## 3.1 Main building blocks

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### Bank balance sheet, instruments, and valuation
- Exchange rate notation: S (units of local currency per unit of foreign currency); S_t = exp[s_t], S_{t+1} = exp[s_{t+1}].
- Balance sheet items (values in domestic currency unless specified):
  - Loans: C, with C_dom and C_for (foreign-currency loans fixed in foreign currency).
  - Deposits: D, with D_dom and D_for (fixed).
  - Wholesale funding (liabilities, net of wholesale lending): F; can be negative. Domestic-currency wholesale funding in period t: W_dom_t. Foreign-currency value of funding in foreign currency: W_for_t.
  - Equity: K (predetermined). Total wholesale funding in period t, W_t, predetermined.
- Forward contract / FX swap participation:
  - Forward rate contract payoff written with F_{t+1} = exp[f_{t+1}] (units domestic currency per foreign currency) known in period t.
  - Buying G_for units of forward contract yields period t+1 payoff:
    G_for(F_{t+1} − S_{t+1}) − α_{t+1}/2 · F_{t+1} · (G_for)^2
  - Gross cost (including principal) of wholesale funding:
    - Domestic: exp[r_{W,dom,t+1}]
    - Foreign: exp[r_{W,for,t+1}]
  - Covered foreign currency funding is treated as risk-free via spot-forward combination; deviations from covered interest parity captured via quadratic balance-sheet cost with parameter α_{t+1}.

### Bank objective and equity evolution
- CRRA utility: u = (1 − γ_t)^{−1} (K_{t+1})^{1−γ_t}.
- Equity identity in period t:
  K_t = C_dom − D_dom − W_dom_t + S_t [C_for − D_for − W_for_t]
- Equity in period t+1:
  K_{t+1} = C_dom − D_dom − W_dom_t R_{W,dom,t+1} + S_{t+1} (C_for − D_for − W_for_t R_{W,for,t+1}) + G_for(F_{t+1} − S_{t+1}) − α_{t+1}/2 (G_for)^2

### First-order optimality conditions
- Wholesale funding FOCs (combined domestic and foreign):
  0 = E(K_{t+1})^{−γ_t} [ S_{t+1}/S_t R_{W,for,t+1} − R_{W,dom,t+1} ]  (Equation (1))
- Forward contract FOC:
  0 = E(K_{t+1})^{−γ_t} ( F_{t+1} − S_{t+1} − α_{t+1} F_{t+1} G_for )  (Equation (2))

### Solution strategy (two-step Taylor expansion)
- Solve highly non-linear FOCs via Taylor expansion (quadratic and cubic approximations), keeping terms proportional to innovations and squares of innovations.
- Two steps:
  1. Derive baseline (steady-state) wholesale funding in foreign currency and forward contract position given exogenous baseline environment (loans, deposits, interest rates, expected exchange rate movements, moments of baseline shocks).
  2. Compute portfolio values in a shifted environment where loans, deposits, interest rates, exchange rates, risk, and risk aversion shift away from baseline; assess impacts on portfolio choice.
- Baseline and shifted variables denoted with base and shift subscripts.

### Baseline portfolio closed-form results
- Definitions:
  - Net_OBS_base = C_for_base − D_for_base − W_for_base  (Equation (3))
  - Net_TOT_base = Net_OBS_base − G_for_base  (Equation (4))
- Parity deviations (for k = base, shift):
  - uip_k = r_{W,for,t+1,k} + s_{t+1,k} − s_{t,k} − r_{W,dom,t+1,k}
  - cip_k = p_k + uip_k, where p is forward premium (forward minus expected spot)
- Baseline portfolio (Equation (5)):
  - G_for_base = cip_base / α_base
  - Net_TOT_base = K_base · uip_base / (γ_base · σ^2_{fx,base})
- Interpretations:
  - G_for_base determined by deviation from covered interest parity adjusted by α_base; unaffected by risk aversion or risk.
  - Net_TOT_base determined by deviation from uncovered interest parity adjusted by σ^2_{fx,base} and γ_base.

### Shifted portfolio sensitivities (first-order deviations)
- Forward contract holdings adjust:
  - g_for_shift = cip_shift / cip_base − α_shift / α_base  (Equation (6))
- Wholesale funding adjustment (Equation (7); preserved in form):
  W_for_shift =
  Net_TOT_base [ σ^2_{fx,shift} + γ_shift ]
  + Net_TOT_base · s_{t,shift}
  − Net_OBS_base · Net_TOT_base / K_base · s_{t,shift}
  + C_for_shift − D_for_shift − K_base uip_shift / (γ_base · σ^2_{fx,base})
  − G_for_base · g_for_shift
- Channel interpretations (exact implications as in source):
  - Net_TOT_base [σ^2_{fx,shift} + γ_shift]: higher exchange rate risk σ^2_{fx,shift} > 0 and higher risk aversion γ_shift > 0 lead to reduction in exchange rate risk exposure; translation into foreign currency wholesale funding depends on sign of Net_TOT_base.
  - Net_TOT_base · s_{t,shift}: portfolio rebalancing in response to level shifts in s_{t,shift}.
  - − Net_OBS_base · Net_TOT_base / K_base · s_{t,shift}: risk-taking channel driven by capital gains/losses and convex utility reducing marginal utility.
  - C_for_shift − D_for_shift: shifts in foreign-currency loans and deposits offset via foreign-currency funding changes.
  - − K_base uip_shift / (γ_base · σ^2_{fx,base}): increased relative cost of foreign funding (uip_shift > 0) reduces reliance on foreign-currency funding.
  - − G_for_base · g_for_shift: adjustments via forward contract position.

### Empirical specification derived from model
- Normalize by total domestic-currency value of bank assets in baseline A_base and take first differences.
- Empirical simplifying assumptions:
  - Cannot distinguish risk and risk aversion empirically; use global factor (GF) and local factor (LF). GF and LF proxies capture higher risk or higher risk aversion when larger.
  - α (cost of FX swaps) not observable; time fixed effects may capture common α across countries. Robustness checks replace GF with time fixed effects.
  - Country fixed effects included.
- Main empirical estimating equation (Equation (9); preserved exactly):
  d ̃W_for_{t,i} = β_0 + β_1 · dlog(GF_{t−1}) + β_2 · Net_i · dlog(GF_{t−1})
  + β_3 · dlog(LF_{t−1}) + β_4 · Net_i · dlog(LF_{t−1})
  + β_5 · dlog(S_{i,t−1}) + β_6 · Net_i · dlog(S_{i,t−1}) + β_7 · Net^2_i · dlog(S_{i,t−1})
  + β_8 · Net_i + β_9 · Net^2_i + β_{10} · d ̃C_for_{t,i} + β_{11} · d ̃D_for_{t,i}
  + β_{12} · duip_{i,t−1} + β_{13} · dcip_{i,t−1} + μ_i + ε_{i,t}
- Variable definitions in empirical equation (exact):
  - d ̃W_for_{t,i} = S_{i,t−1} dW_for_{t,i} / A_{i,t}
  - d ̃C_for_{t,i} = S_{i,t−1} dC_for_{t,i} / A_{i,t}
  - d ̃D_for_{t,i} = S_{i,t−1} dD_for_{t,i} / A_{i,t}
  - Net_i is empirical measure of net foreign currency exposure (see Section 4.4)
- Endogeneity: explanatory variables lagged by one quarter; lagging may not fully remove endogeneity given persistence.
- Model-implied coefficient signs (as stated in source):
  - Implied positive: β_1, β_2, β_4, β_6, β_{10}
  - Implied negative: β_7, β_{11}, β_{12}, β_{13}
  - No priors for remaining βs

*Source: wp1897 - 3.1 Main building blocks (IMF working paper content).*

### Appendix B — data series and stylized facts

### Bank balance sheet data (SFLM)
- Main data source: Swiss National Bank’s Swiss Franc Lending Monitor (SFLM) database of country-level bank balance sheets from 20 participating European central banks.
- Sample notes:
  - Sample reduces to 16 countries with sufficiently complete data.
  - Countries in SFLM listing include: Austria, Bulgaria, Czech Republic, Croatia, Denmark, Estonia, France, Germany, Greece, Hungary, Iceland, Italy, Latvia, Luxembourg, Poland, Romania, Serbia, Slovenia, Slovakia, and the United Kingdom.
  - Exclusions/notes: exclude Iceland and France due to insufficient coverage; exclude Luxembourg as outlier; exclude Poland due to incomplete asset-side data; include Estonia from 2011 when it joined the euro; include Latvia only until 2014 (treated as non-euro country in sample).
- Time coverage: SFLM initiated in 2009; unbalanced sample starts in 2007Q1 to include part of the financial crisis.
- Coverage: quarterly country-aggregated resident banks’ balance sheet positions, broken down by currency (local, Swiss francs, other foreign currencies), assets split between lending and other assets, liabilities split into deposits (including repo and interbank), own securities issuance and other liabilities. Includes subsidiaries of foreign banks (not foreign bank branches).
- Comparison to BIS locational statistics: SFLM covers broader set of European countries (notably eastern European), but does not include flow data — flows computed by adjusting positions for valuation effects.

### Foreign currency wholesale funding (measurement and patterns)
- Definition: change in net wholesale funding in foreign currency (evaluated at constant exchange rates) relative to total bank assets net of domestic interbank positions.
- Measurement: foreign currency wholesale funding = foreign currency liabilities to non-resident bank counterparties − foreign currency denominated claims on non-resident bank counterparties.
- Observed patterns:
  - Most banking systems in sample net debtors in foreign-currency wholesale funding at start of sample.
  - Many countries with large initial net foreign-currency wholesale funding substantially reduced it by end of sample.
  - Bulgaria changed from wholesale debtor to wholesale creditor in foreign currency.
  - Cross-country variation in net foreign-currency wholesale funding flows is substantial.
  - Foreign currency bank flows have smaller amplitudes in euro area countries than in non-euro countries relative to total banking system assets.
- Valuation adjustment:
  - SFLM quotes positions in domestic currency; changes adjusted for direct valuation impact of exchange rate movements. CHF positions straightforward; other foreign currencies require country-specific procedures (Appendix C).

### Risk factors and exposure measures
- Global risk factor (main): gbdl = quarterly growth in US broker dealer leverage multiplied by −1 (increase denotes higher risk aversion / tighter global financial conditions).
  - Other global measures considered: lvix (log of VIX), relative changes in US real and nominal effective exchange rates, other US financial and monetary condition measures.
  - Robustness: main results use gbdl; robustness checks show gbdl results robust relative to other global measures.
- Local risk: lvol_USD proxied by realized intra-quarter volatility of daily exchange rate vs USD.
- Net foreign currency exposure (Net_i): on-balance sheet foreign currency mismatch = total foreign currency assets − total foreign currency liabilities, divided by total bank assets (excluding domestic interbank positions); Net_i ∈ [−1, 1], positive indicates banking sector long foreign currency exposure.
  - Country-specific data caveats: Austria, Denmark, Czech Republic do not report other assets/liabilities; proxy by mismatch in total lending and total deposits for these countries.
  - Decomposition shows negative association between exposure in wholesale funding and exposure in other components — banks offset exposures using wholesale funding currency mix.
  - Off-balance sheet exposures not reported in SFLM; on-balance sheet mismatch used as proxy; omission may cause attenuation bias.

### Other explanatory variables (definitions preserved)
- Exchange rate variable: dUSD measured against US dollar; positive indicates appreciation of the dollar during the quarter. Robustness tests using euro as foreign funding currency for non-euro countries show USD results more significant.
- Flows:
  - d̂C_FX and d̂D_FX: flows of new loans and deposits in foreign currency computed from SFLM positions adjusted for valuation effects.
- Relative funding cost proxies:
  - dUIP: deviations from uncovered interest parity (simple USD-domestic interest differential changes).
  - dCIP: deviations from covered interest parity (computed using 3-month forward premiums and spot rates, and 3-month Libor).
  - Note: swap markets not liquid for all countries; deviations volatile for some countries (Latvia, Romania, Serbia).
- Standard errors clustered on countries.

### Summary statistics (sample unbalanced, Q1 2007 to Q1 2016; variables in percentage or percentage points except Net)
- Panel (a): All Sample Countries (selected entries)
  - d̂F_FX: Mean -0.10, Median -0.10, Max 7.13, Min -5.66, Std. Dev 1.24, Obs 482
  - d̂F_l: Mean -0.02, Median -0.02, Max 8.50, Min -7.86, Std. Dev 1.16, Obs 510
  - Net: Mean 0.02, Median 0.01, Max 0.16, Min -0.15, Std. Dev 0.05, Obs 519
  - lvol_USD: Mean -1.33, Median -1.34, Max 1.37, Min -3.24, Std. Dev 0.79, Obs 576
  - dUSD: Mean 3.03, Median 0.08, Max 90.98, Min -66.12, Std. Dev 22.55, Obs 576
  - d̂C_FX: Mean 0.07, Median -0.06, Max 9.02, Min -15.37, Std. Dev 1.66, Obs 496
  - d̂D_FX: Mean 0.11, Median -0.02, Max 8.92, Min -4.80, Std. Dev 1.41, Obs 494
  - dUIP: Mean 0.00, Median 0.03, Max 7.91, Min -8.04, Std. Dev 1.08, Obs 583
  - dCIP: Mean -0.00, Median -0.01, Max 4.15, Min -3.85, Std. Dev 0.57, Obs 536
- Panel (b): Non-Euro European countries (selected entries)
  - d̂F_FX: Mean -0.15, Median -0.18, Max 7.13, Min -4.49, Std. Dev 1.48, Obs 276
  - d̂F_l: Mean 0.05, Median 0.03, Max 3.42, Min -2.50, Std. Dev 0.76, Obs 298
  - Net: Mean 0.03, Median 0.04, Max 0.16, Min -0.15, Std. Dev 0.05, Obs 303
  - lvol_USD: Mean -1.23, Median -1.21, Max 1.37, Min -3.24, Std. Dev 0.82, Obs 324
  - dUSD: Mean 3.55, Median 0.38, Max 90.98, Min -66.12, Std. Dev 24.36, Obs 324
  - d̂C_FX: Mean 0.17, Median -0.02, Max 9.02, Min -15.37, Std. Dev 2.04, Obs 290
  - d̂D_FX: Mean 0.24, Median 0.08, Max 8.92, Min -4.80, Std. Dev 1.77, Obs 290
  - dUIP: Mean 0.02, Median 0.05, Max 7.91, Min -8.04, Std. Dev 1.39, Obs 324
  - dCIP: Mean 0.00, Median -0.01, Max 4.15, Min -3.85, Std. Dev 0.70, Obs 294
- Panel (c): Euro Countries (selected entries)
  - d̂F_FX: Mean -0.03, Median -0.06, Max 5.30, Min -5.66, Std. Dev 0.81, Obs 206
  - d̂F_l: Mean -0.10, Median -0.09, Max 8.50, Min -7.86, Std. Dev 1.55, Obs 212
  - Net: Mean 0.01, Median 0.00, Max 0.06, Min -0.08, Std. Dev 0.03, Obs 216
  - lvol_USD: Mean -1.47, Median -1.43, Max 0.20, Min -3.24, Std. Dev 0.74, Obs 252
  - dUSD: Mean 2.35, Median -0.59, Max 45.50, Min -39.35, Std. Dev 20.02, Obs 252
  - d̂C_FX: Mean -0.08, Median -0.06, Max 4.94, Min -5.59, Std. Dev 0.86, Obs 206
  - d̂D_FX: Mean -0.08, Median -0.05, Max 1.91, Min -2.26, Std. Dev 0.55, Obs 204
  - dUIP: Mean -0.02, Median 0.01, Max 1.20, Min -2.10, Std. Dev 0.48, Obs 259
  - dCIP: Mean -0.01, Median -0.01, Max 1.24, Min -1.38, Std. Dev 0.36, Obs 242
- Panel (d): Global Factors
  - gbdl: Mean 2.11, Median 1.44, Max 25.96, Min -5.22, Std. Dev 5.83, Obs 37
  - lvix: Mean 332.36, Median 326.77, Max 439.27, Min 283.38, Std. Dev 37.25, Obs 37

### Key stylized empirical findings (Appendix B)
- Substantial cross-country and time variation in net foreign currency wholesale funding stocks and flows.
- Euro area countries exhibit smaller amplitudes in foreign currency bank flows relative to banking system size than non-euro countries.
- Banks use currency mix of wholesale funding to offset exposures in other balance sheet components (negative association between wholesale and non-wholesale FX exposures).
- Heterogeneity in unconditional correlation of funding flows with global risk factors; association with average net FX exposure for some countries (e.g., Estonia and Romania influential).
- On-balance sheet mismatch provides relevant information about total exposure for some countries; omission of off-balance sheet exposures may attenuate estimated associations.
- Robustness checks include alternative exposure proxies (Net_TOT_hat via regression residuals), alternative currency weightings, and alternative global/local factor measures.

*Source: Appendix B, "The date series we consider and some key stylized facts are presented below." (wp1897).*

### 5.1 Full sample — main empirical results

### Global factor (gbdl) interaction with net exposure
- Global factor: gbdl = growth in US broker dealer leverage (multiplied by −1).
- Key empirical finding:
  - gbdl not significant when entered alone, but significant when interacted with lagged net foreign currency exposure (Net−1 · gbdl−1).
  - Higher gbdl increases foreign-currency funding for banks with net long foreign-currency exposure and reduces it for banks with net short exposure — consistent with theoretical prediction.
- Quantitative illustration (based on column V estimates and a one standard deviation increase in gbdl of about 6):
  - Predicted increase in foreign-currency wholesale funding of 0.5% of total bank assets for a country with initial net long foreign-currency exposure of 10% of assets.
  - Predicted contraction of foreign-currency wholesale funding of 0.7% of total assets for a country with initial net short exposure of 10%.
  - Algebraic expressions from source: −0.017·6 + 1.145·0.1·6 = 0.46% and −0.013·6 + 1.145·(−0.1)·6 = −0.68%.

### Role of local risk and other covariates
- lvol_USD (volatility of bilateral USD exchange rate) generally not significant, alone or interacted with Net.
- Realized exchange rate movements (dUSD−1) significant with expected sign only when interacted with Net (Net−1 · dUSD−1) and through risk-taking channel (Net2−1 · dUSD−1); effect driven mainly by Estonia and Romania.
- Wholesale funding responds positively to foreign-currency loan flows ( ̃C_FX−1 ), suggesting partial offsetting of new FX exposure; effect driven mainly by variation during financial crisis and not significant post-crisis.
- No consistent reaction of funding to deposit movements ( ̃D_FX−1 ).
- No consistent evidence that funding responds across countries to deviations from interest parity: coefficients on dUIP−1 and dCIP−1 generally insignificant in full sample regressions.

### Robustness of gbdl × Net interaction
- Interaction significance robust to:
  - Inclusion/exclusion of fixed and time effects.
  - Alternative Net exposure measure (Equation (10)).
  - Allowing country-specific parameters on cip and uip.
  - Using euro exchange rate for non-euro countries (interaction remains significant but weaker).
  - Inclusion of additional macro and US financial controls.
- Interaction effect stronger when crisis period omitted (post-crisis sample).
- Results not driven solely by Estonia and Romania: excluding them reduces estimate but does not eliminate significance.

### Sample and data notes
- Full sample period: Q1 2007 to Q1 2016. Post-crisis (PC) sample starts in 2009Q2.
- Panel includes 16 countries in main analysis.
- Main dependent variable: valuation-adjusted bank wholesale funding flows in percent of total assets.

### Implications emphasized
- Global risk factor shocks (as proxied by gbdl) affect cross-border bank funding flows differently depending on banking sectors’ pre-existing on-balance sheet foreign-currency exposure:
  - Net long exposures → amplify inflows when global conditions ease (gbdl falls).
  - Net short exposures → amplify outflows when global conditions tighten (gbdl rises).
- Local exchange rate volatility appears less important than global factors for driving foreign-currency funding behavior in sampled banking sectors.

*Source: IMF Working Paper (section 5.1, full sample results; sample Q1 2007–Q1 2016; post-crisis from 2009Q2).*

*Source: wp1897 - IMF Working Paper (https://www.imf.org/-/media/files/publications/wp/2018/wp1897.pdf).*

### 3.1    Main building blocks

### 3.1    Main building blocks

### Bank balance sheet and currency notation
- Exchange rate between domestic and foreign currency (units of local currency per unit of foreign currency) is denoted by S.
- Periods: t and t+1. Exchange rate written as S_t = exp[s_t] in period t and S_{t+1} = exp[s_{t+1}] in period t+1.
- Loans (assets): C, issued either in domestic or foreign currency. Values in domestic currency: C_dom; values in foreign currency (fixed in foreign currency): C_for.
- Deposits (liabilities): D, with D_dom and D_for similarly defined and fixed.
- Wholesale funding (liabilities, net of wholesale lending): F, can be negative. Domestic-currency wholesale funding in period t: W_dom_t. Foreign-currency value of funding in foreign currency: W_for_t.
- Equity: K. Equity is predetermined; total wholesale funding in period t, W_t, is predetermined.
- Loans and deposits are exogenous and cannot be adjusted by the bank in the model (assumption flagged as relaxed in other contributions).

### Forward contract and swap market participation
- Bank can take positions in the currency forward market via a forward exchange rate contract.
- Forward rate contract payoff: F_{t+1} = exp[f_{t+1}] units of domestic currency per foreign currency in period t+1; F_{t+1} is known in period t.
- Purchasing the contract entails taking a short position in the spot market in period t+1.
- Covered foreign currency funding is risk-free in the sense of spot-forward combination and comparable to domestic funding in risk profile.
- Gross cost (including principal) of wholesale funding:
  - Domestic currency funding: exp[r_{W,dom,t+1}]
  - Foreign currency funding: exp[r_{W,for,t+1}]
- If f_{t+1} − s_t < r_{W,dom,t+1} − r_{W,for,t+1}, cost of swapped funding is lower than cost of domestic wholesale funding — implying arbitrage absent other costs.
- To capture observed deviations from covered interest parity and limits to arbitrage, positions in forward contracts entail a balance sheet cost quadratic in the amount of positions:
  - Total payoff in period t+1 of buying G_for units of the forward contract:
    G_for(F_{t+1} − S_{t+1}) − α_{t+1}/2 · F_{t+1} · (G_for)^2

### Period structure, valuation, and utility
- Bank values future equity using CRRA utility function: u = (1 − γ_t)^{−1} (K_{t+1})^{1−γ_t}.
- Bank initially endowed with equity position K_t in domestic currency, with identity:
  K_t = C_dom − D_dom − W_dom_t + S_t [C_for − D_for − W_for_t]
- Equity in period t+1:
  K_{t+1} = C_dom − D_dom − W_dom_t R_{W,dom,t+1} + S_{t+1} (C_for − D_for − W_for_t R_{W,for,t+1}) + G_for(F_{t+1} − S_{t+1}) − α_{t+1}/2 (G_for)^2

### Optimality conditions (first-order conditions)
- Combining first-order conditions with respect to wholesale funding in domestic and foreign currency yields:
  0 = E(K_{t+1})^{−γ_t} [ S_{t+1}/S_t R_{W,for,t+1} − R_{W,dom,t+1} ]  (presented in text as)
  0 = E(K_{t+1})^{−γ_t} [ S_{t+1}/S_t R_{W,for,t+1} − R_{W,dom,t+1} ]  (Equation (1) notation)
- First-order condition with respect to holdings of the forward contract:
  0 = E(K_{t+1})^{−γ_t} ( F_{t+1} − S_{t+1} − α_{t+1} F_{t+1} G_for )  (Equation (2))

### Solution approach: two-step Taylor expansion
- Optimality conditions (1) and (2) are highly non-linear; solution computed by Taylor expansion (quadratic and cubic approximations), keeping terms proportional to innovations, square of innovations, etc.
- Solution proceeds in two steps:
  1. Derive baseline values of wholesale funding in foreign currency and forward contract position given exogenous baseline environment (loans, deposits, interest rates, expected exchange rate movements, moments of baseline shocks). Baseline portfolio interpreted as steady state.
  2. Compute portfolio values in a shifted environment where loans, deposits, interest rates, exchange rates, risk, and risk aversion shift away from baseline; assess impact of shifts (e.g., unusually large loans, expected exchange rate movement, or risk) on portfolio choice.
- Variables in baseline and shifted environments denoted with base and shift subscripts.

### Baseline portfolio: definitions and closed-form baseline results
- Define on-balance-sheet baseline exposure (Net_OBS_base) and total baseline exposure including forward position (Net_TOT_base):
  - Net_OBS_base = C_for_base − D_for_base − W_for_base  (Equation (3))
  - Net_TOT_base = Net_OBS_base − G_for_base  (Equation (4))
- Deviations from uncovered and covered interest parity (for k = base, shift):
  - uip_k = r_{W,for,t+1,k} + s_{t+1,k} − s_{t,k} − r_{W,dom,t+1,k}
  - cip_k = p_k + uip_k, where p is the difference between forward exchange rate and expected spot exchange rate (forward premium)
- Baseline portfolio expressions (Equation (5)):
  - G_for_base = cip_base / α_base
  - Net_TOT_base = K_base · uip_base / (γ_base · σ^2_{fx,base})
- Interpretations:
  - G_for_base reflects deviation from covered interest parity adjusted for marginal cost α_base; unaffected by risk aversion or risk because forward + spot offers risk-free arbitrage.
  - Net_TOT_base reflects deviation from uncovered interest parity adjusted by exchange rate risk σ^2_{fx,base} and risk aversion γ_base. Absent uip_base ≠ 0, bank would fully hedge. If funding is relatively expensive in foreign currency (uip_base > 0) bank may accept long exchange rate exposure.

### Shifted portfolio: sensitivity to environment shifts
- First-order deviations of funding from baseline allocation capture sensitivities.
- Forward contract holdings adjust:
  - g_for_shift = cip_shift / cip_base − α_shift / α_base  (Equation (6))
- Wholesale funding adjustment expression (Equation (7)):
  W_for_shift =
  Net_TOT_base [ σ^2_{fx,shift} + γ_shift ]
  + Net_TOT_base · s_{t,shift}
  − Net_OBS_base · Net_TOT_base / K_base · s_{t,shift}
  + C_for_shift − D_for_shift − K_base uip_shift / (γ_base · σ^2_{fx,base})
  − G_for_base · g_for_shift
- Key channels and interpretations:
  - First term (Net_TOT_base [σ^2_{fx,shift} + γ_shift]): higher exchange rate risk (σ^2_{fx,shift} > 0) and higher risk aversion (γ_shift > 0) lead bank to reduce exchange rate risk exposure; translation into foreign currency wholesale funding depends on Net_TOT_base sign.
    - If Net_TOT_base > 0 (normally long foreign currency), reducing risk exposure requires increase in foreign currency wholesale funding.
    - If Net_TOT_base < 0, foreign currency wholesale funding is reduced when reducing exposure.
  - Second-line first term (Net_TOT_base · s_{t,shift}): portfolio rebalancing — if foreign currency stronger than baseline (s_{t,shift} > 0) and bank is long (Net_TOT_base > 0), offset through increase in funding in foreign currency.
  - Second-line second term (− Net_OBS_base · Net_TOT_base / K_base · s_{t,shift}): a risk-taking channel depending on cross-product of on-balance-sheet and total exposures. If s_{t,shift} > 0 produces capital gain and convex utility reduces marginal utility, bank takes more risk. If Net_TOT_base > 0, taking extra risk achieved through increase in exposure and hence reduction of foreign currency funding (W_for_shift < 0).
  - Third-line terms: shifts in foreign currency loans and deposits (C_for_shift > 0 or D_for_shift < 0) increase exchange rate exposure and are offset via higher foreign currency funding.
  - Last-line terms: increased relative cost of foreign funding (uip_shift > 0) leads bank to reduce reliance on foreign currency funding; forward contract position affords an alternative adjustment (− G_for_base · g_for_shift).
- Note: model assumes return on bank loans constant and abstracts from indirect currency risk via credit risk of clients; appendix (available on request) develops richer version where net exposure includes direct and indirect channels.

### Empirical specification
- To derive estimating equation from expression (7):
  - Divide all terms by total domestic currency value of bank assets in baseline A_base to obtain comparable ratios across countries.
  - Take first differences to cancel many baseline values.
- Empirical notation: variables without time subscript refer to country-specific baseline values; variables with time and country subscripts refer to shifted variables; variables without country subscript refer to global time-varying variables.
- Simplifying empirical assumptions:
  - Cannot distinguish risk and risk aversion empirically: test for both jointly and distinguish between a global financial factor (GF) and a local factor (LF).
    - GF captures risk and risk aversion related to global conditions in bank funding markets.
    - LF captures country-specific foreign currency risk and risk aversion.
    - Higher values of GF and LF denote higher risk or higher risk aversion.
    - Include GF and LF on their own and interacted with net currency exposure.
  - Parameters for risk-taking channel and for uip and cip depend on country-specific features; baseline specification restricts them to be same across countries (robustness checks allow cross-country variation).
  - Cost of entering FX swap contracts α is not observable and not included; α may be correlated with cip; time fixed effects can capture common balance sheet cost across countries. Robustness checks run regressions with time fixed effects instead of the global factor, while still including interaction with net currency exposure.
  - Add country fixed effects to capture time-invariant country-specific factors.
- Main empirical estimating equation (Equation (9)):
  d ̃W_for_{t,i} = β_0 + β_1 · dlog(GF_{t−1}) + β_2 · Net_i · dlog(GF_{t−1})
  + β_3 · dlog(LF_{t−1}) + β_4 · Net_i · dlog(LF_{t−1})
  + β_5 · dlog(S_{i,t−1}) + β_6 · Net_i · dlog(S_{i,t−1}) + β_7 · Net^2_i · dlog(S_{i,t−1})
  + β_8 · Net_i + β_9 · Net^2_i + β_{10} · d ̃C_for_{t,i} + β_{11} · d ̃D_for_{t,i}
  + β_{12} · duip_{i,t−1} + β_{13} · dcip_{i,t−1} + μ_i + ε_{i,t}
- Definitions in empirical equation:
  - d ̃W_for_{t,i} = S_{i,t−1} dW_for_{t,i} / A_{i,t} is valuation-adjusted change in net foreign currency wholesale funding as a share of total bank assets.
  - d ̃C_for_{t,i} = S_{i,t−1} dC_for_{t,i} / A_{i,t} is valuation-adjusted change in foreign currency assets net of wholesale assets as a share of total bank assets.
  - d ̃D_for_{t,i} = S_{i,t−1} dD_for_{t,i} / A_{i,t} is valuation-adjusted change in foreign currency liabilities net of wholesale liabilities as a share of total bank assets.
  - Net_i is an empirical measure of net foreign currency exposure (discussed further in Section 4.4).
- Endogeneity considerations:
  - Model treats supply and prices as exogenous; possible endogeneity if demand changes affect prices and supply.
  - Standard approach: lag all explanatory variables by one quarter (e.g., Cerutti et al. [2017a]); lagging may not fully alleviate endogeneity given persistence in bank balance sheet dynamics and explanatory variables.
- Model-implied signs for coefficients:
  - β_1, β_2, β_4, β_6, and β_{10} are implied positive.
  - β_7, β_{11}, β_{12}, and β_{13} are implied negative.
  - No priors for signs of remaining βs.

*Source: wp1897 - 3.1    Main building blocks (IMF working paper content).*

### Appendix B.  The date series we consider and some key stylized facts are presented below.

### Appendix B.  The date series we consider and some key stylized facts are presented below.

### Bank Balance Sheet Data
- Data source: Swiss National Bank’s Swiss Franc Lending Monitor (SFLM) database of country-level bank balance sheets, containing aggregate bank balance sheet data collected from 20 participating European central banks.
- Sample: 16 countries with sufficiently complete data. Countries in SFLM listing: Austria, Bulgaria, Czech Republic, Croatia, Denmark, Estonia, France, Germany, Greece, Hungary, Iceland, Italy, Latvia, Luxembourg, Poland, Romania, Serbia, Slovenia, Slovakia, and the United Kingdom. Exclusions and notes:
  - Exclude Iceland and France due to insufficient data coverage.
  - Exclude Luxembourg as an outlier.
  - Exclude Poland due to incomplete data on the asset side.
  - Include Estonia from 2011 when it joined the euro (treated as a euro area country).
  - Include Latvia only until 2014 (treated as a non-euro country in this sample).
- Time coverage: SFLM initiated in 2009; unbalanced sample starts in 2007Q1 to cover part of the financial crisis period. Individual country charts reflect period covered for each country.
- Coverage and granularity:
  - Quarterly country-aggregated resident banks’ balance sheet positions.
  - Positions broken down by currency: local currency, Swiss francs, and foreign currencies other than the Swiss franc (the latter not further disaggregated).
  - Assets split between lending and other assets; liabilities split between deposits (including repo and interbank borrowing), own securities issuance and other liabilities.
  - Lending and deposits further divided by counterparties: resident banks and non-banks, non-resident banks and non-banks. Resident non-bank counterparties further divided into households, non-bank corporations and government.
  - Includes subsidiaries of foreign banks (not foreign bank branches).
- Comparison to BIS locational banking statistics:
  - SFLM covers a broader set of European countries, notably eastern European countries not reporting to BIS locational statistics.
  - Disadvantage: SFLM does not include flow data — flows must be computed by adjusting positions for valuation changes from exchange rates.

### Foreign Currency Wholesale Funding
- Definition used: change in net wholesale funding in foreign currency (evaluated at constant exchange rates) relative to total bank assets net of domestic interbank positions.
- Measurement: foreign currency wholesale funding = foreign currency liabilities to non-resident bank counterparties minus foreign currency denominated claims on non-resident bank counterparties.
- Observations and patterns:
  - Most banking systems in the sample were net debtors in wholesale funding in foreign currency at the beginning of the sample.
  - Many countries with large initial net foreign currency wholesale funding substantially reduced it by the end of the sample period.
  - Bulgaria changed from being a wholesale debtor to a wholesale creditor in foreign currency.
  - Cross-country variation in net foreign currency wholesale funding flows is substantial.
  - Foreign currency bank flows have smaller amplitudes in euro area countries than in non-euro countries, relative to total banking system assets; sorting by standard deviation of flows separates euro countries (low variation) from non-euro countries (high variation).
- Adjustment for valuation effects:
  - SFLM quotes positions in domestic currency; changes adjusted for direct valuation impact of exchange rate movements.
  - Swiss franc positions adjusted straightforwardly; other foreign currency positions require country-specific data sources and procedures (described in Appendix C).
- Figures referenced:
  - Figure 2(a): Net FX wholesale funding share of liabilities in 2009Q1 and 2016Q1 (with country-specific start/end exceptions).
  - Figure 2(b): Country-specific sample mean net FX wholesale funding flow (valuation adjusted quarterly change in stock) with one country-specific standard deviation bars. Source: SFLM.
- Regression approach: regressions split sample into euro and non-euro area countries.

### Risk Factors
- Global risk factor measures considered:
  - Main measure: quarterly growth in US broker dealer leverage, multiplied by −1 in regressions and denoted gbdl (increase denotes higher risk aversion / more restrictive global financial conditions).
    - Rationale: higher leverage growth among US broker dealers associated with lower global risk aversion and higher liquidity in global wholesale funding markets.
  - Other measures considered: log of the V IX (denoted lvix), relative change in US real and nominal effective exchange rates, and other US financial and monetary condition measures (further described in Section 5.3).
- Empirical choices and robustness:
  - Main results presented using gbdl; robustness checks show results using growth in US broker dealer leverage are very robust, but not so for other global risk measures.
  - gbdl is multiplied by −1 so that an increase corresponds to tighter global financial conditions.
  - Local risk proxied by realized intra-quarter volatility of the daily exchange rate vis-à-vis the US dollar.
- Figure referenced:
  - Figure 3: plots gbdl (quarterly growth in US broker dealer leverage multiplied by minus one) and lvix (log of the highest realized value of daily realizations of the V IX over the quarter), showing they comove but not one-for-one at quarterly frequency.

### Net Foreign Currency Exposure
- Main exposure measure used: on-balance sheet foreign currency mismatch (Net_i) = total foreign currency assets − total foreign currency liabilities, divided by total bank assets (total bank assets exclude domestic interbank positions).
  - Conceptual range: Net_i in [−1, 1]; positive indicates banking sector long foreign currency exposure.
  - Use: lagged Net_i in regressions.
- Country-specific handling and proxies:
  - Austria, Denmark and the Czech Republic do not report other assets and liabilities; for these countries the foreign currency mismatch proxied by mismatch in total lending and total deposits only.
  - Estonia, Greece and Austria: euro countries with relatively small average net foreign currency exposure and little variation (consistent with regulation).
  - Non-euro countries (except Romania) tend to have more persistently positive net currency exposures with more variation.
- Decomposition and evidence of behavior:
  - Net on-balance sheet exposure decomposed into exposure within wholesale funding positions and exposure within other balance sheet components.
  - Figure 4(b): shows a clear negative association between the exposure in wholesale funding and exposure in other components, consistent with banks using wholesale funding currency mix to offset exposures in other positions.
  - Figure 4(c): plots country-specific regression coefficients from simple OLS of foreign currency wholesale funding inflows (percent of total assets) on lagged gbdl against country mean FX exposure — shows substantial heterogeneity in unconditional correlation of funding flows with global factors and a positive association between banking systems that are short foreign currency and a negative association of funding inflows with global risk factors (noting Estonia and Romania as influential).
- Off-balance sheet considerations:
  - SFLM does not report off-balance sheet exposures; on-balance sheet mismatch is used as proxy.
  - Regressions produce significant interaction between on-balance sheet currency mismatch and global factor, suggesting on-balance sheet exposures contain relevant information about total exposures for some countries.
  - Omission of off-balance sheet exposures could cause attenuation bias, implying actual associations might be stronger.
- Alternative proxy for total exposure (robustness):
  - Use relationship Net_OBS = Net_TOT + cip * α_base; regress Net_OBS on deviations from covered interest parity (cip) and use residuals ε_i,t as alternative proxy for Net_TOT: Net_TOT_hat_i,t = ε_i,t where Net_OBS_i,t = β cip_i,t + ε_i,t.
  - This approach relies on volatile deviations from covered interest parity and other omitted factors; used only as robustness check.

### Other explanatory variables
- Exchange rate variable: dUSD measured against US dollar; positive value indicates appreciation of the dollar during the quarter. Robustness tests using the euro as foreign funding currency for non-euro countries show USD results are more significant and robust.
- Flows computed:
  - d̂C_FX and d̂D_FX: flows of new loans and deposits in foreign currency computed from SFLM positions adjusted for valuation effects.
- Relative funding costs proxies:
  - dUIP: deviations from uncovered interest parity, computed as simple interest differential.
  - dCIP: deviations from covered interest parity (computed and discussed in Appendix B).
  - Note: currency swap markets not necessarily liquid for all countries, giving volatile deviations (notably Latvia, Romania, Serbia). Robustness tests using monthly averages instead of last week of quarter produce largely same results; deviations from covered interest parity are rarely significant.
- Standard errors:
  - Regressions cluster standard errors on countries.
- Summary statistics (sample unbalanced, runs from Q1 2007 to Q1 2016). Variables expressed in percentage or percentage points, except Net (ratio of total bank assets).
  - Table 1 (selected entries reproduced exactly as in source):
    - Panel (a): All Sample Countries
      - d̂F_FX: Mean -0.10, Median -0.10, Max 7.13, Min -5.66, Std. Dev 1.24, Obs 482
      - d̂F_l: Mean -0.02, Median -0.02, Max 8.50, Min -7.86, Std. Dev 1.16, Obs 510
      - Net: Mean 0.02, Median 0.01, Max 0.16, Min -0.15, Std. Dev 0.05, Obs 519
      - lvol_USD: Mean -1.33, Median -1.34, Max 1.37, Min -3.24, Std. Dev 0.79, Obs 576
      - dUSD: Mean 3.03, Median 0.08, Max 90.98, Min -66.12, Std. Dev 22.55, Obs 576
      - d̂C_FX: Mean 0.07, Median -0.06, Max 9.02, Min -15.37, Std. Dev 1.66, Obs 496
      - d̂D_FX: Mean 0.11, Median -0.02, Max 8.92, Min -4.80, Std. Dev 1.41, Obs 494
      - dUIP: Mean 0.00, Median 0.03, Max 7.91, Min -8.04, Std. Dev 1.08, Obs 583
      - dCIP: Mean -0.00, Median -0.01, Max 4.15, Min -3.85, Std. Dev 0.57, Obs 536
    - Panel (b): Non-Euro European countries (Bulgaria, Croatia, Czech Republic, Denmark, Hungary, Latvia (before joining the euro), Romania, Serbia and the UK)
      - d̂F_FX: Mean -0.15, Median -0.18, Max 7.13, Min -4.49, Std. Dev 1.48, Obs 276
      - d̂F_l: Mean 0.05, Median 0.03, Max 3.42, Min -2.50, Std. Dev 0.76, Obs 298
      - Net: Mean 0.03, Median 0.04, Max 0.16, Min -0.15, Std. Dev 0.05, Obs 303
      - lvol_USD: Mean -1.23, Median -1.21, Max 1.37, Min -3.24, Std. Dev 0.82, Obs 324
      - dUSD: Mean 3.55, Median 0.38, Max 90.98, Min -66.12, Std. Dev 24.36, Obs 324
      - d̂C_FX: Mean 0.17, Median -0.02, Max 9.02, Min -15.37, Std. Dev 2.04, Obs 290
      - d̂D_FX: Mean 0.24, Median 0.08, Max 8.92, Min -4.80, Std. Dev 1.77, Obs 290
      - dUIP: Mean 0.02, Median 0.05, Max 7.91, Min -8.04, Std. Dev 1.39, Obs 324
      - dCIP: Mean 0.00, Median -0.01, Max 4.15, Min -3.85, Std. Dev 0.70, Obs 294
    - Panel (c): Euro Countries (Austria, Estonia (post euro), Germany, Greece, Slovakia and Slovenia)
      - d̂F_FX: Mean -0.03, Median -0.06, Max 5.30, Min -5.66, Std. Dev 0.81, Obs 206
      - d̂F_l: Mean -0.10, Median -0.09, Max 8.50, Min -7.86, Std. Dev 1.55, Obs 212
      - Net: Mean 0.01, Median 0.00, Max 0.06, Min -0.08, Std. Dev 0.03, Obs 216
      - lvol_USD: Mean -1.47, Median -1.43, Max 0.20, Min -3.24, Std. Dev 0.74, Obs 252
      - dUSD: Mean 2.35, Median -0.59, Max 45.50, Min -39.35, Std. Dev 20.02, Obs 252
      - d̂C_FX: Mean -0.08, Median -0.06, Max 4.94, Min -5.59, Std. Dev 0.86, Obs 206
      - d̂D_FX: Mean -0.08, Median -0.05, Max 1.91, Min -2.26, Std. Dev 0.55, Obs 204
      - dUIP: Mean -0.02, Median 0.01, Max 1.20, Min -2.10, Std. Dev 0.48, Obs 259
      - dCIP: Mean -0.01, Median -0.01, Max 1.24, Min -1.38, Std. Dev 0.36, Obs 242
    - Panel (d): Global Factors
      - gbdl: Mean 2.11, Median 1.44, Max 25.96, Min -5.22, Std. Dev 5.83, Obs 37
      - lvix: Mean 332.36, Median 326.77, Max 439.27, Min 283.38, Std. Dev 37.25, Obs 37

### Key stylized empirical findings (as described in the appendix)
- Substantial cross-country and time variation in net foreign currency wholesale funding stocks and flows.
- Euro area countries exhibit smaller amplitudes in foreign currency bank flows relative to banking system size than non-euro countries.
- Banks appear to use the currency mix of wholesale funding to offset exposures in other balance sheet components (negative association between wholesale and non-wholesale FX exposures).
- Heterogeneity exists in the unconditional correlation between funding flows and global risk factors; the sign and magnitude vary across countries and are associated with average net foreign currency exposure in some cases.
- On-balance sheet foreign currency mismatch provides economically meaningful information about total exposure for some countries; omission of off-balance sheet exposures may attenuate estimated associations.

*Source: Appendix B, "The date series we consider and some key stylized facts are presented below." (wp1897).*

### 5.1    Full sample

### 5.1    Full sample

### Main empirical finding: global factor and net foreign currency exposure
- The global factor used in the baseline is growth in US broker dealer leverage (gbdl).
- gbdl is not significant when entered alone, but becomes significant when interacted with net foreign currency exposure (Net−1 · gbdl−1).
- A higher gbdl increases foreign currency funding for banks with a net long foreign currency exposure and reduces it for banks with a net short exposure, consistent with the theoretical prediction.
- Quantitative illustrations (based on column V estimates and a one standard deviation increase in gbdl of about 6):
  - Predicted increase in foreign currency wholesale funding (funding inflow) of 0.5% of total bank assets in a country with an initial net long foreign currency exposure of 10% of assets.
  - Predicted contraction of foreign currency wholesale funding of 0.7% of total assets in a country with an initial net short exposure of 10%.
  - Footnote algebraic expressions reported in the source: −0.017·6 + 1.145·0.1·6 = 0.46% and −0.013·6 + 1.145·(−0.1)·6 = −0.68%.

### Role of local risk factors and other covariates
- The volatility of the bilateral USD exchange rate (lvol USD−1) is generally not significant, either alone or when interacted with net foreign currency exposure.
- Realized exchange rate movements (dUSD−1) are significant with the expected sign only when interacted with net exposure (Net−1 · dUSD−1) and through the risk-taking channel (Net2−1 · dUSD−1). The exchange rate effect appears driven mainly by Estonia and Romania (see Column VIII).
- Wholesale funding reacts to flows of loans granted in foreign currency ( ̃C FX−1 ) with a positive coefficient, suggesting partial offsetting of new exchange rate exposure; this effect is driven mainly by variation in the financial crisis period and is not significant in the post-crisis sample.
- No consistent reaction of funding to movements in deposits ( ̃D FX−1 ).
- No consistent evidence that funding responds across countries to deviations from interest parity: coefficients on changes in dUIP−1 and dCIP−1 are generally insignificant in the full sample regressions.

### Robustness of the gbdl × exposure interaction
- The significance of the interaction between gbdl and net exposure is robust to:
  - Inclusion or exclusion of fixed and time effects (columns IV and IX).
  - Use of an alternative net exposure measure (Equation (10); columns X and XI).
  - Allowing parameters on deviations from interest parity to vary across countries (appendix Table 6).
  - Using the euro exchange rate for exchange rate appreciation and volatility terms for non-euro countries (appendix Table 9; interaction remains significant but less so).
  - Inclusion of additional controls such as inflation, growth, and US financial and monetary variables (appendix Table 7).
- The interaction effect is stronger when the crisis period is omitted (columns V and X).
- Results are not driven solely by Estonia and Romania: excluding these countries (columns VI and VII) reduces the parameter estimate but does not eliminate significance.

### Sample and data notes
- Full sample period: Q1 2007 to Q1 2016.
- Post financial crisis (PC) sample starts in 2009Q2.
- Panel includes 16 countries in the main analysis.
- Main dependent variable: valuation adjusted bank wholesale funding flows in percent of total assets.

### Implications highlighted in this section
- Global risk factor shocks (as captured by gbdl) affect cross-border bank funding flows differently depending on banking sectors’ pre-existing on-balance sheet foreign currency exposure: net long exposures amplify inflows, net short exposures amplify outflows.
- Local risk measures (exchange rate volatility) appear less important than global factors for driving perceived foreign currency risk in the sampled banking sectors.

*Source: IMF Working Paper (section 5.1, full sample results; sample Q1 2007–Q1 2016; post-crisis from 2009Q2).*

### References

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### Theoretical model — structure and main components (Appendix A)
- Model scope:
  - Bank active over two periods, t and t+1.
  - Assets: risk-free loans in domestic and foreign currency.
  - Liabilities: risk-free deposits in both currencies, wholesale funding in both currencies, and equity.
  - Emphasis on portfolio choice of wholesale funding across the two currencies.
- Order decomposition:
  - Variables split into zero-, first-, second-, and third-order components; for a variable xt: xt = xt(0) + xt(1) + xt(2) + xt(3)...
- Exchange rates:
  - Stated as S (local currency per unit foreign currency).
  - St = S(0) exp[s_t] with S(0) set to 1; s_t = s_t(1) + s_t(2) + s_t(3).
  - s_{t+1} includes shock ε^S_{t+1} with E_t(ε^S_{t+1}) = 0 and Var_t(ε^S_{t+1}) = σ^2_fx [1 + ν_σ t+1(1)].
  - Forward rate F_{t+1} = exp[f_{t+1}] with forward premium p_{t+1} having second- and third-order components: exp[f_{t+1}] = exp[p_{t+1}(2) + p_{t+1}(3) + s_{t+1}(1) + s_{t+1}(2) + s_{t+1}(3)].
- Bank balance sheet components:
  - Loans: C_dom = C_dom(0) exp[c_dom]; C_for = C_for(0) exp[c_for].
  - Deposits: D_dom = D_dom(0) exp[d_dom]; D_for = D_for(0) exp[d_for].
  - Wholesale funding domestic: W_dom_t = W_dom(0) exp[w_dom]; W_dom_{t+1} = W_dom(0) exp[w_dom] exp[r_{W,dom,t+1}].
  - Wholesale funding foreign: W_for_t = W_for(0) exp[w_for]; W_for_{t+1} = W_for(0) exp[w_for] exp[r_{W,for,t+1}].
  - Interest rates include second- and third-order components (r_{W,dom,t+1} = r_{W,dom,t+1}(2) + r_{W,dom,t+1}(3); r_{W,for,t+1} similarly).
- Deviations:
  - uip deviation: uip_{t+1}(r) = r_{W,for,t+1}(r) + s_{t+1}(r) − s_t(r) − r_{W,dom,t+1}(r).
  - cip deviation: cip_{t+1}(r) = p_{t+1}(r) + uip_{t+1}(r).
- Forward contract:
  - Bank buys G_for = G_for(0) exp[g_for].
  - Ex-post payoff: G_for (F_{t+1} − S_{t+1}) − α_{t+1}/2 F_{t+1} (G_for)^2.
  - α_{t+1} introduces quadratic friction allowing deviations from covered interest parity.
- Equity expressions:
  - K_t = C_dom − D_dom − W_dom_t + S_t (C_for − D_for − W_for_t) = expression (11).
  - K_{t+1} = expression (12) including forward contract payoff and quadratic cost.

### Optimization and portfolio Euler conditions (Appendix A.2)
- Objective:
  - Bank maximizes E_t u(K_{t+1}, γ_t) with γ_t = γ(0) exp[ˆγ].
  - Constraint: total wholesale funding ̄W_t = W_dom_t + S_t W_for_t = W_dom(0) exp[w_dom] + W_for(0) exp[s_t + w_for].
- First-order conditions (Euler equations):
  - With respect to W_dom_t and W_for_t lead to:
    - 0 = E_t u_K(K_{t+1}, γ_t) [S_{t+1}/S_t R_{W,for,t+1} − R_{W,dom,t+1}] (equation (13)).
  - With respect to G_for:
    - 0 = E_t u_K(K_{t+1}, γ_t) [F_{t+1} − S_{t+1} − α_{t+1} F_{t+1} G_for] (equation (14)).
- Approximation strategy:
  - Zero-order allocation from quadratic expansion captures baseline.
  - First-order deviations require cubic approximation to capture shifts in risk.

### Expansion and baseline solutions (Appendix A.2.2–A.2.3)
- Zero- and first-order results:
  - Zero-order: 0 = u_K α_{t+1}(0) G_for(0). Zero-order excess returns = 0 and α_{t+1}(0) = 0.
  - First-order: uip holds to first order; α_{t+1}(1) = 0.
- Definitions:
  - Net_OBS(0) = C_for(0) − D_for(0) − W_for(0).
  - Net_TOT(0) = Net_OBS(0) − G_for(0).
- Second-order portfolio solution:
  - Net_TOT(0) = K(0) (u_K − u_KK K) uip_{t+1}(2) σ^2_fx (equation (15)). [As presented in text: Net_TOT(0) = K(0) / (u_K − u_KK K) uip_{t+1}(2) σ^2_fx — preserved exactly as in source formatting]
  - Forward contract baseline: G_for(0) = cip_{t+1}(2) / α_{t+1}(2) (equation (16)).
- First-order (shifted) components:
  - W_for(0) w_for(1) + G_for(0) g_for(1) = [complex expression given in equation (17)].
  - g_for(1) = cip_{t+1}(3)/cip_{t+1}(2) − α_{t+1}(3)/α_{t+1}(2) (equation (18)).
- Special case — CRRA utility:
  - Expression provided for W_for(0) w_for(1) + G_for(0) g_for(1) with terms including γ and σ^2_fx as in text.

### Data (Appendix B) — main variables and sources
- Main data source: Swiss Franc Lending Monitor (SFLM), end-of-quarter bank balance sheet positions. Other sources: Datastream, Bloomberg, US Financial Accounts, national statistical agencies, central banks.
- Variable definitions (exact text preserved):
  - d̄W_FX: quarterly change in net foreign currency wholesale funding (adjusted for exchange rate valuation) in percent of total bank assets. Net wholesale funding proxied by lending to and deposits from foreign banks. For Austria, Czech Republic and Denmark, data on total assets not reported; total lending used instead. Source: SFLM.
  - d̄W_l: quarterly change in net domestic currency wholesale funding in percent of total bank assets. Source: SFLM.
  - Net: total foreign currency assets less total foreign currency liabilities, as a share of total assets. For Austria, Czech Republic and Denmark, total lending and total deposits plus own securities used instead. Source: SFLM.
  - d̃C_FX: quarterly change in foreign currency assets other than wholesale assets (adjusted for exchange rate valuation) in percent of total bank assets. Source: SFLM.
  - d̃D_FX: quarterly change in foreign currency liabilities other than wholesale liabilities (adjusted for exchange rate valuation) in percent of total bank assets. Source: SFLM.
  - lvix: log of the VIX option-based expected stock price volatility based on the S&P 500, calculated by the CBOE. Highest daily realized value of quarter (end of quarter has been used alternatively). Source: Datastream.
  - gbdl: quarterly percentage growth in US Broker Dealer Leverage as defined in Adrian et al. [2014], page 9, multiplied by minus one. Quarterly, based on end-of-quarter accounts. Source: US Financial Accounts (http://www.federalreserve.gov/datadownload/Build.aspx?rel=Z1).
  - lvol_USD: log of the quarterly average of squared daily percentage changes of the nominal exchange rate of the local currency against the USD (quarterly mean of (100 * dlog(S_t))^2). Source: Datastream.
  - dUSD: percentage nominal appreciation of the US dollar over the quarter. Exchange rate measured as average of daily exchange rates of the local currency per foreign currency unit in the last week of the quarter. Due to abnormal move for Romania between last quarter of 2014 and first quarter of 2015, these observations for Romania excluded. Source: Datastream.
  - dUIP: quarterly changes in deviations from uncovered interest parity, computed as changes in the simple USD-domestic interest differential based on 3-month money market interest rates (annualized). Average of daily observations over the last week of the quarter used. Source: Bloomberg.
  - dCIP: quarterly changes in deviations from covered interest parity, computed daily based on 3-month forward premiums and spot rates against the USD, and 3-month Libor rates (annualized). Average for the last week of the quarter used. Source: Bloomberg.
  - dUSRR: US short-term real interest rate. 3-month USD money market interest rate, average for the last week of the quarter, less US realized annualized inflation rate as defined below, annualized. Source: Bloomberg and National Statistical Authorities or central banks.
  - dUSREER: quarterly percentage change (annualized) in the real effective exchange rate of the USD. Last month of quarter. Source: IFS.
  - dUSNEER: quarterly percentage change (annualized) in the nominal effective exchange rate of the USD. Last month of quarter. Source: IFS.
  - Inflation: percentage change in the consumer price index between last month of quarter and same quarter the previous year. Sources: National Statistical Authorities or Central Banks.
  - Growth: percentage change between quarterly real GDP and the same quarter the previous year. Source: National Statistical Authorities (list of national agencies provided in source).

### Adjusting bank positions for exchange rate valuation effects (Appendix C)
- Position dynamics (exact expressions preserved):
  - For wholesale funding position of country c in foreign currency j, value in domestic currency at t is L_{c,j,t}. Total change between t−1 and t:
    - L_{c,j,t} = L_{c,j,t−1} + W_{c,j,t} + L_{c,j,t−1} dln(S_{c,j,t}).
  - Rewritten as:
    - ̇l_{c,j,t} = w_{c,j,t} + ̇s_{c,j,t} where ̇l_{c,j,t} = dL_{c,j,t}/L_{c,j,t−1}, w_{c,j,t} = W_{c,j,t}/L_{c,j,t−1}, ̇s_{c,j,t} = dln(S_{c,j,t}).
  - For CHF and other foreign currencies:
    - ̇l_{c,CH,t} = w_{c,CH,t} + ̇s_{c,CH,t} (equation (19))
    - ̇l_{c,FX,t} = w_{c,FX,t} + ∑_j $_{c,j} ̇s_{c,j,t} (equation (20)) where $_{c,j} are currency shares in non-CHF foreign currency positions.
- Estimation of currency weights $_{c,j}:
  - Three sources used:
    1. ECB annual report on international role of the euro (ECB [2014]) — when one currency dominates.
    2. Regression of ̇l_{c,FX,t} on ̇s_{c,j,t} across euro, USD, GBP, JPY to estimate coefficients then rescaled to sum to one.
    3. Central bank / ECB / national sources when regressions are problematic (e.g., countries pegged to the euro).
  - Acknowledgement: currency weights may have changed over sample; data does not allow accounting for such changes.

### Currency weighting scheme (Table 5 — selected entries)
- Austria:
  - Approach: Austrian National Bank: bulk of non-CHF FX lending to domestic households is in Yen; all non-CHF FX assumed in yen for all subcategories.
  - $USD = 0, $EUR = 0, $JPY = 1, $GBP = 0
- Bulgaria:
  - All non-CHF FX in euro for all subcategories.
  - $USD = 0, $EUR = 1, $JPY = 0, $GBP = 0
- Czech Republic:
  - ECB [2014] based weights:
  - $USD assets = 0.1, $EUR assets = 0.9, $JPY assets = 0.00, $GBP assets = 0.00
  - $USD liab = 0.07, $EUR liab = 0.93, $JPY liab = 0.00, $GBP liab = 0.00
- Germany:
  - Bundesbank data used:
  - $USD assets = 0.78, $EUR assets = 0, $JPY assets = 0.06, $GBP assets = 0.16
  - $USD liab = 0.84, $EUR liab = 0, $JPY liab = 0.03, $GBP liab = 0.14
- Denmark:
  - Danmarks Nationalbank and ECB [2014]:
  - $USD assets = 0.20, $EUR assets = 0.75, $JPY assets = 0.01, $GBP assets = 0.04
  - $USD liab = 0.30, $EUR liab = 0.64, $JPY liab = 0.02, $GBP liab = 0.05
- United Kingdom (regression-based):
  - $USD assets = 0.28, $EUR assets = 0.58, $JPY assets = 0.14, $GBP assets = 0.00
  - $USD liab = 0.26, $EUR liab = 0.58, $JPY liab = 0.16, $GBP liab = 0.00
- Greece (Bank of Greece averages):
  - $USD assets = 0.97, $EUR assets = 0.00, $JPY assets = 0.03, $GBP assets = 0.00
  - $USD liab = 0.83, $EUR liab = 0.00, $JPY liab = 0.12, $GBP liab = 0.04
- Romania (National Bank of Romania):
  - $USD assets = 0.04, $EUR assets = 0.96, $JPY assets = 0.00, $GBP assets = 0.00
  - $USD liab = 0.08, $EUR liab = 0.92, $JPY liab = 0.00, $GBP liab = 0.00
- Serbia (National Bank of Serbia and ECB [2014]):
  - $USD assets = 0.12, $EUR assets = 0.88, $JPY assets = 0.00, $GBP assets = 0.00
  - $USD liab = 0.06, $EUR liab = 0.94, $JPY liab = 0.00, $GBP liab = 0.00
- Slovenia (after joining the euro):
  - $USD = 1, $EUR = 0, $JPY = 0, $GBP = 0
- Notes on table: weights apply to non-Swiss franc foreign currency positions and sum to one. Where only one set of weights is provided, same weights applied to assets and liabilities.

### Appendix figures and tables — selected descriptions and exact numeric references
- Figures 5–10:
  - Series plotted for multiple countries (Austria, Bulgaria, Czech Republic, Germany, Estonia, Greece, Croatia, Hungary, Italy, Romania, Serbia, Slovenia, Slovakia, United Kingdom, Latvia, Denmark).
  - Figure caption (Figure 5 example): "Net foreign currency and domestic currency wholesale funding inflows. The black line is quarterly valuation adjusted change in foreign currency interbank funding less foreign currency interbank lending, divided by total bank assets. The gray line is quarterly domestic currency foreign interbank funding less domestic currency foreign interbank lending, divided by total bank assets. Valuation adjusted described in Appendix C. Source: The Swiss Franc Lending Monitor, SNB."
- Figure 8–10 captions:
  - "Net foreign currency exposure. The black line is the difference between foreign currency assets and foreign currency liabilities as a share of total bank assets. The gray line is foreign currency interbank cross border lending less similarly defined funding, as a share of total bank assets. Source: The Swiss Franc Lending Monitor, SNB."
- Appendix tables (selected exact statistics and regression metadata):
  - Table 6: Robustness to country specific cip and uip parameter estimates. Sample period runs from Q1 2007 to Q1 2016. Nobs = 454 (All), 255 (Non-Euro), 181 (Euro Area). No. cross sections = 16. R^2 values: 0.22 (All), 0.28 (Non-Euro), 0.16 (Euro Area). R^2 adjusted: 0.14, 0.19, 0.02. Fixed effects: Yes. Asterisks denote significance at the 10, 5, 1 percent levels.
  - Table 7: Robustness to inclusion of additional controls. Sample period runs from Q1 2007 to Q1 2016. Nobs = 423 (All), 239 (Non-Euro), 167 (Euro Area). R^2: 0.17, 0.23, 0.15. R^2 adjusted: 0.13, 0.16, 0.03. Fixed effects: Yes.
  - Table 8: Regression for total cross border flows denominated in domestic currency. Sample Q1 2007 to Q1 2016. Nobs = 475 (All), 271 (Non-Euro), 185 (Euro Area). Fixed effects: Yes.
  - Table 9: Regression using EUR as foreign currency for non-euro countries. Sample full runs Q1 2007 to Q1 2016; Post-crisis (PC) sample starts in 2009Q2. Nobs vary by specification (e.g., 462, 401, 444, etc.). No. cross sections = 16 (typical). R^2 values across specifications range from 0.047 to 0.234; R^2 adjusted reported per column.
  - Table 10: Regression using EUR as foreign currency for non-euro countries — subsample results. Nobs = 263 (full) and 225/201/207 (PC variants). R^2 values range from 0.05 to 0.23. Fixed effects: Yes.
  - Table 9–10 include coefficients and interaction terms such as Net_{−1} · gbdl_{−1}, Net_{−1} · lvol_FX_{−1}, Net_{−1} · dFX_{−1}, Net^2_{−1} · dFX_{−1}, and controls ̃C_FX_{−1}, ̃D_FX_{−1}, dUIP_FX_{−1}, dCIP_FX_{−1}. Significance markers *, **, *** reported in tables.

*Italic: Source — wp1897 - References (PDF chapter), https://www.imf.org/-/media/files/publications/wp/2018/wp1897.pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp1897.pdf_