5ptrobust exchange rates and the international entropy...

Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion

Robust Exchange Rates andThe International Entropy Frontier

Ric Colacito & Max Croce

1 / 16N


Motivation

Goal: understand the role of concern for model misspecification ininternational finance

Study an economy with:

complete marketsmultiple goodsrobust preferences

We find that

International Risk Sharing involves variances, skewness, kurtosis,...Endogenous disagreement about distribution of fundamentalsEndogenous time-variation in volatility of FX rates

2 / 16N


Roadmap

Setup of the Economy

The International Mean-Entropy Frontier

Planner’s problem and relevant state variables

Distorted probabilities and endogenous disagreement

Robust Exchange Rates

3 / 16N


Setup of the economy

Each agent i ∈ {h, f} has a preference for robustness

Ui,t = (1−δ) logCi,t + δθ logEt exp

{Ui,t+1

θ

}

where θ < 0 measures the degree of concern about modelmisspecification.

Preferences are defined over the consumption aggregate

Ch,t = (xh,t )α (yh,t )

1−α and Cf ,t = (xf ,t )1−α (yf ,t )

α

Consumption bias: α > 1/2.

Complete markets.

Endowments are i.i.d. homoscedastic

1 Two states: HL = {X = 103,Y = 100} and LH = {X = 100,Y = 103}2 Rare events Details

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Ui,t = (1−δ) logCi,t + δEt [Ui,t+1]

where θ < 0 measures the degree of concern about modelmisspecification. If θ→−∞: Expected Utility case.



1−α and Cf ,t = (xf ,t )1−α (yf ,t )

α


Complete markets.



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{Ui,t+1

θ

}




1−α and Cf ,t = (xf ,t )1−α (yf ,t )

α


Complete markets.



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Ui,t ≈ (1−δ) logCi,t + δEt [Ui,t+1] +δVt [Ui,t+1]

2θ+

δEt (Ui,t+1−Et Ui,t+1)3

6θ2 . . .

where θ < 0 measures the degree of concern about modelmisspecification. Conditional Moments matter.



1−α and Cf ,t = (xf ,t )1−α (yf ,t )

α


Complete markets.



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Ui,t ≈ (1−δ) logCi,t + δEt [Ui,t+1] +δVt [Ui,t+1]

2θ+

δEt (Ui,t+1−Et Ui,t+1)3

6θ2 . . .︸︷︷︸Discounted Entropy




1−α and Cf ,t = (xf ,t )1−α (yf ,t )

α


Complete markets.



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{Ui,t+1

θ

}




1−α and Cf ,t = (xf ,t )1−α (yf ,t )

α


Complete markets.



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{Ui,t+1

θ

}




1−α and Cf ,t = (xf ,t )1−α (yf ,t )

α


Complete markets.


1 Two states: HL = {X = 103,Y = 100} and LH = {X = 100,Y = 103}

2 Rare events Details

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{Ui,t+1

θ

}




1−α and Cf ,t = (xf ,t )1−α (yf ,t )

α


Complete markets.



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The International Mean-Entropy Frontier(Two states)

4.46 4.48 4.5 4.52 4.54 4.560

1

2

3

4

5

6x 10

−5

E[Uh,t+1(st+1|st)]

Ent

ropy

θ=1/(1−25) [More Risk−Sensitive]θ=1/(1−10) [Less Risk−Sensitive]

4.46 4.48 4.5 4.52 4.54 4.560

0.2

0.4

0.6

0.8

1x 10

−3

Vol

atili

ty

4.46 4.48 4.5 4.52 4.54 4.56−1

0

1x 10

−4

Ske

wne

ss

4.46 4.48 4.5 4.52 4.54 4.561

1

1

1

1

E[Uh,t+1(st+1|st)]

Kur

tosi

s

5 / 16N


The International Mean-Entropy Frontier(Rare Events)

4.46 4.48 4.5 4.52 4.544.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

5.5

5.6x 10

−3

E[Uh,t+1(st+1|st)]

Ent

ropy

4.45 4.46 4.47 4.48 4.49 4.5 4.51 4.52 4.533.6

3.8

4

4.2

4.4x 10

−3

Vol

atili

ty

4.45 4.46 4.47 4.48 4.49 4.5 4.51 4.52 4.53−5.4

−5.2

−5

−4.8

−4.6

Ske

wne

ss

4.45 4.46 4.47 4.48 4.49 4.5 4.51 4.52 4.5327

28

29

30

31

E[Uh,t+1(st+1|st)]

Kur

tosi

s

6 / 16N


Planner’s problem

Efficient allocations are the solution to the planner’s problem

choose {xh,t ,xf ,t ,yh,t ,yf ,t}+∞

t=0

to max Q = µhUh,0 + µf Uf ,0

s.t. xh,t + xf ,t = Xt

yh,t + yf ,t = Yt , ∀t ≥ 0

µh and µf correspond to an initial distribution of assets.

Notation: S = µh/µf .

7 / 16N


Allocations

Time Additive Preferences

Let k = α

1−α:

xht =

kSt

1 + kStXt , x f

t =1

1 + kStXt

yht =

St

k + StYt , y f

t =k

k + StYt

where

S = µh/µf

8 / 16N


Allocations

Risk Sensitive Preferences

Let k = α

1−α:

xht =

kSt

1 + kStXt , x f

t =1

1 + kStXt

yht =

St

k + StYt , y f

t =k

k + StYt

where

St = St−1 ·δexp{Uh,t/θ}

Et−1 exp{Uh,t/θ}

/δexp{Uf ,t/θ}

Et−1 exp{Uf ,t/θ}

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Properties of the Pareto weights

Pareto weights are:

1 countercyclical Graph

2 expected to increase (decrease) when they are low (high) Graph

3 stationary Graph

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Distorted Probabilities and Disagreement

Concern for misspecification of the endowments’ distributiontranslates into distorted probabilities

10 / 16N




Worst case distortion is state-specific

π̃HLi,t+1 = π

HLt+1

exp{

UHLi (st+1)/θ

}∑st+1

exp{Ui(st+1|µh,t)/θ}π(st+1)

π̃LHi,t+1 = π

LHt+1

exp{

ULHi (st+1)/θ

}∑st+1

exp{Ui(st+1|µh,t)/θ}π(st+1), ∀i ∈ {h, f}

International Disagreement as an endogenous outcome.

10 / 16N




Worst case distortion is state-specific and country-specific:

π̃HLi,t+1 = π

HLt+1

exp{

UHLi (st+1)/θ

}∑st+1

exp{Ui(st+1|µh,t)/θ}π(st+1)

π̃LHi,t+1 = π

LHt+1

exp{

ULHi (st+1)/θ

}∑st+1

exp{Ui(st+1|µh,t)/θ}π(st+1), ∀i ∈ {h, f}

International Disagreement as an endogenous outcome.

10 / 16N


Distorted Probabilities (Two states)

Home Country

0 0.2 0.4 0.6 0.8 10.49

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0.51X=100, Y=103

0 0.2 0.4 0.6 0.8 10.49

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0.51X=103, Y=100

11 / 16N



Home Country

0 0.2 0.4 0.6 0.8 10.49

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0.51X=100, Y=103

0 0.2 0.4 0.6 0.8 10.49

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0.51X=103, Y=100

→ Distorted probability of high endowment of good X is decreasing;

11 / 16N



Home Country

0 0.2 0.4 0.6 0.8 10.49

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0.51X=100, Y=103

0 0.2 0.4 0.6 0.8 10.49

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0.51X=103, Y=100


→ Distorted probability of high endowment of good Y is increasing;

11 / 16N



Home Country

0 0.2 0.4 0.6 0.8 10.49

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0.51X=100, Y=103

0 0.2 0.4 0.6 0.8 10.49

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0.51X=103, Y=100


→ Distorted probability of high endowment of good Y is increasing;

→ π̂HL Q π̂LH depends on worst case induced by risk-sharing.11 / 16

N



Foreign Country

0 0.2 0.4 0.6 0.8 10.49

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0.51X=100, Y=103

0 0.2 0.4 0.6 0.8 10.49

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0.51X=103, Y=100

→ Foreign country’s distorted probabilities are mirror image

11 / 16N


Distorted Probabilities (Rare Events)

12 / 16N



Home Country

12 / 16N



Home Country

0 0.2 0.4 0.6 0.8 10.01

0.012

0.014

0.016

0.018

0.02X=60, Y=60

0 0.2 0.4 0.6 0.8 10.01

0.012

0.014

0.016

0.018

0.02X=60, Y=100

0 0.2 0.4 0.6 0.8 10.01

0.012

0.014

0.016

0.018

0.02X=60, Y=103

0 0.2 0.4 0.6 0.8 10.01

0.012

0.014

0.016

0.018

0.02X=100, Y=60

0 0.2 0.4 0.6 0.8 10.2355

0.236

0.2365

0.237

0.2375

0.238X=100, Y=100

0 0.2 0.4 0.6 0.8 10.228

0.23

0.232

0.234

0.236

0.238

0.24X=100, Y=103

0 0.2 0.4 0.6 0.8 10.008

0.01

0.012

0.014

0.016

0.018

0.02X=103, Y=60

0 0.2 0.4 0.6 0.8 10.225

0.23

0.235

0.24X=103, Y=100

0 0.2 0.4 0.6 0.8 10.225

0.23

0.235

0.24X=103, Y=103

0 0.2 0.4 0.6 0.8 10.01

0.012

0.014

0.016

0.018

0.02X=60, Y=60

12 / 16N



Home Country

0 0.2 0.4 0.6 0.8 10.01

0.012

0.014

0.016

0.018

0.02X=60, Y=60

0 0.2 0.4 0.6 0.8 10.01

0.012

0.014

0.016

0.018

0.02X=60, Y=100

0 0.2 0.4 0.6 0.8 10.01

0.012

0.014

0.016

0.018

0.02X=60, Y=103

0 0.2 0.4 0.6 0.8 10.01

0.012

0.014

0.016

0.018

0.02X=100, Y=60

0 0.2 0.4 0.6 0.8 10.2355

0.236

0.2365

0.237

0.2375

0.238X=100, Y=100

0 0.2 0.4 0.6 0.8 10.228

0.23

0.232

0.234

0.236

0.238

0.24X=100, Y=103

0 0.2 0.4 0.6 0.8 10.008

0.01

0.012

0.014

0.016

0.018

0.02X=103, Y=60

0 0.2 0.4 0.6 0.8 10.225

0.23

0.235

0.24X=103, Y=100

0 0.2 0.4 0.6 0.8 10.225

0.23

0.235

0.24X=103, Y=103

0 0.2 0.4 0.6 0.8 10.01

0.012

0.014

0.016

0.018

0.02X=60, Y=60

→ Distorted probability of joint disaster is very large.12 / 16

N


Distorted moments (Rare Events)

Home Country, Home Good (X )

0 0.2 0.4 0.6 0.8 115

20

25

30

Con

ditio

nal K

urto

sis

0 0.2 0.4 0.6 0.8 1−5.2

−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

−3.6

Con

ditio

nal S

kew

ness

0 0.2 0.4 0.6 0.8 17

7.5

8

8.5

9

9.5

10

Con

ditio

nal V

olat

ility

0 0.2 0.4 0.6 0.8 199.2

99.4

99.6

99.8

100

100.2

100.4

100.6

Con

ditio

nal M

ean

13 / 16N




0 0.2 0.4 0.6 0.8 115

20

25

30

Con

ditio

nal K

urto

sis

0 0.2 0.4 0.6 0.8 1−5.2

−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

−3.6

Con

ditio

nal S

kew

ness

0 0.2 0.4 0.6 0.8 17

7.5

8

8.5

9

9.5

10

Con

ditio

nal V

olat

ility

0 0.2 0.4 0.6 0.8 199.2

99.4

99.6

99.8

100

100.2

100.4

100.6

Con

ditio

nal M

ean

→ Conditional Mean is decreasing

13 / 16N




0 0.2 0.4 0.6 0.8 115

20

25

30

Con

ditio

nal K

urto

sis

0 0.2 0.4 0.6 0.8 1−5.2

−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

−3.6

Con

ditio

nal S

kew

ness

0 0.2 0.4 0.6 0.8 17

7.5

8

8.5

9

9.5

10

Con

ditio

nal V

olat

ility

0 0.2 0.4 0.6 0.8 199.2

99.4

99.6

99.8

100

100.2

100.4

100.6

Con

ditio

nal M

ean

→ Conditional Volatility is higher than true volatility

13 / 16N




0 0.2 0.4 0.6 0.8 115

20

25

30

Con

ditio

nal K

urto

sis

0 0.2 0.4 0.6 0.8 1−5.2

−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

−3.6

Con

ditio

nal S

kew

ness

0 0.2 0.4 0.6 0.8 17

7.5

8

8.5

9

9.5

10

Con

ditio

nal V

olat

ility

0 0.2 0.4 0.6 0.8 199.2

99.4

99.6

99.8

100

100.2

100.4

100.6

Con

ditio

nal M

ean

→ Conditional Skewness is higher than true one Why?

13 / 16N




0 0.2 0.4 0.6 0.8 115

20

25

30

Con

ditio

nal K

urto

sis

0 0.2 0.4 0.6 0.8 1−5.2

−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

−3.6

Con

ditio

nal S

kew

ness

0 0.2 0.4 0.6 0.8 17

7.5

8

8.5

9

9.5

10

Con

ditio

nal V

olat

ility

0 0.2 0.4 0.6 0.8 199.2

99.4

99.6

99.8

100

100.2

100.4

100.6

Con

ditio

nal M

ean

→ Conditional Skewness is higher than true one Why?

→ Conditional Kurtosis is lower than true one Why?

13 / 16N


Robust FX

Vt [∆et+1] = Vt [mf ,t+1−mh,t+1]

14 / 16N


Robust FX

Vt [∆et+1] = Vt [mf ,t+1] + Vt [mh,t+1]−2ρt ·√

Vt [mf ,t+1] ·√

Vt [mh,t+1]

14 / 16N


Robust FX


Vt [mf ,t+1] ·√

Vt [mh,t+1]

0 0.2 0.4 0.6 0.8 110.5

11

11.5

12

12.5

13

13.5

14

14.5

µh

σ t(∆e t+

1)

→ Average Volatility ≈ 14%

14 / 16N


Robust FX


Vt [mf ,t+1] ·√

Vt [mh,t+1]

0 0.2 0.4 0.6 0.8 110.5

11

11.5

12

12.5

13

13.5

14

14.5

µh

σ t(∆e t+

1)

→ Average Volatility ≈ 14%

→ Time-varying exchange rate volatility

14 / 16N


Conditional Correlations

Introduction The Economy Risk-Sharing Scheme International Pricing Qualitative implications Conclusion


0 0.2 0.4 0.6 0.8 10.37

0.375

0.38

0.385

corr t(Δc t+1h,Δc t+1f)

0 0.2 0.4 0.6 0.8 10.6

0.7

0.8

0.9

μ

corr t(mt+1

h,mt+1

f)

22 / 2915 / 16N





0 0.2 0.4 0.6 0.8 10.37

0.375

0.38

0.385

corr

t(∆c t+

1h

,∆c t+

1f

)

0 0.2 0.4 0.6 0.8 10.6

0.7

0.8

0.9

µ

corr

t(mt+

1h

,mt+

1f

)

→ Low, time-varying correlation of consumption

22 / 29


15 / 16N





0 0.2 0.4 0.6 0.8 10.37

0.375

0.38

0.385

corr

t(∆c t+

1h

,∆c t+

1f

)

0 0.2 0.4 0.6 0.8 10.6

0.7

0.8

0.9

µ

corr

t(mt+

1h

,mt+

1f

)


→ High, time-varying correlation of marginal utilities

22 / 29


→ High, time-varying correlation of marginal utilities

15 / 16N


Concluding Remarks

Robust International Risk Sharing generates

rich dynamics of conditional variance, skewness, kurtosis,...endogenous disagreement about distribution of fundamentalstime-variation in FX volatility

Next steps (in progress):

entropy of FX?→ Co-entropy (codependence of higher moments): Chabi-Yo and Colacito (2013)production?more than 2 countries?heteroskedastic endowments?

16 / 16N


Concluding Remarks




entropy of FX?→ Co-entropy (codependence of higher moments): Chabi-Yo and Colacito (2013)

production?more than 2 countries?heteroskedastic endowments?

16 / 16N


Concluding Remarks




entropy of FX?→ Co-entropy (codependence of higher moments): Chabi-Yo and Colacito (2013)production?

more than 2 countries?heteroskedastic endowments?

16 / 16N


Concluding Remarks




entropy of FX?→ Co-entropy (codependence of higher moments): Chabi-Yo and Colacito (2013)production?more than 2 countries?

heteroskedastic endowments?

16 / 16N


Concluding Remarks




entropy of FX?→ Co-entropy (codependence of higher moments): Chabi-Yo and Colacito (2013)production?more than 2 countries?heteroskedastic endowments?

16 / 16N


Pareto weights: phase diagrams Back

HL LH

0 0.5 1−3

−2

−1

0

1

2

3x 10

−3

µh

∆µhLH

0 0.5 1−3

−2

−1

0

1

2

3x 10

−3

µh

∆µhH

L

1 / 5N



HL LH

0 0.5 1−3

−2

−1

0

1

2

3x 10

−3

µh

∆µhLH

0 0.5 1−3

−2

−1

0

1

2

3x 10

−3

µh

∆µhH

L

→ Abundant X , scarce Y :

1 / 5N



HL LH

0 0.5 1−3

−2

−1

0

1

2

3x 10

−3

µh

∆µhLH

0 0.5 1−3

−2

−1

0

1

2

3x 10

−3

µh

∆µhH

L


→ Good news for home

1 / 5N



HL LH

0 0.5 1−3

−2

−1

0

1

2

3x 10

−3

µh

∆µhLH

0 0.5 1−3

−2

−1

0

1

2

3x 10

−3

µh

∆µhH

L


→ Good news for home

→ Home Pareto weight ↓

1 / 5N



HL LH

0 0.5 1−3

−2

−1

0

1

2

3x 10

−3

µh

∆µhLH

0 0.5 1−3

−2

−1

0

1

2

3x 10

−3

µh

∆µhH

L

→ Scarce X , abundant Y :

→ Bad news for home

→ Home Pareto weight ↑

1 / 5N


Pareto weights: expected change Back

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−5

µh

Et[∆

µ h’]

2 / 5N



0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−5

µh

Et[∆

µ h’]

→ Et [µh,t+1]> µh,t , if µh,t ≤ 1/2

2 / 5N



0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−5

µh

Et[∆

µ h’]

→ Et [µh,t+1]> µh,t , if µh,t ≤ 1/2

→ Et [µh,t+1]< µh,t , if µh,t > 1/2

2 / 5N


Time-invariant distribution of Pareto weights Back

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9x 10

6

µh

→ Ergodic distribution is symmetric around 1/2

3 / 5N


Unscaled Moments (Rare Events) Back


0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

1.1

1.2

1.3x 10

5

Con

ditio

nal F

ourt

h M

omen

t

0 0.2 0.4 0.6 0.8 1−3400

−3200

−3000

−2800

−2600

−2400

−2200

−2000

−1800

Con

ditio

nal T

hird

Mom

ent

0 0.2 0.4 0.6 0.8 17

7.5

8

8.5

9

9.5

10

Con

ditio

nal V

olat

ility

0 0.2 0.4 0.6 0.8 199.2

99.4

99.6

99.8

100

100.2

100.4

100.6C

ondi

tiona

l Mea

n

4 / 5N


Rare Events Back

X Y π

103 103 0.2375

103 100 0.2375

100 103 0.2375

100 100 0.2375

103 60 0.0100

100 60 0.0100

60 60 0.0100

60 103 0.0100

60 100 0.0100

5 / 5N


Rare Events Back

X Y π

103 103 0.2375

103 100 0.2375

100 103 0.2375

100 100 0.2375

103 60 0.0100

100 60 0.0100

60 60 0.0100

60 103 0.0100

60 100 0.0100

-Four equally likely

no-disaster events

5 / 5N


Rare Events Back

X Y π

103 103 0.2375

103 100 0.2375

100 103 0.2375

100 100 0.2375

103 60 0.0100

100 60 0.0100

60 60 0.0100

60 103 0.0100

60 100 0.0100

� Five equally likely

disaster events

5 / 5N

5ptrobust exchange rates and the international entropy...

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