## wpiea2021084-print-pdf

## Source details

**Canonical URL:** [wpiea2021084-print-pdf](https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021084-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2021/english/wpiea2021084-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2021/english/wpiea2021084-print-pdf.pdf.json)

---

### Debt-limit channel: core mechanism and intuition
- Proposed mechanism: "debt limit channel" as alternative to original-sin and demand-side misperception/speculation explanations for household mortgage dollarization in CEE.
- Key idea: when the nominal exchange rate is expected to depreciate (positive interest-rate differential), borrowing in foreign currency backloads real payments, allowing credit-constrained agents to hold more debt over the loan duration.
- Preconditions required for the debt-limit channel to bias mortgage choices toward foreign currency:
  - (i) the difference between domestic and foreign interest rates is positive,
  - (ii) borrowers are credit constrained,
  - (iii) mortgage debt is long-term,
  - (iv) collateral constraint applies to new borrowing rather than outstanding debt (i.e., established on newly purchased assets).

### Illustrative deterministic example (2-period adjustable-rate loan)
- Setup:
  - Borrower can take 100 units (domestic or foreign) at time 0; equal principal payments of 50 units in each period.
  - Real interest rate normalized to zero.
  - Domestic nominal interest rates: 1% for period 1 and 0% for period 2.
  - Foreign nominal interest rates: 0% in both periods.
  - No-arbitrage: nominal exchange rate depreciates by 1% in period 1 and then remains constant.
- Local-currency cash flows:
  - Domestic borrowing: period 1 = 50 + 0.01×100 = 51; period 2 = 50 + 0×50 = 50.
  - Foreign borrowing: period 1 and 2 = 50×1.01 = 50.5.
- Present discounted value: both streams = 101/1.01 = 100.
- Distinction: foreign borrowing backloads real payments and increases outstanding local-currency value of foreign-denominated debt (e.g., outstanding debt end of period 1 = 50 for local borrowing, 50.5 for foreign borrowing).

### Role of uncertainty and risk
- Exchange rate fluctuations under uncertainty make foreign-currency borrowing risky for households earning in local currency: unexpected depreciation worsens balance sheets of foreign-currency debt holders.
- For sufficiently high risk, backloading benefits may be outweighed and dollarization may not be chosen.
- Empirical evaluation requires confronting the debt-limit channel strength with realistic risk descriptions.

### General equilibrium model: structure and assumptions
- Environment:
  - Small open economy with stochastic endowment, nominal long-term debt collateralized on housing, and nominal exchange rate risk.
  - International investors are risk-neutral and price mortgages as nominal perpetuities with principal declining geometrically at rate 0 < δ ≤ 1; variable-rate mortgages (interest adjusted every period).
  - No-arbitrage mortgage pricing: q_t = R_{t-1} − 1 (domestic) and q_t = R^*_{t-1} − 1 (foreign).
  - UIP: R_t = R^*_t E_t{S_{t+1}/S_t}.
- Representative household:
  - Maximizes E_0 Σ_{t=0}^∞ β^t u(c_t,h_t), with 0 < β < 1 and impatience relative to rest of world so β r^* < 1.
  - Nominal budget constraint (compact form used):
    - P_t c_t + P_{h,t} h^{new}_t + R_{t-1} L_{H,t-1} + R^*_{t-1} S_t L_{F,t-1} ≤ P_t y_t + L_{H,t} + S_t L_{F,t}.
- Collateral and accumulation:
  - Collateral constraint on new borrowing: L^{new}_{H,t} + S_t L^{new}_{F,t} ≤ m P_{h,t} h^{new}_t, with 0 < m < 1.
  - Debt accumulation: L_{·,t} = L^{new}_{·,t} + (1−δ) L_{·,t-1}.
  - Housing accumulation: h_t = h^{new}_t + (1−δ_h) h_{t-1}, 0 < δ_h ≤ 1.
- Monetary policy and Fisher relationship:
  - R_t = r^* ̄π (π_t / ̄π_t)^ν and R^*_t = r^* ̄π^* (π^*_t / ̄π^*_t)^ν with ν > 1.
  - E_t{R^*_t π^*_{t+1}} = r^*.
- Housing market clearing: h_t = ̄h.

### Debt-limit channel inside the model
- Collateral constraint in real terms, with total real debt l_t = l_{H,t} + l_{F,t}, foreign share ω_t = l_{F,t} / l_t, exchange rate depreciation γ_t:
  - l_t ≤ m δ_h p_{h,t} ̄h + (1−δ) l_{t-1} / [π_t (1 + γ_t ω_{t-1})].
- Multi-period loans (δ < 1) imply past debt composition matters for current debt capacity.
- Steady-state expression:
  - l = (m δ_h ̄π p_h ̄h) / [̄π − (1−δ) (1 + γ ω)].
- Implication: if R > R^* implying γ > 0, a higher foreign share ω increases total debt l for given collateral. When collateral constraints bind and γ > 0, financially constrained households optimally borrow in foreign currency (degenerate portfolio choice).
- Channel relies on collateral constraint applying to new mortgages; alternative formulations (e.g., L_{H,t} + S_t L_{F,t} ≤ m P_{h,t} h_t) would remove the channel absent risk considerations.

### Quantitative and normative implications (model outcomes and welfare)
- Calibration to Poland yields a sizable share of foreign currency mortgage debt and predictions consistent with CEE empirical evidence.
- Presence of housing in the collateral constraint creates a pecuniary externality: decentralized outcome exhibits overborrowing and bias toward foreign currency relative to constrained-optimal time-consistent policy.
- Novel finding: dollarization of mortgages may be inefficient and warrant regulatory intervention.

### Risk considerations — definitions and equilibrium conditions
- One-period gross real returns:
  - r_{H,t+1} = R_t / π_{t+1}
  - r_{F,t+1} = R^*_t / π^*_{t+1} = R^*_t / [π_{t+1}(1+γ_{t+1})]
  - Excess return: r_{x,t+1} = r_{F,t+1} − r_{H,t+1}
- Euler equations with multipliers Θ_t (collateral tightness) and μ_{H,t}, μ_{F,t} (non-negativity):
  - u_{c,t} − Θ_t = βE_t{u_{c,t+1} r_{H,t+1}} − β(1−δ)E_t{Θ_{t+1}/π_{t+1}} − μ_{H,t}
  - u_{c,t} − Θ_t = βE_t{u_{c,t+1} r_{F,t+1}} − β(1−δ)E_t{Θ_{t+1}/[π_{t+1}(1+γ_{t+1})]} − μ_{F,t}

### Decomposition: balance sheet effect and debt limit channel
- Right-hand sides separate:
  - Balance sheet effect: expected financial cost of repaying a loan, driven by risk and covariances with u_{c,t+1} and returns.
  - Debt limit channel: expected value of rollover commitment from multi-period loans (present only if δ < 1); positive unless collateral is known to be slack in the future (Θ_{t+1} = 0).
- Fisher relationship implies E_t{r_{H,t+1}} = E_t{r_{F,t+1}} = r^*; thus balance sheet effect focuses on risk (covariances) not expected return differences.
- Exchange rate volatility increases the balance sheet channel against foreign-currency borrowing for households earning in local currency.

### Debt limit channel subcomponents and determinants of portfolio choice
- Debt limit subcomponents:
  - Expected exchange rate direction: expected depreciation favors foreign borrowing; expected appreciation favors domestic borrowing.
  - Risk: positive comovement between collateral tightness and exchange rate when households hold foreign debt tightens constraints and affects choices.
- Key determinants:
  - Nominal interest rate spread R_t − R^*_t: higher spreads strengthen debt-limit channel and favor foreign borrowing.
  - Nominal exchange rate volatility: higher volatility strengthens balance sheet channel and discourages dollarization.
  - Exogenous forces (e.g., stochastic endowment z_t) affect marginal utility and collateral tightness.
- Corner solutions possible: non-negativity multipliers can imply all debt denominated in one currency.
  - In zero-risk limit, choice depends only on sign of interest rate spread: full dollarization occurs for R_t > R^*_t with μ_F = 0 and μ_H > 0.

### Constrained-efficient allocations — planner comparison implications
- Social planner internalizes collateral-price general-equilibrium effects; planner’s housing Euler implementability constraint:
  - p_{h,t}(u_{c,t} − mΘ_t) = u_{h,t} + β(1−δ_h)E_t{p_{h,t+1}(u_{c,t+1} − mΘ_{t+1})}
- Market outcomes display socially inefficient bias toward foreign currency because private agents fail to internalize how portfolio choices increase exposure to exchange-rate risk.
- Calibration (Poland, 1998q1–2008q2) indicates planner chooses lower total debt and a safer currency composition:
  - Increases average consumption by 0.08%
  - Reduces consumption standard deviation by 0.25%

### Model fit to Polish time series (period of non-discriminatory foreign currency mortgages)
- Fit highlights:
  - Model matches volatility of foreign currency share in mortgages well.
  - Model persistence of foreign currency mortgage share is substantial but "falls short of that observed in the data."
  - Comovement signs with endogenous variables correctly replicated.
  - Model predicts nearly zero comovement with excess return correctly.
  - Model overemphasizes negative correlation of foreign currency loan share with output, consumption, and house prices—likely due to too little house price variation in baseline model.
- Experiment with stochastic housing weight A_h:
  - Calibrating A_h process to match house price inertia and volatility improves fit for correlations with consumption and house prices, and somewhat for output.
  - Predictions about degree of dollarization under decentralized and constrained-efficient equilibria barely differ from baseline.

### Key empirical statistics and calibration notes
- Debt write-offs due to default on housing loans in Poland: averaging to merely 0.35% of outstanding mortgage debt over 2009-2014 (NBP, 2015).
- This ratio never exceeded 0.5% during that period despite large zloty depreciation.
- Correlation reported by Skibinska (2018) for nominal interest rate volatility and share of foreign currency loans: 0.21.

### Qualitative predictions and empirical comparisons
- Empirical regularities confronted and model responses:
  - Debt dollarization positively correlated with interest rate differential: model explains via debt-limit channel.
  - Positive correlation with level of inflation: model links through average interest rate differential.
  - Exchange rate volatility discourages foreign-currency debt: model reproduces negative relationship.
  - Domestic inflation volatility raises dollarization when included via inflation target shocks.
  - Remittances as exogenous foreign-currency endowment increase foreign share via natural hedging.
- Empirical relationship not matched:
  - Positive correlation between nominal interest rate volatility and foreign currency debt: model yields zero because nominal interest rate is constant in equilibrium; movements would operate via UIP and exchange rate.
- Preconditions empirical relevance in CEE:
  - Condition (i) holds in all CEE economies except Czechia and Slovakia (these two have virtually non-existent foreign currency mortgages).
  - Condition (ii) supported by evidence that young/less patient households chose foreign currency loans.
  - Conditions (iii) and (iv) hold: mortgages long-term and typically issued at property purchase; home equity lines and multiple mortgages not dominant.

### Additional discussion of model assumptions
- Solution complexity: endogenous portfolio choice and occasionally binding constraints require global solution methods and value-function-based algorithms.
- Foreign lenders assumed risk-neutral:
  - This assumption means foreign-currency mortgage share reflects only domestic risk considerations; risk-averse foreign lenders would bias toward domestic-currency lending, so risk-neutrality acts against obtaining a bias toward foreign currency.
- UIP imposed exactly in baseline; allowing time-varying risk premia yields similar roles to foreign inflation shocks.
- Monetary policy determination:
  - Asset market segmentation and simple policy rules effectively keep home nominal interest rate from responding to domestic conditions; nominal exchange rate exogenous to households.
  - Experiments with Taylor-like rules and sticky prices: less aggressive central bank responsiveness can raise equilibrium foreign-currency loan share, but holding exchange rate volatility constant leaves bias largely unaffected.

### Conclusions (selected)
- Debt limit channel: when loans are multi-period and collateralized on newly purchased assets, domestic and foreign currency borrowing are imperfect substitutes even under certainty equivalence.
- Mechanism: higher domestic nominal interest rates lead borrowers to prefer foreign currency to backload real repayments and increase borrowing capacity under collateral limits.
- Quantitative findings:
  - Bias toward foreign currency is empirically relevant and survives realistic exchange rate risk.
  - Dollarization from the debt-limit channel is socially inefficient when credit conditions depend on collateral prices; planner would choose lower foreign share.
  - Banning foreign currency mortgages completely can be a reasonable substitute for more sophisticated regulation accounting for the debt-limit channel.
- Scope and limitations:
  - Analysis abstracts from default; justification: Polish housing debt write-offs averaged 0.35% over 2009-2014 and never exceeded 0.5%.
  - Extensions including default and shadow debt prices in the collateral constraint could produce different outcomes, especially for firms.

### Computational algorithms, calibration and data
- Numerical setup:
  - Endogenous state grid: 50 points for l and 20 points for ω → total 1000 grid points.
  - Shock discretization: Tauchen and Hussey with 3 realizations for each shock → 9 exogenous states with s={y, ̄π, ̄π*} in baseline.
  - Optimization: Matlab function fmincon used at each grid point; linear interpolation for l′ outside grid; ω′ constrained to grid for numerical stability.
- Decentralized equilibrium algorithm highlights iterative computation of Θ(z,s) using appropriate Euler equations depending on ω′ (ω′ < 1 or ω′ > 0).
- Constrained-efficient equilibrium algorithm involves outer-loop iteration over future planner policy functions ̃c, ̃l′, ̃ω′, ̃p_h, ̃Θ until time-consistent convergence.
- Model calibration structural parameters (exact values preserved):
  - r* 1.005 — World real interest rate
  - β 0.99 — Discount factor
  - ̄π* 1.005 — Steady-state inflation abroad
  - ̄π 1.012 — Steady-state inflation at home
  - δ 0.015 — Loan decay parameter
  - m 0.85 — LTV ratio on mortgage originations
  - A_h 0.09 — Weight of housing in utility
  - δ_h 0.009 — Housing depreciation rate
  - ̄h 5.24 — Steady-state housing stock
- Shock discretization parameters:
  - std(y) 0.007 — Standard dev. of endowment process
  - corr(y,y−1) 0.87 — Autocorrelation of endowment process
  - std(̄π) 0.035 — Standard deviation of foreign inflation target
- Model and data moments (percent shares) — simulated vs data (exact table entries preserved):
  - mean(ω) Data 57.86 / Decentralized 71.97 / Constrained-efficient 7.1
  - std(ω) Data 29.44 / Decentralized 32.81 / Constrained-efficient 14.1
  - corr(ω,ω−1) Data 0.95 / Decentralized 0.79 / Constrained-efficient 0.80
  - corr(ω,l) Data 0.06 / Decentralized 0.41 / Constrained-efficient 0.58
  - corr(ω,c) Data -0.32 / Decentralized -0.76 / Constrained-efficient -0.83
  - corr(ω,p_h) Data -0.04 / Decentralized -0.64 / Constrained-efficient -0.70
  - corr(ω,y) Data -0.33 / Decentralized -0.82 / Constrained-efficient -0.55
  - corr(ω,r_x) Data -0.04 / Decentralized -0.05 / Constrained-efficient -0.02
- Table 3 and table notes, regional statistics and country-level shares preserved as provided in the source.
- Data sources and processing (definitions preserved exactly):
  - Interest rate -- short term (3-month money market) interest rate; Eurostat
  - Nominal exchange rates -- bilateral exchange rates of the Polish zloty; Eurostat
  - Output -- real gross domestic product at market prices, chain-linked volumes; Eurostat
  - Consumption -- real household and NPISH final consumption expenditure, chain-linked volumes; Eurostat
  - Inflation -- log-difference in all-items HICP
  - Loans -- other monetary financial institutions (other MFIs) loans to households for house purchase, also by currency; Narodowy Bank Polski.
  - House prices -- residential property prices of existing flats in big cities per square meter; Bank of International Settlements.
- Data processing:
  - All series are quarterly.
  - Except for the interest and exchange rates, all data are seasonally adjusted.
  - When used in real terms, house prices and loans are deflated with HICP.
  - Output, consumption, real loans and real house prices are logged and filtered with the Hodrick-Prescott filter, using 1600 as the smoothing parameter.
- Mortgage currency composition:
  - Framework allows currency conversion of loans at no cost; testing uses data on originations. Originations data: ZBP AMRON-SARFiN reports back to 2006; stock-based estimates constructed via formulas (4), (5) and nominal exchange rate data correlate at 0.95 with ZBP.

*Source: IMF working paper content (wpiea2021084-print-pdf).*

### 2014.  A couple of months later, a controversial law was passed in Croatia that facilitated the conversion of

### 2014.  A couple of months later, a controversial law was passed in Croatia that facilitated the conversion of loans denominated in Swiss franks into loans denominated in euro.

### Debt-limit channel: core mechanism and intuition
- The paper proposes the "debt limit channel" as an alternative explanation for household mortgage dollarization in CEE countries, distinct from original-sin and demand-side misperception/speculation explanations.
- Key idea: when the nominal exchange rate is expected to depreciate (positive interest-rate differential), borrowing in foreign currency effectively backloads real payments, allowing credit-constrained agents to hold more debt over the loan duration.
- Conditions required for the debt-limit channel to bias mortgage choices toward foreign currency:
  - (i) the difference between domestic and foreign interest rates is positive,
  - (ii) borrowers are credit constrained,
  - (iii) mortgage debt is long-term,
  - (iv) collateral constraint applies to new borrowing rather than outstanding debt (i.e., established on newly purchased assets).

### Illustrative deterministic example (2-period adjustable-rate loan)
- Borrower can take 100 units (domestic or foreign) at time 0; equal principal payments of 50 units in each period.
- Real interest rate normalized to zero.
- Domestic nominal interest rates: 1% for period 1 and 0% for period 2.
- Foreign nominal interest rates: 0% in both periods.
- Financial markets enforce no-arbitrage: nominal exchange rate depreciates by 1% in period 1 and then remains constant.
- Cash flows in local currency:
  - Domestic borrowing: period 1 = 50 + 0.01×100 = 51; period 2 = 50 + 0×50 = 50.
  - Foreign borrowing: period 1 and 2 = 50×1.01 = 50.5.
- Present discounted value of both streams = 101/1.01 = 100.
- Distinction arises from distribution over time: exchange rate depreciation backloads real loan payments, increasing outstanding local-currency value of foreign-denominated debt (e.g., outstanding debt end of period 1 = 50 for local borrowing, 50.5 for foreign borrowing).

### Role of uncertainty and risk
- Under uncertainty, exchange rate fluctuations make foreign-currency borrowing relatively risky for households earning in local currency: unexpected depreciation worsens balance sheets of foreign-currency debt holders.
- For sufficiently high risk, backloading benefits may be outweighed and dollarization may not be chosen.
- Empirical evaluation requires confronting debt-limit channel strength with realistic risk descriptions.

### General equilibrium model: structure and assumptions
- Small open economy with stochastic endowment, nominal long-term debt collateralized on housing, and nominal exchange rate risk.
- International investors are risk-neutral and price mortgages as nominal perpetuities with principal declining geometrically at rate 0 < δ ≤ 1; variable-rate mortgages (interest adjusted every period).
- No-arbitrage implies mortgage pricing q_t = R_{t-1} − 1 (domestic) and q_t = R^*_{t-1} − 1 (foreign).
- Uncovered interest parity (UIP) condition: R_t = R^*_t E_t{S_{t+1}/S_t}.
- Representative household maximizes E_0 Σ_{t=0}^∞ β^t u(c_t,h_t), with 0 < β < 1 and impatience relative to rest of world so β r^* < 1.
- Budget constraint (nominal form):
  - P_t c_t + P_{h,t} h^{new}_t + (R_{t-1} − 1 + δ) L_{H,t-1} + (R^*_{t-1} − 1 + δ) S_t L_{F,t-1} ≤ P_t y_t + L^{new}_{H,t} + S_t L^{new}_{F,t}.
- Equivalent compact budget (using debt evolution) independent of δ:
  - P_t c_t + P_{h,t} h^{new}_t + R_{t-1} L_{H,t-1} + R^*_{t-1} S_t L_{F,t-1} ≤ P_t y_t + L_{H,t} + S_t L_{F,t}.
- Collateral (borrowing) constraint on new borrowing:
  - L^{new}_{H,t} + S_t L^{new}_{F,t} ≤ m P_{h,t} h^{new}_t,  with 0 < m < 1 (LTV).
- Debt accumulation:
  - L_{H,t} = L^{new}_{H,t} + (1−δ) L_{H,t-1}
  - L_{F,t} = L^{new}_{F,t} + (1−δ) L_{F,t-1}
- Housing accumulation: h_t = h^{new}_t + (1−δ_h) h_{t-1},  0 < δ_h ≤ 1.
- Monetary policy rules (feedback):
  - R_t = r^* ̄π (π_t / ̄π_t)^ν
  - R^*_t = r^* ̄π^* (π^*_t / ̄π^*_t)^ν
  - π_t = P_t / P_{t-1}; π^*_t = P^*_t / P^*_{t-1}; ν > 1 ensures determinacy.
- Fisher relationship (world real interest rate constant r^*):
  - E_t{R^*_t π^*_{t+1}} = r^*.
- Housing market clearing: h_t = ̄h with ̄h > 0 fixed.

### Debt-limit channel inside the model
- Rewriting collateral constraint using real variables, total real debt l_t = l_{H,t} + l_{F,t}, foreign share ω_t = l_{F,t} / l_t, exchange rate depreciation γ_t = (S_t − S_{t-1})/S_{t-1} = π_t / π^*_{t-1}:
  - l_t ≤ m δ_h p_{h,t} ̄h + (1−δ) l_{t-1} / [π_t (1 + γ_t ω_{t-1})].
- Past debt composition matters for current debt when loans are multi-period (δ < 1).
- Steady-state expression:
  - l = (m δ_h ̄π p_h ̄h) / [̄π − (1−δ) (1 + γ ω)].
- If R > R^* (nominal interest rates in emerging economies usually higher than in developed countries) implying γ > 0, a higher foreign share ω increases the total amount of debt l that can be held for given collateral.
- When binding constraints and positive γ, financially constrained households optimally borrow in foreign currency (degenerate portfolio choice).
- The channel relies on collateral constraint applying to new mortgages (loans for house purchase) rather than total outstanding debt; alternative collateral formulations (e.g., L_{H,t} + S_t L_{F,t} ≤ m P_{h,t} h_t) would remove the debt-limit channel absent risk considerations.

### Quantitative and normative implications (model outcomes and welfare)
- Embedding the mechanism into a quantitative general equilibrium model calibrated to Poland yields a sizable share of foreign currency mortgage debt and generates predictions consistent with CEE empirical evidence.
- Presence of housing in the collateral constraint creates a pecuniary externality: decentralized outcome exhibits overborrowing and a bias toward foreign currency relative to constrained-optimal time-consistent policy.
- The finding that dollarization of mortgages may be inefficient and warrant regulatory intervention is presented as novel in the literature.

*Source: IMF working paper content (model and analysis excerpt).*

### 2.4    Risk considerations

### 2.4    Risk considerations

### Definitions and equilibrium conditions
- One-period gross real rates of return:
  - r_{H,t+1} = R_t / π_{t+1}
  - r_{F,t+1} = R^*_t / π^*_{t+1} = R^*_t / [π_{t+1}(1+γ_{t+1})]
  - Excess return: r_{x,t+1} = r_{F,t+1} − r_{H,t+1}
- Euler equations for borrowing in local and foreign currency (with Lagrange multipliers Θ_t on the collateral constraint and μ_{H,t}, μ_{F,t} on non-negativity of L_{H,t}, L_{F,t}):
  - u_{c,t} − Θ_t = βE_t{u_{c,t+1} r_{H,t+1}} − β(1−δ)E_t{Θ_{t+1}/π_{t+1}} − μ_{H,t}
  - u_{c,t} − Θ_t = βE_t{u_{c,t+1} r_{F,t+1}} − β(1−δ)E_t{Θ_{t+1}/[π_{t+1}(1+γ_{t+1})]} − μ_{F,t}

### Decomposition: balance sheet effect and debt limit channel
- Right-hand sides decompose into two effects:
  - Balance sheet effect: expected financial cost of repaying a loan; driven by risk and captured by covariances with u_{c,t+1} and returns.
  - Debt limit channel: expected value of the rollover commitment guaranteed by multi-period loans (shows up only if δ < 1); positive unless the collateral constraint is known to be slack in the future (Θ_{t+1} = 0).
- Rewritten Euler equations separate these components (covariance decomposition):
  - Each equation contains terms βE_t{u_{c,t+1}}E_t{r_{·,t+1}} + βCov_t{u_{c,t+1}, r_{·,t+1}} for the balance sheet effect, and debt-limit terms involving E_t{Θ_{t+1}/π_{t+1}} and covariances with γ_{t+1} for the foreign-currency case.

### Role of Fisher relationship and risk
- Fisher relationship implies E_t{r_{H,t+1}} = E_t{r_{F,t+1}} = r^*, so the balance sheet effect is essentially about risk (covariances) rather than differences in expected returns.
- Increase in nominal exchange rate volatility discourages debt dollarization (households typically have income in local currency): higher volatility strengthens the balance sheet channel against foreign-currency borrowing.

### Debt limit channel details
- The debt limit channel has two subcomponents:
  - Expected direction of exchange rate movements: expected depreciation (appreciation) of the exchange rate favors foreign (domestic) currency borrowing via its impact on effective rollover commitment.
  - Risk: expected comovement between tightness of the collateral constraint and the exchange rate. This comovement is positive when households hold foreign currency debt because exchange rate depreciation worsens balance sheets and house prices, tightening collateral constraints and biasing choices toward foreign currency if it carries lower nominal interest.

### Determinants of portfolio choice and possible corner solutions
- Key determinants:
  - Nominal interest rate spread R_t − R^*_t: higher values strengthen the debt limit channel and favor foreign-currency borrowing.
  - Nominal exchange rate volatility: higher volatility strengthens the balance sheet channel and discourages dollarization.
  - Other exogenous forces (e.g., stochastic endowment z_t) affect household marginal utility and collateral tightness, thus altering channel strengths.
- Non-negativity Lagrange multipliers imply corner (non-internal) solutions are possible: households may choose to denominate all debt in one currency.
  - In the limit of zero risk, choice depends only on sign of interest rate spread: full dollarization occurs for R_t > R^*_t; in that case μ_F = 0 while μ_H > 0 so both Euler conditions can hold simultaneously.

### Implications for constrained-efficient allocations (previewing planner comparison)
- Social planner internalizes collateral-price general-equilibrium effects but cannot commit to future decisions; planner’s problem includes the household housing Euler as an implementability constraint and the house-price equilibrium condition:
  - p_{h,t}(u_{c,t} − mΘ_t) = u_{h,t} + β(1−δ_h)E_t{p_{h,t+1}(u_{c,t+1} − mΘ_{t+1})}
- In the richer framework where currency composition is endogenous, market outcomes display a socially inefficient bias toward foreign currency because private agents fail to internalize how portfolio choices increase exposure to exchange-rate risk.
- Quantitative simulations (calibration to Poland, 1998q1–2008q2) indicate the planner chooses both lower total debt and a safer currency composition, which:
  - Increases average consumption by 0.08%
  - Reduces consumption standard deviation by 0.25%

*Source: 2.4 Risk considerations (excerpted from the supplied IMF content).*

### 3.4    Model fit

### 3.4    Model fit

### Fit of the baseline model to Polish time series (period of non-discriminatory foreign currency mortgages)
- The model does a very good job at matching the volatility of the foreign currency share in mortgages.
- The model implies substantial persistence of the foreign currency mortgage share, but this persistence "falls short of that observed in the data."
- Comovement with key endogenous variables:
  - The model correctly replicates all signs of comovement reported in the data.
  - The model correctly predicts nearly zero comovement of foreign currency loan share with excess return.
  - The model overemphasizes the negative correlation of foreign currency loan share with output, consumption, and house prices.
- A likely reason for the misfit is that the model generates too little variation in house prices, a common feature in this class of models unless an additional exogenous source of house price variation is included (references: Iacoviello and Neri (2010), Justiniano et al. (2015)).
- Experiment with stochastic housing weight in utility (A_h):
  - Calibrating the process to better match inertia and volatility of house prices improves the fit for:
    - Correlation of foreign currency loan share with consumption and house prices (very good fit).
    - Some improvement for correlation with output.
  - This variant takes longer to solve.
  - Predictions about the degree of dollarization under both decentralized and constrained-efficient equilibrium barely differ from the baseline without housing preference shocks.
- Overall assessment:
  - The theoretical framework is consistent with time series evidence on how the foreign currency share of mortgages behaves over the business cycle.
  - Caveat: inference is based on a rather short sample of data for just one CEE economy.

### Key empirical statistics referenced
- Debt write-offs due to default on housing loans in Poland: averaging to merely 0.35% of outstanding mortgage debt over the period 2009-2014 (NBP, 2015).
- This ratio has never exceeded 0.5% during that period despite large zloty depreciation.
- Correlation reported by Skibinska (2018) for nominal interest rate volatility and share of foreign currency loans: 0.21 (about half of the correlation coefficient between the share of foreign currency loans and inflation volatility, per the text).

---

### 3.5    Qualitative predictions

### Model qualitative predictions vs empirical evidence on mortgage debt dollarization in CEE
- Empirical regularities the model confronts (based on Cuaresma et al. (2011), Fidrmuc et al. (2013), Skibinska (2018)):
  - Debt dollarization is positively correlated with interest rate differential.
    - Model explanation: interest rate differential biases household choices toward foreign currency.
  - Positive correlation with the level of inflation.
    - Model explanation: domestic inflation affects the average interest rate differential.
  - Exchange rate volatility discourages foreign currency denominated debt.
    - Model reproduces this negative relationship.
  - Domestic inflation volatility (not in baseline) raises dollarization when included via domestic inflation target shocks.
    - Intuition: volatile home inflation increases riskiness of domestic-currency loans (covariance term in equation (20) becomes larger).
  - Remittances (not included in baseline) when added as an exogenous foreign-currency fixed endowment increase equilibrium foreign currency share in mortgage debt.
    - Intuition: natural hedging—foreign currency loans insure against fluctuations in home currency value of remittances.
- Empirical relationship not matched:
  - Positive correlation between nominal interest rate volatility and foreign currency denomination of debt.
    - In the model this correlation is zero because the nominal interest rate is constant in equilibrium.
    - Even allowing movements in nominal interest rate would affect household choices only via impact on the nominal exchange rate through UIP.
    - Households assess domestic-currency loan attractiveness based on the level of nominal interest rate and inflation risk (affecting volatility of ex post return).
- Preconditions for the debt limit channel to bias choices toward foreign currency (must hold):
  - (i) the nominal interest rate differential is positive,
  - (ii) borrowers are credit constrained,
  - (iii) mortgage debt is long-term,
  - (iv) collateral constraint is established on newly purchased assets.
- Empirical relevance of prerequisites in CEE:
  - Condition (i) holds in all CEE economies except Czechia and Slovakia (these two countries have virtually non-existent foreign currency mortgages).
  - Condition (ii) supported by Fidrmuc et al. (2013): foreign currency loans were chosen mainly by young households (relatively impatient).
  - Macro evidence (Falk et al. (2018)): agents in CEE are less patient than in more developed countries.
  - Conditions (iii) and (iv): mortgages are usually issued at longer maturities and are typically taken when a property is purchased in CEE; home equity lines and multiple mortgages on a single property are not dominant.
  - These features help explain popularity of foreign currency mortgages for housing, but not necessarily for consumer credit.

---

### 3.6    Additional discussion of model assumptions

### Solution method and complexity
- Endogenous portfolio choice and occasionally binding constraints require solving the model with global methods, imposing natural restrictions on complexity.
- Older, richer DSGE extension (Kolasa, 2018) solved with perturbation methods served as reference; it required simplifying assumptions (e.g., collateral constraint always binding; certainty equivalence effects eliminated).

### Key modeling assumptions and their impacts
- Foreign lenders assumed risk-neutral:
  - Implies equilibrium foreign currency mortgage share reflects only domestic household risk considerations.
  - If foreign investors were risk averse, they would view home-currency mortgages as more risky (emerging market exchange rates tend to depreciate in bad times), which would increase bias toward domestic currency—thus risk-neutrality assumption acts against obtaining a bias toward foreign currency in the model.
- UIP holds exactly in the model:
  - Empirically UIP fails; a natural extension would allow time-varying risk premia.
  - Experiments show UIP shocks play essentially the same role as foreign inflation shocks for determining equilibrium share of foreign currency mortgages.
  - Exchange rate volatility is the key driver; it is captured by calibrating variance of foreign inflation shocks in baseline.
- Determination of nominal interest rate:
  - Asset market segmentation and a simple monetary feedback rule effectively rule out home nominal interest rate response to domestic conditions.
  - Result: nominal exchange rate is exogenous to home households' decisions.
  - Experiments with richer model featuring a Taylor-like monetary policy rule and sticky prices:
    - Equilibrium share of foreign currency loans can increase when central bank responsiveness to inflation is less aggressive.
    - However, holding exchange rate volatility constant (via adjustments to UIP or foreign inflation shock variances) keeps equilibrium bias toward foreign currency largely unaffected by reasonable monetary policy rule modifications.

---

### 4    Conclusions (selected summary)
- Proposed explanation: the "debt limit channel"—when loans are multi-period and collateralized on newly purchased assets, borrowing in domestic and foreign currency are imperfect substitutes even under certainty equivalence.
- Mechanism: if domestic interest rates are higher than abroad, borrowers prefer foreign currency to backload real repayments and effectively increase the debt limit imposed by the collateral constraint.
- Quantitative findings:
  - The bias toward foreign currency is empirically relevant and survives realistic exchange rate risk (which opposes dollarization).
  - Dollarization resulting from the debt limit channel in a decentralized equilibrium is socially inefficient when credit conditions depend on collateral prices.
  - A social planner internalizing the pecuniary externality would choose a much lower share of foreign currency debt.
  - Model suggests that banning foreign currency mortgages completely (as done by some CEE authorities) can be a reasonable substitute for more sophisticated regulation that explicitly accounts for the debt limit channel.
- Scope and limitations:
  - The analysis abstracts from default; rationale: in Poland debt write-offs on housing loans averaged 0.35% of outstanding mortgage debt over 2009-2014 and never exceeded 0.5% in that period.
  - Extension to include default and shadow debt prices in the collateral constraint could produce interesting outcomes, especially for firms where long-term foreign currency loan composition is less understood.

*Source: wpiea2021084-print-pdf - 3.4    Model fit*

### References

### wpiea2021084-print-pdf - References

### Major themes in the references
- Dollarization and financial dollarization: Barajas and Mendez Morales (2003); Ize and Levy Yeyati (2003); Terrones and Catao (2000); Geng, Scutaru, and Wiegand (2018).
- Foreign currency lending, exchange-rate and currency-mismatch risk: Benigno et al. (2013); Brown and De Haas (2012); Brzoza-Brzezina et al. (2017); Korinek (2011); Ranciere, Tornell, and Vamvakidis (2010).
- Housing, mortgages, household leverage and monetary transmission: Campbell and Hercowitz (2009); Iacoviello (2005); Iacoviello and Neri (2010); Garriga, Kydland, and Sustek (2017); Greenwald (2018).
- Macroprudential policy and time-consistent planner problems: Bianchi (2011); Bianchi and Mendoza (2018); Jeanne and Korinek (2010); Gelain, Lansing, and Natvik (2018).
- Empirical determinants and evidence on foreign-currency loans and credit dollarization in transition and emerging economies: Cuaresma et al. (2011); Fidrmuc, Hake, and Stix (2013); Rosenberg and Tirpak (2008); Luca and Petrova (2008); Skibinska (2018).
- Modeling and numerical methods: Kehoe and Levine (1993); Tauchen and Hussey (1991); Elenev, Landvoigt, and Van Nieuwerburgh (2016); Landvoigt, Van Nieuwerburgh, and Greenwald (2020).

### Tables and figures — key numeric content and notes
- Figure 1 (description): plots share of foreign currency mortgage debt (1998–2018) and nominal interest rate differential (difference between 3-month money market rate in Czechia, Hungary, Poland and euro area). Countries shown: Czechia, Hungary, Poland.
- Figure 2 (description): household portfolio choices as a function of current level and composition of debt for three realizations of exogenous processes.

- Table 1: Share of foreign currency debt in total outstanding debt (percent)
  - Region / Number of countries / Foreign currency loans (% of total)
  - All / 66 / 29.4
  - Euro area / 14 / 16.2
  - Non-euro area CEE / 12 / 49.4
  - Latin America / 5 / 26.0
  - Africa / 13 / 20.7
  - Asia / 14 / 34.4
  - Notes: Data from IMF Financial Soundness Indicators (FSIs); numbers are for 2014 except Uruguay and Zambia (2013).

- Table 2: Share of foreign currency debt in CEE countries by institutional sector (percent)
  - Non-euro area members (2014)
    - Bulgaria: Total private 58.9 / Non-financial corporations 70.6 / Households (for housing) 37.0 / Households non-financial 53.9
    - Croatia: 71.4 / 68.8 / 73.2 / 92.6
    - Czech Republic: 9.0 / 21.4 / 0.2 / 0.0
    - Hungary: 51.9 / 50.1 / 53.6 / 52.9
    - Lithuania: 72.2 / 75.1 / 69.3 / .
    - Poland: 28.8 / 26.1 / 30.3 / 46.3
    - Romania: 58.3 / 53.6 / 63.3 / 85.1
  - Euro area members prior to euro adoption (year indicated)
    - Estonia (2010): 88.5 / 92.0 / 85.2 / 93.9
    - Latvia (2013): 86.6 / 84.7 / 88.8 / .
    - Slovakia (2008): 19.0 / 32.2 / 2.7 / 2.9
    - Slovenia (2006): 59.4 / 66.2 / 42.8 / 68.6
  - Notes: Shares based on official central bank statistics; non-euro area shares for 2014; euro area members cover year prior to euro adoption.

- Table 3: Model calibration — structural parameters and shock processes
  - Structural parameters
    - r* 1.005 — World real interest rate
    - β 0.99 — Discount factor
    - ̄π* 1.005 — Steady-state inflation abroad
    - ̄π 1.012 — Steady-state inflation at home
    - δ 0.015 — Loan decay parameter
    - m 0.85 — LTV ratio on mortgage originations
    - A_h 0.09 — Weight of housing in utility
    - δ_h 0.009 — Housing depreciation rate
    - ̄h 5.24 — Steady-state housing stock
  - Shock processes for discretization
    - std(y) 0.007 — Standard dev. of endowment process
    - corr(y,y−1) 0.87 — Autocorrelation of endowment process
    - std(̄π) 0.035 — Standard deviation of foreign inflation target

- Table 4: Model and data moments (mean and standard deviation of share of foreign currency loans expressed in percent)
  - Moment / Data / Decentralized / Constrained-efficient
    - mean(ω) 57.86 / 71.97 / 7.1
    - std(ω) 29.44 / 32.81 / 14.1
    - corr(ω,ω−1) 0.95 / 0.79 / 0.80
    - corr(ω,l) 0.06 / 0.41 / 0.58
    - corr(ω,c) -0.32 / -0.76 / -0.83
    - corr(ω,p_h) -0.04 / -0.64 / -0.70
    - corr(ω,y) -0.33 / -0.82 / -0.55
    - corr(ω,r_x) -0.04 / -0.05 / -0.02
  - Notes: Data-based moments calculated for period 1998q1-2008q2. Model-based moments simulated for 1000,000 periods.

- Table 5: Model predictions versus cross-country evidence from empirical studies
  - FCL dependence on: Empirical studies / Model
    - Interest rate differential: ++ / +
    - Interest rate volatility: + / 0
    - Inflation: ++ / +
    - Inflation volatility: ++ / +
    - Exchange rate volatility: -- / -
    - Remittances balance: -- / -
  - Notes: Based on Cuaresma et al. (2011), Fidrmuc et al. (2013) and Skibinska (2018).

### Appendix — analytical and computational structure (A.1–A.3)
- A.1 Recursive competitive equilibrium
  - Nominal block solved first; with inflation target shocks i.i.d., monetary policy rules (9) and (10), UIP (1) and constant real interest rate assumption (11) imply π_t = ̄π_t and π*_t = ̄π*_t, leading to R_t = r* ̄π and R*_t = r* ̄π*.
  - Exogenous shocks collected in vector s = {y, ̄π, ̄π*}. Individual debt positions l_H and l_F separate from aggregates ̃l_H and ̃l_F.
  - Recursive household problem (equation A.1) with value function V(l_H, l_F, h, ̃l_H, ̃l_F, s) and constraints including budget, collateral, non-negativity, and law-of-motion perceptions ̃l′_H = Γ_H(̃l_H, ̃l_F, s), ̃l′_F = Γ_F(̃l_H, ̃l_F, s).
  - Definition 1: A recursive competitive equilibrium is defined by value function V(z, ̃z, s), decision rules c, l′_H, l′_F, h′, house pricing p_h(̃z,s), perceived laws Γ_H, Γ_F, satisfying optimality, housing market clearing h′ = ̄h, and consistency of perceived laws with aggregate choices.

- A.2 Recursive constrained-efficient equilibrium
  - Social planner maximizes representative household utility subject to resource, collateral, and implementability constraints (housing Euler equation from competitive equilibrium).
  - Planner's recursive problem (equation A.2): V(l_H, l_F, s) = max_{c,l′_H,l′_F,p_h,Θ} { u(c, ̄h) + βE_{s′|s} V(l′_H, l′_F, s′) } subject to budget, collateral constraints, complementary slackness Θ ≥ 0 and Θ(l′_H − (1−δ)l_H ̄π + l′_F − (1−δ)l_F ̄π* − mδ_h p_h ̄h) = 0, and non-negativity.
  - Definition 2: A recursive time-consistent constrained-efficient equilibrium is value function V(z,s), policy functions c, l′_H, l′_F, p_h, Θ and matching future-policy functions ̃c, ̃l′_H, ̃l′_F, ̃p_h, ̃Θ such that they solve the planner’s problem and are time-consistent (future policies match current policy functions).

- A.3 Computational algorithms
  - Reformulation in terms of total debt l = l_H + l_F and foreign-currency share ω = l_F / l. Budget and collateral constraints rewritten in (l, ω); non-negativity implies l ≥ 0 and 0 ≤ ω ≤ 1.
  - A.3.1 Computing decentralized equilibrium — algorithm outline
    1. Choose grid for endogenous states z and exogenous state space s with transition matrix.
    2. Guess pricing function p_h(z,s).
    3. Solve household problem (A.1) with value function iteration, set h′ = ̄h; obtain policy functions c(z,s), l′(z,s), ω′(z,s). Mark states with l′ = 0 or ω′ = 0 or ω′ = 1.
    4. Calculate tightness of collateral constraint consistent with solution:
       - (a) Initialize Θ(z,̃z,s) = 0 for all states.
       - (b) For states with l′ ≠ 0 iterate Θ(z,s) to convergence using Euler equation (18) if ω′ < 1 (implying μ_H = 0) or equation (19) if ω′ > 0 (and μ_F = 0):
         - Θ(z,s) = u_c(c(z,s), ̄h) − βE_{s′|s}{ u_c(c(z′,s′), ̄h) R ̄π′ } + β(1−δ)E_{s′|s}{ Θ(z′,s′) ̄π′ }  (when ω′ < 1)
         - Θ(z,s) = u_c(c(z,s), ̄h) − βE_{s′|s}{ u_c(c(z′,s′), ̄h) R* (̄π*)′ } + β(1−δ)E_{s′|s}{ Θ(z′,s′) (̄π*)′ }  (when ω′ > 0)
    - Note: Multi-period debt and occasionally binding collateral constraint imply equilibrium conditions depend on future Lagrange multipliers (see house pricing equation (22)); this precludes pure time-iteration algorithms and motivates the use of value-function-based methods.

*Italicized: Content derived solely from the "References" and accompanying appendix, tables and figure notes in the provided PDF content unit.*

### 5.  Proceed similarly to calculate the house pricing functionp

### 5.  Proceed similarly to calculate the house pricing function p_h(z,s)

### Decentralized equilibrium: computing p_h(z,s)
- Iterate to convergence the housing Euler equation:
  p_h(z,s) =
  u_h(c(z,s), ̄h)
  +β(1−δ_h)E_{s′|s}{
  p_h(z′,s′)
  [
  u_c(c(z′,s′), ̄h)
  −mΘ(z′,s′)
  ]}
  u_c(c(z,s), ̄h)
  −mΘ(z,s)

- Algorithmic steps (summary of implementation choices):
  - Choose 50 points on the grid for l and 20 points for ω, which gives in total 1000 grid points.
  - Increasing the number of grid points did not produce significantly different results.
  - Optimization at each grid point during value function iteration uses Matlab function fmincon that can handle inequality constraints.
  - Use linear interpolation while evaluating the value function for l′ lying outside the grid for l, but force ω′ to lie on the grid for ω.
  - Allowing interpolation for ω′ produced numerically unstable results because outcomes of optimization produced by fmincon were very sensitive to starting values.
  - Shock state space and probability transition matrix obtained using the Tauchen and Hussey (1991) algorithm, with 3 possible realizations for each shock, which gives 9 possible exogenous states in the baseline application with s={y, ̄π∗}.

### Constrained-efficient equilibrium: algorithm
- The constrained-efficient equilibrium is computed using value function iteration with additional iteration over future planners’ policy functions in the outer loop, accounting for occasionally binding collateral constraints.
- Conventions: endogenous state z={l,ω}; follow section A.2 conventions otherwise.
- Numerical algorithm steps:
  1. Choose a grid for endogenous states z and the state space for exogenous states s together with the associated matrix describing the probability of transition from s to s′.
  2. Guess the policy rules of future planners  ̃c(z,s), ̃l′(z,s), ̃ω′(z,s), ̃p_h(z,s), ̃Θ(z,s).
  3. Solve the social planner problem (A.2) with value function iteration to produce policy functions c(z,s), l′(z,s), ω′(z,s), p_h(z,s), Θ(z,s).
  4. Update the guess of future planners’ policy functions  ̃c(z,s), ̃l′(z,s), ̃ω′(z,s), ̃p_h(z,s), ̃Θ(z,s), and repeat step 3 until convergence.
- Implementation notes:
  - Use the same discretization of the state space as in the decentralized equilibrium computation, applying linear interpolation only for l′.
  - Optimization at each grid point during the value function iteration uses Matlab function fmincon that can handle inequality constraints.

### Numerical and implementation details
- Grid and discretization:
  - 50 points for l; 20 points for ω; total 1000 grid points.
  - Shocks discretized via Tauchen and Hussey with 3 realizations each → 9 exogenous states.
- Optimization and interpolation:
  - fmincon for constrained optimization at each grid point.
  - Linear interpolation for l′ outside the grid; ω′ constrained to grid to maintain numerical stability.

### Data used for calibration and validation
- Data series and sources (definitions preserved exactly as in source):
  - Interest rate -- short term (3-month money market) interest rate; Eurostat
  - Nominal exchange rates -- bilateral exchange rates of the Polish zloty; Eurostat
  - Output -- real gross domestic product at market prices, chain-linked volumes; Eurostat
  - Consumption -- real household and NPISH final consumption expenditure, chain-linked volumes; Eurostat
  - Inflation -- log-difference in all-items HICP
  - Loans -- other monetary financial institutions (other MFIs) loans to households for house purchase, also by currency; Narodowy Bank Polski.
  - House prices -- residential property prices of existing flats in big cities per square meter; Bank of International Settlements.
- Data processing and transformations:
  - All series are quarterly.
  - Except for the interest and exchange rates, all data are seasonally adjusted.
  - When used in real terms, house prices and loans are deflated with HICP.
  - Output, consumption, real loans and real house prices are logged and filtered with the Hodrick-Prescott filter, using 1600 as the smoothing parameter.
- Mortgage currency composition:
  - The framework allows currency conversion of loans at no cost; testing model predictions about currency composition of mortgages uses data on originations rather than outstanding amounts.
  - Originations data available from the Polish Bank Association (ZBP) AMRON-SARFiN reports go back only to 2006.
  - Estimates of mortgage originations are also calculated from stocks using formulas (4), (5) and nominal exchange rate data.
  - The constructed series is very similar to the ZBP data over the overlapping period (the correlation coefficient is 0.95).
  - These estimates of the foreign currency loan share are used when testing the model against the data.

*Source: wpiea2021084-print-pdf (section 5 and A.3.2–A.4 as provided)*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021084-print-pdf.pdf_