium Stage 2 game: investors with ˆf Default conditions for each country, and individual bond prices pi Almost same as previous analyses Stage 1 game sets investment in common bond f ˆf based on: E[R c ] = E[R stage2 ] 28 / 40

Equilibrium as function of α s=0.5_z=1. δ * α HL α * 0.4 0.2 0.0-0.2 0.2 0.4 0.6 0.8 1.0 α -0.4 High α > α joint safety equilibrium always Low α < α HL single reserve asset, threshold equilibrium For α [α HL, α ] both equilibria are possible 29 / 40

Conclusion US government debt is reserve asset because: Good relative fundamentals Debt size is large, and world is in high demand for reserve asset (savings glut) Nowhere else to go Economics of reserve asset suggest that there can be gains from coordination Eurobonds as coordinated security-design We also discuss the country s incentives to change its bond issuance size 30 / 40

More details on the oscillating equilibrium The non-monotone equilibrium: Let φ (y) be investment in country 1 of agent with signal y Consider φ (y) oscillating on (δ L, δ H ) 0, y < δ L 1, y [δ L, δ L + kσ] 0 y [δ L + kσ, δ L + 2σ] φ (y) = 1, y [δ L + 2σ, δ L + (2 + k) σ].. 0, y [δ H (2 k) σ, δ H ] 1, y > δ H Suppose joint safety for δ (δl, δ H ). Then a no-arbitrage condition between bond prices has to hold k = 2 1+s 31 / 40

An oscillating strategy 1.0 0.8 0.6 0.4 0.2 ϕ(δ) -0.6-0.4-0.2 0.2 δ Example: z = 1, s = 1/4 Thresholds: δ L = 0.37, δ H = 0.12 Full oscillations on (δ L, δ H ): n = 2 Joint safety: no-arbitrage condition requires investment splits of 1 s 1+s = 80% to country 1 and 1+s = 20% to country 2 on (δ L, δ H ) Let us look at the incentives of different investors... 32 / 40

An oscillating strategy x=0. ϕ(δ) 1.0 0.8 0.6 0.4 0.2-0.6-0.4-0.2 0.2 δ Consider marginal agent δ j = δ L who thinks he has the most positive signal, i.e. x = 0 Investment in country 1 conditional on x, δ+x2σ increasing in x δ (1 x)2σ φ(y) 2σ dy, 33 / 40

An oscillating strategy x=0.5 ϕ(δ) 1.0 0.8 0.6 0.4 0.2-0.6-0.4-0.2 0.2 δ Consider marginal agent δ j = δ L who thinks he has the median signal, i.e. x = 1/2 Investment in country 1 conditional on x, δ+x2σ increasing in x δ (1 x)2σ φ(y) 2σ dy, 34 / 40

An oscillating strategy x=1. ϕ(δ) 1.0 0.8 0.6 0.4 0.2-0.6-0.4-0.2 0.2 δ Consider marginal agent δ j = δ L who thinks he has the most negative signal, i.e. x = 1 Investment in country 1 conditional on x, δ+x2σ increasing in x δ (1 x)2σ φ(y) 2σ dy, 35 / 40

An oscillating strategy x=0. ϕ(δ) 1.0 0.8 0.6 0.4 0.2-0.6-0.4-0.2 0.2 δ Consider interior agent δ j = δ L + kσ who thinks he has the most positive signal, i.e. x = 0 Investment in country 1 conditional on x: δ+x2σ φ(y) δ (1 x)2σ 2σ dy = 1+s 1 36 / 40

An oscillating strategy x=0.5 ϕ(δ) 1.0 0.8 0.6 0.4 0.2-0.6-0.4-0.2 0.2 δ Consider interior agent δ j = δ L + kσ who thinks he has the median signal, i.e. x = 1/2 Investment in country 1 conditional on x: δ+x2σ φ(y) δ (1 x)2σ 2σ dy = 1+s 1 37 / 40

An oscillating strategy x=1. ϕ(δ) 1.0 0.8 0.6 0.4 0.2-0.6-0.4-0.2 0.2 δ Consider interior agent δ j = δ L + kσ who thinks he has the most negative signal, i.e. x = 1 Investment in country 1 conditional on x: δ+x2σ φ(y) δ (1 x)2σ 2σ dy = 1+s 1 Interior agent s expectation of investment in country 1 independent of relative position x 38 / 40

An oscillating strategy: Incentives Π 1 (δ)-π 2 (δ) -0.6-0.4-0.2 0.2 δ 1.0 0.5-0.5 1.0 0.8 0.6 0.4 0.2 ϕ(δ) -0.6-0.4-0.2 0.2 δ Indifference only on (δ L + kσ, δ H (2 k) σ) 39 / 40

An oscillating strategy: Equilibrium Indifference of marginal agent δ L requires Π 2 (δ L ) = Π 1 (δ L ): Country 2 safe for sure, so always certain return Country 1 not always safe in the eyes of marginal agent (strategic uncertainty), so require higher expected return conditional on survival Solving, we have { s s δ L = z ln (1 + s) (1+s) { } 1 δ H = z + ln (1 + s) 1+s s Note that both δ H and δ L are independent of σ (need specific σ s to make sure that there exists n s.t. δ H = n 2σ + k) As we are taking σ 0, this is wlog Define unique z HL so that δ L (z HL ) = δ H (z HL ). Then oscillating equilibrium possible for z > z HL. } 40 / 40