5ptrobust exchange rates and the international entropy...
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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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Setup of the economy
Each agent i ∈ {h, f} has a preference for robustness
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.
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Setup of the economy
Each agent i ∈ {h, f} has a preference for robustness
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.
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Setup of the economy
Each agent i ∈ {h, f} has a preference for robustness
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
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
4 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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}
8 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
9 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Distorted Probabilities and Disagreement
Concern for misspecification of the endowments’ distributiontranslates into distorted probabilities
10 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Distorted Probabilities and Disagreement
Concern for misspecification of the endowments’ distributiontranslates into distorted probabilities
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Distorted Probabilities and Disagreement
Concern for misspecification of the endowments’ distributiontranslates into distorted probabilities
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Distorted Probabilities and Disagreement
Concern for misspecification of the endowments’ distributiontranslates into distorted probabilities
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Distorted probability of high endowment of good X is decreasing;
11 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Distorted probability of high endowment of good X is decreasing;
→ Distorted probability of high endowment of good Y is increasing;
11 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Distorted probability of high endowment of good X is decreasing;
→ Distorted probability of high endowment of good Y is increasing;
→ π̂HL Q π̂LH depends on worst case induced by risk-sharing.11 / 16
N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Distorted Probabilities (Two states)
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Distorted Probabilities (Rare Events)
12 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Distorted Probabilities (Rare Events)
Home Country
12 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Distorted Probabilities (Rare Events)
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Distorted Probabilities (Rare Events)
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Conditional Mean is decreasing
13 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Conditional Volatility is higher than true volatility
13 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Conditional Skewness is higher than true one Why?
13 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Conditional Skewness is higher than true one Why?
→ Conditional Kurtosis is lower than true one Why?
13 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Robust FX
Vt [∆et+1] = Vt [mf ,t+1−mh,t+1]
14 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Robust FX
Vt [∆et+1] = Vt [mf ,t+1−mh,t+1]
14 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Robust FX
Vt [∆et+1] = Vt [mf ,t+1] + Vt [mh,t+1]−2ρt ·√
Vt [mf ,t+1] ·√
Vt [mh,t+1]
14 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Robust FX
Vt [∆et+1] = Vt [mf ,t+1] + Vt [mh,t+1]−2ρt ·√
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Robust FX
Vt [∆et+1] = Vt [mf ,t+1] + Vt [mh,t+1]−2ρt ·√
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Conditional Correlations
Introduction The Economy Risk-Sharing Scheme International Pricing Qualitative implications Conclusion
Conditional Correlations
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Conditional Correlations
Introduction The Economy Risk-Sharing Scheme International Pricing Qualitative implications Conclusion
Conditional Correlations
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
→ Low, time-varying correlation of consumption
15 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Conditional Correlations
Introduction The Economy Risk-Sharing Scheme International Pricing Qualitative implications Conclusion
Conditional Correlations
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
→ High, time-varying correlation of marginal utilities
22 / 29
→ Low, time-varying correlation of consumption
→ High, time-varying correlation of marginal utilities
15 / 16N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Abundant X , scarce Y :
1 / 5N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Abundant X , scarce Y :
→ Good news for home
1 / 5N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Abundant X , scarce Y :
→ Good news for home
→ Home Pareto weight ↓
1 / 5N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
→ Scarce X , abundant Y :
→ Bad news for home
→ Home Pareto weight ↑
1 / 5N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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’]
→ Et [µh,t+1]> µh,t , if µh,t ≤ 1/2
2 / 5N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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’]
→ Et [µh,t+1]> µh,t , if µh,t ≤ 1/2
→ Et [µh,t+1]< µh,t , if µh,t > 1/2
2 / 5N
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
Unscaled Moments (Rare Events) Back
Home Country, Home Good (X )
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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
Introduction The Economy Mean-Entropy Frontier Pareto problem Disagreement Robust FX Conclusion
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