## wpiea2019059

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### Key findings and methodological positioning
- Two complementary approaches developed to assess house prices:
  - Borrowing capacity approach (prudential): computes house prices implied by household maximum safe borrowing given income, market interest rate, and down payment.
  - Intrinsic value approach (valuation): computes net-present value of the property for a prospective owner-occupier household given current and expected market conditions.
- Indicators: simple-to-use, available in real time, structural, suffer from no ex-post revisions, and explicitly forward-looking (particularly the investment/valuation approach).
- The paper avoids time-series or panel-data econometrics; parameters and inputs are structural and easy to understand and calibrate.
- Empirical summary for Prague:
  - House prices were on average overvalued with respect to economic fundamentals at the time of writing, but the degree of overvaluation was smaller than in 2008 due to increases in household income and low mortgage interest rates.
  - A sizable portion of the increase in attainable house prices (borrowing-capacity approach) can be attributed to a decline in mortgage rates.
- Limitations:
  - Focus on averages and aggregated values; may not detect more overvalued segments of the market.
  - Tools can be applied to more granular data (individual household or property) to increase precision.
- Relationship to literature:
  - Contrasts with regression/historical-ratio approaches and co-integration/regression methods; argues against reliance on unstable coefficients, ex-post zero-mean residuals, and backward-looking estimation.

### Static Borrowing Capacity (SBC): formulation and interpretation
- Key definitions and equations (notation preserved):
  - A_t = α × Y_t. (1)
  - Given nominal net mortgage interest rate per month, i_{m,t}, and mortgage payment A_t, mortgage loan for N years:
    - L_t = A_t × [ z_t (1 − z_t^{N×12}) / (1 − z_t) ] where z_t ≡ 1 / (1 + i_{m,t}). (2)
  - Attainable housing value:
    - P_{h,t} × H_t = L_t + D_t. (3)
  - Rewriting using observed or maximum values of α and loan-to-value ratio φ:
    - P_{h,t} × H_t = (1/φ) [ z_t (1 − z_t^{N×12}) / (1 − z_t) ] × α Y_t. (4)
- Interpretations and implications:
  - With constant nominal interest rates, stable leverage and constant share of mortgage payments for new borrowers, the price-to-income ratio is uniquely determined and nominal house prices should grow at the same rate as nominal disposable income: ̇PH_t = ̇Y_t.
  - Relationship between house prices and interest rates is nonlinear; non-linearity increases with mortgage duration N. With average mortgage durations usually exceeding 20 years, non-linear impacts can be pronounced.
  - SBC is static and does not reflect expected future changes in mortgage rates or household income, increasing sensitivity to current mortgage rates and potentially making implied house prices more pro-cyclical than forward-looking valuation measures.
- Counterfactual and prudential-use cautions:
  - SBC can gauge effects of changes in mortgage rates; debt-service-to-income ratio α; loan-to-value ratio φ; average duration N.
  - Naive use of (4) can yield misleading counterfactuals (e.g., lowering φ may not mechanically reduce attainable prices if down payment and α adjust).
  - Example logic: if α_0 = α_max and φ_0 = φ_max with D_0 predetermined, imposing φ_1 = L_1/(L_1 + D_0) leads to L_1 ≤ L_0 and α_1 ≤ α_0 for given income and rates; over time income and down payment changes can relax constraints.

### Dynamic Borrowing Capacity (DBC): concept and implementation
- Definition and key relations:
  - Define κ_t = D_t / Y_t and rewrite borrowing constraint as P_{h,t} × H_t = α f(z_t) Y_t + κ_t Y_t; parameters α, φ, and κ are related and cannot all be set independently.
- DBC concept:
  - Extends SBC to find maximum sustainable mortgage loan so future debt-service burden does not exceed prudential threshold α_max.
  - For adjustable-rate or floating-rate mortgages, DBC finds an initial loan size and initial debt-service-to-income ratio α_{init} so α_t ≤ α_{max} at all future periods.
  - DBC is asymmetric and satisfies L_{DBC} ≤ L_{SBC}. SBC is an upper bound for any DBC metric.
- Types implemented:
  - Baseline DBC.
  - DBC with an ‘offset account’: households deposit the difference between α_{t+k} × Y_{t+k} and α_max × Y_{t+k} for K periods; in period K households use saved deposits and interest proceeds to lower debt and debt-service burden.
- Practical construction and optimization:
  - Problem: find amortization schedule that leads to maximum affordable loan without violating α_max in any period.
  - Procedure: start from SBC, calculate amortization schedule given expected paths of income growth and mortgage rates, check for α violations, lower loan if violated, iterate until feasible loan equals DBC.
  - Implemented via simple grid search with scaling coefficients applied to SBC or via optimizers with heavy penalties on constraint violations.
- Adjustments and extensions:
  - Distinguish recommended initial α and maximum allowed α_max.
  - If mortgage rates are expected to decline, DBC can allow a larger loan.

### Intrinsic-Value Approach (IV): valuation framework and implementation
- Concept:
  - Asset value equals discounted net present value (NPV) of income stream from housing services; explicitly forward-looking and sensitive to long-horizon expectations of income, rents, and mortgage rates.
  - Valuation is subjective and motivates sensitivity and scenario analysis.
- Renting versus buying setup:
  - Renting: invest D_t at return i^e_t, pay rent_t for housing services.
  - Buying: down payment D_t, borrow L_t at interest i^m_t, P_{H,t} = D_t + L_t, LTV φ ≡ L_t / P_{H,t}; pay mortgage payment approximated by i^m_t L_t with interest tax deductibility at marginal rate τ; pay maintenance and property taxes (δ + τ_p) P_{H,t}.
- No-arbitrage pricing and NPV relationships:
  - PH_t = rent_t + (1 / (1+z_t)) PH_{t+1|t}.
  - Composite discount rate: (1+z_t) ≡ (1+[(1−φ)i^e_t + φ×(1−τ)i^m_t] + δ + τ^p).
  - NPV formula: PH_t = Σ_{i=0}^∞ rent_{t+i|t} / ∏_{j=0}^i (1+z_{t+j}).
  - Steady-state: PH_{ss} = rent / (z − g_n) = αY / (z − g_n).
  - Long-run price-to-income ratio driven by nominal income growth g_n and effective net discount factor z; not affected by long-run inflation level.
  - Increased leverage φ can lower cost of capital when i^e > i^m and push house prices up.
  - Caveat: using PH = αY_t / (z_t − g_n_t) with current values is misleading if current income growth and z_t are cyclical.
- Implementation: multi-step NPV model and mortgage structure
  - Mortgage duration N years (application uses N = 25); rate reset frequency K ≤ N (application K = 5).
  - Mortgage payments constant within each K-year block; payments change at resets until repayment at N.
  - Valuation formula (flow to owner, discounted by i^e) preserved in source text (multi-term sum plus terminal perpetuity).
  - Modeling choices:
    - Growth rate of nominal rental income assumed identical to household nominal disposable income growth; rent-to-income kept constant in projection.
    - Monthly mortgage computations aggregated to annual totals; model frequency is annual.
  - LTV circularity solution: choose down payment and loan size flexibly so LTV is set in valuation terms; solve by grid search of scaling coefficients to find fixed point satisfying LTV.

### Prague application: parameterization, calibration, and results
- Income and scaling:
  - Use nominal disposable income per capita, scaled up by 30% to match average Prague wage; further scaled by 1.65 to reflect relationship between per-capita disposable income and household income in mortgage applications.
- Calibration values used:
  - Debt-service-to-income ratio baseline: α = 35%.
  - Loan-to-value ratio: φ = 80%.
  - Rent set at 35% of household income for valuation compatibility.
  - Mortgage duration: N = 25 years; renegotiation horizon: K = 5 years.
  - Offset-deposit DBC assumed interest rate lower than mortgage rate by 100 basis points (bps).
  - Pre-2008 income forecast projection: AR(1) with persistence ρ = 0.25 and steady-state growth g_n = 4 %.
  - Steady-state mortgage interest rate chosen: 5%.
  - Opportunity cost (discount rate for equity flows): i^e = 6.5%.
  - Effective marginal tax rate assumed: τ = 15%; property tax τ^p = 0 assumed negligible.
  - Average apartment size assumed: 68 m^2.
- Maintenance/depreciation:
  - Not included in user cost of capital; typical range under moderate assumptions is 1.5% to 2.5% of house value.
- Prague findings:
  - SBC and DBC:
    - Strong overvaluation in 2008 and overvaluation in 2002–2004 and from 2017 onward.
    - Dynamic borrowing capacity mostly tracked static borrowing capacity except for a short deviation in late 2016 / early 2017.
    - Attainable house prices often act as a lower support for observed prices; pricing gap typically positive or zero.
  - Counterfactuals and decomposition:
    - Mortgage rate decline from 3.42 percent to 2.3 percent (112 basis points) could have raised attainable prices by 15 percent (roughly close to a half-million CZK).
    - Decomposition isolates contributions from income growth and interest rate changes (interest rate contribution measured relative to 2002Q2 level).
  - Sensitivity to α:
    - Static borrowing capacity for α = 30%, 35%, 40%, 45% yields substantially different attainable prices.
    - Example values (Figure 6, Prague, Mil. CZK): 3.78, 4.41, 5.04, 5.67 correspond to α = 30%, 35%, 40%, 45% respectively.
    - Moving α from 30% to 45% increases affordable housing by 50% in the calibration.
  - Intrinsic-value valuation outcomes:
    - Intrinsic (fair) value increasing, driven mainly by robust nominal household income growth.
    - Overvaluation episodes relative to fundamentals:
      - ~10% in 2003.
      - ~30% in 2008 (historically highest).
      - ~15% from end-2015 to 2017.
    - Observed market prices rarely fall below attainable/value-implied prices; attainable prices tend to be a lower bound.
    - Increasing AR(1) persistence from ρ = 0.25 to ρ = 0.85 or ρ = 1 increases valuation volatility and can produce valuations that track market booms more closely.
    - Static/steady-state valuation using current variables as permanent can overvalue property when low interest rates and high income growth are temporary.

### Policy implications, extensions, and toolbox
- Practical attributes:
  - Structural, forward-looking, usable at macro or individual property/household level.
  - Available in real time, not reliant on historical estimation, not subject to revisions.
  - Suitable for scenario and sensitivity analysis; linkable to official macro forecasts.
- Policy-relevant insights:
  - Attainable house prices often form a lower bound for market prices; price gaps are asymmetric (average positive), implying regression models forcing zero-mean residuals may overestimate equilibrium prices.
  - Key parameters for policy: α (debt-service-to-income) and LTV φ; prudential limits on α materially affect attainable prices.
  - Short-run vs long-run interpretation is crucial when using SBC for counterfactuals; down-payment dynamics and distance to φ_max matter.
- Extensions and suggested analysis:
  - Apply framework to other jurisdictions, regions, or cities.
  - Investigate causes of persistent valuation gaps (demand-supply mismatches, demographics, zoning, supply constraints); be cautious of reverse causality between completed dwellings and registered households.
  - Embed the framework into broader assessments of prudential regulation effects on household and corporate credit.
- Toolbox and implementation:
  - Implementation environments: R and Matlab; user interface implemented in Shiny/R for interactive exploration.
  - Figure 12 in source shows interactive interface to the toolbox.

*Source: wpiea2019059 - References and appendices (excerpted content provided in input).*

### References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### References

### Document structure and appendices
- Appendices listed: A.Why We Do Not Use a Regression Model; B.Building Prudential Scenarios and Counterfactuals with the SBC Approach; C.Toolbox.
- Figures and Tables inventory includes numbered items such as:
  - Figures: 1.Prague: Housing Market Snapshot; 2.Loan-Service-to-Income and Loan-to-Value Ratios for 2017H2; 3.Borrowing Capacity Approach: Results; 4.SBC Counterfactual; 5.Income and Interest Rate Effects; 6.Sensitivity Analysis with Respect toα; 7.Average Intrinsic Value vs. Observed Prices; 8.Price-to-Value Estimate for Prague Housing Prices; 9.Effects of Over-Optimistic Income Expectations; 10.Prague: Completed Dwellings vs. Increase in Number of Households; 11.Co-Integration Estimates: Results; 12.Interactive Interface to the Toolbox.

### Key findings (from Introduction and empirical discussion)
- The paper develops two complementary approaches to assess house prices:
  - Borrowing capacity approach (prudential): computes house prices implied by household maximum safe borrowing given income, market interest rate, and down payment.
  - Intrinsic value approach (valuation): computes net-present value of the property for a prospective owner-occupier household given current and expected market conditions.
- Indicators are:
  - Simple-to-use, available in real time, structural, and suffer from no ex-post revisions due to their structural nature.
  - Explicitly forward-looking (particularly the investment/valuation approach).
- The approach avoids time-series or panel-data econometrics; parameters and inputs are structural and easy to understand and calibrate.
- The analysis for Prague indicates house prices were on average overvalued with respect to economic fundamentals at the time of writing, but the degree of overvaluation was smaller than in 2008 due to increases in household income and low mortgage interest rates.
- A sizable portion of the increase in attainable house prices (in the borrowing-capacity approach) can be attributed to a decline in mortgage rates.
- Limitations noted:
  - Focus on averages and aggregated values; may not detect more overvalued segments of the market.
  - Tools can be applied to more granular data (individual household or property) to increase precision.

### Relationship to the literature (methodological positioning)
- Common alternatives in the literature include:
  - Regression models or historical averages of price-to-income or price-to-rent indexes (e.g., Philiponnet and Turrini (2017); Igan and Loungani (2012); Geng (2018); ECB (2015, 2011); OECD (2005); McCarthy and Peach (2004); Warisse (2017)).
  - Co-integration/error-correction regressions using income, interest rates, credit stock, demographic and supply-side factors.
  - Structural models (DSGE) and SVARs to estimate shocks (e.g., housing preference shocks, housing credit shocks).
- Critiques of regression/historical-ratio approaches highlighted:
  - Dependence on abundant historical data over multiple cycles; estimated coefficients may be unstable (see Hejlová, Hlaváček, and Komárek (2017)).
  - Regression residuals summing to zero can bias valuation when overvaluation is asymmetric.
  - Downward trend in nominal mortgage rates undermines implicit mean-reversion assumptions in ratios.
  - Risk of overfitting when many explanatory variables are included without proper cross-validation.
- This paper departs by deriving attainable house prices from fundamentals without referencing observed prices, relying on structural models and later contrasting with observed prices.

### The borrowing-capacity approach (concept and prudential use)
- Purpose:
  - Derives maximum mortgage loan a household can safely borrow given income, market interest rates, and share of income allocated to mortgage payments.
  - Prudential (liquidity-constrained) rather than valuation-focused; useful for macroprudential authorities and counterfactual scenario analysis.
- Behavioral assumptions:
  - Average household is credit constrained and will reach for maximum housing they can afford, respecting regulatory DSTI and LTV limits.
  - Observed data often show LTV and DSTI ratios bunching around prudential limits.

### Static Borrowing Capacity (SBC): formulation and interpretation
- Key definitions and equations (preserve notation as in source):
  - A_t = α × Y_t. (1)
  - Given nominal net mortgage interest rate per month, i_m_t, and mortgage payment A_t, mortgage loan for N years:
    - L_t = A_t × [ z_t (1 − z_t^{N×12}) / (1 − z_t) ] where z_t ≡ 1 / (1 + i_m_t). (2)
  - Attainable housing value:
    - P_h_t × H_t = L_t + D_t. (3)
  - Rewriting using observed or maximum values of α and loan-to-value ratio φ:
    - P_h_t × H_t = (1/φ) [ z_t (1 − z_t^{N×12}) / (1 − z_t) ] × α Y_t. (4)
- Interpretations and implications:
  - In an economy with constant nominal interest rates, stable leverage and constant share of mortgage payments for new borrowers, the price-to-income ratio is uniquely determined and nominal house prices should grow at the same rate as nominal disposable income of households: ̇PH_t = ̇Y_t.
  - The relationship between house prices and interest rates is nonlinear; non-linearity increases with mortgage duration N. Given average mortgage durations usually exceed 20 years, non-linear impacts can be pronounced and should not be ignored.
  - SBC is static and does not reflect expected future changes in mortgage rates or household income, increasing sensitivity to current mortgage rates and potentially making implied house prices more pro-cyclical than forward-looking valuation measures.

### Counterfactual analysis and prudential regulation effects (SBC considerations)
- The SBC formula can be used to gauge effects of changes in:
  - Mortgage rates; debt-service-to-income ratio α; loan-to-value ratio φ; average duration N.
- Caution in interpreting short-run vs. long-run responses:
  - Short-run and long-run responses may differ markedly depending on which variables are fixed or slow-moving.
  - Naive implementation of (4) can yield misleading counterfactuals: e.g., enforcing a lower loan-to-value ratio φ (lower leverage) could naively imply higher attainable house prices unless dynamics of down payment D_t and borrowing behavior are properly accounted for.
  - Example scenario logic from the source:
    - If mortgage-payment-to-income ratio is at regulatory maximum α_0 = α_max, LTV at maximum φ_0 = φ_max, and down payment predetermined at D_0, implementing a new lower LTV φ_1 = L_1/(L_1 + D_0) will lead to a new lower loan L_1 ≤ L_0, with a lower implied α_1 ≤ α_0 for given income, interest rates, and available funds for down payment. Over time, income and available down payment may change and relax constraints, affecting attainable house price dynamics.
- Appendix B and Andrle and Plašil (2019) provide more detail on counterfactuals and short-run vs long-run assumptions.

*Italic: Source: wpiea2019059 - References (excerpted content provided in input).*

### Appendix B illustrates that we can always defineκ

### Appendix B illustrates that we can always defineκ

### Definition and parameter relations
- Define κ_t = D_t / Y_t and rewrite the borrowing constraint as P_h_t × H_t = α f(z_t) Y_t + κ_t Y_t.
- Key point: the parameters α, φ, and κ are related and cannot all be set independently.

### Dynamic Borrowing Capacity (DBC) — concept
- DBC extends the static borrowing-capacity approach to seek the maximum sustainable mortgage loan such that the risk of excessive debt-service burden in the future is avoided or lowered.
- For adjustable-rate or floating-rate mortgages the DBC finds a new initial loan size and initial debt-service-to-income ratio, α_init, so that the household’s debt-service capacity does not exceed the prudential threshold α_max at any future period.
- If expected household income increases do not compensate for expected increases in mortgage rates, the initial loan size must be lowered.
- The DBC is asymmetric and satisfies L_DBC ≤ L_SBC.
- Static borrowing capacity is an upper bound for any DBC metric; DBC constraints become binding under rather extreme circumstances where current mortgage rates are unusually low given generally positive expected growth in nominal income.

### Types of DBC implemented
- Two types implemented:
  - Baseline DBC.
  - DBC with an ‘offset account’:
    - Households deposit the difference between α_{t+k} × Y_{t+k} and α_max × Y_{t+k} for K periods into an offset deposit account.
    - In period K where the mortgage rate is re-negotiated, households use saved deposits and interest proceeds to lower debt and debt-service burden.
- For the offset-account DBC annuities are adjusted for saved deposits and interest proceeds.

### Practical construction and optimization
- Construction is an optimization problem: find the amortization schedule that leads to a maximum affordable loan without violating α_max in current or any future period.
- Practical solution approach:
  - Use static borrowing capacity as a starting value.
  - Calculate amortization schedule consistent with expected paths of income growth, mortgage rates, and the adjustable-rate scheme.
  - Check for violations of the debt service-to-income constraint in any period.
  - If violated, lower the affordable loan amount and re-check; continue until an acceptable solution is found and set it equal to the DBC metric.
- Implementation notes:
  - Finding the maximum loan differs between baseline and offset-account DBC only in amortization-schedule construction.
  - In practice, a simple grid search with reasonable scaling coefficients applied to static borrowing capacity can be used.
  - An alternative is to use traditional optimizers where constraint violations are heavily penalized.

### Adjustments and extensions to DBC
- DBC can be adjusted along many dimensions, for example:
  - Distinguish between the recommended initial debt-service-to-income ratio, α, and the maximum allowed one, α_max.
  - If there are reasonable expectations of mortgage rates declining, the DBC could allow a larger loan.

### Transition: Intrinsic-Value Approach (IV)
- The intrinsic-value approach views asset value as the discounted net present value of the income stream the asset provides; it provides an explicit forward-looking valuation of real-estate property.
- While conceptually motivated for empirical work, valuation critically depends on long-horizon expectations of household income, rents, or mortgage rates and is more involved than prudential borrowing-capacity approaches.
- Valuation is subjective and sensitive to perceived opportunity cost and the economic outlook; this motivates sensitivity and scenario analysis.
- Authors refine simplified theoretical models from introductory finance, noting computations can be performed in a spreadsheet even if no simple closed-form expression is obtained.
- Distinction between investor types:
  - Professional investors (e.g., REITs) tend to stay leveraged indefinitely and have different opportunity costs, lower transaction costs, and better credit access.
  - Retail investors (owner-occupiers) are assumed to plan to repay the mortgage loan in full rather than stay leveraged indefinitely; to stabilize leverage, retail investors would need to borrow up as property value increases.

### IV.A. Motivation — renting versus buying framework
- Setup for owner-occupier choice between renting and buying on a period-by-period basis; indifference pins down asset value.
- Renting:
  - Invest available funds, D_t, with period return, i^e_t.
  - Pay the rent_t for housing services.
- Buying:
  - Use available funds as down payment, D_t, and borrow L_t at interest i^m_t to buy property for P_H_t = D_t + L_t.
  - Consistent with SBC, define LTV ratio φ ≡ L_t / P_H_t.
  - Pay mortgage payment approximated by i^m_t L_t, deducting interest payments from taxable income at marginal rate τ.
  - Pay maintenance cost and property taxes, (δ + τ_p) P_H_t.

*Source: wpiea2019059 - Appendix B illustrates that we can always defineκ*

### 4.  Sell the property at the end of the period for an expected price,PH

### 4.  Sell the property at the end of the period for an expected price,PH

### No-arbitrage pricing and key valuation relationships
- No-arbitrage condition equating renting and buying:
  - PH_t = rent_t + (1 / (1+z_t)) PH_{t+1|t}.
- Composite discount rate definition:
  - (1+z_t) ≡ (1+[(1−φ)i^e_t + φ×(1−τ)i^m_t] + δ + τ^p).
- Net present value (NPV) principle for real-estate asset:
  - PH_t = Σ_{i=0}^∞ rent_{t+i|t} / ∏_{j=0}^i (1+z_{t+j}).
  - Steady-state: PH_ss = rent / (z − g_n) = αY / (z − g_n).
- Drivers and intuition:
  - Long-run price-to-income ratio driven by nominal income growth (g_n) and effective net discount factor z.
  - Long-run price-to-income ratio is not affected by long-run inflation level.
  - Effective cost of funds term: [(1−φ)i^e_t + φ×(1−τ)i^m_t].
  - Increased leverage φ can lower the cost of capital when i^e > i^m, ceteris paribus, and push house prices up.
- Caveats on steady-state (static) use:
  - Using PH = αY_t / (z_t − g_n_t) with current values is misleading: assumes current income growth persists at g_n and exaggerates sensitivity to cyclical z_t.

### Implementation: multi-step NPV model and mortgage structure
- Mortgage and payment structure assumptions:
  - Mortgage duration: N years (in application N = 25).
  - Rate reset frequency: every K ≤ N years (in application K = 5).
  - Mortgage payments constant within each K-year block; payments change at resets until repayment at N.
- Valuation formula (flow to owner, discounted by opportunity cost i^e):
  - V_{t|t} = Σ_{i=0}^{K−1} [(1−τ)rent_{t+i|t} − mpay_{t+i|t} + τ×intcost_{t+i|t}] / ∏_{j=0}^i (1+i^e_{t+j|t})
    + Σ_{i=K}^{N−1} [(1−τ)rent_{t+i|t} − mpay_{t+i|t} + τ×intcost_{t+i|t}] / ∏_{j=0}^i (1+i^e_{t+j|t})
    + (1 / ∏_{j=0}^N (1+i^e_{t+j|t})) × ((1+g_n)×(1−τ)rent_{t+N|t} / (i^e − g_n)).
- Key modeling choices:
  - Growth rate of nominal rental income assumed identical to household nominal disposable income growth; rent-to-income kept constant in projection.
  - Monthly mortgage computations aggregated to annual totals; frequency of model is annual.
- LTV circularity and solution:
  - Initial loan size must satisfy regulatory LTV (φ); valuation depends on loan size and vice versa.
  - Solution: choose down payment and loan size flexibly so LTV is set in terms of valuation; solve by grid search of scaling coefficients to find fixed point satisfying LTV.

### Application to Prague — parameterization and data inputs
- Household income and scaling:
  - Use nominal disposable income per capita, scaled up by 30% to match average Prague wage; further scaled by 1.65 to reflect relationship between per-capita disposable income and household income in mortgage applications.
- Key calibration values used in application:
  - Debt-service-to-income ratio baseline: α = 35% (sensitivity examined).
  - Loan-to-value ratio: φ = 80%.
  - Rent set at 35% of household income for valuation compatibility.
  - Mortgage duration: N = 25 years; renegotiation horizon: K = 5 years.
  - Offset-deposit DBC assumed to yield an interest rate lower than mortgage rate by 100 basis points (bps).
  - Pre-2008 income forecast projection: AR(1) with persistence ρ = 0.25 and steady-state growth g_n = 4 %.
  - Steady-state mortgage interest rate chosen: 5%.
  - Opportunity cost (discount rate for equity flows): i^e = 6.5%.
  - Effective marginal tax rate assumed: τ = 15%; property tax τ^p = 0 assumed negligible.
  - Average apartment size assumed: 68 m^2.
- Maintenance/depreciation note:
  - Maintenance and depreciation not included in user cost of capital; typical range under moderate assumptions is 1.5% to 2.5% of house value.

### Prague results — borrowing-capacity and intrinsic-value findings
- Borrowing-capacity approach (SBC, DBC variants):
  - Indicated strong overvaluation in 2008 and overvaluation in 2002–2004 and from 2017 onward.
  - Decline in mortgage rates and robust nominal income growth increased attainable prices.
  - Dynamic borrowing capacity mostly tracked static borrowing capacity except for a short deviation in late 2016 / early 2017.
  - Attainable house prices often act as a lower support for observed prices; pricing gap typically positive or zero.
- Counterfactual and decomposition:
  - Counterfactual: mortgage rate decline from 3.42 percent to 2.3 percent (112 basis points) could have raised attainable prices by 15 percent (roughly close to a half-million CZK).
  - Decomposition shows contributions to attainable price from income growth and interest rate changes (interest rate contribution measured relative to 2002Q2 level).
- Sensitivity to α (debt-service-to-income):
  - Static borrowing capacity for α = 30%, 35%, 40%, 45% yields substantially different attainable prices.
  - Example values in Figure 6 (Borrowing Capacity Angel Chart, Prague, Mil. CZK): 3.78, 4.41, 5.04, 5.67 correspond to α = 30%, 35%, 40%, 45% respectively.
  - Moving α from 30% to 45% increases affordable housing by 50% in the calibration.
- Intrinsic-value (valuation) approach:
  - Intrinsic (fair) value is increasing, driven mainly by robust nominal household income growth.
  - Periods of overvaluation relative to fundamentals:
    - ~10% in 2003.
    - ~30% in 2008 (historically highest).
    - ~15% from end-2015 to 2017.
  - Observed market prices rarely fall below attainable/value-implied prices; attainable prices tend to be a lower bound.
  - Over-optimistic income persistence (e.g., increasing AR(1) persistence from ρ = 0.25 to ρ = 0.85 or ρ = 1) increases valuation volatility and can produce valuations that track market booms more closely.
  - Static/steady-state valuation (using current variables as permanent) risks overvaluing property when low interest rates and high income growth are temporary.

### Implications, extensions, and conclusions
- Practical attributes of the proposed indicators:
  - Structural, forward-looking, usable at macro or individual property/household level.
  - Available in real time, not reliant on historical estimation, not subject to revisions.
  - Suitable for scenario and sensitivity analysis; linkable to official macro forecasts.
- Policy and analytical insights:
  - Attainable house prices often form a lower bound for market prices; price gaps are asymmetric (average positive), implying standard regression models that force zero-mean residuals may overestimate equilibrium prices.
  - Key policy-relevant parameters: α (debt-service-to-income) and LTV φ; prudential limits on α materially affect attainable prices.
- Extensions noted by authors:
  - Apply framework to other countries, regions, or cities.
  - Investigate causes of persistent valuation gaps (demand-supply mismatches, demographics, zoning, supply constraints); caution on reverse causality between completed dwellings and registered households.
  - Embed analysis into broader assessment of prudential regulation on household and corporate credit.

*Source: IMF working paper section "4.  Sell the property at the end of the period for an expected price,PH" (wpiea2019059).*

### REFERENCES

### REFERENCES

### Bibliographic citations
- Andrle, M., and M. Plašil, 2019, “A Framework for Assessing the Effects of Macroprudential Policies,” mimeo, Czech National Bank, International Monetary Fund.
- Damodaran, A., 2012, Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, Third Edition (New York: John Wiley & Sons, Inc.).
- ECB, 2011, “Box 3: Tools for Detecting a Possible Misalignment of Residential Property Prices From Fundamentals,” Financial stability review, European Central Bank, Frankfurt.
- ECB, 2015, “Box 3: A Model-Based Valuation Metric for Residential Property Markets,” Financial stability review, European Central Bank, Frankfurt.
- Fox, R., and P. Tulip, 2014, “Is Housing Overvalued?” Research Discussion Paper RDP 2014-06, Reserve Bank of Australia, Sydney.
- Geng, Nan, 2018, “Fundamental Drivers of House Prices in Advanced Economies,” Working Paper 18/164, International Monetary Fund, Washington DC.
- Hejlová, H., M. Hlaváček, and L. Komárek, 2017, “A Comprehensive Method for House Price Sustainability Assessment in the Czech Republic,” Prague Economic Papers, Vol. 26, No. 3, pp. 269–285.
- Hejlová, H., L. Holub, and M. Plašil, 2018, “The introduction and calibration of macroprudential tools targeted at residential real estate exposures in the Czech Republic,” in CNB Financial Stability Report 2017/2018, Occasional Publications - Chapters in Edited Volumes, chap. 0, pp. 126–135 (Czech National Bank, Research Department).
- Himmelberg, Ch., Ch. Mayer, and T. Sinai, 2005, “Assessing High House Prices: Bubbles, Fundamentals and Misperceptions,” Journal of Economic Perspectives, Vol. 19, No. 4 (Fall), pp. 67–92.
- Iacoviello, M., and S. Neri, 2010, “Housing Market Spillovers: Evidence from an Estimated DSGE Model,” American Economic Review, No. April, pp. 125–164.
- Igan, Deniz, and Prakash Loungani, 2012, “Global Housing Cycles,” Working Paper WP/12/217, International Monetary Fund, Washington, D.C.
- McCarthy, J., and R.W. Peach, 2004, “Are Home Prices the Next “Bubble”?” Federal Reserve Bank of New York, Economic Policy Review, No. December, pp. 1–17.
- OECD, 2005, “Recent House Price Developments: The Role of Fundamentals,” OECD Economic Outlook No. 78, Organization for Economic Cooperation and Development, Paris.
- Philiponnet, N., and A. Turrini, 2017, “Assessing House Price Developments in the EU,” Discussion Paper 048/2017, European Commission, Brussels.
- Silver, M., 2012, “Why House Prices Indexes Differ: Measurement and Analysis,” Working Paper WP/12/125, International Monetary Fund, Washington, D.C.
- Warisse, Ch., 2017, “Analysis of the Developments in Residential Property Prices: Is the Belgian Market Overvalued,” Economic Review 4, National Bank of Belgium (NBB), Brussels.

*References continued in source.*

### Appendix A — WHY WE DO NOT USE A REGRESSION MODEL: key arguments and findings
- Primary motivation: preference for a more structural model with a few clearly interpretable parameters and inputs (household income, mortgage rates, loan-to-value ratio) without full general-equilibrium complexity.
- Limitations of regression analysis listed:
  - Regression models are often only vaguely motivated by economic theory in their structure and variable choice.
  - Inclusion of multiple variables (income, multiple interest rates, stock of credit, stock market indicators, demographic variables) can implicitly reintroduce constructs (e.g., loan-to-value ratio) without explicit use for scenario analysis.
  - Explanatory variables typically capture both fundamental and transitory factors, complicating their separation.
  - Econometric models require significant historical data, preferably over multiple business cycles or financial cycles, to avoid parameter instability; short samples can yield severely biased coefficients without theory restrictions.
  - Long-run trends in nominal and real interest rates make it nontrivial to obtain theory-consistent coefficient signs and values without tight priors.
  - Regression models are static and backward-looking, whereas the authors favor dynamic borrowing capacity and present-value computations driven by current and future conditions.
- Empirical illustration using Prague:
  - Estimated co-integrating relationship:
    - logPH_t = α_0 + α_1 logY_t + α_2 (1/i_t) + ε_t, estimated over sample running from 2002:Q1 to 2017:Q4. (Equation (10) in source.)
  - Findings from this Prague exercise:
    - Coefficients are not stable throughout the sample; fitted values for an average apartment change as coefficients are re-estimated.
    - Under no transformation of the interest-rate term do unconstrained coefficients follow the theoretical sign.
    - Coefficient instability limits sensitivity analysis and counterfactual scenarios.
    - As the sample accrues, ex-post views differ from real-time views; the most recent period is especially affected by coefficient instability.
    - Linear regression models with an intercept produce residuals averaging to zero by construction, flagging both under- and over-valuation, which differs from the paper’s approaches that indicate predominance of over-valuation and fair valuation in Prague, with limited undervaluation.
    - Growth-rate specifications with an intercept introduce an autonomous drift in house prices that may exceed fundamentals’ dynamics.
  - Acknowledged limitation: the estimated model is rudimentary; adding more variables with the available sample risks over-parameterization and would require shrinkage estimation with careful cross-validation to avoid over-fitting.
- Objective clarification:
  - The authors’ aim is not to build a forecasting model but to find an ‘equilibrium’ toward which house prices will revert over time.
- Cross-reference:
  - Hejlová, Hlaváček, and Komárek (2017) also point to instability of estimated coefficients in a much richer Czech house price model.

### Appendix B — BUILDING PRUDENTIAL SCENARIOS AND COUNTERFACTUALS WITH THE SBC APPROACH: analysis and policy-relevant mechanism
- Key considerations when using the static borrowing capacity (SBC) approach for policy or counterfactual simulations:
  - Decide whether the scenario targets short-run, medium-run, or long-run effects; assume some variables fixed or slow-moving accordingly.
  - Identify which prudential constraints are binding and their distance from the bound (see Andrle and Plašil (2019) for details).
- Mechanics illustrated (interest-rate decline example):
  - A decline in mortgage interest rates relaxes the borrowing constraint; for unchanged debt-service-to-income (DSTI) ratio, α, the loan amount available increases.
  - Short-run attainable prices depend on down payment behavior and distance from maximum loan-to-value ratio, φ_max:
    - If down payment stays constant and φ < φ_max, households can borrow more and ceteris paribus warranted house prices rise; attainable prices correspond to P_H1 = L_1 + D_0 with L_1 ≥ L_0 and higher leverage until LTV_max binding.
    - If φ = φ_max, lower mortgage rates will not allow households to borrow more unless down payment increases, assuming flexible house prices immediately absorb potential loan increases.
    - Larger house-price increases occur if down payments can increase and accommodate higher mortgage loans while stabilizing leverage.
  - Short-term dynamics of funds available for the down payment are important for assessing prudential policy effects.
- Fundamental relationships provided (equations preserved as in source):
  - P_h_t × H_t = L_t + D_t, s.t. α ≤ α_max, φ ≤ φ_max. (Equation (11))
  - Alternative borrowing-capacity expression (explicit down payment):
    - P_h_t × H_t = [z_t (1 − z^(N×12)_t) / (1 − z_t)] × α Y_t + D_t. (Equation (12))
  - For credit-constrained households linking down payment to income:
    - P_h_t × S_t = [z_t (1 − z^(N×12)_t) / (1 − z_t)] × α Y_t + κ_t × Y_t. (Equation (13))
    - Relationship among κ_t, LTV_t, α_t, and f(z_t):
      - κ_t = (1 − LTV_t) / LTV_t × α_t × f(z_t). (Equation (14))
- Modeling down-payment accumulation for credit-constrained households:
  - Decompose D_t = D_Y_t + D_W_t; for macro scenarios set D_W_t = 0 and focus on past savings accumulation.
  - Simple law of motion (saving s × Y each year for R years, zero interest on liquid saving):
    - D_t = s × Y_t + s × Y_t−1 + ... + s × Y_t−(R−1). (Equation (15))
    - Rewritten with gross income growth g_t:
      - D_t = s × (Y_t + Y_t / g_t + ... + Y_t / ∏_{i=0}^{R−2} g_{t−i}).
    - Steady-state down payment to income ratio:
      - D_t = s × (1 − g^R) / ((1 − g) g^{R−1}) × Y_t. (Equation (16))
    - Emphasis on dependence on R (number of years saving) and income growth g.
- Scenario construction implications:
  - Bounds for prudential-policy effects can be devised using SBC, but a more comprehensive structural model is needed for more advanced tasks.
  - Core assumption: households are constrained and their perceived housing value makes them willing buyers up to their DSTI and LTV potential.

### Appendix C — TOOLBOX: implementation and user interface
- Implementation environments:
  - Toolbox developed using R and Matlab computing environments; other platforms would be straightforward to use.
- User interface:
  - A user-friendly web-based graphic user interface (GUI) has been developed (Shiny/R environment) for flexible and interactive exploration of the borrowing capacity approach.
- Figure reference:
  - Figure 12 in the source shows the interactive interface to the toolbox (Source: Czech National Bank).

*Source: wpiea2019059 - REFERENCES (PDF chapter/section).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019059.pdf_