## Appendix III. Calculation of Point-in-Time Loan Value, LTVs, and DSTIs in Section V ## Source details **Canonical URL:** [Appendix III. Calculation of Point-in-Time Loan Value, LTVs, and DSTIs in Section V](https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020165-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2020/english/wpiea2020165-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2020/english/wpiea2020165-print-pdf.pdf.json) --- ### I. Objectives and conceptual framing - Two main objectives: - Provide a conceptual framework of systemic risk and risk resilience to operationalize the macroprudential policy goal linked to readily available metrics and assess policy success by benchmarking projected stressed credit losses against peak NPLs during a systemic banking crisis (Laeven and Valencia (2018)). - Apply the framework to mortgage loan markets in two countries (Switzerland and Austria) to assess appropriateness of the macroprudential policy stance and quantify necessary adjustments in calibration of macroprudential instruments. - Policy framing: - Characterize macroprudential policy as minimizing a loss function that would erode a significant fraction of banking system capital conditional on a distribution of risk factors. - Use stress-testing via a semi-structural mortgage-default model to produce forward-looking measures of banking system resilience and to guide calibration of borrower-based tools and sectoral capital buffers. ### II. Data approach and drivers of mortgage risk - Data strategy: - Use banks’ portfolio/regulatory reporting data segmented by risk-vintage buckets rather than micro-level borrower data. - PSD performance data (PSD007) mortgage requirements introduced in 2015; HFCS covers 20 countries since 2013 at three-year frequency and lacks loan performance information. - Fundamental drivers included: - State variables: debt, income, collateral value. - Shocks: idiosyncratic (demographic, unemployment) and systematic (income, real estate, interest rate). - Parameters: transaction costs, foreclosure discount, risk premia. - Mortgage payment schedule: type of mortgage, repricing schedule, tenor. - Rationale for risk-vintage simulation: - Controls for life-cycle effects, changes in lending standards across the cycle, and regulatory adjustments across vintages. ### III. Macroprudential instruments and operational policy space - Instrument types: - Bank capital requirements (e.g., sectoral CCyB) applied to outstanding portfolios. - Borrower-based tools (caps on LTVs, limits on DSTIs, amortization requirements) applied to new loan production. - Operational distinction: - CCyB increases tK to absorb loan losses on outstanding portfolio. - Borrower-based limits affect new loan production; calibration uses probability thresholds (λ) and quantiles (p). Hard limits set λ = 0; speed limits allow small λ (example interpretation preserved). ### IV. Baseline semi-structural mortgage-default model (concept and key equations) - Conceptual basis: "ability to pay" — default driven by borrower capacity to service payments. - Predictors: macroeconomic conditions, loan characteristics, borrower idiosyncratic shocks, regulatory environment. - Two default triggers: - (1) Financial distress: Pr(FD_it) = Φ( ... function of DSTI_{i,t-1}, ΔDSTI, U_{t-1}, ΔU, D with parameters β1, β3, α, γ ... ). Non-linearities via γ and Φ_{DSTI^{-1}} capture threshold effects. - (2) Inability to prepay by selling collateral net of transaction costs C: P^~_{i,t} − C < NPV(L_{i,t}, r_{type,M,t}, r_{f,t}, T_{s,t}). - Loss given default (LGD): - LGD_{i,t} = max(0, NPV(…) − (1 − δ) × P^_{i,t+n} (1 + r_{f,t} + sstr)^{n} ). - Notes: transaction costs increase in crisis; foreclosure discount δ > 0; discount rate for foreclosed assets exceeds risk-free. - Affordability & collateral effects: - DSTI affects Pr(FD) (equation (1)); LTV affects prepayment constraint (equation (2)) and LGD (equation (4)). - In Switzerland, LTV enters distress through margin-call mechanics. ### V. Scenario construction and parameter calibration - Scenario approaches: - Worst-plausible-distribution search (Breuer and Summer (2018)) replaced relative-entropy plausibility with a DSGE-based plausibility measure. - Conditional "at-risk" approach: use GaR and HaR (Adrian et al. (2019)) to design Ω when conditioning on current conditions and macroprudential stance. - Benchmarking projected losses: - Compare EX under Ω and policy τ to crisis loss level L observed in crisis C; compare EX to existing banking resilience tK using policy risk tolerance tβ. - Calibration groups: - (1) Sensitivity parameters to macro drivers. - (2) Housing loan characteristics and market structure. - (3) Parameters matching current/past mortgage losses. - (4) Parameters reflecting macroprudential regulation. - Horizon: - Adverse scenario assumed to last two or three years for default realization; bank stress testing used two- or three-year horizons; Swiss bank stress test uses five-year horizon (2019–23) for collateral realization in simulations. ### VI. Swiss application — market structure, calibration, and mechanics - Market structure and exposures: - Accumulated real estate price growth 1999–2018: 50 percent (investment-led), 75 percent (owner-occupied). - Share of new mortgages where imputed costs > one-third of income at 5 percent interest: about 50 percent. - Mortgage loans ≈ three quarters of Swiss banks’ lending portfolio (parent level). - Shares for domestically focused banks (DFBs): Raiffeisen and regional/savings banks 95 percent; cantonal banks 87 percent. - Aggregate loss rates on mortgage claims recent three years: about 10 basis points. - Mortgage types and maturities: - Fixed-rate: 75 percent; Money-market: about 15 percent; Redeemable adjustable-rate: 10 percent. Floating-share pooled: 25 percent. - Average maturity: ≈ five years (fixed-rate), ≈ three years (floating-rate). - Mortgage structure and policy timeline: - Many mortgages split into first mortgage (up to two-thirds; interest-only) and second mortgage (amortizing). - Margin-call option: banks may request margin call; cancellation period six months; discretionary execution. - Sectoral CCyB activated February 2013: initial 1 percent of risk-weighted assets; reset at 2 percent in 2014; reduced to 0 percent in March 2020. - FINMA/SBA self-regulation: from July 1, 2012, LTV on new mortgages must be reduced to at most two-thirds within at most 20 years; at least 10 percent hard equity required. June 2014 SBA revision shortened max amortization to 15 years for relevant vintages. - Amortization, repricing, margin call mechanics in simulations: - Mortgages issued after June 2014: amortize to two-thirds within max 15 years. - 2012–2014 vintages: second mortgage amortized within max 20 years. Pre-2012: 20-year rule. - Fixed-rate repricing 1–10 years; floating linked to one-year CHF libor; stressed market interest applied to non-maturing floating-rate mortgages as loans reprice annually. - Margin-call simulation: recalculate PiT LTV after realized house price; if amortization insufficient to meet two-thirds outstanding requirement over remaining maturity, bank tightens amortization schedule; this raises DSTI and Pr(FD), acting as amplification channel. ### VII. Swiss model calibration and stress-test results (core parameter values preserved) - Core parameter values (Swiss calibration from Table 2 preserved): - C 10% of house value - L 0.2 Ts - 훽1 4.1 - rf 0.05% - γ 2 - spread 2% - 훽2 0.06 - 훿 15% - 훽3 2.7 if 훥훥훥훥t>0 and 0 otherwise - Q 2 - 훼 1 - σP 15% - Scenario (Swiss adverse two-year key cumulative changes from Table 1): - Real disposable income: Baseline 3.6% ; Adverse -4.4% - Real house price level: Baseline 0% ; Adverse -25.4% - Unemployment rate: Baseline -0.11% ; Adverse 1.4% - Mortgage rate: Baseline 1.25% ; Adverse 3.75% - Additional outcomes: deviation of real GDP from baseline by 7.7 percent in 2020; 3.3 standard deviation move in two-year cumulative real GDP growth; peak-to-trough real estate price decline 25.4 percent during 2018–20 with further 8.6 percent correction by 2023; recession with GDP growth negative for seven quarters and peak decline of -4.75 percent in 2019. - Simulation implementation (per bucket): - Number of borrower draws per bucket, N = 2000. - Number of simulation iterations per bucket, K = 10,000. - Bucket-specific PD = (number of draws where expected sales price < loan NPV) / N × bucket-specific probability of financial distress from equation (1). - Bucket-specific LGD = average LGDs among defaulted loans. - Final bucket PDs and LGDs = averages across K iterations; portfolio-wide PD, LGD, loss rate = weighted combination by portfolio shares. - Swiss stress-test results (baseline LTV-vintage specification): - Average annualized default rate of aggregate mortgage portfolio: 4.2 percent in 2019–20. - Average LGD: 32.3 percent. - Loss rate of mortgage claims: 1.77 percent. - Banking system CET1 ratio decline: 135 basis points. - Stressed cumulative default rate over two-year stress: 12.1 percent. - Median peak NPL ratio across systemic banking crises in advanced countries (1970–2017): about 13 percent. - Default rate implied in the 1.1 percent peak NPL ratio for Switzerland during global financial crisis: 0.55 percent. - Comparison: projected EX (4.2 percent) exceeds reference 0.55 percent (notation preserved). - Sensitivity and robustness: - Real estate price severe test: cumulative 40 percent decline by 2020 → loss rates increase to 2.3 percent; CET1 depletion 173 basis points. - Interest-rate sensitivity: nonlinear/exponential relations; example equivalence: 2.3 percent default rate reachable by either (i) lending rates to 6 percent + moderate real estate price shock 25.4 percent, or (ii) lending rate 3 percent + abrupt price correction 40 percent. - Vintage assumption alternative (origination LTV distribution equal across vintages) increases mortgage loss rates from 1.77 percent to 2.33 percent; PDs from 4.22 percent to 5.25 percent; adds 43 basis points to CET1 depletion. - Margin-call deactivation: if banks do not execute margin calls, average loss rate decreases from 1.77 percent to 1.10 percent; aggregate capital increases by 50 basis points. ### VIII. Swiss model validation and historical replication - Historical magnitudes motivating calibration: - Unemployment increased by 60 percent during past severe episodes. - House prices fell by about 30 percent over three years. - Stock of mortgage arrears of six months or more increased by a maximum of 600 percent. - Loan write-offs increased between one hundred and two hundred times. - Replication of early 1990s Swiss event: - Historical: policy rate rose from 6.0 percent in 1989 to 7.8 percent in 1991; real estate prices fell 16.4 percent peak-to-trough; unemployment increased by 52 basis points by 1991; mortgage portfolio loss rate increased from 0.53 percent to 1.03 percent (estimate). - Model simulated 1991 recession with inputs as of 1989 and produced projected loss rate 1.05 percent (close to observed 1.03 percent). - Benign-conditions validation (1999 and 2018 comparisons): - 1999: simulated loss rate 0.53 percent vs observed 0.61 percent. - Baseline 2019–20: expected loss rate 0.06 percent; average "expected" loan loss rate reported by Swiss banks in 2018: 0.10 percent; incurred credit loss measure 0.04 percent. ### IX. Austria application — data, adverse scenario (GaR/HaR), calibration, and results - Austrian dataset and construction: - Covers about two-thirds of new housing loan issuances at quarterly frequency, back to 2011Q1–2018Q4. - Data include distributions by LTV, DSTI, DTI, joint LTV-DSTI, joint LTV-DTI, share of variable-rate loans, average maturity. - PiT LTVs and DSTIs of outstanding loans computed as of 2018Q4; absolute shares by LTV-vintage bucket derived using new mortgage flows and mortgage stocks from OeNB. - Mortgages originated before 2011 assumed similar risk characteristics as earliest vintages (2011Q1–2012Q4). - GaR/HaR adverse two-year calibration (Table 3 cumulative adverse changes): - Real disposable income: -2.5% - Real house price level: -11.0% - Unemployment rate: 1.9% - Mortgage rate: 1.4% - Median annualized quarterly GDP growth over 2019 median ≈ 2.1 percent; 5 percent likelihood GDP growth falls to -1.25 percent; median real house price growth ≈ 3.2 percent; 5th percentile house price growth -5.5 percent. - Robustness alternative: 20 percent real house price decline over two years considered. - Core Austria calibration parameters (Table 4 preserved): - Equation (1) parameters: - 휙(LLDTD) = 0 if LLDTD < 15% - = (LLDTD − 15%) / (30% − 15%) if LLDTD ∈ [15%, 30%] - = 1 if LLDTD > 30% - 퐿 0.2 - 훽1 0.0003 - γ 2.5 - 훽2 0.006 - 훽3 0.007 if 훥훥훥훥t >0 and 0 otherwise - Q 1.25 - 훼 1 - Equations (2)-(4) parameters: - C 5% of house value - Ts 25 - rf 0.1% - spread 2% - 훿 20% - σP 15% - Baseline timing: macroprudential tools introduced k = 8 quarters before adverse scenario; sensitivity with k = 4 increases PDs and portfolio losses somewhat but ordering of calibrations unchanged. - Absence of macroprudential limits — adverse scenario annualized results (percent): - System-wide mortgage portfolio: - 1-Year PD 1.9 - LGD 26.6 - Expected Annualized Losses 0.8 - New mortgages (originated in eight quarters before adverse): - 1-Year PD 3.9 - LGD 34.0 - Expected Annualized Losses 1.6 - Sensitivity analyses (selected exact results): - Accounting for borrower financial wealth: - System-wide: PD 1.6; LGD 28.5; Expected Annualized Loss 0.7 - New mortgages: PD 3.4; LGD 33.7; Expected Annualized Loss 1.5 - Higher interest rate increase (real increase 200 basis points): - System-wide: PD 3.4; LGD 32.0; Expected Annualized Loss 1.5 - New mortgages: PD 6.9; LGD 40.0; Expected Annualized Loss 3.1 - Larger house price decline (real decline 20 percent): - System-wide: PD 2.3; LGD 29.2; Expected Annualized Loss 0.9 - New mortgages: PD 4.5; LGD 37.7; Expected Annualized Loss 2.0 - Benchmarking: 20 percent real house price decline → annualized loss rate for new mortgages implies three-year cumulative NPL ratio 12.9 percent (assuming no write-offs/recoveries); matches median peak NPL across past crises (Laeven and Valencia (2018)). - Macroprudential limits on new mortgages — selected entries from Table 6 (exact preserved): - No limits: PD 3.9; LGD 34.0; Exp. Annual Loss 1.6 - Hard LTV = 80, no DSTI: PD 2.7; LGD 31.9; Exp. Annual Loss 1.0 - Hard DSTI = 40, no LTV: PD 3.2; LGD 31.8; Exp. Annual Loss 1.3 - Hard LTV = 80 & hard DSTI = 30: PD 1.3; LGD 31.8; Exp. Annual Loss 0.5 - Hard LTV = 80 & hard DSTI = 40: PD 2.2; LGD 31.8; Exp. Annual Loss 0.8 - Hard LTV = 90 & hard DSTI = 40: PD 2.7; LGD 32.1; Exp. Annual Loss 1.0 - Speed limit (20%) none: PD 3.3; LGD 32.9; Exp. Annual Loss 1.3 - Speed limit (20%) LTV = 80, DSTI = 40: PD 4.0; LGD 34.1; Exp. Annual Loss 1.6 - Speed limit (20%) LTV = 80, DSTI = 30: PD 2.3; LGD 32.9; Exp. Annual Loss 0.9 - Speed limit (20%) LTV = 80, DSTI = 40: PD 3.3; LGD 32.9; Exp. Annual Loss 1.3 - Speed limit (20%) LTV = 90, DSTI = 40: PD 3.7; LGD 33.4; Exp. Annual Loss 1.5 - Key observations (Austria): - Hard LTV-DSTI = 80-40 yields annualized loss 0.8 percent on new mortgages; speed limit 20% with LTV-DSTI = 80-30 yields 0.9 percent — similar outcomes. - Standalone DSTI = 40 is relatively ineffective: hard DSTI = 40 reduces PD from 3.9 to 3.2 and annualized losses from 1.6 to 1.3 (modest). - Historical low lending rates depressed DSTIs of new loans; system-wide average DSTI for new flows somewhat below 25 percent in 2018. - Appropriately calibrated speed limits can approximate hard limits while allowing flexibility. - Rule-of-thumb selection guidance: - Compare expected losses on new mortgages (subject to limits) with losses on older vintages; select calibration that makes new-loan loss rate similar to reference portfolio. - Austria example: annualized loss on total portfolio = 0.8 percent; calibrations achieving similar rate for new loans: LTV-DSTI = 80-30 with speed limit 20 percent, or hard LTV-DSTI = 80-40. - Formal objective: minimize Pr{...} comparing TnewX to TrefX (exact expression preserved in source). ### X. Methodologies for vintage flows, PiT LTV distribution, and DSTI/LTV computations (implementation details) - Vintage flow approximation (Swiss procedure): - Compute average inflow rate and portfolio growth using SNB data to back out stock at t−1; apply inflow rate to obtain new mortgage volumes by vintage; split initial mortgage into first (interest-only) and second (fully amortized) using share α; back out α by matching outstanding stock by vintage to reported outstanding stock in 2018. - Caveats: recreated first/second distribution may reflect staggered refinancing; data exclude indirect amortization (Pillar 3) that do not reduce outstanding mortgage amount. - Alternative LTV-vintage distribution approach (Appendix II): - Solve for issuance distribution {α̂b} that, after amortization and house-price revaluation, generates observed PiT LTV shares {βb} at time t. - PiT LTV for a loan computed from origination LTV, amortization schedule A, and real estate price growth g. - Aggregation: β̂_{t,b} = Σ_s Σ_k γ_s α̂_{s,k} I_{PiT_{s,k,b}}, solve system {βb} = function({α̂b}). - Point-in-time loan value, LTVs, DSTIs as of 2018Q4 (Appendix III equations preserved): - Outstanding value of loan in bucket h as of 2018Q4: L_{h,2018Q4} = Σ_i LTN_{h,i} × P_{i} × T_{h,2018Q4}/T_h × P_t (equation A.2; notation preserved). - Borrower income at origination from DTI at origination: DTI_income_{s,h,orig} = L_{h,orig} / DTI_{h,orig} (A.3). - Borrower income as of 2018Q4: DTI_income_{s,h,2018Q4} = DTI_income_{s,h,orig} × (1 + g_{orig:2018Q4,cum}) (A.4), where g_{orig:2018Q4,cum} uses OECD cumulative real household income growth. - Outstanding principal used to compute LTVs: LTP_{h,2018Q4} = L_{h,2018Q4} / P_t (A.5). - Debt-service for DSTI as of 2018Q4: LDT_{h,2018Q4} = Σ_i LTN_{h,i} × P_i × [1/T_h + r × T_h] × Dnct_{h,2018Q4} (A.6) (notation and formula components preserved). - Implementation notes: - Average maturity at origination T_h used in computations. - Quarterly average nationwide house prices used for outstanding value. - LTVs and DSTIs computed per bucket using above formulas and OECD income growth scaling. ### XI. Appendix I — penalty for early prepayment (key formula structure) - Net present value of prepayment (equation A.1) decomposes the lender’s foregone interest into: - Geometric sum component (A). - Arithmetic–geometric sequence sum component (B). - Final algebraic expression for loan NPV combines these components (exact algebra preserved in source). - Assumptions: - Linear repayment schedule during remaining maturity T_{i,t}. - Lender reinvests recovered principal at r_f; foregone return deducted from amount owed. ### XII. Main policy-relevant findings and implications (preserving quantitative magnitudes) - Swiss findings: - Baseline adverse stress (2019–20) → average annualized default rate 4.2 percent; LGD 32.3 percent; loss rate 1.77 percent; CET1 decline 135 basis points. - Sectoral CCyB at 2 percent would be exhausted under this stress; 135 basis points equals 4.2 times CCyB; unexpected losses exceed CCyB by 3.8 times after netting average provisions. - Margin-call activation materially amplifies losses: deactivating margin calls reduces average loss rate from 1.77 percent to 1.10 percent (aggregate capital +50 basis points). - Recalibrating amortization (shorter amortization from 20 to 10 years for post-2014 vintages) increases overall loss rate from 1.77 percent to 2.45 percent and adds ≈ 52 basis points to capital depletion; short amortization tightens affordability raising Pr(FD) notably for high LTI loans (e.g., LTI > 7: 46.8% → 66.5% for new mortgages). - Austria findings: - Without limits (adverse two-year GaR/HaR): system PD 1.9; LGD 26.6; expected annualized losses 0.8; new mortgages PD 3.9; LGD 34.0; expected annualized losses 1.6. - Accounting for household financial wealth reduces expected annualized losses by about 8 percent for total portfolio (0.8 → 0.7) and by 6 percent for new mortgages (1.6 → 1.5). - High share of variable-rate new loans (45 percent) makes interest-rate shocks particularly damaging: real interest rate increase of 200 basis points → new mortgages expected annualized loss 3.1. - Effective calibrations: combined LTV-DSTI limits (e.g., hard 80-40 or speed-limit 80-30 with 20% speed) can reduce expected annualized losses on new mortgages to levels similar to reference portfolio (0.8–0.9). - Standalone DSTI = 40 has modest effect given low DSTI of new loans (average DSTI < 25 percent in 2018). - Cross-cutting implications: - Semi-structural model is suitable where micro data scarcity or benign recent cycles hinder statistical methods. - Policy calibration should weigh general-equilibrium and second-round effects; rule-of-thumb: align expected losses on new loan production with reference (“old”) portfolio losses. - Importance of collecting granular vintage disclosures and risk-bucket regulatory data emphasized to improve stress testing and macroprudential calibration. - Model calibrated on pre-COVID information; application to COVID-19 shocks and pandemic-related policy effectiveness left for future work. _Italic: Source — wpiea2020165-print-pdf (IMF working paper excerpt provided in the prompt)_ ### REFERENCES _________________________________________________________________ 49 ### wpiea2020165-print-pdf - REFERENCES _________________________________________________________________ 49 ### Figures - Figure 1. Bank Mortgage Portfolio and Real Estate Prices in Switzerland ____________________ 20 - Figure 2. Stress Test Scenario Paths for Real Estate Prices in Switzerland ____________________ 23 - Figure 3. Nonlinear Effects of Financial Distress and Negative Equity as a Function of LTI, LTV Buckets, and Vintages ____________________________________________________________ 29 - Figure 4. Stress Test Results in Switzerland ___________________________________________ 31 - Figure 5. Sensitivity Tests in Switzerland _____________________________________________ 32 - Figure 6. Robustness Exercise: Expected Loss (EL) Rate and PD by Assumption (in percent) ____ 33 - Figure 7. Mortgage Loss Rates and Model Backtesting in Switzerland_______________________ 35 - Figure 8. House Prices in Austria ____________________________________________________ 38 - Figure 9. GaR and HaR Estimates for Austria __________________________________________ 39 ### Tables - Table 1. Key Macroeconomic Assumptions in the Stress Scenario for Switzerland _____________ 22 - Table 2. Model Calibration for Switzerland ____________________________________________ 28 - Table 3. Key Macroeconomic Assumptions in the Adverse Scenario for Austria _______________ 40 - Table 4. Model Calibration for Austria _______________________________________________ 43 - Table 5. Losses on Mortgage Portfolios in the Absence of Macroprudential Limits (Percent) _____ 44 - Table 6. Losses on new mortgages in the presence of macroprudential limits (Percent) __________ 45 ### Appendices - Appendix I. Calculation of Penalty for Early Prepayment _________________________________ 52 - Appendix II. Calculation of LTV Distribution under Alternative Approach in Section IV ________ 53 *https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020165-print-pdf.pdf* ### Appendix III. Calculation of Point-in-Time Loan Value, LTVs, and DSTIs in Section V ________ 54 ### Appendix III. Calculation of Point-in-Time Loan Value, LTVs, and DSTIs in Section V ### I. Objectives and conceptual framing - Two main objectives of the paper: - Provide a conceptual framework of systemic risk and risk resilience to operationalize the macroprudential policy goal linked to readily available metrics and assess policy success by benchmarking projected stressed credit losses against peak NPLs during a systemic banking crisis (Laeven and Valencia (2018)). - Apply the framework to mortgage loan markets in two countries (Switzerland and Austria) to assess appropriateness of the macroprudential policy stance and quantify necessary adjustments in calibration of macroprudential instruments. - Key elements of the policy framing: - Characterize macroprudential policy as minimizing a loss function that would erode a significant fraction of banking system capital conditional on a distribution of risk factors. - Use stress-testing via a semi-structural mortgage-default model to produce forward-looking measures of banking system resilience and to guide calibration of borrower-based tools and sectoral capital buffers. ### II. Data approach and drivers of mortgage risk - Data strategy: - Use banks’ portfolio/regulatory reporting data segmented by risk buckets (risk-vintage buckets) rather than micro-level borrower data to simulate default rates and expected losses. - Regulatory reporting data is typically more readily available and aligned with supervisory risk buckets; PSD performance data (PSD007) mortgage requirements were introduced in 2015 but had to be reported for all live regulated mortgage contracts, including those opened before 2015. - Limitations of micro datasets: HFCS covers 20 countries, is available since 2013 at a three-year frequency, and lacks loan performance information. - Fundamental drivers included in the model: - State variables: debt, income, collateral value. - Shocks: idiosyncratic (demographic, unemployment) and systematic (income, real estate, interest rate). - Parameters: transaction costs, foreclosure discount, risk premia. - Mortgage payment schedule: type of mortgage, repricing schedule, tenor. - Rationale for risk-vintage simulation: - Controls for life-cycle effects (LTV decreases over time via repayments and house price appreciation). - Controls for changes in lending standards across the cycle (e.g., an 80 percent LTV issued when prices are low is tighter than one issued when prices are high). - Allows incorporation of regulatory adjustments that affect loan size and amortization schedules across vintages. ### III. Macroprudential instruments and policy space - Two broad instrument types considered: - Bank capital requirements (for example, sectoral CCyB) to absorb loan losses on the outstanding portfolio. - Borrower-based tools (caps on LTVs, limits on DSTIs, amortization requirements) that apply to new loan production. - Operational distinction: - CCyB applies to outstanding portfolios (increasing tK reduces net systemic risk). - Borrower-based limits affect new loan production; calibration requires quantifying riskiness of new production relative to a reference portfolio and setting probability thresholds (λ) and quantiles (p) for allowed breaches (hard limits vs. speed limits). ### IV. Country applications — Switzerland and Austria (motivation and salient differences) - Switzerland: - Experienced a real estate bubble in the 1980s that burst in the early 1990s, providing a stress episode for calibration and backtesting. - Binding macroprudential policies activated starting in 2012; sectoral CCyB first introduced in 2013 and reassessed in 2014. - Loans tend to have short tenors (from two to 10 years) and are typically rolled over. - Model enhancements account for regulatory changes with differential impacts by vintage (e.g., margin calls affecting DSTI). - Counterfactual analysis used to test sufficiency of the CCyB and sensitivity to amortization requirements. - Austria: - Has not experienced a sharp house price correction historically, but evidence of increasing overvaluation in recent years. - High share of variable-rate loans (debt servicing sensitive to increases in lending rates). - Typical maturity of loans is long (between 25 and 30 years). - In 2017 authorities added borrower-based tools (LTV, DSTI, DTI limits); these tools had not been activated as of the paper. - In September 2018, Austrian authorities published a guide on sustainable lending standards in real estate financing. - Model used to simulate forward-looking portfolio losses under stress and assess effectiveness of different calibrations of borrower-based instruments if activated. ### V. Loss modeling framework and stress-scenario design - Two alternative approaches to define stress scenario distribution Ω: - Worst-plausible-distribution search (Breuer and Summer (2018) formulation): sup over Ω of EX conditional on a distance D(Ω, Ψ) measure of plausibility relative to historical distribution Ψ. The paper replaces relative-entropy plausibility with a DSGE-based plausibility measure. - Conditional "at-risk" approach: use conditional distribution of growth at risk (GaR) and house price at risk (HaR) to design Ω (Adrian et al. (2019) approach), appropriate when conditioning on current economic/financial conditions and macroprudential stance. - Benchmarking projected losses: - Compare expected cumulative losses EX under Ω and policy τ to a crisis loss level L observed in a relevant crisis C: (EX) ≥ L. - Compare EX to existing banking resilience tK using policy risk tolerance parameter tβ: EX −> t K t β (notation in source; assessment determines whether policy stance is non-neutral). - Guidance for borrower-based tool calibration: - Operationalized via minimization: - min_{B,p,λ} Pr(new_t,L_new > L_ref) ≤ λ subject to choice of borrower-based tool B and quantile p. - Interpretation: hard limits set λ = 0; speed limits allow small λ (e.g., an LTV limit of 80 percent with a speed limit of 10 percent means max 10 percent of new loans can have LTV > 80 percent). ### VI. Baseline semi-structural mortgage-default model (concept and key equations) - Conceptual basis: "ability to pay" principle — default driven by borrower capacity to service payments. - Four categories of predictors: - Macroeconomic conditions: interest rates, aggregate income, real estate valuations. - Loan characteristics: type, tenor, rollover rate, size relative to collateral. - Borrower characteristics: idiosyncratic income shocks, demographic shocks. - Regulatory environment: minimum down payment, affordability caps, amortization requirements. - Default modeled as satisfying two conditions (Harrison and Mathew (2008) base, augmented here): - (1) Financial distress (liquidity inability to service loan): probability of financial distress for borrower i in period t is specified as equation (1) in the source: - Pr(FD_it) = Φ( ... function of DSTI_{i,t-1}, ΔDSTI, U_{t-1}, ΔU, D with parameters β1, β3, α, γ ... ) - The function Φ and non-linearities (parameter γ, function Φ_{DSTI^{-1}}) are used to capture threshold and non-linear DSTI impacts and idiosyncratic shock non-linearities. - (2) Borrower cannot prepay by selling collateral net of transaction costs C: P^~_{i,t} − C < NPV(L_{i,t}, r_{type,M,t}, r_{f,t}, T_{s,t}) (equation (2) in source). - NPV includes outstanding loan and penalty for early prepayment approximated as net present value of foregone future interest payments (equation (3) in source). - Loss given default (LGD): - Bank loss when default occurs and collateral sold at time t+n with foreclosure discount δ and sale price P^_{i,t+n} discounted at risk premium sstr: - LGD_{i,t} = max(0, NPV(…) − (1 − δ) × P^_{i,t+n} (1 + r_{f,t} + sstr)^{n} ) (equation (4) in source). - Notes: transaction costs tend to increase in crisis; discount rate for foreclosed assets exceeds risk-free rate. - Key affordability metrics embedded: - DSTI affects probability of financial distress in equation (1). - LTV affects the ability to prepay (equation (2)) and bank losses upon default (equation (4)). - In Switzerland, LTV also enters the distress equation through margin-call mechanics. ### VII. Scenario construction and parameter calibration - Scenario horizon and motivation: - Adverse scenario assumed to last two or three years to reflect time needed for adverse shocks to materialize and trigger defaults. - For bank stress testing (Section IV), scenarios are designed to be sufficiently severe but plausible for Swiss banks. - For policymaker calibration of borrower-based tools (Section V), the adverse scenario is calibrated to reflect current cyclical downside risks using GaR and HaR methods. - Parameter groups in calibration: - (1) Sensitivity parameters of default and losses to macro drivers (unemployment, lending rates, house prices). - (2) Parameters reflecting housing loan characteristics and market structure. - (3) Parameters calibrated to match current or past levels of banks’ mortgage losses. - (4) Parameters reflecting the macroprudential regulatory framework. - Specific calibration notes: - The function Φ_{DSTI^{-1}} and parameters β1, β3, α, γ in equation (1) are calibrated to past loan distress episodes (Harrison and Mathew (2008) calibrated to UK early-1990s housing loan loss experience when interest rates rose by over [text truncated in source at end of excerpt]). - Nonlinearities: γ allows non-linear impact of ΔDSTI and ΔU; DSTI threshold function makes idiosyncratic shocks ineffective below a DSTI threshold and progressively effective above it. - Model flexibility: - Can incorporate borrowers’ financial assets (sensitivity analysis in Section V reforms equation (2) to account for Austrian households’ financial assets). - Can be adjusted to reflect country-specific early prepayment penalty rules and non-recourse vs. full-recourse loan regimes. *Source: Appendix III. Calculation of Point-in-Time Loan Value, LTVs, and DSTIs in Section V — wpiea2020165-print-pdf* ### 3.5 percentage points, unemployment increased by 60 percent, and house prices fell by about ### wpiea2020165-print-pdf - 3.5 percentage points, unemployment increased by 60 percent, and house prices fell by about ### Scenario and key macroeconomic assumptions - Stress horizon: five years, 2019–23; model produces default rates over 2019–20 but requires longer horizon for collateral realization. - Scenario trigger: global financial cycle downturn with abrupt repricing of risk premia; calibrated using IMF’s Global Macrofinancial Model (GFM). - Key two-year cumulative changes (Table 1): - Real disposable income: 3.6% (Baseline); -4.4% (Adverse) - Real house price level: 0% (Baseline); -25.4% (Adverse) - Unemployment rate: -0.11% (Baseline); 1.4% (Adverse) - Mortgage rate: 1.25% (Baseline); 3.75% (Adverse) - Additional scenario outcomes: - Deviation of real GDP from baseline by 7.7 percent in 2020. - A 3.3 standard deviation move in two-year cumulative real GDP growth rate. - Peak-to-trough cumulative real estate price decline of 25.4 percent during 2018–20, with a further 8.6 percent price correction by 2023. - Macrofinancial stress triggers a recession with GDP growth negative for seven quarters and a peak decline of -4.75 percent in 2019. ### Simulation of portfolio losses and numerical implementation - Portfolio segmentation: vintage (origination year/quarter), LTI bucket, and LTV bucket (alternative bucketing possible depending on data). - Simulation process per vintage-LTI-LTV bucket: - Number of borrower draws per bucket, N = 2000. - Number of simulation iterations per bucket, K = 10,000. - For each of the N draws, generate house price realization with mean at scenario house price level; check default condition (condition (2)); calculate LGD from equation (4) if defaulted. - Bucket-specific PD = (number of draws where expected sales price < loan NPV) / N × bucket-specific probability of financial distress from equation (1). - Bucket-specific LGD = average LGDs among defaulted loans. - Final bucket PDs and LGDs = averages across K iterations. - Portfolio-wide PD, LGD, and loss rate = weighted combination of vintage-LTI-LTV bucket outputs by portfolio shares. - Use of risk-bucket level average DSTI instead of borrower-specific DSTIs due to data availability; micro-level data noted as preferable but often unavailable at required frequency. ### Model calibration, parameters, and structural features - Non-linear default dependence on DSTI: function Φ(DSTI_{t-1}) calibrated to reflect that default probability rises substantially above DSTI threshold of 50 percent (as documented in Nier et al. (2019)). - Country-specific housing market parameters: - Cost parameter C (equation (2)) can reflect average taxes and notary fees. - Discount parameter 훿: Harrison and Mathew (2008) estimate ≈ 15 percent for New Zealand; Pennington (2003) estimates ≈ 25 percent for the US. - Time to realize sale of collateral, n: Harrison and Mathew (2008) estimate 1.25 years in New Zealand. - Demographic parameters D and 훽2 set to match average bank loan loss rates on mortgage claims in “normal” times. - Regulatory framework effects incorporated: required amortization schedule changes in Switzerland in 2012 and 2014 affect affordability risk and remaining time to maturity T_{s,t} across vintages. - Calibration example: model parameters and other inputs are shown calibrated to capture the Swiss real estate crisis in the early 1990s (Section IV). ### Swiss mortgage market structure and regulatory environment - Mortgage market exposures and price growth: - Accumulated real estate price growth over 1999–2018: 50 percent in investment-led segment; 75 percent for owner-occupied property. - Share of new mortgages where imputed costs would exceed one-third of income at an interest rate of 5 percent: about 50 percent. - Mortgage loans account for about three quarters of Swiss banks’ lending portfolio (parent level). - Share for domestically focused banks (DFBs): 95 percent for Raiffeisen and regional and savings banks; 87 percent for cantonal banks. - Aggregate loss rates on mortgage claims in recent three years: about 10 basis points, leading to low starting portfolio PDs. - Mortgage types and distribution: - Fixed-rate mortgages: 75 percent of outstanding mortgage volume. - Money-market mortgages: about 15 percent. - Redeemable adjustable-rate mortgages: 10 percent. - For analysis, money-market and redeemable categories pooled and labeled floating-rate; overall floating share used in examples: 25 percent. - Average maturity: about five years for fixed-rate mortgages; about three years for floating-rate mortgages. - Mortgage structure under Swiss rules: - Many mortgages split into first mortgage (up to two-thirds of property value; interest-only, non-amortizing) and second mortgage (amortizing). - Under current tax system, leverage is tax-advantaged; most mortgages are not fully amortized. - Margin call option: banks may request margin call if collateral value decline leads to violation of amortization requirements; cancellation period six months but no automatic execution mechanism, implying discretion and possible amplification via fire sales. - Regulatory timeline and requirements: - Switzerland activated sectoral CCyB in February 2013, applied to entire domestic residential mortgage book. - Initial CCyB calibration: 1 percent of risk-weighted assets. - Sectoral CCyB reset at 2 percent of risk-weighted assets in 2014. - As reaction to the Covid pandemic, sectoral CCyB reduced to 0 percent in March 2020. - July 1, 2012 FINMA declaration made SBA’s self-regulation a minimum standard for all banks: - LTV on new mortgages must be reduced to at most two-thirds within at most 20 years. - Homebuyers required at least 10 percent of house value as “hard equity” (own funds excluding pension assets). - June 2014 SBA revision shortened maximum amortization period from 20 to 15 years for relevant vintages. - Typical owner-occupied LTV is below 80 percent despite no quantitative cap. - Data composition and limitations: - Supervisory residential mortgage data used for the full real-estate-backed loan portfolio due to commercial real estate data quality concerns. - Real estate-backed loan portfolio composition: about 80 percent residential (of which 50 percent is owner-occupied), 20 percent commercial real estate (CRE). - Supervisory templates: distribution of mortgages by LTV and LTI ratios (FINMA’s Building Block Analysis templates) as of end-2018 applied across banks. ### Model mechanics for Swiss application: amortization, repricing, margin calls - Amortization rules by origination year: - Mortgages issued after June 2014: amortize to two-thirds of collateral value within maximum 15 years. - Mortgages between 2012 and 2014: second mortgage amortized within a maximum of 20 years. - Mortgages issued before 2012: assumed to follow 20-year amortization rule. - Repricing and rollover: - Fixed-rate mortgages: repricing schedule ranges 1–10 years; rolled over at maturity to prevailing market rates consistent with self-regulation. - Floating-rate mortgages: linked to one-year CHF libor; stressed market interest applied to all non-maturing floating-rate mortgages as loans reprice annually. - Rollover calculation: assume mortgages rolled over at maturity to match original contract maturity at prevailing market rates; number of rollover periods depends on origination year and initial maturity. - Margin call mechanism in simulations: - For each simulated loan draw, recalculate LTV using realized house price. - If initial amortization is insufficient to repay the outstanding loan in excess of two-thirds of collateral over remaining maturity, bank applies margin call tightening amortization schedule; otherwise initial schedule applies. - Updated amortization under margin call can increase DSTI and thereby raise default probability; margin call can act as an amplification channel when rates rise and house prices decline simultaneously. ### Empirical and calibration notes, limitations - Use of risk-bucket average DSTI is a simplification that abstracts from borrower-level heterogeneity; micro-level borrower-loan data would enhance realism but are often unavailable or infrequent. - Supervisory data limitations: lack of vintage disclosure in Swiss supervisory stocks as of end-2018 required reconstruction of vintages (focus on vintages reconstructed back to 1999 for analysis). - Statistical approaches based solely on historical regressions may be hindered by benign recent cycle (aggregate loss rates ≈ 10 basis points) and structural changes; semi-structural model offers advantages for stress testing given scenario-driven tail behavior and regulatory changes. ### Findings and observed historical tail-event magnitudes (from empirical evidence cited) - During past severe episodes noted in the text: - Unemployment increased by 60 percent. - House prices fell by about 30 percent over a period of three years. - Stock of mortgage arrears of six months or more increased by a maximum of 600 percent. - Loan write-offs increased between one hundred and two hundred times. - These historical magnitudes motivate calibration choices and functional forms to capture non-linear default responses. _Italic: Source — wpiea2020165-print-pdf (IMF working paper excerpt provided in the prompt)_ ### 2018. We approximate vintage flows using the following procedure. ### wpiea2020165-print-pdf - 2018. We approximate vintage flows using the following procedure. ### Methodology: vintage flows, LTV-vintage distribution, and portfolio segmentation - Vintage flow approximation: - Compute average inflow rate and portfolio growth using Swiss National Bank (SNB) data to back out stock at time t-1. - Apply the inflow rate for each period to obtain the volume of new mortgages issued in past vintages. - Compute outstanding balance of each vintage at end-2018 by splitting the initial mortgage into first mortgage (interest payments only) and second mortgage (fully amortized). - Assume a share α is the first mortgage and (1- α) is the second mortgage; apply amortization rule at origination to the second mortgage; back out α by matching calculated outstanding stock by vintage to the reported outstanding stock in 2018. - Caveats: recreated distribution of first/second mortgages may reflect staggered refinancing rather than original tenor; data exclude material “indirect amortization” flows (Pillar 3 accounts) that do not reduce outstanding mortgage amount. - LTV-vintage distribution: - Baseline: assume the same PiT distribution of LTV for each vintage (PiT = point-in-time). - Alternative (robustness): assume same distribution of LTV shares at origination (rather than PiT); this is more conservative (results in higher losses) because it generates higher PiT-to-LTV ratios in recent vintages that are riskier given lower lending rates and rising real estate valuations. - Portfolio segmentation and outputs: - Segment portfolio by LTV-vintage buckets. - Model calculates PDs, LGDs, and expected loss rates under the stress scenario by combining outputs for each risk bucket weighted by their share in the bank’s outstanding portfolio. - Nonlinearity noted: - Probability of financial distress across LTI-vintage buckets and probability of economic default across LTV-vintage buckets are highly nonlinear (referenced Figures showing heat maps). ### Model calibration (Switzerland) - Calibration target: - Recalibrate model to match historical loan losses on mortgage claims observed during the early 1990s Swiss real estate crisis. - Historical context (1989–91): - Policy rate: increased from 6.0 percent in 1989 to 7.8 percent in 1991. - Real estate prices: fell 16.4 percent peak-to-trough. - Unemployment: increased by 52 basis points by 1991. - Estimated mortgage portfolio loss rate: increased from 0.53 percent to 1.03 percent (subject to uncertainty). - Aggregate default/distress calibration procedure: - Generate aggregate rate of financial distress by dividing average loss rate over 1990–92 by average empirical LGD to obtain aggregate default rate. - Use average simulated share of loans with negative equity to back out portfolio share in distress (Pr[FFLD i,t ] in equation (1)). - Allocate share of financial distress to DSTI tightening (80 percent) and to rise in unemployment (20 percent) following Harrison and Mathew (2008). - Calibrate DSTI parameters using rise in lending rates during Swiss crisis and estimate 75 percent share of interest payments in the DSTI ratio in 1989. - Calibrate Φt−1DSTI() and γ in equation (1); calibrate D and β2 to match observed loss rate as of end-2018; validate across housing cycle. - Core parameter values (as presented in Table 2; values preserved exactly): - C 10% of house value - L 0.2 Ts - 훽1 4.1 - rf 0.05% - γ 2 - spread 2% - 훽2 0.06 - 훿 15% - 훽3 2.7 if 훥훥훥훥t>0 and 0 otherwise - Q 2 - 훼 1 - 휎P 15% - Note: phi(LDTD) functional form shown in table (preserved as in source). ### Stress test results (baseline LTV-vintage specification) and key statistics - Aggregate mortgage portfolio projections under baseline LTV-vintage distribution: - Average annualized default rate of the aggregate mortgage portfolio reaches 4.2 percent in 2019–20. - Average LGD: 32.3 percent. - Loss rate of mortgage claims: 1.77 percent. - Banking system CET1 ratio decline: 135 basis points. - Comparison to historical/systemic crisis metrics: - Stressed cumulative default rate projected over a two-year stress period: 12.1 percent. - Median peak NPL ratio across systemic banking crises in advanced countries (1970–2017): about 13 percent. - Default rate implied in the 1.1 percent peak NPL ratio for Switzerland during the global financial crisis (Laeven and Valencia 2018): 0.55 percent. - In notation of Section II, ( ),,t TC EXL τΩ ≥ , where LHS at 4.2 exceeds RHS value of 0.55 percent. - Default concentration by vintage and LTV: - For most recent vintages, loss rates range between 2.7 percent and 8.8 percent. - For oldest vintages: - Default rates range between 0.01 percent for mortgages with LTV ratios below 60 percent and 0.14 percent for mortgages exceeding 90 percent LTV limits. - Projection beyond immediate stress (2021–23): - Despite assumed V-shape recovery, loss rates remain material due to deterioration in borrowers’ balance sheets. - Model capability note: - Semi-structural approach can generate higher default rates than regression-based satellite models because it controls for borrower-level state variables affecting creditworthiness. ### Sensitivity analysis: real estate prices and interest rates - Real estate price sensitivity: - Test: same adverse paths for real disposable income and unemployment but milder interest rate shock (175-basis point shift in five-year mortgage rate to 3 percent); vary decline in real estate prices. - Loss rate rises exponentially with real estate price shock due to nonlinear impact on liquidity condition (equation 1) and negative equity condition (equation 2). - Example: cumulative decline in real estate prices by 40 percent by 2020 leads to: - Increase in loss rates to 2.3 percent. - CET1 capital depletion of 173 basis points. - Interest-rate sensitivity: - Test: same paths for income, unemployment, and real estate prices as adverse scenario in Table 1 but vary lending rate shocks. - Relationship between default rates and interest rates is exponential due to nonlinear entries in financial distress and economic default equations (penalty for early mortgage termination). - Example equivalence: a 2.3 percent default rate can be achieved by either: - Sharp increase in lending rates to 6 percent and a moderate real estate price shock of 25.4 percent; or - Softer lending rate of 3 percent but an abrupt real estate price correction of 40 percent. ### Robustness tests: vintage assumptions and margin calls - Vintage LTV distribution alternative: - Alternative assumption: equal distribution of LTV shares at origination (versus equal PiT LTV shares across vintages baseline). - Rationale: declining lending rates (from 3.9 percent in 1999 to 1.24 percent in 2018 for a five-year fixed-rate mortgage) and cumulative real estate price increase of 55 percent in 1999–2018 make recent vintages relatively more risky. - Results (Figure 6 confirmation): - Mortgage loss rates increase from 1.77 percent to 2.33 percent. - PDs increase from 4.22 percent to 5.25 percent. - Adds 43 basis points to CET1 capital depletion. - Interpretation: even with same underwriting at origination, matching outstanding stock to current PiT LTV distributions generates higher PiT LTV ratios for more recent vintages, increasing sensitivity to price and rate shocks. - Activation of margin calls: - If Swiss banks decide not to execute margin calls: - Average loss rate decreases from 1.77 percent to 1.10 percent. - Aggregate capital increases by 50 basis points. - Margin calls act as an amplification mechanism: activation can fuel a downward spiral as LTVs are revalued, triggering new rounds of margin calls and potential fire-sale effects. ### Model validation and out-of-sample replication - Validation approach: - Assess model’s ability to replicate three episodes: “bad times” (early 1990s), “good times” (late 1990s recovery), and “benign conditions” (baseline 2019–20). - Caveats: no historical data on mortgage portfolio structure by LTV-vintage buckets; assume past distribution resembles current distribution; historical loss rates are not directly observed and are backed out using accounting provision composition and judgment. - Replication of early 1990s Swiss mortgage event: - Historical observed loss rates: doubled in 1991 from 0.53 percent in 1989 to 1.03 percent. - Model inputs as of 1989 and simulated 1991 recession (50 basis point increase in unemployment to 1.1 percent; 180 basis points increase in mortgage rates to 7.8 percent; peak-to-trough real estate price decline of 16.4 percent) produce: - Projected loss rate of 1.05 percent, close to observed 1.03 percent. *Source: wpiea2020165-print-pdf - 2018.* ### 1.32 percent to 0.61 percent over 1997–99. The model simulates the macroeconomic ### wpiea2020165-print-pdf - 1.32 percent to 0.61 percent over 1997–99 ### Model backtesting: Switzerland - Historical context: - Real estate prices declined between 1992 and 1996, with a cumulative price correction of 19 percent. - Model simulation for 1999 (using 1997 macroeconomic conditions and assumed changes): - Unemployment rate declines by 250 basis points to 2.7 percent. - Interest rates contract by 50 basis points to 3.9 percent. - Real estate prices recover by 1.4 percent. - Loss rate results: - Simulated loss rate: 0.53 percent. - Observed loan loss rate in 1999: 0.61 percent. - Benign-conditions validation (baseline scenario for 2019–20): - Unemployment improves by 10 basis points to 2.6 percent. - Mortgage rates increase from 1.5 percent to 2.8 percent over two years. - House prices remain stable over 2019–20. - Resulting expected loss rate for the average bank mortgage portfolio: 0.06 percent. - Average "expected" loan loss rate on mortgage claims reported by Swiss banks in 2018: 0.10 percent. - Incurred credit loss measure for the Swiss mortgage portfolio in 2018: 0.04 percent. - Note on measures of default: - The model’s measure is point-in-time and expected to be lower during benign conditions (like in 2018) than through-the-cycle loss rates estimated by banks. ### Assessment of macroprudential policy measures (Switzerland) - Existing macroprudential measures discussed: 1. Sectoral CCyB at 2 percent of risk-weighted positions secured by residential property situated in Switzerland. 2. Mandatory amortization to two-thirds of the property collateral value at origination over a 15-year period, with a linear repayment schedule. - CCyB sufficiency under stress scenario: - Under the stress scenario, capital depletion reaches 135 basis points. - 135 basis points equals 4.2 times the size of the CCyB. - Netting out average provisions on mortgage loans, "unexpected losses" would exceed the CCyB by 3.8 times. - Projected mortgage losses would exhaust the sectoral CCyB and deplete about 50 percent of banks’ capital conservation buffer (CCB). - The CCyB’s capacity to absorb unexpected mortgage losses depends crucially on the risk-weight density of mortgage loans. - Interaction with risk-weight density: - For standardized approach exposures, risk-weight density ranges between 35 and 100 percent—the latter for tranches larger than 80 percent of collateral value or if mortgage loan does not meet SBA self-regulation requirements. - Banks using an internal ratings-based approach can apply lower risk-weights to non-income producing mortgages. - Complementary evidence: - Capital-constrained banks with low capital cushions raise mortgage rates relatively more than competitors, shifting new mortgage production from less to more resilient banks (as found by Basten and Koch (2015)). ### Counterfactual analysis: Amortization requirement change - Policy simulated: - Lowering the maximum amortization period in 2014 from 20 to 10 years, applied to loan vintages issued in or after 2014. - Aggregate effect by 2020: - Overall loss rate increases from 1.77 percent to 2.45 percent. - Additional capital depletion: about 52 basis points. - Mechanisms and trade-offs: - Shorter amortization reduces outstanding balance of recent vintages (reducing negative home equity risk) but tightens amortization schedule (increasing affordability risk). - Shorter amortization decreases probability of negative home equity by reducing outstanding balance, lessening prepayment constraint under borrower distress. - Impact on margin call channel ambiguous: reduces risk of margin call being triggered (by increasing PiT LTV ratio) but exacerbates affordability risk if margin call is activated. - Heterogeneous effects by LTI: - For new mortgages with LTI ≤ 3: - Probability of financial distress increases from 7.6 percent to 9.6 percent. - For new mortgages with LTI > 7: - Probability of financial distress increases from 46.8 percent to 66.5 percent. - For mortgages issued before 2013: - Low LTI mortgages: financial distress increases from 7.2 percent to 7.3 percent. - High LTI mortgages: financial distress increases from 46.9 percent to 49.0 percent. - Policy implication: - Early activation of macroprudential measures is crucial to increase policy effectiveness. - Note: - The counterfactual analysis applies to newly generated mortgages only; cost in terms of aggregate effect on credit supply is outside the scope of the analysis. ### Calibration of macroprudential policy tools in Austria - Context and toolkit: - Borrower-based tools added in September 2017: LTV, DSTI, and DTI limits. - Mortgage market characteristics: - Loans with LTV ≥ 80 percent accounted for nearly 38 percent of new mortgages in 2018 (compared to 24 percent in 2012). - Average DSTI increasing but still below 30 percent in recent years. - Share of variable-rate household loans remains high at 45 percent of new loans. - House prices estimated overvalued by about 10–15 percent nationally and by over 20 percent in Vienna. - Real estate transaction volume growth: 13 percent in 2018. - Housing loans in total assets of Austrian banks rose from 8 percent in 2008 to 16 percent in 2018. - Austrian insurance sector allocates about 8 percent of their total assets to real estate. - Over 45 percent of the population rents rather than owns a house or an apartment. - Use of structural model for calibration: - Structural models helpful where short time series or limited past macroprudential use constrain statistical approaches. - Four-step approach outlined: reflect market characteristics in model, construct adverse scenario, introduce macroprudential policies into model, present sample results based on Austrian mortgage data. ### Defining the adverse scenario for Austria (GaR and HaR approach) - Methodology: - Use GaR (Growth-at-Risk) and HaR (House price-at-Risk) methodology to link tightness of macroprudential tools to current level of key risks. - At-Risk approach uses bottom 5th percentile of forecast distributions of future GDP and house prices. - Quantile regression estimation on macrofinancial data from 1993Q3–2018Q4. - Explanatory variables for HaR: house price growth, financial conditions index, private sector credit growth, house price misalignment, GDP growth. - Explanatory variables for GaR: lagged GDP growth and financial conditions index. - Predicted distributions as of end-2018: - Median annualized quarterly GDP growth over 2019 projected to be about 2.1 percent. - 5 percent likelihood that GDP growth would fall to -1.25 percent over the same horizon. - Median real house price growth predicted around 3.2 percent. - 5th percentile for house price growth: -5.5 percent. - Adverse scenario calibration (two-year horizon, GaR/HaR 5th percentile continued for two years): - Table 3: Cumulative percentage change over two years (adverse): - Real disposable income: -2.5% - Real house price level: -11.0% - Unemployment rate: 1.9% - Mortgage rate: 1.4% - Robustness exercise: - Given house price overvaluation of 10–15 percent nationwide in 2018, HaR-implied -11 percent decline seen as relatively mild. - Alternative robustness assumption: 20 percent decline in real house prices during the two-year period (see Section V.E). - Calibration notes: - Increase in unemployment rate estimated based on past relationship with GDP growth. - Increase in mortgage interest rate calibrated based on evidence from past recessions. *Source: IMF working paper content provided in the supplied PDF excerpt.* ### 3.65 percent, while the real rates on housing loans increased by 200 basis points between 2008Q2 and ### wpiea2020165-print-pdf - 3.65 percent, while the real rates on housing loans increased by 200 basis points between 2008Q2 and ### Introducing Macroprudential Tools - Macroprudential borrower-based limits are introduced k quarters before the adverse scenario materializes and affect LTVs and DSTIs of new mortgage flows. - Baseline assumption for new mortgage flows in the next k quarters: similar in volume and in LTV, DTI, and DSTI distributions to the average flows observed in the last four quarters of available data. - Outstanding loans mature at volumes similar to recent quarters. - Macroprudential limits reduce the relative mass of loans with LTVs or DSTIs above regulatory limits: - To zero if “hard” limits are introduced. - To a certain percentage of total new loans if “speed limits” are allowed. - “Bunching” effect: among loans complying with regulatory limits, the relative share of loans with LTV/DSTI ratios just below the limits increases; the bunching effect is assumed proportional to the decline in the mass of loans above the limits. - During the k quarters, household income and house prices grow at the median values (adjusted to quarterly frequency) predicted from the GaR and HaR exercises. - Austria-specific assumptions: no changes in the unemployment rate and in the mortgage interest rate before the adverse scenario materializes. - Model flexibilities and options: - Calibrations considered: (1) range of LTV limits; (2) range of DSTI limits; (3) combination of LTV-DSTI limits. - DSTI limits affect DSTI in the stressed sale formula and can affect LTV distribution where joint LTV-DSTI distribution is available. - DTI limits could be introduced but were difficult to incorporate given Austria’s DTI reporting. - Distinction between hard limits and speed limits (pre-defined share of loans allowed not to comply). - Analysis uses banking sector-wide aggregate mortgage portfolio data instead of bank-level data. - For each macroprudential choice, compare losses on aggregate portfolio in adverse scenario with losses in absence of any limits. ### Calibration and Data Transformations (Austria) - Austrian dataset: - Covers about two-thirds of all new housing loan issuances at quarterly frequency. - Data back to 2011; latest datapoint 2018Q4. - Data include distributions by: (1) LTV; (2) DSTI; (3) DTI; (4) LTV-DSTI jointly; (5) LTV-DTI jointly; (6) share of variable-rate loans in new loans; (7) average maturity of new loans. - Constructing PiT (point-in-time) LTVs and DSTIs of outstanding loans by LTV-vintage bucket as of 2018Q4 via additional calculations (Appendix III referenced). - Absolute shares of mortgages in each LTV-vintage bucket as of 2018Q4 derived using time series of new mortgage flows and mortgage stocks from the National Bank of Austria (OeNB). - Mortgages originated before 2011 captured by assuming similar risk characteristics as loans in earliest eight vintages (2011Q1–2012Q4) on average. - Calibration approach for distress probabilities: - Parameters driving probability of financial distress in equation (1) calibrated following Harrison and Mathew (2008) (UK early 1990s) due to Austria’s lack of severe historical house price downturns. - Parameter D set to match expected probabilities of default reported by banks over past three years; other parameters capture Austria-specific housing market characteristics. - DSTI specification: - DSTI includes both amortization and interest payments. - Lower DSTI threshold (where Φ starts to increase) set somewhat higher than 10 percent in Harrison and Mathew (2008). - Upper DSTI threshold where Φ = 1 set at 30 percent. - Parameter 훽3 positive only when 훥훥훥훥t >0 (only increases in unemployment assumed to have nonlinear and positive impact on DSTI). - Interest rate and foreclosure calibrations: - Historical shares of fixed- vs floating-rate mortgages at origination used to apply interest rate changes only to fraction of loans equal to historical floating-rate share. - Transaction cost C calibrated to match sum of average real estate transfer tax rates and registration fees reported for Austria in Schneider and Wagner (2015). - Average loan maturity at origination Ts from OeNB. - Discount rate for foreclosed house sales 훿 set at 20 percent based on OeNB expert discussions. - Risk-free rate for discounting set close to zero; apply a 200-basis point risk premium for discounting revenues from foreclosed assets. - Time to sale of a foreclosed property Q set at 1.25 years (sensitivity tested). - House prices in equation (2) drawn from a normal distribution with mean adjusted by assumed house price decline in adverse scenario; house price distribution standard deviation σP set at 15 percent. - Table 4 core parameter values (as presented): - Equation (1) parameters: - 휙(LLDT D) = 0 if L L D T D < 15% - = (LLDT D − 15%) / (30% − 15%) if L L D T D ∈ [15%, 30%] - = 1 if L L D T D > 30% - 퐿 0.2 - 훽1 0.0003 - γ 2.5 - 훽2 0.006 - 훽3 0.007 if 훥훥훥훥t >0 and 0 otherwise - Q 1.25 - 훼 1 - Equations (2)-(4) parameters: - C 5% of house value - Ts 25 - rf 0.1% - spread 2% - 훿 20% - σP 15% ### Results (k = 8 quarters unless stated) - Baseline: k = 8 (new mortgages originated for eight quarters before adverse scenario materializes). - Sensitivity: when k = 4, PDs and portfolio losses increase somewhat; ordering of macroprudential calibrations remains the same. - Absence of macroprudential limits — adverse scenario results (annualized; percent): - System-wide mortgage portfolio: - 1-Year PD 1.9 - LGD 26.6 - Expected Annualized Losses 0.8 - New mortgages (originated in eight quarters before adverse scenario): - 1-Year PD 3.9 - LGD 34.0 - Expected Annualized Losses 1.6 - Sensitivity Analysis (alternative specifications): - Model with borrower financial wealth: - System-wide mortgage portfolio: PD 1.6; LGD 28.5; Expected Annualized Loss 0.7 - New mortgages: PD 3.4; LGD 33.7; Expected Annualized Loss 1.5 - Interpretation: accounting for household financial assets reduces annualized losses by about 8 percent for total portfolio (0.8 → 0.7) and by 6 percent for new mortgages (1.6 → 1.5). - A higher interest rate increase (real interest rate increase of 200 basis points): - System-wide mortgage portfolio: PD 3.4; LGD 32.0; Expected Annualized Loss 1.5 - New mortgages: PD 6.9; LGD 40.0; Expected Annualized Loss 3.1 - Interpretation: large interest rate increases materially raise expected losses, especially for new mortgages due to high share of variable-rate loans. - A larger house price decline (real house price decline of 20 percent): - System-wide mortgage portfolio: PD 2.3; LGD 29.2; Expected Annualized Loss 0.9 - New mortgages: PD 4.5; LGD 37.7; Expected Annualized Loss 2.0 - Benchmarking and interpretation: - Under the scenario of a 20 percent real house price decline, the annualized loss rate for newly issued mortgages implies a three-year cumulative NPL ratio of 12.9 percent (assuming no write-offs or recoveries). - The NPL ratio of 12.9 percent matches the median peak NPL across past banking crises in advanced economies reported in Laeven and Valencia (2018). ### Macroprudential Limits — Effects on New Mortgages (Table 6 summary) - Table 6 reports PD, LGD, and Expected Annualized Loss for new mortgages under ten alternative calibrations (combinations of LTV and DSTI, hard limits and speed limits of 20 percent). - Selected exact entries from Table 6 (rows: PD, LGD, Exp. Annual Loss): - No limits: PD 3.9; LGD 34.0; Exp. Annual Loss 1.6 - Hard LTV = 80, no DSTI: PD 2.7; LGD 31.9; Exp. Annual Loss 1.0 - Hard DSTI = 40, no LTV: PD 3.2; LGD 31.8; Exp. Annual Loss 1.3 - Hard LTV = 80 & hard DSTI = 30: PD 1.3; LGD 31.8; Exp. Annual Loss 0.5 - Hard LTV = 80 & hard DSTI = 40: PD 2.2; LGD 31.8; Exp. Annual Loss 0.8 - Hard LTV = 90 & hard DSTI = 40: PD 2.7; LGD 32.1; Exp. Annual Loss 1.0 - Speed limit (20%) LTV = none, DSTI = none: PD 3.3; LGD 32.9; Exp. Annual Loss 1.3 - Speed limit (20%) LTV = 80, DSTI = 40: PD 4.0; LGD 34.1; Exp. Annual Loss 1.6 - Speed limit (20%) LTV = 80, DSTI = 30: PD 2.3; LGD 32.9; Exp. Annual Loss 0.9 - Speed limit (20%) LTV = 80, DSTI = 40: PD 3.3; LGD 32.9; Exp. Annual Loss 1.3 - Speed limit (20%) LTV = 90, DSTI = 40: PD 3.7; LGD 33.4; Exp. Annual Loss 1.5 - Key observations: - Different combinations of LTV and DSTI limits can have similar impacts on expected losses. Example: hard LTV-DSTI = 80-40 yields annualized loss 0.8 percent on new mortgages, similar to speed limit of 20 percent with LTV-DSTI = 80-30 which yields 0.9 percent. - Standalone DSTI limit of 40 percent is not very effective at reducing default frequency: - With speed limits, expected loss for LTV-DSTI = 80-40 equals that for standalone LTV = 80 (both 1.3 vs 1.3 in certain calibrations). - Hard DSTI = 40 reduces PD from 3.9 to 3.2 and annualized losses from 1.6 to 1.3 (modest declines). - Reason: historically low lending rates have depressed DSTIs of new loans; system-wide average DSTI for new mortgage flows was somewhat below 25 percent in 2018. - Appropriately calibrated speed limits can achieve similar objectives to hard limits while allowing banks flexibility and affecting business models less. ### Which Macroprudential Limits to Choose? - Selection depends on factors outside the credit-default model, including regulatory tolerance for risk and general equilibrium effects (impact on bank credit, house prices, GDP, feedback to PDs and LGDs). - Empirical evidence referenced (summarized from cited studies): - Laliotis et al. (2019): LTV limits have non-marginal impact on house prices and credit only once they fall below 80 percent. - Alam et al. (2019): Impact of LTV limits on credit growth is non-linear relative to the size of tightening vs initial average LTV. - Gross and Población (2017) and Gross et al. (2020): quantify second-round effects and emphasize need for nonlinear macrofinancial simulation frameworks for two-way dependencies. - Proposed rule of thumb to guide selection after accounting for second-round general-equilibrium effects: - Compare expected losses on new mortgages (subject to macroprudential limits) with losses on less risky older mortgage vintages. - If loss rate on new mortgages is considerably lower than losses on older vintages, calibration might be too tight relative to reference portfolio. - In the Austria example: - Annualized loss rate on total mortgage portfolio (proxy for “old” mortgages) = 0.8 percent. - Calibrations achieving similar rate for new loans: LTV-DSTI = 80-30 with speed limit 20 percent, or hard LTV-DSTI = 80-40. - Formal objective (as presented): - Minimize Pr{...} expression comparing riskiness of new loan production TnewX to reference portfolio TrefX under constraints (exact formula provided in text). - If reference EX calibrated to peak NPL observed in Austria during global financial crisis (4.1 percent), corresponding reference default rate = 2.1 percent. - Objective reaches zero if macroprudential instruments adjusted to either combined LTV-DSTI limit 80-30 with speed limit 20 percent, or hard LTV-DSTI 80-40. ### Conclusions - Proposed a semi-structural approach for credit risk analysis when data scarcity, absence of tail events, or structural shifts make statistical models unreliable. - Approach suitable for countries without large historical house price or economic downturns and where regulatory changes have altered default processes. - Advantages: - Does not require micro-level data; relies on regulatory information at risk bucket and bank portfolio level with quarterly frequency. - Emphasizes importance of collecting granular vintage disclosures and risk-bucket regulatory data to safeguard financial stability. - Extends Harrison and Mathew (2008) double-trigger ability-to-pay default model to capture mortgage characteristics, housing finance modalities, and macroprudential measures. - Calibrations, backtests, sensitivity tests, and robustness checks conducted. - Policy relevance: - Supports macroprudential authorities in using stress test results to guide changes in borrower-based limits via a rule to minimize authorities’ loss function relative to benchmark portfolio. - Model simulations assess effectiveness and calibration of macroprudential instruments and inform timing of macroprudential policy. - Flexibility demonstrated: calibration can adapt to data availability, country specifics, and structural changes. - Note on COVID-19: model calibrated on pre-COVID information but still useful due to structural approach; calibration to COVID-19 shocks and assessment of pandemic-related policy effectiveness left for future research. *Italic: Source: wpiea2020165-print-pdf - 3.65 percent, while the real rates on housing loans increased by 200 basis points between 2008Q2 and (IMF PDF content provided).* ### REFERENCES ### REFERENCES ### Bibliographic citations - Adrian, T., N. Boyarchenko, and D. Giannone, 2016. “Vulnerable Growth.” American Economic Review, 109(4), 1263-89. - Albacete, N., and P. Linder, 2017. “Simulating the impact of borrower-based macroprudential policies on mortgages and the real estate sector in Austria—evidence from the Household Finance and Consumption Survey 2014,” OeNB Financial Stability Report Vol. 33, 52-68. - Albacete, N., P. Fessler, and P. Linder, 2018. “One policy to rule them all? On the effectiveness of LTV, DTI and DSTI ratio limits as macroprudential policy tools,” OeNB Financial Stability Report Vol. 35, 52-68. - Alam, Z., Alter, A., Eiseman, J., Gelos, G., Kang, H., Narita, M. Nier, E. and N. Wang, 2019. “Digging Deeper—Evidence on the Effects of Macroprudential Policies from a New Database,” IMF Working Paper 19/66. - Bank for International Settlements, 2018. “Stress Testing Principles,” Basel Committee on Banking Supervision, October. - Bank for International Settlements, 2012. “Operationalizing the selection and application of macroprudential instruments,” Committee on the Global Financial System, CGFS Paper No. 48, December. - Basten, C., and C. Koch, 2015. “Higher Bank Capital Requirements and Mortgage Pricing: Evidence from the Countercyclical Capital Buffer (CCB),” BIS Working Paper No. 511, September. - Bhutta, N., J. Dokko, and H. Shan, 2010. “The depth of negative equity and mortgage default decisions,” Finance and Economics Discussion Series 2010-35, Federal Reserve Board. - Breuer, T., and M. Summer, 2018. “Systematic Systemic Stress Tests,” OeNB Working Paper 225, December. - Elul, R., Souleles, N., Chomsisengphet, S., Glennon, D., and R. Hung, 2010. “What ‘Triggers’ Mortgage Default?” The American Economic Review, Vol. 100, No. 2, Papers and Proceedings, 490-94, May. - European Central Bank, 2007. “The impact of short-term interest rates on bank credit risk taking,” Financial Stability Review, December. - Financial Conduct Authority, 2015. “Methods for analyzing mortgage markets,” Occasional Paper No. 11. - FINMA, 2019a. “Mortgages for investment properties: FINMA recognizes adjustments to self-regulation,” August. - FINMA, 2019b. “Action is now needed on investment property,” speech by Mark Branson at the annual media conference, April. - Foote, C., K. Gerardi, and P. Willen, 2009. “Negative equity and foreclosure: No easy answers,” in NBER Macroeconomic Annual 2009. Chicago: University of Chicago Press. - Gross, M., L. Mindaugas, and L. Valderrama, 2020. “The Macro-Financial System Simulator (MASS),” IMF Working Paper (forthcoming). - Gross, M., and J. Población, 2017. “Assessing the efficacy of borrower-based macroprudential policy using an integrated micro-macro model for European households,” Economic Modelling Vol. 61, 510-28. - Harrison, I., and C. Mathew, 2008. “Project TUI: A Structural Approach to the Understanding and Measurement of Residential Mortgage Lending Risk,” Reserve Bank of New Zealand. - International Monetary Fund, 2020a. “Austria Financial Sector Assessment Program: Technical Note—Macroprudential Policy Framework and Tools,” IMF Country Report 20/64, March. - International Monetary Fund, 2020b. “Canada Financial Sector Assessment Program: Technical Note—Stress Testing and Financial Stability Analysis,” IMF Country Report 20/16, January. - International Monetary Fund, 2019a. “Switzerland Financial Sector Assessment Program: Technical Note—Stress Testing the Banking Sector,” IMF Country Report 19/189, June. - International Monetary Fund, 2019b. “Lower for Longer,” Global Financial Stability Report, Chapter 1, October. Washington: International Monetary Fund. - International Monetary Fund, 2017. “Is Growth at Risk?” Global Financial Stability Report, Chapter 1, October. Washington: International Monetary Fund. - International Monetary Fund, 2014. “Staff Guidance Note on Macroprudential Policy,” IMF Policy Paper, December. - Jurca, P., Klacso J., Tereanu, E., Forletta, M. and M. Gross, M., 2019. “The effectiveness of borrower-based macroprudential measures: a quantitative analysis for Slovakia,” IMF Working Paper (forthcoming). - Laeven, L., and F. Valencia, 2018. “Systemic Banking Crises Revisited,” IMF Working Paper 18/206. - Laliotis, D., Buesa, A., Leber, M., and J. Poblacion, 2019. “An Agent-Based Model for the Assessment of LTV Caps,” ECB Working Paper 2294, July. - Lazarov, V., and M. Hinterschweiger, 2018. “Determinants of distress in the UK owner-occupier and buy-to-let mortgage markets,” Bank of England Staff Working Paper No. 760, October. - Nier, E., Popa, R., Shamloo, M., and L. Voinea, 2019. “Debt Service and Default: Calibrating Macroprudential Policy Using Micro Data,” IMF Working Paper 19/82. - Pennington-Cross, A., 2003. “Subprime and Prime Mortgages: Loss Distributions,” OFHEO Working Papers 03-1. - Schneider, M., and K. Wagner, 2015. “Housing Markets in Austria, Germany and Switzerland,” Monetary Policy and the Economy, Oesterreichische Nationalbank, Issue 1, 42-58. *References list concludes before Appendix I.* --- ### Appendix I. Calculation of Penalty for Early Prepayment ### Key formula and definitions - Net present value of terminating ahead of schedule: - NPPPrepay notation given in text as: - 푁푁푃푃푃푃 푖푖,푡푡 =퐿퐿 푖푖,푡푡 + ∑ 푟푟 푖푖,푡푡 퐿퐿 푖푖,푡푡 (1− 푗푗 푇푇 푖푖, 푡푡 ) (1+푟푟 푓푓 ) 푗푗 푇푇−1 푗푗=0 , (A.1) - Variables: - 퐿퐿 푖푖,푡푡 is the outstanding principal. - 푟푟 푖푖,푡푡 is the interest rate of the mortgage at time t (for floating-rate mortgages it denotes the libor rate at time t; for rollover mortgages, it is the interest rate prevailing at the last resetting period). - 푇푇 푖푖,푡푡 is the remaining maturity of the loan. - 푟푟 푓푓 is the risk-free return the lender can earn by reinvesting the recovered principal immediately. - Outstanding principal under linear amortization: - 퐿퐿 ,, it = ⋅ , , is i T LL M where iM denotes the maturity at origination and ,isL represents the face value of the loan at origination. - Decomposition of foregone interest net present value: - Geometric sum (A): - () , ,, 1 1 1 1 T f it itit ff itit r A rL rr   + = ⋅ ⋅⋅−     +   - Arithmetic–geometric sequence sum (B): - () () ( ) () , , ,, 2 ,, , , 11 1 1 1 1 T T f f it it itit ff f itit it f it r r B rL rr r T r    +−  + =⋅⋅⋅ −⋅    +   ⋅  +   - Final algebraic expression for NPV of the loan: - () () ( ) () , , ,,, 1 , ,, 11 1 1 1 T T f T it f titititit T f ff it itit r NPVLrLr Tr rr −  +−   = + ⋅ ⋅⋅+−⋅   ⋅ ⋅+  ### Procedural notes - The equation (A.1) assumes a linear repayment schedule during the remaining maturity 푇푇 푖푖,푡푡. - The lender recovers the full amount owed at time t and can reinvest immediately at 푟푟 푓푓; this foregone return is deducted from the amount owed by the borrower. - The decomposition isolates a geometric component (A) and an arithmetic–geometric component (B) to compute the net present value effect of foregone interest. --- ### Appendix II. Calculation of LTV Distribution under Alternative Approach in Section IV ### Objective - Generate PiT LTV shares for each outstanding vintage in the mortgage portfolio under the assumption that LTV distributions are equal at origination. ### Notation and setup - LTV buckets: {b1, b2, b3, b4, b5} denoted {} 12345 ,,,,B bbbbb= with buckets labeled in increasing order of LTV. - Observed PiT LTV shares at time t: {βb}5 1 =. - Candidate distribution at issuance: {α̂b}5 1 =, denoted orig s LTVˆ . - For each LTV bucket, compute the “second mortgage” and apply the amortization rule (A in the formula below). Amortization schedule is a function of orig s LTV and year of issuance s (which determines maximum amortization period). - Revalue collateral at time t at current market prices using growth rate g of real estate prices. ### PiT LTV calculation for a loan - PiT LTV expression provided as: - ( ) () ,, 1 ,, 1 1,, 1 ts orig sbsb i PiT stb ts ssk k A LTVs iL LTV HPg − = − + =  −⋅   = ⋅+ ∑ ∏ ### Aggregation to compute shares at time t - Share of outstanding mortgages with PiT LTV = b at time t: - 0 5 ,,, , 1 ˆ ˆ t btskstbk stk β γα == =⋅⋅Ι ∑∑ - Definitions: - sγ is the share of vintage s in the outstanding stock at time t. - ,, ,. PiT stbkstbk LTVb [ Ι ≡ Ι∈ ] is a time-varying indicator function taking value 1 if the PiT LTV of a loan falls into bucket kb, and 0 otherwise. ### Solving for origination distribution - The vector {α̂b}5 1 = is the solution to the multiple-equation system: - 0 5 ,,, , 1 t btskstbk stk β γα == =⋅⋅Ι ∑∑ - Where {βb}5 1 = is the observed LTV distribution at time t. - Once {α̂b}5 1 = is identified, compute PiT shares for each vintage-LTV bucket applying the amortization rule described above. --- ### Appendix III. Calculation of Point-in-Time Loan Value, LTVs, and DSTIs (Section V) ### Reference date - These values are computed as of 2018Q4. ### Equations and definitions - Outstanding value of loan in bucket h as of 2018Q4: - 퐿퐿 ℎ,2018 푄푄4 = � 퐿퐿푇푇푁푁 ℎ,표표표표 푖푖표표 푖푖푛푛 ×푃푃 표표표표 푖푖표표 푖푖푛푛 × 푇푇 ℎ,2018 푄푄4 푇푇 � ℎ � 푃푃 푡푡 , (A.2) - Borrower’s income at origination (from DTI at origination): - 퐷퐷퐼퐼퐼퐼퐼퐼퐼퐼 푠푠 ℎ,표표푟푟푖푖 표표푖푖푛푛 = 퐿퐿 ℎ,표표표표 푖푖표표 푖푖푛푛 퐷퐷푇푇퐷퐷 ℎ,표표표표 푖푖표표 푖푖푛푛 (A.3) - Borrower’s income as of 2018Q4: - 퐷퐷퐼퐼퐼퐼퐼퐼퐼퐼 푠푠 ℎ,2018 푄푄4 =퐷퐷퐼퐼퐼퐼퐼퐼퐼퐼 푠푠 ℎ,표표푟푟푖푖 표표푖푖푛푛 ×(1 +푔푔 표표푟푟푖푖 표표푖푖푛푛:2018 푄푄4 푐푐푐푐푐푐 ) (A.4) - Where 푔푔 표표푟푟푖푖 표표푖푖푛푛:2018 푄푄4 푐푐푐푐푐푐 is the cumulative growth rate of real household income (using OECD data). - Outstanding principal used to compute LTVs: - 퐿퐿푇푇푃푃 ℎ,2018 푄푄4 = 퐿퐿 ℎ,2018 푄푄4 푃푃 푡푡 , (A.5) - Debt-service measure for DSTI calculation as of 2018Q4: - 퐿퐿퐷퐷 푇푇퐷퐷 ℎ,2018 푄푄4 = �퐿퐿푇푇푁푁 ℎ,표표표표 푖푖표표 푖푖푛푛 ×푃푃 표표표표 푖푖표표 푖푖푛푛 × � 1 푇푇 � ℎ +푟푟× 푇푇 ℎ,2018 푄푄4 푇푇 � ℎ � � 퐷퐷푛푛푐푐표표푐푐푡푡 ℎ,2018 푄푄4 , (A.6) ### Implementation notes - 푇푇 � ℎ denotes the average maturity at origination of loans in the vintage-LTV bucket h. - Equation (A.2) assumes current LTV equals LTV at origination net of repayments made. - Quarterly average nationwide house prices are used to compute outstanding values in (A.2). - Equations (A.3)–(A.4) compute borrower income at origination and income as of 2018Q4 by scaling past income by cumulative real household income growth from OECD data. - LTVs and DSTIs on outstanding loans in bucket h are computed using equations (A.5)–(A.6). *End of content unit.* --- _Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020165-print-pdf.pdf_