## wpiea2025153-source-pdf

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### 2.1 Global bank (intermediary) — balance sheet, exposures, and pricing
- Balance sheet (accounting identity):
  - B∗t + H∗t + A∗t = W∗t + D∗t. (1)
  - Size of balance sheet measured by H∗t (risky investments).
- Assets and liabilities:
  - Liabilities: net worth W∗t and external funding D∗t (assumed in dollars) with external funding cost R∗t.
  - Assets: dollar reserves B∗t ≥ 0 (pay R∗t), risky investments H∗t ≥ 0 (ex-post return ˜R∗t+1), local-currency assets A∗t (net of LC borrowing) with LC return Rt converting ex-post into dollar return Rt Et / Et+1.
  - Off-balance-sheet: Forwards F∗t (commitment scaled so t + 1 dollar payout = (Ft / Et+1 − 1) Rt Et F∗t / Ft) and Swaps S∗t with payout (R∗t − Rt Et St) S∗t.
- Net worth evolution (aggregating payoffs):
  - W∗t+1 = R∗t B∗t + ˜R∗t+1 H∗t + Rt Et / Et+1 A∗t − R∗t D∗t + Rt Et (Ft / Et+1 − 1) F∗t + (R∗t − Rt Et St) S∗t.
- Lemma 1 (assuming St = Ft) — net worth decomposition:
  - W∗t+1 = R∗t (W∗t − H∗t − B∗t) + R∗t B∗t + ˜R∗t+1 H∗t + CIPt · X∗t + UIPt+1 · Z∗t. (2)
  - Definitions:
    - X∗t ≡ F∗t + S∗t (off-balance-sheet exposure).
    - Z∗t ≡ A∗t + F∗t (currency-risk exposure).
    - UIPt+1 ≡ Rt Et / Et+1 − R∗t.
    - CIPt ≡ R∗t − Rt Et Ft. (3)
- Exchange-rate pass-through into net worth volatility:
  - ∂(W∗t+1 / W∗t) / ∂(Et / Et+1) = Rt Z∗t / W∗t (pass-through proportional to Z∗t).
- Forwards vs swaps:
  - Forwards F∗t affect both X∗t and Z∗t and thus collect both UIP and CIP premia.
  - Swaps S∗t carry only the CIP premium.

### 2.1 Optimal portfolio, reserves constraint, and supply schedules
- Pricing and objective:
  - Banks price returns with stochastic discount factor Θt+1 with Et Θt+1 = 1 / R∗t. Risk-neutral case Θt+1 ≡ 1 / R∗t admissible.
  - Bank portfolio problem:
    - max over B∗t ≥ 0, H∗t ≥ 0, X∗t, Z∗t of Et Θt+1 W∗t+1 subject to (2), (4)–(5) and |X∗t| ≤ X̄∗. (6)
- Reserves requirement (binding in equilibrium):
  - B∗t ≥ at H∗t + bt |Z∗t| + δ |X∗t|. (4)
  - where at ≡ α 2 H∗t / W∗t and bt ≡ γ σt2 |Z∗t| / W∗t with σt2 ≡ R2t · vart (Et / Et+1) and α, γ, δ > 0. (5)
- Assumption 1 ensures constraint binding and H∗t > 0:
  - R∗t < R∗t and Et [Θt+1 ˜Rt+1] > 1.
- Proposition 1 (under Assumption 1):
  - μt = R∗t − R∗t > 0 and the balance sheet constraint is binding.
  - (a) St = Ft.
  - (b) Supply schedules (per bank):
    - UIPt = γ μt σt Z∗t / W∗t
    - CIPt = δ μt · sign(X∗t) for |X∗t| ≤ X̄∗. (7)
  - Off-balance-sheet position decision:
    - X∗t = X̄∗ · sign(CIPt) when |CIPt| > δ μt and X∗t = 0 when |CIPt| < δ μt.
- Economic interpretation:
  - Shadow cost of expanding exposures that require one dollar increase in reserves equals μt = R∗t − R∗t.
  - UIP premium is linear in σt Z∗t / W∗t; CIP premium is a step function in X∗t and constant per unit.

### 2.1 Aggregation to currency market equilibrium
- Aggregate definitions per currency:
  - Z∗t ≡ Σi (A∗it + F∗it) (aggregate unhedged demand to sell currency risk).
  - X∗t ≡ Σi (F∗it + S∗it) (aggregate hedged demand to buy currency forwards and swaps). (9)
- Proposition 2 — equilibrium premia as functions of aggregates:
  - UIPt ≡ Rt Et bEt − R∗t = γ̄t μt σt · Z∗t / W∗t. (10)
  - CIPt ≡ R∗t − Rt Et Ft = δ̄t μt · sign(X∗t). (11)
  - Definitions:
    - μt = R∗t − R∗t.
    - W∗t ≡ Σi W∗it (aggregate net worth).
    - γ̄t ≡ [Σi W∗it / γit / W∗t]−1 (hyperbolic weighted average).
    - δ̄t = δit (tightness parameter of marginal bank).
- Key implications:
  - Both UIP and CIP premia expand when μt is large.
  - UIP depends on aggregate unhedged demand Z∗t relative to W∗t with slope γ̄t μt σt / W∗t.
  - CIP depends only on market-wide tightness δ̄t μt and sign(X∗t); CIP does not respond to Z∗t in the same way.
  - UIP supply curve slope = μt σt [Σi W∗it / γit]−1; greater aggregate net worth increases elasticity and reduces pass-through from demand shocks.

---

### 2.3 Testable implications — cross section and dynamics
- Cross-section supply schedules (currency index k):
  - UIPkt ≡ Rkt Ekt bEkt − R∗t = μt ̄γkt σkt · Z∗kt W∗kt.
  - CIPkt ≡ R∗t − Rkt Ekt Fkt = μt ̄δkt · sign(X∗kt).
- Assumptions for empirical tests:
  - Assumption 2: long-run E{Ekt/Ek,t+1} ≈ 1.
  - Assumption 3: currencies with A∗kt > 0 feature (a) Rkt > R∗t, (b) F∗kt > 0, (c) S∗kt > 0; vice versa for A∗kt < 0.
- Proposition 3 (under Assumptions 2 and 3):
  - (a) A∗kt > 0 → Rkt > R∗t, positive UIP and CIP premia, expensive forward dollars (Fkt < Ekt).
  - (b) A∗kt < 0 → low Rkt, negative UIP and CIP premia, cheap forward dollars (Fkt > Ekt).
- Dynamics predictions:
  1. Small local-currency demand shock df∗kt > 0 (sales of local currency) with dZ∗kt/df∗kt > 0 and dX∗kt/df∗kt > 0 yields:
     - dUIPkt/df∗kt > 0 and dCIPkt/df∗kt = 0. (16)
  2. Aggregate shocks to intermediation cost dμt > 0 increase both UIP and CIP premia in absolute values:
     - d|UIPkt|/dμt > 0 and d|CIPkt|/dμt > 0. (17)
     - Impact on UIP premia increases in A∗kt: dUIPkt/dμt ∝ A∗kt. (18)
  3. Comparable dynamics apply to shocks to ̄γkt/W∗kt (UIP) and to ̄δkt (CIP).
- Nonlinearity note:
  - Large aggregate shifts in currency demand may change CIPkt via changes in ̄δkt when additional intermediaries with tighter constraints must participate.

### 2.3 Empirical setup and stylized facts (preview)
- Sample: G7+ advanced economies and 4 EM currencies (JPY, CHF, EUR, GBP, CAD, AUS, USD base, NZD, MXN, BRL, ZAR, RUB).
- UIP and CIP premia measured at 3-month horizon (monthly frequency), annualized with one-period = 3 months:
  - CIPkt = −(rkt − rUSt) + 4· log(Fkt/Ekt).
  - UIPk,t+3 = (rkt − rUSt) − 4· log(Ek,t+3/Ekt).
  - UIPkt = (rkt − rUSt) − 4· log(bEkt/Ekt).
- Stylized facts:
  - The UIP premium is several orders of magnitude larger than the CIP premium, especially for G7+ currencies.
  - Among the G7+, the average monthly change in expected UIP premium is 4.8 percent (480 bps) on average (in either direction), while it is only 4 basis points for the CIP premium.
  - UIP and CIP for MXN is more volatile than for G7+ currencies.
  - Most time-series volatility comes from exchange rates, very little from interest rate dynamics.
- Dealer banks’ net FX futures position (CFTC TFF):
  - f∗kt = 100 · Dealer Net Positionkt / (1/12 Σ11 j=0 Open Interestk,t−j). (20)
  - TFF series for G7+ currencies go back to June 2006 (218 monthly observations per currency).
  - One standard deviation change in dealer net positions ≈ 20 percent of total futures market size per currency.

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### 3.2 Cross section of currency premia — empirical cross-sectional patterns
- Positive correlation between average funding gap ̄A∗k and average interest rate differential ̄Rk − ̄R∗: currencies with large funding gaps feature high local-currency interest rates.
- Table 2 — currency-level averages (January 2000 to December 2020; shorter for MXN, RUB, BRL):
  - JPY: Funding gap −31.53 (7.51); Interest rate gap −1.78 (1.71); Carry return −7.07 (19.58); Survey UIP −2.13 (9.42); CIP premium −0.22 (0.17)
  - CHF: Funding gap −50.36 (23.51); Interest rate gap −1.39 (1.19); Carry return −2.76 (18.10); Survey UIP −3.12 (10.13); CIP premium −0.21 (0.21)
  - EUR: Funding gap −2.61 (5.60); Interest rate gap −0.39 (1.32); Carry return −0.60 (19.49); Survey UIP −1.07 (9.74); CIP premium −0.20 (0.25)
  - GBP: Funding gap −1.10 (5.64); Interest rate gap 0.51 (1.26); Carry return 1.31 (18.18); Survey UIP −1.48 (7.94); CIP premium −0.11 (0.17)
  - CAD: Funding gap 11.02 (6.30); Interest rate gap 0.18 (0.75); Carry return 1.38 (15.92); Survey UIP 0.06 (7.30); CIP premium −0.05 (0.14)
  - AUD: Funding gap 24.12 (6.13); Interest rate gap 2.02 (1.66); Carry return 9.02 (25.47); Survey UIP −0.14 (12.63); CIP premium 0.04 (0.17)
  - NZD: Funding gap 17.64 (8.19); Interest rate gap 2.31 (1.55); Carry return 10.99 (24.94); Survey UIP −1.98 (13.33); CIP premium 0.09 (0.20)
  - MXN: Funding gap 3.83 (3.29); Interest rate gap 4.69 (1.36); Carry return 14.58 (23.59); Survey UIP 4.07 (8.54); CIP premium −0.02 (0.69)
  - ZAR: Funding gap 1.87 (9.13); Interest rate gap 5.71 (2.07); Carry return 18.71 (34.82); Survey UIP 2.18 (16.50); CIP premium 0.40 (0.40)
  - RUB: Funding gap −15.12 (9.36); Interest rate gap 6.47 (4.41); Carry return 21.30 (33.80); Survey UIP 6.52 (9.62); CIP premium 0.04 (3.09)
  - BRL: Funding gap 8.88 (8.11); Interest rate gap 10.32 (3.33); Carry return 29.80 (34.46); Survey UIP 9.10 (12.20); CIP premium −1.42 (3.06)
- Magnitudes and associations:
  - Expected UIP deviations (survey) large: on order of 200 basis points (2%) annualized; time-series std ≈ 10% annualized.
  - Average realized currency returns are about five-fold larger than expected UIP deviations.
  - CIP deviations on average ≈ 20 basis points (0.2%) annualized; time-series std ≈ 20 basis points.
  - Strong positive cross-sectional association between average CIP deviations, average UIP deviations, and the average interest rate differential.
- Classification:
  - Funding currencies: JPY, CHF (large negative funding gaps; low interest rates; negative UIP and CIP).
  - Balanced: EUR, GBP (small funding gaps; negative CIP deviations).
  - Investment/commodity and EM: CAD, AUD, NZD, EMs (positive funding gaps; high interest rates; positive UIP and CIP deviations).
- Long-run properties align with Proposition 3:
  - Funding currencies (A∗k < 0) → negative average interest rate differentials.
  - Investment currencies (A∗k > 0) → positive average interest rate differentials.
  - Forward premium puzzle: forward premium does not predict expected depreciation; forward premium offsets interest rate differential to deliver equilibrium CIP premium.

---

### 3.3 Dynamics of currency premia — identification and pooled results
- Dynamic panel specification (distributed lag) for premia v_kt:
  - ∆v_kt = θ_k + θ_t + Σ_{j=0,1,2} β_j ∆f^∗_{k,t−j} + γ∆w_kt + ρ v_{k,t−1} + ε^v_kt. (21)
- First-stage for dealer positions (time-series process):
  - ∆f^∗_kt = θ_k + θ_t − 0.130 [4.50] · f^∗_{k,t−1} + 0.183 [5.20] · ∆f^∗_{k,t−1} − 0.119 [3.83] · ∆f^∗_{k,t−2} + ε^f_kt.
  - Std dev of innovation = 15.6; half-life ≈ 6 months.
- Identification: reduced-form first-stage ∆Z^∗_kt = α_k + η_k ∆f^∗_kt + u_kt requires η_k > 0 and significant R^2 (holds strongly).
- Limits: (21) is reduced-form projection; not structural supply elasticities.

Pooled G7+ findings (Table 3):
- Strong contemporaneous response and mean reversion for survey-expected and realized UIP premia to ∆f^∗_kt shocks.
- CIP premium shows no response to ∆f^∗_kt shocks.
- Quantitative magnitudes:
  - Coefficients on ∆f^∗_kt for survey-expected UIP: 0.203, 0.171, 0.226, 0.189 (t-stats [16.61], [15.35], [12.15], [11.83]).
  - ∆f^∗_kt coefficients for CIP: −0.0000, −0.0000 (t-stats [0.01], [0.06]).
  - Lag coefficient v_{k,t−1} estimates: −0.462 ∗∗∗ to −0.272 ∗∗∗.
  - One standard deviation innovation in ∆f^∗_kt (15.6) → nearly 300 basis point increase in UIP premia (400 bps if time FE not absorbed).
- Explained variation:
  - Within R^2 for survey UIP: 0.466 (without time FE); 0.705 (with time FE).
  - CIP premia: most variation absorbed by time FE; R^2 much lower.
- Comovement:
  - Negative comovement between UIP and CIP at monthly frequency disappears when time FE included → driven by broad financial/dollar cycles.

Decomposition of premia responses (Table 4):
- ∆f^∗_kt associated with large significant effect on UIP but not CIP.
- Interest rate differential effectively unchanged with ∆f^∗_kt (decline < 1 basis point per one standard deviation ∆f^∗_kt).
- Both spot and forward depreciate strongly and in sync with increase in ∆f^∗_kt (∼350 basis points annualized), explaining stable CIP.

Currency-specific dynamics (Table 5, Figure 5):
- Per-currency survey UIP coefficients on ∆f^∗_kt positive and significant (e.g., JPY 0.191 [8.07]; EUR 0.252 [5.33]; NZD 0.166 [8.19]; MXN 0.075 [3.55]).
- ∆f^∗_kt coefficients for CIP effectively zero and precisely estimated across currencies (e.g., JPY −0.0004 [0.93]; EUR −0.0006 [0.67]).
- VIX and broad dollar explain significant variation; VIX effects vary across funding gaps.
- CIP premia move with lower-frequency global financial/dollar cycles; idiosyncratic monthly demand shocks do not move CIP materially.

Synthesis:
- Empirical findings consistent with model: idiosyncratic month-to-month currency demand shocks → UIP changes; CIP stable to such shocks and driven by market-wide tightness μt and δ̄t.
- UIP premia two orders of magnitude more volatile than CIP premia at monthly frequency.

---

### 3.4 Exchange rate dynamics — spot responses, persistence, and fit
- Raw pooled correlation between ∆f∗kt and ∆ logEkt for G7+: 0.6.
- Dynamic projection (individual currencies incl. MXN):
  - ∆ logEkt = θk + Σj=0,1,2,3 ∆f∗k,t−j + γk ∆wkt + ρk logEk,t−1 + εekt.
- Contemporaneous responses:
  - Strong contemporaneous response of spot to dealer position shocks; t-stats ≈ 10 overall (range: MXN 4.0 to NZD 15.9).
  - One standard deviation innovation in ∆f∗kt → typical depreciation ≈ 1.3% (MXN 0.9%; CAD 0.9%; JPY 1.3%; CHF 1.3%; NZD 1.6%).
  - Panel with time FE: one standard deviation → 0.9% depreciation.
- Persistence and mean reversion:
  - Estimated ρ̂k ≈ 0 (no exogenous mean reversion); mean reversion arises from mean reversion in dealer positions.
  - Half-life of f∗kt shock ≈ 5–6 months; half-life of exchange rate ≈ 6–8 months (individual currencies 6–13 months).
- Model fit and explained variation:
  - Model explains 45–60% of variation in spot changes for individual currencies; 70% in pooled panel with time FE.
  - Contribution of dealer positions to R2 (selected):
    - JPY: 0.812; CHF: 0.890; EUR: 0.686; GBP: 0.636; CAD: 0.515; AUD: 0.416; NZD: 0.628; MXN: 0.352.
  - R2 values (individual currencies): 0.450, 0.436, 0.473, 0.485, 0.592, 0.606, 0.607, 0.444; panel: 0.464 and 0.700.
- Role of macro-finance controls:
  - Interest rate differential associated with appreciations for most currencies except CHF and MXN.
  - Treasury basis negative effect: reduction in basis → depreciation.
  - VIX: weak appreciating impact on funding currencies; depreciating for commodity/investment currencies.
  - Dealer positions retain large independent explanatory power for most currencies.
- Impulse responses for a one standard-deviation innovation in f∗kt:
  - Half-life for dealers’ position innovation: 5 months; exchange rate half-life: 6 months.
  - Impact depreciation (panel with time FE): 0.87% (87 log basis points); alternative specification (no time FE) impact: 1.29%.
  - Over following year: gradual appreciation (≈ 6 log basis points monthly over first 12 months).
  - CIP and interest rate differential: virtually no change following ∆f∗kt innovation.

---

### Impulse responses, returns mechanics, and policy implications
- Impulse responses for uncovered currency returns to one standard deviation positive innovation in f∗kt:
  - Large negative returns for positions open before the shock: up to 540 basis points (5.4%) annualized loss on a three-month position.
  - Positions entered after the shock earn predictable positive returns: up to 175 basis points (1.75%) in first quarter; over 100 basis points on average over the full year.
  - Entire currency return driven by predictable spot path; interest rate response effectively zero.
- Dynamic profile labeled “steamroller and pennies”:
  - Sharp unexpected depreciation on impact (steamroller) producing large instantaneous losses.
  - Predictable gradual appreciation over the following year (pennies) delivering modest expected returns.
  - High volatility and modest Sharpe ratio for intermediaries collecting expected returns.
- Cross-section interpretation:
  - Funding currencies (e.g., JPY, CHF): low local rates, negative UIP and CIP, cheap forward dollars.
  - Investment/commodity/EM currencies: opposite patterns.
  - UIP premia reflect long-run compensation equal to interest rate differentials; forward premiums adjust to generate required CIP premia (forward premium puzzle).
- Comparison of shocks:
  - Dealer futures position shocks strongly correlate with contemporaneous depreciation and large realized losses for pre-shock holders; predictable positive returns thereafter.
  - Interest-rate shocks have weak predictive power for future depreciations (near-zero R2 in Fama regressions).
- Operational measurement:
  - Weekly/monthly dealer futures positions provide a real-time proxy for expected UIP deviations and shifts in currency demand intermediated by dealers.
- Policy implications:
  - Frictional intermediation suggests rationale for policy interventions that lean against currency demand shocks by stabilizing UIP premium and spot exchange rate.
  - Stabilizing CIP premium implies leaning against global financial and dollar cycles (stabilizing forward premium).
  - Combination of objectives can be operational targets depending on shock (e.g., “basis control”).
- Open questions and future work:
  - Allocation of currency risk among market participants and shifts in their demand.
  - Connection between dealer futures positions and broader financial/macroeconomic variables to resolve exchange rate disconnect.
  - Optimal policy analysis requires a fully-specified general equilibrium model; paper identifies critical elements for such analysis.

---

*Source: wpiea2025153-source-pdf*

### 2.1  Global bank (intermediary)

### 2.1  Global bank (intermediary)

### Balance sheet and associated returns
- Liability side: net worth W∗t and external funding D∗t (assumed in dollars) with external funding cost R∗t.
- Asset side:
  - Dollar reserves B∗t ≥ 0 that pay R∗t.
  - Risky investments H∗t ≥ 0 with ex-post realized dollar return ˜R∗t+1.
  - Local-currency (LC) assets (net of LC borrowing) A∗t with LC return Rt which converts ex-post into dollar return Rt Et / Et+1, where Et is the spot exchange rate defined as the local-currency price of the dollar (an increase in Et denotes depreciation of the local currency).
- Off-balance-sheet positions:
  - Forwards F∗t (scaled so that F∗t denotes a commitment to buy Rt Et F∗t units of local currency in exchange for Rt Et F∗t / Ft dollars at t + 1). The t + 1 dollar payout on this zero-capital position is (1/Et+1 − 1/Ft) Rt Et F∗t = (Ft / Et+1 − 1) Rt Et F∗t / Ft.
  - Swaps S∗t defined as simultaneous spot and forward transactions with forward leg at swap rate St. Per unit of swap position S∗t the payout at t + 1 is (R∗t − Rt Et St) S∗t. Swaps result in no additional currency risk exposure since payoffs are pre-determined in dollars.
- Balance sheet identity (as accounting):
  - B∗t + H∗t + A∗t = W∗t + D∗t. (1)
- Size of balance sheet measured by H∗t (risky investments), leaving out B∗t, A∗t and D∗t. A∗t and D∗t can be positive or negative (e.g., A∗t < 0 corresponds to net LC liability).

### Bank net worth dynamics and currency exposures
- Aggregating payoffs yields period t + 1 net worth:
  - W∗t+1 = R∗t B∗t + ˜R∗t+1 H∗t + Rt Et / Et+1 A∗t − R∗t D∗t + Rt Et (Ft / Et+1 − 1) F∗t + (R∗t − Rt Et St) S∗t.
- Lemma 1 (assuming St = Ft): net worth evolution can be written as
  - W∗t+1 = R∗t (W∗t − H∗t − B∗t) + R∗t B∗t + ˜R∗t+1 H∗t + CIPt · X∗t + UIPt+1 · Z∗t. (2)
  - Definitions:
    - X∗t ≡ F∗t + S∗t (off-balance-sheet exposure).
    - Z∗t ≡ A∗t + F∗t (currency-risk exposure).
    - UIPt+1 ≡ Rt Et / Et+1 − R∗t.
    - CIPt ≡ R∗t − Rt Et Ft. (3)
- Pass-through of exchange rate volatility into net worth volatility:
  - ∂(W∗t+1 / W∗t) / ∂(Et / Et+1) = Rt Z∗t / W∗t, so pass-through is proportional to exposure Z∗t.
- Forwards F∗t affect both X∗t and Z∗t and thus collect both UIP and CIP premia; swaps S∗t carry only the CIP premium.

### Balance sheet constraints and optimal portfolio problem
- Banks price returns with stochastic discount factor Θt+1 with property Et Θt+1 = 1 / R∗t. Non-stochastic Θt+1 ≡ 1 / R∗t admissible (risk-neutral bank).
- Balance sheet constraint (reserves requirement):
  - B∗t ≥ at H∗t + bt |Z∗t| + δ |X∗t|. (4)
  - where at ≡ α 2 H∗t / W∗t and bt ≡ γ σt2 |Z∗t| / W∗t with σt2 ≡ R2t · vart (Et / Et+1) and α, γ, δ > 0. (5)
  - Interpretation: more risky investment H∗t and greater currency exposure |Z∗t| require higher reserves; off-balance-sheet exposure |X∗t| requires reserves linear in |X∗t| due to counterparty risk.
  - Additional assumption: |X∗t| ≤ X̄∗ for some X̄∗.
- Bank portfolio problem:
  - max over B∗t ≥ 0, H∗t ≥ 0, X∗t, Z∗t of Et Θt+1 W∗t+1 subject to (2), (4)–(5) and |X∗t| ≤ X̄∗. (6)
- Assumption 1 to ensure constraint binding and H∗t > 0:
  - R∗t < R∗t and Et [Θt+1 ˜Rt+1] > 1.
- Proposition 1 (under Assumption 1):
  - μt = R∗t − R∗t > 0 and the balance sheet constraint is binding.
  - (a) St = Ft.
  - (b) Supply schedules for spot and forward currency:
    - UIPt = γ μt σt Z∗t / W∗t
    - CIPt = δ μt · sign(X∗t) for |X∗t| ≤ X̄∗. (7)
  - Thus X∗t = X̄∗ · sign(CIPt) when |CIPt| > δ μt and X∗t = 0 when |CIPt| < δ μt.
- Economic interpretation:
  - Shadow cost of expanding exposures that require one dollar increase in reserves equals μt = R∗t − R∗t.
  - Off-balance-sheet exposure X∗t carries counterparty risk: CIP premium is constant per unit and a step function in X∗t.
  - Risky exposures H∗t and Z∗t are convex in reserves; UIP premium is linear in σt Z∗t / W∗t.
  - Expected UIP premium:
    - UIPt ≡ Rt Et bEt − R∗t, where bEt = [Et (Θt+1 Et Θt+1 E−1t+1)]−1 is the risk-neutral expectation of future spot exchange rate. (8)
  - Forward positions are compensated by UIPt + CIPt = Rt Et (1 / bEt − 1 / Ft).

### Currency market equilibrium (aggregation and implications)
- Aggregate variables for a given currency: Z∗t (aggregate demand to sell currency risk / unhedged demand) and X∗t (aggregate demand to buy currency forwards and swaps / hedged demand), where Z∗t ≡ A∗t + F∗t and X∗t ≡ F∗t + S∗t.
- Market clearing (aggregating bank positions):
  - Z∗t = Σi (A∗it + F∗it), X∗t = Σi (F∗it + S∗it). (9)
- Aggregation of supply schedules (using (7)) and definitions yields Proposition 2:
  - Equilibrium expected UIP and CIP premia as functions of aggregate unhedged and hedged demand:
    - UIPt ≡ Rt Et bEt − R∗t = γ̄t μt σt · Z∗t / W∗t. (10)
    - CIPt ≡ R∗t − Rt Et Ft = δ̄t μt · sign(X∗t). (11)
  - Definitions and aggregation:
    - μt = R∗t − R∗t.
    - W∗t ≡ Σi W∗it is aggregate net worth of dealer banks.
    - γ̄t ≡ [Σi W∗it / γit / W∗t]−1 is the hyperbolic weighted average of γit.
    - δ̄t = δit is the tightness parameter of the marginal bank supplying forwards and swaps.
- Key contrasts and testable implications:
  - Both UIP and CIP premia expand when μt = R∗t − R∗t is large (tightened funding constraints).
  - UIPt depends on the size of aggregate unhedged demand Z∗t relative to aggregate net worth W∗t with elasticity γ̄t μt σt / W∗t.
  - CIPt depends only on aggregate financial conditions δ̄t μt and the sign of X∗t; CIPt does not respond to local currency demand shocks Z∗t in the same way UIPt does.
  - For off-balance-sheet hedged demand X∗t, CIPt is a step function in X∗t: CIP shifts only with market-wide tightness rather than local demand movements.
  - The UIP supply curve slope equals γ̄t μt σt / W∗t = μt σt [Σi W∗it / γit]−1; greater aggregate intermediary net worth or more slack in balance-sheet constraints increases market elasticity and reduces pass-through from currency demand shocks to UIP.
- Additional notes:
  - Conditions (10)–(11) apply both in levels and in changes, characterizing average signs and dynamics of UIP and CIP premia.
  - The model separates currency demand shocks (Z∗t, X∗t) from shifts of currency supply curves (γ̄t, δ̄t, μt, W∗t).
  - The role of intermediaries can be played by any agent whose currency supply responds according to (7).
  - Levels of exchange rates are also determined by other forces outside this partial equilibrium model (e.g., goods market equilibrium and inter-temporal budget constraint).

*Source: wpiea2025153-source-pdf - 2.1  Global bank (intermediary)*

### 2.3  Testable implications

### 2.3  Testable implications

### Cross section of currencies
- Currency supply schedules in the model (reintroduced currency index k):
  - UIPkt ≡ Rkt EktbEkt−R∗t = μt ̄γkt σkt · Z∗kt W∗kt.
  - CIPkt ≡ R∗t − Rkt EktFkt = μt ̄δkt · sign(X∗kt).
- Key terms and interpretations:
  - Ekt (Fkt) is the spot (forward) exchange rate of currency k in units of currency k for one dollar.
  - ̂Ekt is the (risk-neutral) expectation of the future currency k spot exchange rate.
  - Rkt is the local currency k interest rate; R∗t is the dollar interest rate.
  - X∗kt and Z∗kt are demand shifters for (“risky” and “hedged”) exposure to currency k intermediated by banks.
  - μt (tightness of the balance-sheet constraint) is common across currencies.
- Aggregate local-currency funding gap A∗kt:
  - A∗kt > 0: excess demand for local-currency funding (insufficient local-currency savings).
  - A∗kt < 0: excess supply of local-currency savings (need to convert local funding into dollar investments).
- Assumptions made for empirical analysis:
  - Assumption 2: In the long-run equilibrium, the nominal exchange rate does not systematically drift, i.e., E{Ekt/Ek,t+1} ≈ 1.
  - Assumption 3: Currencies with A∗kt > 0 feature (a) Rkt > R∗t, (b) F∗kt > 0, and (c) S∗kt > 0; and vice versa for A∗kt < 0.
- Proposition 3 (under Assumptions 2 and 3):
  - (a) Currencies with excess demand for local-currency investment (A∗kt > 0) feature:
    - High local-currency interest rates (Rkt > R∗t).
    - Positive UIP and CIP premia.
    - Expensive forward dollars (Fkt < Ekt).
  - (b) Currencies with excess supply of local-currency savings (A∗kt < 0) feature:
    - Low local-currency interest rates.
    - Negative UIP and CIP premia.
    - Cheap forward dollars (Fkt > Ekt).
- Analytical relations and implications:
  - Unconditional expectation of the UIP premium (Assumption 2):
    - E{UIPkt} = E{Rkt − R∗t}. (14)
  - For countries with A∗kt > 0, Z∗kt = A∗kt + F∗kt > 0; by Proposition 2 the associated intermediation requires a positive CIP premium:
    - CIPkt = R∗t − Rkt EktFkt > 0  ⇒  Fkt Ekt > Rkt R∗t > 1. (15)
    - Forward dollars are expensive for countries with local-currency funding gaps (cross-sectional forward premium puzzle).
  - For countries with A∗kt < 0, X∗kt is negative, requiring CIPkt < 0 and hence Fkt/Ekt < Rkt/R∗t < 1.

### Dynamics of currency premia and exchange rates
- Model predictions (from currency supply schedules (12)–(13)) for dynamics:
  1. Small local-currency demand shock df∗kt with df∗kt > 0 (sales of local currency and purchases of dollars, spot or forward), with dZ∗kt/df∗kt > 0 and dX∗kt/df∗kt > 0, yields:
     - dUIPkt/df∗kt > 0 and dCIPkt/df∗kt = 0. (16)
     - CIP remains unchanged provided df∗kt is small enough not to affect the identity of the marginal bank leaving ̄δkt unchanged.
  2. Aggregate shocks to the (shadow) cost of intermediation dμt > 0 increase both UIP and CIP premia in absolute values:
     - d|UIPkt|/dμt > 0 and d|CIPkt|/dμt > 0. (17)
     - For investment currencies with A∗kt > 0 (Z∗kt > 0, X∗kt > 0), premia become more positive; for funding currencies with A∗kt, Z∗kt, X∗kt < 0, premia become more negative (spanning out of both currency premia).
     - Impact of financial shocks on UIP premia is increasing in A∗kt:
       - dUIPkt/dμt ∝ A∗kt. (18)
  3. Comparable dynamics apply to shocks to ̄γkt/W∗kt for UIP premia and to shocks to ̄δkt for CIP premia, both proxying for shocks to supply of currency intermediation.
- Note on nonlinearity:
  - Large aggregate shifts in currency demand may affect CIPkt via changes in ̄δkt when market clearing requires participation of additional intermediary banks with tighter constraints.

### Empirical setup and data constructs (preview of tests)
- Sample:
  - G7+ advanced economies and 4 emerging market currencies.
  - G7+ currencies: JPY, CHF, EUR, GBP, CAD, AUS, plus USD base, and NZD.
  - EM currencies: MXN, BRL, ZAR, RUB.
- UIP and CIP premia measurement horizon: 3 months (monthly frequency).
- UIP and CIP empirical definitions (annualized, one-period = 3 months):
  - CIPkt = −(rkt − rUSt) + 4· log(Fkt/Ekt).
  - UIPk,t+3 = (rkt − rUSt) − 4· log(Ek,t+3/Ekt).
  - UIPkt = (rkt − rUSt) − 4· log(bEkt/Ekt).
  - rkt and rUSt are annualized net 3-month money market or deposit interest rates.
  - Ekt is the spot exchange rate in month t (local currency per US dollar).
  - Fkt is the 3-month outright forward exchange rate quoted in t.
  - bEkt is the 3-month ahead consensus survey exchange rate forecast at t.
- Stylized facts from Appendix Figures and tables:
  - The UIP premium is several orders of magnitude larger than the CIP premium (level and changes), especially for G7+ currencies.
  - Among the G7+, the average monthly change in the expected UIP premium is 4.8 percent (480 bps) on average (in either direction), while it is only 4 basis points for the CIP premium.
  - UIP and CIP for MXN is more volatile than for G7+ currencies.
  - Decompositions show most time-series volatility comes from exchange rates, and very little from interest rate dynamics.
- Dealer banks’ net FX futures position data (CFTC TFF weekly report):
  - TFF provides weekly aggregate net long/short futures positions by trader category: Dealer/Intermediary, Asset Manager, Leveraged Funds, Other.
  - Dealers (sell side) are broker-dealer arms of major global banks.
  - Dealers’ net futures position scaled measure:
    - f∗kt = 100 · Dealer Net Positionkt / (1/12 Σ11 j=0 Open Interestk,t−j). (20)
    - Scaling yields a unitless percentage of the market; denominator is 12-month moving average of monthly open interest.
  - Interpretation: positive f∗kt implies dealers are long the FX currency future and short the US dollar future.
  - Empirical notes:
    - TFF series for G7+ currencies go back to June 2006, providing 218 monthly observations per currency.
    - One standard deviation change in dealer net positions amounts to nearly 20 percent of the total futures market size for each currency.
    - Dealer positions co-move but display substantial currency-specific variation.
- Variables affecting slope of currency supply schedules (model counterparts and proxies):
  - Marginal dollar funding cost μt proxied by VIX (baseline), also correlated with global financial cycle factor, intermediary net worth (He, Kelly, and Manela, 2017), and broad dollar index.
  - Other slope determinants: W∗t, σ2kt (expected exchange rate volatility), ̄γkt, ̄δkt.
  - Sample correlation matrix (Appendix Table A3) confirms strong co-movement among these global financial cycle variables.

*Source: wpiea2025153-source-pdf - 2.3  Testable implications*

### 3.2  Cross section of currency premia

### 3.2  Cross section of currency premia

### Cross-sectional patterns: funding gaps and interest rate differentials
- The analysis ranks currencies by the average local-currency interest rate differential relative to the dollar,
 ̄Rk − ̄R∗, and relates it to the average local-currency funding gap ̄A∗k.
- There is a clear positive correlation between the average funding gap ̄A∗k and the average interest rate differential ̄Rk − ̄R∗: currencies with a large funding gap feature high local-currency interest rates, and vice versa.
- Figure 3 (G7+ currencies) illustrates the positive association between ̄A∗k and ̄Rk − ̄R∗.

### Table 2 — currency premia and statistics (averages and standard deviations)
- Table entries are averages with standard deviations in parentheses over January 2000 to December 2020 (shorter for MXN, RUB and BRL).
- Column labels: (1) Funding gap, Ā∗k; (2) Interest rate gap, R̄k − R̄∗; (3) Carry return, UIPk,t+1; (4) Survey UIP, [UIPk]; (5) CIP premium [UIPkt CIPkt].
- Currency-level averages (average (standard deviation)):
  - JPY: Funding gap −31.53 (7.51); Interest rate gap −1.78 (1.71); Carry return −7.07 (19.58); Survey UIP −2.13 (9.42); CIP premium −0.22 (0.17)
  - CHF: Funding gap −50.36 (23.51); Interest rate gap −1.39 (1.19); Carry return −2.76 (18.10); Survey UIP −3.12 (10.13); CIP premium −0.21 (0.21)
  - EUR: Funding gap −2.61 (5.60); Interest rate gap −0.39 (1.32); Carry return −0.60 (19.49); Survey UIP −1.07 (9.74); CIP premium −0.20 (0.25)
  - GBP: Funding gap −1.10 (5.64); Interest rate gap 0.51 (1.26); Carry return 1.31 (18.18); Survey UIP −1.48 (7.94); CIP premium −0.11 (0.17)
  - CAD: Funding gap 11.02 (6.30); Interest rate gap 0.18 (0.75); Carry return 1.38 (15.92); Survey UIP 0.06 (7.30); CIP premium −0.05 (0.14)
  - AUD: Funding gap 24.12 (6.13); Interest rate gap 2.02 (1.66); Carry return 9.02 (25.47); Survey UIP −0.14 (12.63); CIP premium 0.04 (0.17)
  - NZD: Funding gap 17.64 (8.19); Interest rate gap 2.31 (1.55); Carry return 10.99 (24.94); Survey UIP −1.98 (13.33); CIP premium 0.09 (0.20)
  - MXN: Funding gap 3.83 (3.29); Interest rate gap 4.69 (1.36); Carry return 14.58 (23.59); Survey UIP 4.07 (8.54); CIP premium −0.02 (0.69)
  - ZAR: Funding gap 1.87 (9.13); Interest rate gap 5.71 (2.07); Carry return 18.71 (34.82); Survey UIP 2.18 (16.50); CIP premium 0.40 (0.40)
  - RUB: Funding gap −15.12 (9.36); Interest rate gap 6.47 (4.41); Carry return 21.30 (33.80); Survey UIP 6.52 (9.62); CIP premium 0.04 (3.09)
  - BRL: Funding gap 8.88 (8.11); Interest rate gap 10.32 (3.33); Carry return 29.80 (34.46); Survey UIP 9.10 (12.20); CIP premium −1.42 (3.06)

### UIP and CIP deviations: magnitudes and associations
- Expected UIP deviations (survey-based) are large on the order of 200 basis points (2%) annualized, with a standard deviation over time about 10% annualized (reported in brackets in Table 2).
- Average realized currency returns (UIPk,+1) are considerably larger — about five-fold relative to expected UIP deviations — and still show a strong positive association with both expected UIP deviations and the average interest rate differential.
- CIP deviations on average are an order of magnitude smaller (20 basis points, or 0.2%, annualized, or less) with a time-series standard deviation of about 20 basis points.
- For emerging market currencies, standard deviations of CIP deviations can be considerably larger and conventional CIP deviations are measured less reliably.
- There is a strong positive cross-sectional association between average CIP deviations, average UIP deviations, and the average interest rate differential, as illustrated in Figure 4 (left panel: expected UIP vs. interest rate differential; right panel: CIP deviations vs. interest rate differential).

### Classification of currencies in the sample
- Funding currencies: JPY and CHF feature large negative funding gaps (excess supply of local-currency savings), lowest local-currency interest rates, and pronounced negative UIP and CIP deviations.
- Balanced currencies: EUR and GBP exhibit more balanced supply of savings and small funding gaps; they behave similarly to funding currencies with negative CIP deviations and less pronounced negative UIP deviations.
- Investment/commodity and emerging market currencies: CAD, AUD, NZD and emerging market currencies feature excess demand for local-currency investment (positive funding gaps), high local-currency interest rates, and generally positive UIP and CIP deviations.
  - Over time, CAD becomes more of a balanced currency like EUR and GBP; AUD shows a similar but less pronounced shift; NZD remains a robust proxy for EM-currency premia among the G7+ currencies.

### Long-run equilibrium properties (theoretical alignment)
- Observations align with theoretical predictions summarized in Proposition 3:
  - Funding currencies (A∗k < 0) → negative average interest rate differentials.
  - Investment currencies (A∗k > 0) → positive average interest rate differentials.
  - The expected UIP premium is approximately equal to the interest rate differential in steady state.
- The local-currency interest rate differential compensates for currency-risk exposure arising from a currency’s average funding gap, limiting interest rate equalization across countries in the long run.
- Investment in high-interest-rate currencies generates demand to partially sell these currencies forward to hedge currency risk:
  - For funding currencies, intermediaries must demand forward dollars that are sufficiently cheap (negative CIP premium: forward premium smaller than the interest rate differential).
  - For investment currencies, intermediaries must supply forward dollars that are sufficiently expensive (positive CIP premium: forward premium exceeds the interest rate differential).
- The forward premium puzzle in the cross section is reflected by the forward premium not predicting expected depreciation but instead ensuring the required equilibrium CIP premium by more than offsetting the interest rate differential.

*Source: 3.2 Cross section of currency premia (selected tables and figures) from the provided IMF PDF content.*

### 3.3  Dynamics of currency premia

### 3.3  Dynamics of currency premia

### Empirical strategy and identification
- Estimated dynamic panel (distributed lag) specification for currency premia v_kt:
  - ∆v_kt = θ_k + θ_t + Σ_{j=0,1,2} β_j ∆f^∗_{k,t−j} + γ∆w_kt + ρ v_{k,t−1} + ε^v_kt (equation (21)).
  - Currency and time fixed effects: θ_k and θ_t. Time interval: month.
  - Premia considered: tri-monthly (quarterly) UIP premium ∆[UIP]_kt (survey-expected), realized tri-monthly UIP deviation UIP_{kt,t+3}, and tri-monthly CIP premium CIP_kt (all annualized and in log points, as defined in (19)).
  - In some specifications time fixed effects θ_t are dropped and aggregate controls (VIX, other financial-condition proxies) included.
- Key right-hand-side variable: ∆f^∗_kt (dealer banks’ position relative to currency market size), defined in equation (20).
  - Interpretation: an increase in f^∗_kt implies dealer banks absorb currency k futures and sell dollars forward; the rest of the market demands dollars forward.
- Time-series process estimated for ∆f^∗_kt (equation (22)):
  - ∆f^∗_kt = θ_k + θ_t − 0.130 [4.50] · f^∗_{k,t−1} + 0.183 [5.20] · ∆f^∗_{k,t−1} − 0.119 [3.83] · ∆f^∗_{k,t−2} + ε^f_kt.
  - |t|-stats in brackets. Standard deviation of innovation = 15.6.
  - Impulse response: considerable persistence with eventual mean reversion; half-life = 6 months.
- Identification argument (first-stage proxy):
  - Unobserved first-stage: ∆Z^∗_kt = α_k + η_k ∆f^∗_kt + u_kt (equation (23)).
  - Require η_k > 0 and significant R^2 for reduced-form (21) to have explanatory power; shown to hold strongly in the data.
- Limits of inference:
  - Specification (21) is a reduced-form projection (prices on a proxy for intermediaries’ currency exposure); not a structural equation and cannot infer structural supply elasticities.
  - When regression (21) features a high R^2, ∆f^∗_kt is capturing well overall shifts in currency demand that shape equilibrium premia.

### Dynamics of currency premia — pooled G7+ findings (Table 3)
- Main empirical patterns:
  - Strong contemporaneous response and ensuing mean reversion for both survey-expected and realized UIP premia to ∆f^∗_kt shocks.
  - No response of the CIP premium to ∆f^∗_kt shocks.
- Quantitative magnitudes (G7+ pooled sample):
  - Coefficients on ∆f^∗_kt for survey-expected UIP: 0.203, 0.171, 0.226, 0.189 (columns reported; t-stats [16.61], [15.35], [12.15], [11.83]).
  - Coefficients on ∆f^∗_kt for CIP: −0.0000, −0.0000 (t-stats [0.01], [0.06]).
  - Lag coefficient v_{k,t−1} estimates range: −0.462 ∗∗∗ to −0.272 ∗∗∗ (|t|-stats shown).
  - A one standard deviation innovation in ∆f^∗_kt (equal to 15.6) is associated with nearly 300 basis point increase in both UIP premia (or 400 basis points if time fixed effects are not absorbed).
- Explained variation:
  - Within R^2 for survey UIP premium: 0.466 before time fixed effects; 0.705 after including time fixed effects.
  - CIP premia: most variation absorbed by time fixed effects; CIP premia are driven by aggregate (common) shocks rather than currency-specific shocks.
- Comovement between UIP and CIP:
  - Negative comovement between UIP and CIP premia at monthly frequency in pooled regressions without time FE.
  - This negative comovement disappears when time fixed effects are included, implying it is driven by broad financial/currency market conditions rather than currency-specific dynamics.
  - Quantitative example: a hundred basis points increase in survey UIP premium comoves with a reduction in CIP premium of only 0.2 basis points (orders-of-magnitude difference noted).

### Decomposition of premia responses into components (Table 4)
- Exact aggregation identities hold across columns: (1) = (3) − (5) + (6) and (2) = (4) − (3) − (6).
- Components analyzed: 4·∆ log E_kt (spot), 4·∆ log F_kt (forward), 4·∆ log b^E_kt (3-month-ahead survey expectations), ∆ log(R_kt/R^∗_t) (tri-monthly interest rate differential). All variables annualized (multiplied by 4).
- Key decomposition findings:
  - ∆f^∗_kt associated with large and significant effect on UIP premium but not on CIP premium.
  - Interest rate differential effectively unchanged with ∆f^∗_kt (declines by less than 1 basis point in response to a one standard deviation ∆f^∗_kt).
  - Both spot and forward exchange rates depreciate strongly and in sync with an increase in ∆f^∗_kt (by 350 basis points when annualized), explaining the lack of change in CIP premium.
  - Spot movement stronger than UIP (by 270 basis points) because survey expected exchange rate 3 months ahead depreciates by a considerably smaller amount.
- Interpretation:
  - Much of the movement in UIP premium comes from on-impact change in spot exchange rate, not from interest rate changes.
  - CIP premium remains stable because synchronized spot and forward depreciations offset each other.

### Currency-specific dynamics and role of global factors (Table 5 and Figure 5)
- Panel A (survey UIP) — per-currency regression results (G7+ plus MXN):
  - Coefficients on ∆f^∗_kt are consistently positive and significant across currencies:
    - Examples: JPY 0.191 [8.07]; CHF 0.113 [5.05]; EUR 0.252 [5.33]; GBP 0.153 [6.49]; CAD 0.107 [6.71]; AUD 0.162 [5.46]; NZD 0.166 [8.19]; MXN 0.075 [3.55].
  - Controls included: ∆ log VIX and ∆ log E_USD (broad dollar index).
  - Dealer positions, broad dollar, and VIX explain between 53% and 67% of variation in currency-specific UIP premia (R^2 range for ∆[UIP]_kt: 0.529–0.671).
- Panel B (CIP) — per-currency regression results:
  - ∆f^∗_kt coefficients are effectively zero and precisely estimated for every currency (examples: JPY −0.0004 [0.93]; CHF 0.0004 [0.86]; EUR −0.0006 [0.67]; GBP 0.0003 [0.74]; etc.).
  - VIX and broad dollar co-move significantly with CIP for many currencies.
  - R^2 for CIP regressions: around 0.165–0.293 (much lower than for UIP).
- Role of VIX (global financial cycle):
  - Effect of ∆ log VIX on UIP premia varies across currencies, from negative for funding currencies (JPY, CHF) to positive for investment/EM/commodity currencies (AUD, NZD, MXN).
  - Estimated impact of VIX on UIP premium varies continuously along funding gaps (local-currency funding imbalance).
  - For CIP premia, VIX exerts similar differentiated impacts: increases in VIX make CIP premia more negative for funding currencies and more positive for investment/EM currencies.
- Role of broad dollar:
  - Broad dollar appreciation associated with increasing UIP premia for every currency in the sample.
  - Broad dollar tends to comove strongly and negatively with CIP premia for most G7+ currencies (exception: NZD and MXN).
  - Controlling for broad dollar and VIX generally removes the observed negative comovement between UIP and CIP for individual currencies.
- Figure 5 summary (CIP premia by currency bins):
  - Currency bins: ‘funding’ = JPY, CHF, EUR; ‘emerging’ = NZD, MXN, ZAR; ‘balanced’ = GBP, CAD, AUD.
  - Patterns:
    - Funding currencies: CIP premia negative and small most of the time, occasionally spike downwards during global cycles.
    - Emerging currencies: mirror image — larger positive CIP deviations and more volatile upward spikes.
    - Balanced currencies: CIP premia close to zero in normal times and occasionally spike downwards with funding currencies; comovement with funding currencies intensifies toward the end of sample.
  - Implication: CIP premia do not respond to month-to-month idiosyncratic demand for individual currencies; they move with lower-frequency global financial and dollar cycles.

### Synthesis and theoretical consistency
- Empirical findings consistent with model predictions (Proposition 2 and Figure 2):
  - Month-to-month shifts in currency demand translate into UIP premia changes required for intermediation by global banks (movement along ‘unhedged’ currency supply schedule).
  - CIP premia remain stable to such idiosyncratic month-to-month demand shifts because hedged supply schedule features flat steps for that frequency (no significant CIP response).
  - CIP premia are driven by broader financial conditions (μ_t), not by idiosyncratic currency flows.
- Additional notes:
  - UIP premia are two orders of magnitude more volatile than CIP premia (Table 2 reference).
  - At higher frequency (weekly), small but statistically significant association between ∆f^∗_kt and ∆CIP_kt exists, yet quantitatively small and low R^2 (Appendix Table A8).
  - Interactions between funding gap Ā^∗_k and VIX (and other financial-condition proxies) significantly predict the response of both premia (Appendix Tables A6 and A7).

*Source: IMF Working Paper — section "3.3 Dynamics of currency premia" from wpiea2025153-source-pdf*

### 3.4  Exchange rate dynamics

### 3.4 Exchange rate dynamics

### Overview and raw correlation
- Raw pooled correlation between ∆f∗kt and ∆ logEkt across all G7+ currencies: 0.6 (highly statistically significant).
- Scatter evidence: pooled currency-months binned into 100 bins based on ∆f∗kt; corresponding ∆ logEkt averaged within every bin (Figure 6).

### Dynamic projection and estimation (equation)
- Estimated dynamic projection for individual currencies (including MXN):
  ∆ logEkt = θk + Σj=0,1,2,3 ∆f∗k,t−j + γk ∆wkt + ρk logEk,t−1 + εekt,
- wkt includes ikt − i∗t, the Treasury basis (difference between the 12-month US Treasury yield and the average of 12-month Treasury yields in G10 currencies swapped into dollar using the corresponding 12-month forward exchange rate), and the VIX.
- Panel results reported with and without time fixed effects.

### Contemporaneous response: magnitudes and significance
- Strong contemporaneous response of spot exchange rate to dealer position shock for every currency; t-statistics roughly around 10 overall, ranging from:
  - 4.0 for MXN
  - 7.6 for GBP
  - 12.6 for CHF
  - 15.9 for NZD
- Effect of a one standard deviation innovation in individual ∆f∗kt on currency k (depreciation against the dollar):
  - Typical depreciation: around 1.3%.
  - MXN: 0.9%.
  - CAD: 0.9%.
  - JPY: 1.3%.
  - CHF: 1.3%.
  - NZD: 1.6%.
- Panel of G7+ currencies with time fixed effects: one standard deviation innovation → 0.9% depreciation.
- Persistence / mean reversion:
  - Estimated ρ̂k on lagged exchange rate ≈ 0 for all k → virtually no exogenous mean reversion.
  - Mean reversion in exchange rates arises from mean reversion in dealer positions.
  - Specification (22) implies half-life of 5 or 6 months for a shock to f∗kt and corresponding half-life of 6 or 8 months for the exchange rate in the panel (with or without time fixed effects, respectively).
  - Half-lives for individual currencies vary from 6 to 13 months; GBP half-life reported as ∞.

### Model fit and explained variation
- Model (24) accounts for:
  - 45 to 60% of variation in spot exchange rate changes for individual currencies.
  - 70% explained variation in pooled panel with time fixed effects.
- Contribution of dealer positions to R2 (share of total R2 due to ∆f∗kt), by currency (selected values reported in Table 6):
  - JPY: 0.812
  - CHF: 0.890
  - EUR: 0.686
  - GBP: 0.636
  - CAD: 0.515
  - AUD: 0.416
  - NZD: 0.628
  - MXN: 0.352
  - G7+ Panel (without/with time FE columns present): 0.776 and 0.602 reported in Table 6
- R2 values reported in Table 6 (individual currencies, columns 1–8): 0.450, 0.436, 0.473, 0.485, 0.592, 0.606, 0.607, 0.444; panel columns: 0.464 and 0.700.
- Standard deviation of innovation (std of innovation (%)) reported in Table 6 (by currency):
  - JPY: 1.32
  - CHF: 1.30
  - EUR: 1.16
  - GBP: 1.09
  - CAD: 0.93
  - AUD: 1.23
  - NZD: 1.61
  - MXN: 0.91
  - G7+ Panel: 1.29 (column 9) and 0.87 (column 10)

### Role of conventional macro-finance variables
- Interest rate differential (∆(ikt − i∗t)):
  - Strong association with exchange rate appreciations for all currencies except CHF and MXN.
- Treasury basis (∆T-basis):
  - Negative effect on exchange rates: reduction in the basis (increased demand for dollar safe assets) → depreciation of other currencies.
  - Effect increases in magnitude and significance moving from funding toward investment currencies.
- VIX (∆ log VIX):
  - Weak appreciating impact on funding currencies; depreciating impact for commodity and investment currencies.
- Changes in log dealer wealth (∆ log W∗t):
  - Decrease in dealer wealth associated with depreciation of all currencies except CHF and JPY.
- Despite these associations, dealer positions maintain a large and often majority independent contribution to model fit for most currencies (except MXN and AUD where contribution is smaller).

### Fit diagnostics and predictive components
- Correlation in changes between exchange rates and predicted component based on dealer positions alone: around 0.6 for most currencies (MXN: 0.5).
- Correlations for the full model in changes: around 0.7.
- Fit in levels is strong but can be misleading for near-non-stationary variables; focus recommended on fit in changes.
- Figures plotting full model and partial fit due to dealer positions show high-quality fit and high correlation coefficients (EUR and GBP shown in introduction; other currencies in appendices).

### Impulse responses and dynamics following a shock to dealer positions
- Impulse responses constructed for a one standard-deviation innovation to f∗kt using panel point estimates with time fixed effects (specification reported in (22) and column 10 of Table 6).
- Persistence:
  - Half-life for dealers’ position innovation: 5 months.
  - Half-life for the exchange rate response: 6 months.
- Impact and subsequent path for a typical exchange rate following a one standard deviation innovation to f∗kt:
  - Impact depreciation: 0.87% (87 log basis points).
  - Following two years: gradual appreciation, by an average of 6 log basis points monthly over the first 12 months.
  - Distinctive pattern: large unexpected devaluation on impact followed by a sequence of predictable small appreciations.
- Alternative specification (without time fixed effects):
  - Impact response: 1.29% depreciation.
  - Mean reversion half-life: 8 months.
- Currency premia, CIP and interest rates following a one standard deviation innovation to ∆f∗kt:
  - Virtually no change in covered interest rate premium CIPk,t+j.
  - Virtually no change in interest rate differential rk,t+j − rUSt+j at any horizon j ≥ 0.
  - Estimated effects on these quantities are of the order of magnitude of one (text truncated at end of provided content).

*Source: IMF working paper chapter — 3.4 Exchange rate dynamics.*

### Appendix Figure A14 shows impulse responses for individual currencies.  Appendix Figure A15 plots impulse re-

### Appendix Figure A14 shows impulse responses for individual currencies. Appendix Figure A15 plots impulse re-

### Impulse responses and currency returns
- The figure plots the impulse response to uncovered currency returns at t + j conditional on a one standard deviation innovation to dealers’ futures positions f∗kt at t.
- The outcome variable is the realized (for j ≤ 0) and predicted (for j > 0) returns on a 3-month zero-capital carry trade at t + j, which is long currency k and short US dollar, per dollar of gross exposure.
- The units on the y-axis are annualized basis points, i.e., 200 corresponds to 2% extra return.

### Mechanics and magnitudes of response to a one standard deviation innovation in f∗kt
- A one standard deviation positive innovation in f∗kt in month t is associated with a very large negative return on currency k for positions open before t (j < 0).
  - These negative returns are as large as 540 basis point (5.4%) annualized loss on a three-month position.
  - One such standard monthly shock during the three-month tenure of the contract instantaneously destroys value (net worth) equal to 1.3% of the net currency exposure.
- For positions taken after the innovation (j > 0), expected returns are positive and realized over a longer period:
  - Up to 175 basis points (1.75%) annualized in the first quarter.
  - Over 100 basis points on average over the full year.
- The entire currency return described is driven by predictable movement in the spot exchange rate; the interest rate response is effectively zero.

### Dynamic profile of exchange rate and UIP premia
- The exchange rate impulse response features:
  - A sharp depreciation on impact (creating an unanticipated value loss, the “steamroller”).
  - A predictable gradual appreciation of currency k over the following year (providing expected returns, the “pennies”).
- The exchange rate process exhibits persistent innovations and long half-lives with a small predictable mean reversion over multiple quarters, producing:
  - Large unpredictable innovations (high volatility).
  - Modest Sharpe ratio for intermediaries collecting expected returns.
- UIP premia:
  - Vary at high frequency in response to currency-specific shocks tightly correlated with dealer banks’ currency futures positions.
  - Respond strongly to shifts in currency-specific demand (proxied by dealer banks’ futures positions).
- CIP premia:
  - Stay stable in response to currency-specific dealer shocks.
  - Change relatively infrequently with aggregate financial conditions; when aggregate conditions tighten (e.g., spike in VIX), CIP premia widen—becoming more positive for investment currencies and more negative for funding currencies.
- Forward exchange rates move in lock-step with spot exchange rates, keeping CIP premium stable in response to shifts in currency demand; only when aggregate financial conditions tighten do forward dollars become cheaper (more expensive) relative to spot dollars for funding (investment) currencies.

### Cross-section interpretation
- Local-currency interest rate differentials and covered and uncovered currency premia reflect the local-currency funding gap of the country (whether a net supplier of savings or a net destination for investment).
- Funding currencies (excess local savings), e.g., Japan and Switzerland:
  - (i) low local-currency interest rates,
  - (ii) negative UIP premia that reflect the interest rate differential,
  - (iii) negative CIP premia (cheap forward dollars relative to spot).
- Investment, commodity and emerging-market currencies (insufficient local-currency savings) require international (dollar) funding and exhibit opposite patterns.
- Interest rate differentials translate one-for-one into long-run UIP premium charged by intermediary banks for holding currency risk; forward premiums more than offset interest rate differentials to compensate intermediary banks supplying currency swaps—consistent with empirical forward premium puzzle.

### Comparison of shocks: dealer positions vs interest rates
- Shocks to dealer banks’ futures positions:
  - Strongly correlated with contemporaneous depreciation of currency k against the dollar.
  - Drive significant financial loss for holders before the shock and predictable positive returns for those entering after the shock.
  - Currency returns following these shocks come entirely from predictable dynamic path of the exchange rate; interest rates do not respond.
- Shocks to interest rates:
  - Have very weak predictive power for future exchange rate depreciations with a nearly zero R2 in the Fama regression.
  - Conditional on interest rate differential, most currency returns are the interest rate differential itself in the panel of currencies.

### Operational measurement and implications for intermediation
- Weekly or monthly shifts in dealer banks’ futures positions offer a real-time way to measure expected UIP deviations that emerge with shifts in currency futures positions of dealer banks.
- The model treats the intermediary sector generically (global bank with affiliated dealer) but extends to any agent with a stable currency supply schedule responding to UIP premium.
- The framework does not require a strict hedger/speculator distinction; agents can act on either motive and equilibrium UIP results from interactions of demand and supply shifts.

### Policy implications and avenues for future work
- Policy rationale:
  - Frictional intermediation suggests a rationale for policy interventions that lean against currency demand shocks by partially or fully eliminating currency premia and stabilizing the exchange rate.
  - Stabilizing the UIP premium amounts to accommodating or dampening currency-specific demand shocks and largely stabilizing the spot exchange rate.
  - Stabilizing the CIP premium means leaning against global financial and dollar cycles, stabilizing the forward premium without eliminating spot exchange rate fluctuations.
  - A combination of these objectives can serve as operational targets depending on the shock to be managed (e.g., “basis control”).
- Open questions:
  - Allocation of currency risk exposure among market participants and nature of shifts in their currency demand.
  - Connection between dealer banks’ futures positions and other financial and macroeconomic variables to resolve exchange rate disconnect.
  - Optimal policy analysis requires a fully-specified general equilibrium model; the paper identifies critical model elements for such analysis.

*Source: wpiea2025153-source-pdf*

### References

### References

### Cited literature
- Comprehensive bibliography of empirical and theoretical work on exchange rates, covered interest parity (CIP), uncovered interest parity (UIP), FX markets, dealer behavior, global financial cycles, and international banking. Representative authors and works include:
  - Abbassi, P., and F. Bräuning (2021): “Demand effects in the FX forward market: Micro evidence from banks’ dollar hedging,” The Review of Financial Studies, 34(9), 4177–4215.
  - Avdjiev, S., W. Du, C. Koch, and H. S. Shin (2019): “The Dollar, Bank Leverage, and Deviations from Covered Interest Parity,” American Economic Review: Insights, 1(2), 193–208.
  - Bacchetta, P., K. Benhima, and B. Berthold (2025): “Foreign exchange intervention with UIP and CIP deviations,” Swiss Finance Institute Research Paper, (23-71).
  - Du, W., and J. Schreger (2022): “CIP deviations, the dollar, and frictions in international capital markets,” in Handbook of International Economics, vol. 6, pp. 147–197. Elsevier.
  - Gourinchas, P.-O., W. Ray, and D. Vayanos (2025): “A preferred-habitat model of term premia, exchange rates, and monetary policy spillovers,” American Economic Review, forthcoming.
  - Itskhoki, O., and D. Mukhin (2025a): “Mussa Puzzle Redux,” Econometrica, 93(1), 1–39.
  - Dao, M. C., and P.-O. Gourinchas (2025): “Covered Interest Parity Deviations in Emerging Markets: Measurement and Drivers,” IMF Working Paper No. 2025/057.
- The bibliography spans journal articles, NBER and IMF working papers, central bank staff working papers, and discussion papers addressing:
  - Dealer risk limits and currency returns.
  - FX market microstructure and order flow.
  - Intermediary asset pricing, dealer balance sheets, and global liquidity.
  - Empirical decompositions of UIP and CIP premia and their links to macrofinancial variables.

### Appendix A — Additional tables and figures (captions and substantive notes)
- Figure A1: Illustration for balance sheet in Table 1: possible currency trades.
- Figure A2: Illustration of international intermediary bank and its dealer subsidiary: FX exposure passthrough.
- Table A1: Interest rates and forward rates: tickers and descriptions.
  - Source: Refinitiv/LSEG.
  - Note: Interest rate day count and forward point quoting conventions differ by currency and period.
  - Important operational detail: "The USD 3-month LIBOR rate is replaced by the synthetic LIBOR rate by the ICE Benchmark Administration after July 2023, following cessation of the official Dollar LIBOR panel rate publication, until September 30 2024. This synthetic rate is calculated using the 3-month SOFR reference rate and a spread reflecting large banks’ credit risk and liquidity conditions. The synthetic LIBOR rate represents a consistent extension of the historical LIBOR rate to facilitate the settlement of legacy contracts, as required by the UK Financial Conduct Authority (FCA). A similar transition requirement and synthetic LIBOR rate methodology applies to the GBP 3-Month ICE LIBOR rate starting after December 2021 until March 2024 (using the 3-month SONIA reference rate)."
- Table A2: Summary statistics for monthly changes in dealer net futures positions ∆f∗kt
  - Notes: "Table shows summary statsitics for the monthly change in net (long) futures position of FX dealers in percent of 12-month moving average open interest by currency. Sample period: June 2006 to August 2024. Source: CFTC TFF Report (weekly reports aggregated to monthly frequency)."
- Table A3: Pairwise correlation coefficients for global financial conditions
  - Note: "Entries show monthly pairwise correlation coefficients. * indicate statistical signifcance at 5 percent. EUSDt is the Broad Dollar Index. log W∗t is the log capital ratio of global intermediaries from He et al. (2017). T-basist is the 12-month Treasury basis as defined in JKL. GFCy is the global financial cycle factor from Miranda-Aggreppino and Rey (2022). EFFR− IORB is the spread between effective federal funds rate and the interest on reserve balances."
- Table A4: UIP for Emerging Market currencies
  - Note: "Regression specification and variable definitions follow Table 3, applied to monthly spot exchange rate changes of Emerging Marlet currencies with available data on dealer futures positions from the CFTC’s TFF database.  All regressions additionally include ∆f∗k,t−2 that in all cases are estimated to be close to zero and insignificant. ∗∗∗ (∗∗ and ∗) denotes statistical significance at the 1-percent (5-percent and 10-percent) level."
  - Reported regression diagnostics (as shown): Within R2 values include 0.280, 0.748, 0.442, 0.529, 0.589, 0.433, 0.396.
  - Observations row (as reported in the table): 55655655621688151101
  - "# currency FE" row (as reported): 444
  - Time FE: ✓
- Table A5: Spot exchange rate regressions for EM currencies
  - Note: "Regression specification and variable definitions follow Table 6, applied to monthly spot exchange rate changes of EM currencies with available data on dealer futures positions from the CFTC’s TFF database. |t|-stats computed using Newey West standard errors in columns 1-4 and Driscoll-Kraay autocorrelation and heteroskedacticity robust standard errors in column 5 shown in brackets. ∗∗∗ , ∗∗ , ∗ denote statistical significance at the 1, 5 and 10 percent level respectively."
  - Reported Within R2 values (as shown): 0.444, 0.598, 0.209, 0.328, 0.374, 0.798.
  - Observations row (as reported in the table): 20914913478570588
  - "# Currency FE" row (as reported): 44
  - Time FE: ✓
- Table A6: UIP panel regressions with alternative measures of μ
  - Note: Panel regressions for G7 currencies with survey-based UIP change as the dependent variable and using alternative measures of μt (marginal cost of Dollar funding): baseline VIX index, the inverse of log Dealer leverage ratio (−DealerW), the negative of the (pro-cyclical) Global Financial Cycle index (−GFCy) from Miranda-Agrippino and Rey and the spread between effective federal funds rate and the interest on reserve balance rate (EFFR− IORB). Measures of μ are interacted with the currency-specific local-currency funding gap  ̄A∗kt, which equals external dollar-debt liabilities minus external dollar-debt assets in percent of GDP, and changes only at the annual frequency. ∆CIPk,t and lagged UIP level [UIPk,t−1 are included in all columns but not reported. Currency fixed effects included in all regressions. |t|-stats computed using Driscoll-Kraay autocorrelation and heteroskedacticity robust standard errors shown in brackets."
  - Observations and fixed-effect dimensions (as shown): Observations1,2041,2041,2041,0641,0641,0221,022 ; # currency FE7777777
  - Within R2 values (as shown): 0.5700, 0.5620, 0.5740, 0.5770, 0.5810, 0.5350, 0.567
- Table A7: CIP panel regressions with dollar gap interaction and alternative measures of μ
  - Note: Panel regressions for G7 currencies with CIP change as dependent variables and using alternative measures of μt (marginal cost of Dollar funding). Measures of μ are interacted with the local-currency funding gap  ̄A∗kt. A constant term, ∆UIPk,t and lagged CIP level CIPk,t−1 included in all columns but not reported. Currency fixed effects included in all regressions. |t|-stats computed using Driscoll-Kraay autocorrelation and heteroskedacticity robust standard errors shown in brackets.
  - Observations and fixed-effect dimensions (as shown): Observations1,2041,2041,2041,0641,0641,0221,022 ; # currency FE7777777
  - Within R2 values (as shown): 0.1930, 0.1920, 0.1990, 0.2130, 0.2140, 0.2170, 0.224
- Table A8: Weekly CIP panel
  - Note: Panel regressions for G-7 and EM currencies with weekly CIP change as dependent variable and weekly change in FX futures dealer position as main explanatory variable. EM Panels include the same four EM currencies as in Table A5. All regressions additionally include ∆f∗t−2 that in all cases are estimated to be close to zero and insignificant. Currency fixed effects included in all regressions and time fixed effects included in columns (2) and (4). |t|-statsitics computed using Driscoll-Kraay autocorrelation and heteroskedacticity robust standard errors shown in brackets. ∗∗∗ , ∗∗ , ∗ denote statistical significance at the 1, 5 and 10 percent level respectively.
  - Reported Within R2 values (as shown): 0.060, 0.575, 0.051, 0.452.
  - Observations (as shown): 6,3286,3282,2922,292
  - "# currency FE" (as shown): 7744
- Figures (selected captions and notes with measurement units and sources):
  - Figure A3: Realizations of logEkt and its fitted values — cumulated fitted values from an empirical model of ∆ logEkt using only dealer currency futures positions ∆f∗kt and from the full model (based on Table 6).
  - Figure A4: Time series of survey UIP and CIP premia: G7 and MXN. Notes: "Left figure plots the 3-month survey UIP premium for G7 currencies (gray) and Mexican Peso (blue), in percent annualized. Right figure plots the 3-month CIP premium for G7 currencies (gray) and Mexican peso (blue), in percent annualized. Source: Bloomberg, Refinitiv, IMF IFS, Consensus Economics."
  - Figure A5: Dealers net futures position by G7 currency. Notes: "The figure plots the time series for net futures positions of dealer banks on the Chicago Mercantile Exchange by currency, in percent of 12-month moving average total open positions in each currency. Data are monthly averages of weekly numbers. Source: CFTC Traders in Financial Futures (TFF) Weekly Reports."
  - Figure A6: UIP and CIP decomposition. Notes: Panel (a) decomposes the survey UIP premium into the local currency vs. dollar interest rate differential (irdiff) and expected appreciation (de). Panel (b) decomposes the CIP premium into the interest differential (irdiff) and forward premium (rho).
  - Figure A7/A8: Cross sections relating funding gaps, interest differentials, UIP and CIP deviations for a wide set of AE and EM currencies. Notes include averaging period: "All variables are averaged over Jan 2012- Dec 2020 monthly observations."
  - Figure A16: Impulse responses of CIPk,t+j (left) and rk,t+j − rUSt+j (right) to a ∆f∗kt innovation. Note: "Both panels are in basis points, that is, 0.5 on the y-axis is half a basis points (or 0.005%), annualized."

### Appendix B — Proofs (high-level content and properties shown)
- Proof of Proposition 1:
  - Uses net worth evolution, expectation E_t Θ_{t+1} = 1/R∗_t, and complementary slackness to derive first-order conditions for optimal {B∗_t ≥ 0, H∗_t ≥ 0, X∗_t, Z∗_t, S∗_t}.
  - Key result: the balance sheet constraint binds (μ_t > 0), reserves satisfy R∗_t = 1/E_t Θ_{t+1}, and S_t = F_t. The proof yields expressions for CIP_t and UIP_t consistent with the proposition.
- Proof of Proposition 2:
  - Aggregation of individual bank positions Z∗_it = A∗_it + F∗_it leads to condition (10) and market-clearing implications for UIP_t.
  - For forward/swap positions X∗_it, sign(X∗_it) = sign(CIP_t) = sign(X∗_t); heterogeneity in δ_it implies three equilibrium cases for which (11) holds, with  ̄δ_t μ_t = |CIP_t|.

### Appendix C — Time-series identification
- The appendix introduces the identification argument behind the local projection regressions (21) and (22). (Text of the identification argument continues beyond the supplied excerpt.)

*Source: References and appendices content from wpiea2025153-source-pdf - References*

### Section 3.3 and (24) in Section 3.4.

### Section 3.3 and (24) in Section 3.4

### Model setup and baseline specification
- Consider the simple case where both f∗_kt and variables of interest v_kt (currency premia, exchange rates) follow exact random walks; lags are not needed in dynamic projections in changes and contemporaneous correlations suffice.
- Baseline specification (in changes):
  - ∆v_kt = θ_k + θ_t + β∆f∗_kt + γw_kt + ε_kt . (A1)
- Empirical implementations include lags to address potential dynamic correlations.
- Focus first on expected UIP premium: v_kt = UIP_kt, then generalize to proxies for expected UIP (e.g., CIP premium, exchange rate).

### Market structure and agent exposures
- Set of agents active in currency market: I_k = I^D_k ∪ I^N_k, with I^D_k ∩ I^N_k = ∅.
  - I^D_kt: agents classified as dealers.
  - I^N_kt: non-dealers (everyone else).
- Z^*_it denotes the currency k risk exposure of agent i ∈ I_k.
- Market clearing condition:
  - Σ_{i∈I_k} Z^*_it = 0 . (A2)
- Total market volume: V^*_kt ≡ Σ_{i∈I_k} |Z^*_it| > 0 (market active).
- Net exposure of all dealers:
  - Z^D_kt ≡ Σ_{i∈I^D_k} Z^*_it . (A3)
  - By clearing, Σ_{i∈I^N_k} Z^*_it = −Z^D_kt.
- Reduced-form decomposition of any agent i's exposure:
  - Z^*_it = −ξ_it + ρ_i · UIP_kt,   ξ_it ⟂ UIP_kt . (A4)
  - Interpretation: projection onto expected premium (supply portion) plus orthogonal residual ξ_it (demand portion).

### Structural interpretation for dealers
- For dealer banks (Proposition 1), rewrite optimality condition (7) as:
  - Z_it = ρ_it · UIP_kt for i ∈ I^D_k where ρ_it ≡ W^*_it/[γ_it μ_t σ_kt].
- Taylor expansion around unconditional mean ̄Z_i = ̄ρ_i · E UIP_kt:
  - dZ_it = −ξ_it + ̄ρ_i · dUIP_kt , where ξ_it ≡ ̄Z_i ̄ρ_i · dρ_it = ̄Z_i · d log ρ_it . (A5)
- dρ_it need not be orthogonal to dUIP_kt; ξ_it can be defined as the residual from projecting dρ_it on dUIP_kt.
- Use structural non-orthogonal expansion (A5) for dealers and reduced-form orthogonal expansion (A4) for non-dealers.

### Equilibrium characterization (Proposition A1)
- Aggregate definitions:
  - Ξ_kt ≡ ξ^D_kt + ξ^N_kt = Σ_{i∈I^D_k} ξ_it + Σ_{i∈I^N_k} ξ_it (currency k demand shock).
  - ρ^J_k ≡ Σ_{i∈I^J_k} ̄ρ_i for J ∈ {D, N}.
- Equilibrium expected UIP premium:
  - UIP_kt = 1/(ρ^D_k + ρ^N_k) · Ξ_kt .
- Equilibrium net dealers’ position:
  - Z^D_kt = −ξ^D_kt + ρ^D_k · UIP_kt
    = −ξ^D_kt + ρ^D_k/(ρ^D_k + ρ^N_k) · Ξ_kt
    = 1/(ρ^D_k + ρ^N_k) · (ρ^D_k ξ^N_kt − ρ^N_k ξ^D_kt).
- Interpretation: UIP_kt reflects aggregated currency demand shock Ξ_kt with pass-through proportional to the inverse of aggregate supply slope ρ^D_k + ρ^N_k.

### Regression of UIP on dealers’ net position (pairwise regression) and R^2
- Consider regression (levels, implemented in changes to allow for unit roots):
  - UIP_kt = α^D + β^D Z^D_kt + ε^D_kt . (A6)
- Corollary A1: expressions for β^D and R^2_D:
  - β^D = 1/ρ^D_k · cov(ξ^N_kt + ξ^D_kt, ξ^N_kt − (ρ^N_k/ρ^D_k)·ξ^D_kt) / var(ξ^N_kt − (ρ^N_k/ρ^D_k)·ξ^D_kt)
  - R^2_D = [cov(ξ^N_kt + ξ^D_kt, ξ^N_kt − (ρ^N_k/ρ^D_k)·ξ^D_kt)]^2 / [ var(ξ^N_kt + ξ^D_kt) · var(ξ^N_kt − (ρ^N_k/ρ^D_k)·ξ^D_kt) ]
- Limits and special cases:
  - When var(ξ^D_kt)/var(ξ^N_kt) → 0 and ρ^D_k > 0:
    - β^D → 1/ρ^D_k and R^2_D → 1.
    - Intuition: dealers’ net positions become a near-perfect instrument for aggregate demand shock Ξ_kt when dealers move mainly along supply curve and have little demand-shift variation.
  - Perfect correlation case: ξ^D_kt = φ ξ^N_kt yields R^2_D = 1 but biased β^D:
    - β^D = 1/ρ^D_k · (1+φ)/(1−φ ρ^N_k/ρ^D_k)
    - Condition for correct sign: ρ^D_k > φ ρ^N_k.
  - Case ρ^N_k = 0 (non-dealers exhibit only demand shifts; Z^D_kt = ξ^N_kt):
    - β^D = 1/ρ^D_k · [ 1 + corr(ξ^D_kt, ξ^N_kt) · (var(ξ^D_kt)/var(ξ^N_kt)) ]^2
    - R^2_D = 1 − [1 − corr(ξ^D_kt, ξ^N_kt)]^2 · var(ξ^D_kt)/var(ξ^N_kt + ξ^D_kt).
    - R^2_D → 1 when corr(ξ^D_kt, ξ^N_kt) → 1 or var(ξ^D_kt)/var(ξ^N_kt) → 0, but bias in β^D disappears only in latter case.
- General statement: R^2_D measures proximity of Z^D_kt to Ξ_kt (how well net dealers’ position approximates aggregate demand shock), regardless of β^D bias.

### Regression with controls for dealer demand shifts
- Alternative specification controlling for ξ^D_kt (as defined in (A5)), corresponding to empirical controls for ̄γ_kt μ_t and W^D_kt:
  - UIP_kt = α^D + β^D Z^D_kt + γ^D ξ^D_kt + ε^D_kt . (A7)
- In practice estimated in changes; β^D equals pairwise regression coefficient of UIP_kt residualized on ξ^D_kt onto residualized Z^D_kt.
- This case corresponds to var(ξ^D_kt)/var(ξ^N_kt) = 0 in pairwise specification, hence β^D = 1/ρ^D_k and R^2_D = 1.

### Empirical proxies and identification for Z^D_kt and UIP_kt
- UIP_kt and Z^D_kt are not directly observable; proxies used:
  - UIP_kt proxies: survey expectations ([UIP_kt]) or realized currency returns (UIP_kt,t+3).
    - Using proxies yields lower observed R^2 relative to perfect observability.
  - Z^D_kt proxy: constructed from futures market positions f∗_kt of affiliated dealer‑bank arms of large international banks.
- Identifying assumption for first-stage:
  - ∆Z^D_kt = α_k + η_k ∆f∗_kt + u^D_kt . (A8)
  - Requires η_k > 0 and sufficiently high R^2 for ∆f∗_kt to be a useful proxy in the reduced-form regression of ∆UIP_kt on ∆f∗_kt.
- Rationale:
  - Overall intermediary exposure: Z^D_kt = A^D_kt + F^D_kt (spot plus forward exposure).
  - Futures and forwards are highly substitutable; shifts in futures market (∆f∗_kt) indicate broader shifts in forward demand including OTC forwards, making ∆f∗_kt a proxy for ∆F^D_kt.
  - Assume net spot position A^D_kt is slow-moving (structural local-currency savings gap), so monthly ∆f∗_kt proxies monthly ∆Z^D_kt; α_k absorbs slower-moving currency-specific trends.

### CIP regressions and interpretation
- Empirical finding: regressions of ∆CIP_kt on ∆f∗_kt show precisely estimated zero coefficients and near-zero R^2 at monthly frequency (some effects detectable at weekly frequency).
- Mapping to model:
  - For CIP, relevant unobserved variable is X^D_kt = S^D_kt + F^D_kt (net swap positions plus forwards), not Z^D_kt = A^D_kt + F^D_kt.
  - ∆f∗_kt is a direct proxy for ∆F^D_kt and a strong predictor of UIP premium, but lack of correlation with CIP premium suggests the second-stage coefficient of ∆CIP_kt on ∆X^D_kt is zero.
- Interpretation and conclusion:
  - The alternative (that ∆f∗_kt and ∆X^D_kt are uncorrelated for every currency) is implausible because it would require near-perfect offset of ∆S_kt and ∆F^D_kt for every currency.
  - Conclude β_CIP_k = 0 in (A1) for every k is a consequence of locally elastic supply of hedged dollars, ρ^D_k → ∞ for CIP (applying Proposition A1 to CIP).

### Decomposition of currency premia dynamics
- Using empirical definitions of currency premia in (19), estimated impulse responses from specifications in (A1) (and generalizations (21) allowing more dynamics) can be decomposed into impulse responses of components:
  - Interest rate differential (forward‑looking object known at t).
  - Forward premium (known at t).
  - Current and future (expected) spot exchange rate (requires care because realized carry trade return depends on both current and future exchange rates).
- Procedure: construct individual impulse response for exchange rate component and aggregate into currency premium component-by-component.

*Source: wpiea2025153-source-pdf - Section 3.3 and (24) in Section 3.4.*

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