a model of safe asset determination1.negative b of safe asset 2.euro-bonds: i german bund currently...

A Model of Safe Asset Determination

Zhiguo He (Chicago Booth and NBER)

Arvind Krishnamurthy (Stanford GSB and NBER)

Konstantin Milbradt (Northwestern Kellogg and NBER)

February 2016∼75min

1 / 45

Motivation

0

1

2

3

4

5

63m OIS

3m Tbill

Fed Funds Target

I US Treasury bonds have been the world safe asset for a long time

I Safe asset portfolios tilted towards US Treasury bondsI “Convenience yield” on US Treasury bondsI Higher premium in bad states (“negative β”) & flight to quality

I Persists despite a high US debt/GDP ratio

I Main idea: Safety is endogenous—when investors believe an assetis safe, their actions can make that asset safe

2 / 45

Preview

Model:I Lack of common knowledge to model coordination gameI Key trade-off between

I default/rollover risk (strategic complementarity): more investorsbuying bond makes country safer, so repayment more likely

I market depth/liquidity (strategic substitutability): for fixed issuancesize, more investors buying lowers returns conditional on repayment

I Main results: for safe asset,I size (float) matters and can lead to an advantage, andI relative rather than absolute fundamentals matter

Some Applications:

1. Negative β of safe asset2. Euro-bonds:

I German bund currently safe asset within the Euro areaI Many proposals to create a Euro-bond to serve as a Euro safe assetI Euro-bonds can help overcome coordination problems

3. Strategic behavior in debt issuance to influence safe-asset statusI If size advantageous, possible rat-race to issue more and more bonds

3 / 45

Literature

I International finance, economic history literature on reserve currencyI Eichengreen (many), Krugman (1984), Frankel (1992)I Store of value, medium of exchange, unity of account, multiple

equilibria

I Shortages of store of valueI Safe assets literature (Holmstrom and Tirole, 1998, Caballero, Farhi,

Gourinchas 2008, Caballero and Krishnamurthy 2009, Krishnamurthyand Vissing-Jorgensen 2012)

I Multiplicity: Samuelson (1958) on money, Diamond (1965) on govtdebt

I No formal models of endogenous determination of which asset ischosen as store of value

I Sovereign debt rollover risk and global gamesI Calvo (1988), Cole and Kehoe (2000), Morris and Shin (1998), He

and Xiong (2012), Milbradt and Cheng (2012)I Highlight strategic substitution in asset market

I Complement to Goldstein and Pauzner (2005) in bank runs

4 / 45

Model Setup

Investors (j):I Measure 1 + f of investors with one unit of funds eachI Risk neutral, each investor must invest his funds in sovereign debt

Countries/debt (i):I Two countries, debt size s1 = 1 and size s2 = s < 1I Debt of face value of si (exogenous) issued at endogenous price pi

Default:I Default if surplus plus bond proceeds insufficient for obligations

si θi + sipi︸︷︷︸total funds available

< si︸︷︷︸debt obligations

⇒ Given price pi , default decision depends on θiI Fundamental (“surplus”) si θi

I Foreign denominated debt: true surplus plus foreign reservesI Domestic denominated debt: true surplus plus any resources CB is

willing to provide to forestall a rollover crisis

I Recovery in default=0; Static model; Real model

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Model Setup

Investors (j):I Measure 1 + f of investors with one unit of funds eachI Risk neutral, each investor must invest his funds in sovereign debt

Countries/debt (i):I Two countries, debt size s1 = 1 and size s2 = s < 1I Debt of face value of si (exogenous) issued at endogenous price pi

Default:I Default if surplus plus bond proceeds insufficient for obligations

si θi + sipi︸︷︷︸total funds available

< si︸︷︷︸debt obligations

⇒ Given price pi , default decision depends on θiI Fundamental (“surplus”) si θi

I Foreign denominated debt: true surplus plus foreign reservesI Domestic denominated debt: true surplus plus any resources CB is

willing to provide to forestall a rollover crisis

I Recovery in default=0; Static model; Real model5 / 45

Multiple Equilibria in a Special Case

I No default (i.e. repayment of 1 for each unit of bond) if,

sipi ≥ si (1− θi )

I Suppose sufficient funding for both countries:

1 + f︸︷︷︸total funds available

≥ (1− θ1) + s(1− θ2)︸︷︷︸funding needs

Possible equilibria1. Country 1 is safe (=safe asset), country 2 defaults:

p1 = 1 + f , p2 = 0 Investor return =1

1 + f

2. Country 2 is safe (=safe asset), country 1 defaults:

p1 = 0, s · p2 = 1 + f Investor return =s

1 + f

3. Both countries safe, two safe assets with equalized returns4. If s = 0, country 1 is safe (Japan?)

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1 + f



1 + f

3. Both countries safe, two safe assets with equalized returns

4. If s = 0, country 1 is safe (Japan?)

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1 + f



1 + f

3. Both countries safe, two safe assets with equalized returns4. If s = 0, country 1 is safe (Japan?) 6 / 45

Lack of Common Knowledge & Coordination Failure

Lack of common knowledge:

I Unobserved relative fundamentals (higher δ̃⇒ country 1 stronger):

1− θ1 = (1− θ) exp(−δ̃)

;

1− θ2 = (1− θ) exp(+δ̃)

.

I Each investor receives a noisy private signal before investing

δj = δ̃ + εj

I Take δ̃ ∈[−δ̄,δ̄

](any cdf, but wide support) & εj ∼ U [−σ, σ]

I We will (mostly) look at σ→ 0: fundamental uncertainty vanishes,but strategic uncertainty remains (i.e. relative position of signal)

Timing assumptions:

I Investors place market orders to buy debt

I Country default decision after orders are submitted

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Returns for δ̃ = 0 (Default or Rollover Risk)

measure of agents investing in country

0

11 − 𝜃

1 − 𝜃1 + 𝑓

𝑠 1 − 𝜃1 + 𝑓

11 + 𝑓

𝑠1 + 𝑓

Country 1 return

Country 2 return

Return to investing in bonds

I Given proportion x investing in country 1, no default if:

x ≥1− θ1

(δ̃)

1 + f(country 1) 1− x ≥

s(1− θ2

(δ̃))

1 + f(country 2)

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Returns for δ̃ = 0 (Liquidity)


0

11 − 𝜃

1 − 𝜃1 + 𝑓

𝑠 1 − 𝜃1 + 𝑓

11 + 𝑓

𝑠1 + 𝑓

Country 1 return

Country 2 return


I Liquidity/market depth: country 2 price-rises/return-falls fasterI Given proportion x investing in country 1, conditional returns are

1

p1=

1

(1 + f ) xand

1

p2=

s

(1 + f ) (1− x)

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Strategy Space

Threshold Equilibrium:

I Let φ (δj ) be investment in country 1 of agent with signal δjI A threshold strategy is given by

If δj > δ∗ invest in country 1 i.e. φ=1; otherwise country 2 i.e. φ=0

I The equilibrium cutoff δ∗ is determined by indifference of marginalinvestor with signal δj = δ∗

How restrictive are threshold strategies?

I We can prove that the threshold equilibrium is the uniqueequilibrium among monotone strategies φ (·) (i.e., φ′ (·) ≥ 0)

I For non-monotone strategies, other equilibria might exist (we willcome back to this)

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Indifference & Expected Returns

Marginal Investor Indifference:

I Marginal investor δj = δ∗ does not know other investors’ signals

I He asks, suppose fraction x ∈ [0, 1] of investors have signals> δjI Marginal agent backs out true δ̃ for given x as follows

δ̃ = δ∗ + (2x − 1)σ

I When σ→ 0 only strategic uncertainty (uncertainty about relativeposition x) remains

I Global games result: x ∼ U [0, 1] from the view of marginal investor,for any prior of δ̃

Expected returnsI Integrating over possible values of x ∼ U [0, 1] gives expected profits

Π1 =∫ 1

(1−θ)e−δ∗1+f

1

(1 + f ) xdx =

1

1 + f

(ln

1 + f

1− θ+ δ∗

)

Π2 =∫ 1+f−s(1−θ)eδ∗

1+f

0

s

(1 + f ) (1− x)dx =

s

1 + f

(− ln s + ln

1 + f

1− θ− δ∗

)

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Indifference & Expected Returns

Marginal Investor Indifference:

I Marginal investor δj = δ∗ does not know other investors’ signals

I He asks, suppose fraction x ∈ [0, 1] of investors have signals> δjI Marginal agent backs out true δ̃ for given x as follows

δ̃ = δ∗ + (2x − 1)σ

I When σ→ 0 only strategic uncertainty (uncertainty about relativeposition x) remains

I Global games result: x ∼ U [0, 1] from the view of marginal investor,for any prior of δ̃

Expected returnsI Integrating over possible values of x ∼ U [0, 1] gives expected profits

Π1 =∫ 1

(1−θ)e−δ∗1+f

1

(1 + f ) xdx =

1

1 + f

(ln

1 + f

1− θ+ δ∗

)

Π2 =∫ 1+f−s(1−θ)eδ∗

1+f

0

s

(1 + f ) (1− x)dx =

s

1 + f

(− ln s + ln

1 + f

1− θ− δ∗

)11 / 45

Expected Returns


0

11 − 𝜃

1 − 𝜃1 + 𝑓

𝑠 1 − 𝜃1 + 𝑓

11 + 𝑓

𝑠1 + 𝑓

Country 1 return

Country 2 return


I For any agent, expected returns linked to graph:

I Π1 = Integral under green curveI Π2 = Integral under red curve

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Equilibrium Threshold

I Threshold δ∗ characterized by equalized expected returns

Π1 (δ∗) = Π2 (δ

∗)

I Solving for δ∗ (recall that s ∈ (0, 1])

δ∗ = −1− s

1 + s· z︸︷︷︸

negative, liquidity

+−s ln s

1 + s︸︷︷︸positive, rollover

where we define aggregate funding conditions

z ≡ ln1 + f

1− θ> 0

I High z means high savings (“savings glut”), good averagefundamentals

I The lower δ∗, the ex-ante safer is country 1

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High z World: Liquidity Effect dominates

I Country 1 is safe asset if fundamental δ̃ > δ∗

I δ∗ < 0 implies that the large country enjoys premium

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Low z World: Rollover Risk Effect dominates

I Country 1 is safe asset if fundamental δ̃ > δ∗

I δ∗ < 0 implies that the large country enjoys premium

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When will world switch?

Currently: In high z world (savings glut)

I US Treasury size: Debt = $12.7tn, (CB money ≈ $4.6tn) :maximum liquidity for the world

I Even if US fiscal position is worse than others (i.e. δ∗ < 0)I ... Switch not on the horizon

Unless macro moves to low z world

I US Treasury size becomes a concern – can the country rollover sucha large debt?

I Investors may start coordinating on countries with (a bit) smallerdebt size

I Germany? Debt = $1.5tn

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A Historical Perspective: Size helps

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

160.0

180.0

200.0

0

1

2

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1850 1860 1870 1880 1890 1900 1910 1920UK Consol Yield UK Debt/UK GDP US Debt/UK GDP

I UK government debt was safe asset until sometime after WWII US GDP exceeds UK GDP by 1870I In 1890, UK Govt Debt ≈3 × US Govt DebtI UK Debt/UK GDP = 0.43 vs. US Debt/US GDP = 0.10 (so safer)

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Relative Fundamentals vs Absolute Fundamentals

Relative fundamentals/GE in safe assets is central to our model

I Take model with no coordination, where repayment is equal tosurplus (θ) and world interest rate is normalized to zero.

p1 = E [min (θ1, 1)] , p2 = E [min (θ2, 1)]

I Our model without safe alternative (e.g. for threshold δ∗ = 0)

θ1 > θ2 ⇒ p1 = 1 + f , p2 = 0θ1 < θ2 ⇒ s · p2 = 1 + f , p1 = 0

I US fiscal position is weaker now than before, but still better thaneveryone else

I Same for Germany within Eurozone

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Negative β

Introducing recoveryI Take an extreme case where country 1 is a.s. safe asset, δ̃� δ∗

I Say s = 1 & suppose recovery in default is liI Country 1 bond price and return (R)

p1 = 1 + f − p2 R = 1p1

= 11+f−p2

I Country 2 bond price p2 = l2R (has to offer same return)

I Solving:

p1 =1 + f

1 + l2and p2 =

1 + f

1 + l2l2

Shocks to recovery values:I Shock to l1 ↓ has no effect on anything, but l2 ↓⇒ p1 ↑ and p2 ↓I Say shocks to average fundamentals hurt l1, l2 equally:

I Reduces p2, increases p1 ⇒ Safe asset has negative β

I Lehman shock: Negative shock to US and world fundamentalsI Treasury yields fall (alternatives rise)

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Negative β: Uncertainty about θ

0.015 0.020 0.025 0.030 0.035 0.040 0.045δ

-3

-2

-1

1

2

β1

θ∼U[0.1,0.6], s=0.9, f=0.1, l=0.7

Country 1 β1 = Cov (p1,θ)Var (θ)

, as function of relative fundamental δ.

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Really safe assets OR What about Switzerland?

“Full-commitment” assets:

I What if there were “full-commitment” safe assets available?

I Switzerland: Debt = $127bn, (CB money ≈ $500bn)I Denmark: Debt = $155bn

I US: Debt = $12.7tn, (CB money ≈ $4.6tn)

I Implicit assumption in our analysis is that substantially all of safeasset demand is satisfied by debt subject to rollover risk

Tweaking the model:

I Say s is quantity of “full-commitment” assets and define

f̂ ≡ f − pss

I Suppose s = 1, so return of “winner” is 11+f̂

I Price ps set using the expected return from investing in country 1,2

I Thus, same analysis as before with adjusted total demand of f̂

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Mixed strategies & Joint Safety

I Monotone strategies: Only one safe asset with threshold equilibriumI φ(δj ) = 1 if δj > δ∗, otherwise 0 (recall φ investment in country 1)

I Non-monotone strategies: Two safe assets (joint safety) possible⇒ Approximate mixed strategies via “oscillating” strategy:

I φ(δj ): 0,1,0,1,0,1,... in a non-monotone fashion (approximates“mixing” strategies when σ→ 0)

I Then, for high z > zHL, joint safety for values of δ̃ in GRAY

0.0 0.5 1.0 1.5 2.0-1.0

-0.5

0.0

0.5

1.0

z

δ*

s=0.25

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Mixed strategies & Joint Safety

I Monotone strategies: Only one safe asset with threshold equilibriumI φ(δj ) = 1 if δj > δ∗, otherwise 0 (recall φ investment in country 1)

I Non-monotone strategies: Two safe assets (joint safety) possible⇒ Approximate mixed strategies via “oscillating” strategy:

I φ(δj ): 0,1,0,1,0,1,... in a non-monotone fashion (approximates“mixing” strategies when σ→ 0)

I Then, for high z > zHL, joint safety for values of δ̃ in GRAY

0.0 0.5 1.0 1.5 2.0-1.0

-0.5

0.0

0.5

1.0

z

δ*

s=0.25

zHL

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Mixed strategies & Joint Safety: Discussion

Interpretation:

I In limit, “oscillating” strategy equivalent to everyone on [δL, δH ]investing φ = 1

1+s in country 1

I Mixed strategy a quite natural outcome

I Everything “normal” (both countries survive) for δ̃ ∈ [δL, δH ]I Only in extreme cases δ̃ /∈ [δL, δH ] are we in a “winner takes all”

world

I Mixed strategy needs joint survival possibility (high enough z!) tomaterialize

Extensions

I Mixed strategy used for positive recovery result above

I Can also incorporate positive prices from inelastic local demand

24 / 45

Sovereign Choices

I Security design as coordination

I Debt size (s), fundamentals (θ), are choice variables

I Externalities in modelI Role for coordination

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Common Bond as Coordination Device

Policy proposals to create a Euro-area safe asset

I Proceeds to all countries, so all countries get some seignorage

I Flight to quality is a flight to all, rather than just German Bund

Countries issue two bonds:

I A common bond of total size α (1 + s) made up of shares 1/ (1 + s)large and s/ (1 + s) small bonds

I Each country issues individual bond of size (1− α)si⇒ Total amount of face-value offered still (1 + s)

I Common bond is pooled bond (essentially a size-based “bundle”),for which each country is responsible for paying only its respectiveshare

I No cross-default provisions (structure is closest to Euro-Safe-Bonds,or ESBies)

Moral hazard:

I We set aside moral hazard considerations which are likely first-order

26 / 45

Common Bond as Coordination Device

Policy proposals to create a Euro-area safe asset

I Proceeds to all countries, so all countries get some seignorage

I Flight to quality is a flight to all, rather than just German Bund

Countries issue two bonds:

I A common bond of total size α (1 + s) made up of shares 1/ (1 + s)large and s/ (1 + s) small bonds

I Each country issues individual bond of size (1− α)si⇒ Total amount of face-value offered still (1 + s)

I Common bond is pooled bond (essentially a size-based “bundle”),for which each country is responsible for paying only its respectiveshare

I No cross-default provisions (structure is closest to Euro-Safe-Bonds,or ESBies)

Moral hazard:

I We set aside moral hazard considerations which are likely first-order26 / 45

Why might this work?

I In basic model (α = 0) no default if,


I Suppose global funds exceeds funding need:


≥ (1− θ1) + s(1− θ2)︸︷︷︸sum of individual funding needs

I When α = 1, joint survival / no country defaults if


≥ (1− θ1) + s(1− θ2)︸︷︷︸funding need of common bond

I Security design coordinates investor actions

I Flight to the safe asset generates stable funding for both countriesI Note: investors cannot coordinate privately (e.g., diversified mutual

fund)

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Common Bond Equilibrium

2-Stage game:

1. Investors buy common bonds on offer with funds f − f̂ > 0

2. Investors receive private signal and use remaining funds 1 + f̂ to buyindividual bonds

Equilibrium by working backwards:

I Stage 2: investors with f̂

I Game almost same as before, except country fiscal surplus nowincludes common bonds proceeds from Stage 1

I Stage 1: Investors equalize expected returns between common bondand their informed investment in Stage 2

E[Rc ] = E[Rstage2]

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Common Bond Equilibria as Function of α

0.2 0.4 0.6 0.8 1.0α

-0.4

-0.2

0.0

0.2

0.4

δ*s=0.5_z=1.

αHL α*

I High α > α∗ ⇒ joint safety equilibrium alwaysI Low α < αHL ⇒ single safe asset, threshold equilibriumI For α ∈ [αHL, α∗] both equilibria are possible

29 / 45

Sovereign Choices

I Security design as coordination

I Debt size (s), fundamentals (θ), are choice variables

I Externalities in modelI Role for coordination

30 / 45

Debt Size & Crowding Out

I Suppose country i can choose (float) size, SiI Surplus is adjusted to θiSi - i.e. keep tax revenues to debt constant

I Given Si , easy to solve for threshold equilibrium:

δ∗ =S2 − S1S1 + S2

z +S1 lnS1 − S2 lnS2

S1 + S2

I Effect of increasing size S1 on threshold δ∗:

h(S1,S2; z) ≡ ∂δ∗ (S1,S2)

∂S1=

[S1 + S2 (lnS1 + lnS2 + 1− 2z)]

(S1 + S2)2

I Decreasing in z , negative for large z

I Expanding US debt can increase US safe asset statusI Decreases other country’s positionI Conjecture: Expansion of US debt/QE 2007-2009 precipitated

European debt crisis

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Endogenous Size Choices:

I Suppose country 1,2 have “natural” debt size (s∗1 , s∗2 )

I Country i chooses new size Si subject to adjustment costs

I Objective: reduce default probability subject to adj costs:

I Country 1:maxS1

−δ∗ (S1,S2)− c(S1 − s∗1 )

I Country 2:maxS2

δ∗ (S1,S2)− c(S1 − s∗1 ).

I Equilibrium:

h(S1,S2; z) = c ′(S1 − s∗1 ) and, h(S2,S1; z) = c ′(S2 − s∗2 ).

32 / 45

Equilibrium via Phase Diagram

I High z case; δ∗ = 0 along diagonal

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Equilibrium via Phase Diagram

I δ∗ = 0 along diagonal

34 / 45

Conclusion

I With a shortage of truly safe assets (Switzerland), safety isendogenous

I US govt debt is safe asset because

I Good relative fundamentalsI Debt size is large, world in high demand for safe asset (savings glut)

I Nowhere else to go

I Economics of safe asset suggest that there can be gains fromcoordination

I Eurobonds as coordinated security-design

I Theoretical contribution: non-monotone (oscillating) equilibrium inglobal games setting

I Future work: one large country, n small countries

35 / 45

More details on the oscillating equilibrium

The non-monotone equilibrium:

I Let φ (y) be investment in country 1 of agent with signal y

I Consider φ (y) oscillating on (δL, δH )

φ (y) =

0, y < δL1, y ∈ [δL, δL + kσ]

0 y ∈ [δL + kσ, δL + 2σ]

1, y ∈ [δL + 2σ, δL + (2 + k) σ]...

...

0, y ∈ [δH − (2− k) σ, δH ]

1, y > δH

I Suppose joint safety for δ ∈ (δL, δH ). Then a no-arbitrage conditionbetween bond prices has to hold ⇒ k = 2

1+s

36 / 45

An oscillating strategy

-0.6 -0.4 -0.2 0.2δ

0.20.40.60.81.0

ϕ(δ)

I Example: z = 1, s = 1/4 ⇒ Thresholds: δL = −0.37, δH = −0.12

I Full oscillations on (δL, δH ): n = 2

I Joint safety: no-arbitrage condition requires investment splits of1

1+s = 80% to country 1 and s1+s = 20% to country 2 on (δL, δH )

I Let us look at the incentives of different investors...

37 / 45


-0.6 -0.4 -0.2 0.2δ

0.20.40.60.81.0

ϕ(δ)x=0.

I Consider marginal agent δj = δL who thinks he has the mostpositive signal, i.e. x = 0

I Investment in country 1 conditional on x ,∫ δ+x2σ

δ−(1−x)2σφ(y )2σ dy ,

increasing in x

38 / 45


-0.6 -0.4 -0.2 0.2δ

0.20.40.60.81.0

ϕ(δ)x=0.5

I Consider marginal agent δj = δL who thinks he has the mediansignal, i.e. x = 1/2


δ−(1−x)2σφ(y )2σ dy ,

increasing in x

39 / 45


-0.6 -0.4 -0.2 0.2δ

0.20.40.60.81.0

ϕ(δ)x=1.

I Consider marginal agent δj = δL who thinks he has the mostnegative signal, i.e. x = 1


δ−(1−x)2σφ(y )2σ dy ,

increasing in x

40 / 45


-0.6 -0.4 -0.2 0.2δ

0.20.40.60.81.0

ϕ(δ)x=0.

I Consider interior agent δj = δL + kσ who thinks he has the mostpositive signal, i.e. x = 0

I Investment in country 1 conditional on x :∫ δ+x2σ

δ−(1−x)2σφ(y )2σ dy = 1

1+s

41 / 45


-0.6 -0.4 -0.2 0.2δ

0.20.40.60.81.0

ϕ(δ)x=0.5

I Consider interior agent δj = δL + kσ who thinks he has the mediansignal, i.e. x = 1/2


δ−(1−x)2σφ(y )2σ dy = 1

1+s

42 / 45


-0.6 -0.4 -0.2 0.2δ

0.20.40.60.81.0

ϕ(δ)x=1.

I Consider interior agent δj = δL + kσ who thinks he has the mostnegative signal, i.e. x = 1


δ−(1−x)2σφ(y )2σ dy = 1

1+s

I Interior agent’s expectation of investment in country 1 independentof relative position x

43 / 45

An oscillating strategy: Incentives

-0.6 -0.4 -0.2 0.2δ

-0.5

0.5

1.0

Π1(δ)-Π2(δ)

-0.6 -0.4 -0.2 0.2δ

0.20.40.60.81.0

ϕ(δ)

Indifference only on (δL + kσ, δH − (2− k) σ)44 / 45

An oscillating strategy: Equilibrium

I Indifference of marginal agent δL requires Π2 (δL) = Π1 (δL):I Country 2 safe for sure, so always certain returnI Country 1 not always safe in the eyes of marginal agent (strategic

uncertainty), so require higher expected return conditional on survival

I Solving, we have

δL = −z − ln

{ss

(1 + s)(1+s)

}

δH = z + ln

{1

(1 + s)1+ss

}I Note that both δH and δL are independent of σ (need specific σ’s

to make sure that there exists n s.t. δH = n · 2σ + k)I As we are taking σ→ 0, this is wlog

I Define unique zHL so that δL (zHL) = δH (zHL). Then oscillatingequilibrium possible for z > zHL.

45 / 45

a model of safe asset determination1.negative b of safe asset 2.euro-bonds: i german bund currently...

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