How do fixed repayment schedules in mortgages affect saving and wealth?

This paper: model + data

Model matches life-cycle moments of liquid wealth and mortgage debt and reveals effects of fixed repayments

Calibration strategy exploits that only homebuyers after 2013 were exposed to policy

• Data on pre-policy homebuyers identifies preferences; post-policy identifies policy friction (\(\tau^\textrm{post}\))
  • Policy change equivalent to imposing a \(67\%\) premium on under-payment

Main results: what if homeowners had always been subject to the policy?

• 1. Homeowners save more, reducing consumption by \(10\%\) of income, but are more exposed to shocks
  • All additional saving flows into mortgage repayment, not liquid buffers
• 2. \(\uparrow\) Total wealth/income, but \(\downarrow\) liquid wealth \(\Rightarrow\) 15 p.p. \(\uparrow\) \(\%\) HtM
  • \(\Rightarrow\) \(\uparrow\) MPCs, \(C\) volatility
• 3. Restriction to consumption smoothing very costly: equivalent to \(\approx\) 2-3% of lifetime \(C\)
  • Young, lower-income homeowners penalized – those who value flexibility most
• 4. Effects on total/liquid saving rate stronger at bottom of the wealth distribution
  • \(\downarrow\) total wealth inequality but HH hold less liquidity, \(\uparrow\) financial wealth inequality

Contribution

Paper fits in literature modelling life-cycle and welfare effects of mortgage structure
Boar et al. (2022); Balke et al. (2024), Boutros et al. (2025); Ferreira et al. (2024), Greenwald et al. (2018), Karlman et al. (2021), Vestman (2018)

Agenda

1. Introduction
2. Model framework and insights
3. Data: stylized facts and calibration
4. Model results
5. Conclusion

Model framework

Overview

Standard incomplete markets model + mortgage debt

• First‐time homebuyer life‐cycle
  • From origination to maturity of the mortgage
• Basic features:
  • Two asset types: liquid safe asset (risk‐free) vs. mortgage debt. Housing fixed
  • Idiosyncratic income risk (permanent + transitory)
    
• Key addition: mortgage contract transaction costs
  • Mandatory amortization schedule: cost to delay repayment
  • How does this wedge affect saving and wealth accumulation?

Model

Household life cycle endowments and decisions

• A home worth \(P_{0}\) (normalized) and a 30‐year fixed‐rate mortgage with initial balance \(M_{0}\)
• Some initial financial wealth: \(A_{0}\) and exogenous risky earnings \(Y_{t}\) over the life cycle
• Decide each period on how much to:
  • consume \(c_t\) and save each period
  • repay \(d_t\) of their mortgage debt

Households in the model maximise utility from non-housing consumption:

\[ U(c_{t}) = \frac{c_{t}^{\,1-\gamma}}{1 - \gamma} \]

• Only non‐housing consumption enters utility (housing \(H\) fixed)
  • Assumption: prefs separable, so \(\text{argmax} \sum_t u(C_t) = \text{argmax} \sum_t u(C_t, \bar{H})\) (Campbell-Cocco 2015)

Model framework

Assets & mortgage frictions

Liquid saving and mortgage debt

• Savings in the liquid asset (\(a_{t}\)) earn risk‐free interest
  • Borrowing limit \(\,a_{t} \ge 0\,\) (no unsecured debt)
  • Household cannot increase mortgage debt, only repay \(\,d_{t} \ge 0\,\)
• Outstanding mortgage debt demands interest \(r + s\)

Mortgage repayment schedule

• Mandatory amortization: \(D^{*}(m_{t-1},\,t)\) from standard annuity formula
  • Deviating from repayment schedule \(d_t < d_t^*\), then incurs transaction cost \(\tau_t ≥ 0\)
• If default, lose house and keep low consumption \(\underline{c}\) until end
  • Repayment usually feasible under calibration \(y: y > D^{*}(m_{t-1},\,t) + m_{t-1} (r+s)\)

Data from euro area countries, focus on NL

The Eurosystem HFCS - Household Finance and Consumption Survey

• Harmonized survey of households in Euro area. Three waves (2013-14; 2016-17; 2020-21)
• Compare avg. of Euro area versus Netherlands (NL): mostly interest-only mortgages
• Netherlands policy reform in 2013:
  • From 2013, MID restricted to fully amortizing loans – high cost of deferred payment
  • New borrowers forced to amortize → sharp rise in repayment flows

• Data on saving rates from consumption and net income
• Amortization backed out from regular payment: \(12\times\text{mthly pmt}_{i}-i_{i}\times\text{debt}_{i},\) for HH \(i\)
  • The median household in EA amortizes ~10% of yearly income; 2.5% in NL Amortization histograms
  • Various checks on amortization measure Regular payment / income
    Histogram by wavesAnnuity formula Interest rates
• Exclude elderly/retired: \(\textrm{Age} > 70\)

Composition of first-time homebuyers stable across pre- and post-policy

Initial conditions by cohort

• Age, income levels, education mix essentially unchanged
• Interest rates fell: 4.5% → 2% (macro trend)
• Liquid wealth higher, but still low

Cohort of homebuyers subject to policy is repaying faster

Remaining loan balance over time since origination

Cohort of homebuyers is building less liquid wealth

Liquid wealth over time since origination

Calibration

Calibration strategy

1. Fixed/externally calibrated params from data
   • Income process: life-cycle profiles + stochastic properties - 2 education types
   • Initial conditions: empirical distributions of wealth, debt, house value
   • Interest rates (fixed + variable), potentially risky
2. 4 estimated parameters
   • \(\beta\): discount factor
   • \(b_0\): bequest motive strength
     
   • \(\tau^{pre}\) pre-2013 cost of delaying repayment
   • \(\tau^{post}\) post-2013 cost of delaying repayment
3. Target moments: 6 moments from HFCS (x 2 types)
   • Post-2013 and pre-2013 early repayment
   • Pre-2013 late repayment and liquid wealth

Model framework

Full dynamic household problem

In practice, solved in terms of consumption \(c_t\) and a transformed repayment share \(\eta_t\), where:

\[ \eta_t \equiv \frac{d_t}{y_t - (r + s)m_t - \tau_t - c_t} \quad \text{(share of saving used for mortgage repayment)} \]

The household solves the dynamic problem:

\[ V(t, s_t) = \max_{\{c_k, \eta_k\}_{k=t}^{T}} \; \mathbb{E}_t \left[ \sum_{k=t}^{T-1} \beta^{k-t} \frac{c_k^{1 - \gamma}}{1 - \gamma} + \beta^{T - t} B(a_T - m_T) \right], \; \textrm{s.t.} \]

\[ \begin{aligned} d_t &= \eta_t \cdot \left(y_t - (r + s)m_t - \tau_t - c_t \right) \\ a_{t+1} &= (1 + r)\bigl[a_t + y_t - (r + s)m_t - d_t - \tau_t - c_t\bigr] \\ m_{t+1} &= m_t - d_t \\ \tau_t &= \tau \cdot \max\{0, d_t^* - d_t\}, \quad a_t \ge 0, \quad m_t \ge 0, \quad d_t \ge 0 \end{aligned} \]

• Solution: deep learning algorithm proposed by Duarte et al. (2022), Barrera & Silva (2024)

Model framework

Full dynamic household problem

Terminal conditions: bequest motive at retirement to match end-of-life wealth and mortgage debt: \[B\bigl(a_{T} - m_{T}) = b_0 \frac{\bigl(a_{T} - m_{T} + b_1\bigr)^{\,1-\gamma}}{1 - \gamma},\textrm{\; $b_0 , b_1$ params}\]

• Mortgage must be fully repaid by retirement \(\Leftrightarrow\) bequest is net wealth \(a_T - m_T\)
• Parameters (following de Nardi et al. 2004):
  • \(b_0\) intensity of bequest motive
  • \(b_1\) extent to which bequest is luxury good

Calibration

Income process

Calibration

Initial conditions drawn from empirical distributions

• Source: Netherlands HFCS, mortgage holders with ≤ 2 years since origination (purchase ≈ observation)
  • Working on extending with DNB data
• Estimate by education type to account for differences in life-cycle growth

Households simulated from empirical distribution:

• Sample each obs. so the model inherits the observed distribution directly for:
  • Purchase age (25–40)
  • House value / income (P₀ / Y₀)
  • Loan‑to‑value (M₀ / P₀)
  • Liquid wealth / income (A₀ / Y₀)
  • Interest rate on mortgage (assumed fixed)

Older buyers tend to have higher liquid wealth; wealthier buyers select lower LTVs

Calibration strategy

Four preference and transaction-cost parameters estimated to match lifecycle patterns:

Target moments from HFCS span early and late mortgage lifecycle:

Model matches debt repayment dynamics

Model vs. data: mean debt repayment over loan life cycle

• Pre-2013: slower early repayment (flexible contracts)
• Post-2013: faster early repayment despite lower rates
• Both education groups captured by common preferences

Model matches liquid wealth dynamics

Model vs. data: mean liquid wealth over loan life cycle

• Pre-2013 calibration target, model fits well
• Post-2013 targeted only in identifying \(\tau^{post}\)

Calibrated parameters

Estimated parameters are reasonable

Policy effect significant

\(\Delta\tau \approx 0.66\) represents the effect of policy restricting repayment flexibility

• Reflects tax penalty and potential changes in bank products
• Large enough to overcome low rates boost (post-2013 rates fell below liquid returns)
• Identified by sharp observed increase in early-stage repayment

Calibration

Externally calibrated parameters

Model HHs forced to amortize cut consumption until late in the loan lifecycle

Average age profiles of consumption and saving

Income-poorer model HHs save more in total, but less into liquid wealth

Means of model population across income quintiles (conditional on age)

• Saving rate increases, but \(\downarrow\) \(C\), liquid savings
• More exposed to shocks

Flattening of saving rate differences reproduces pattern in the data

Households are more exposed to shocks due to forced amortization

• Share of homeowners with liquid wealth \(< 1\) month of income: \(4\%\) \(\rightarrow\) \(19\%\)
  • \(\Rightarrow\) higher MPCs, \(C\) volatility

Amortization policy imposes large welfare costs

Consumption-equivalent variation by education level

• Median welfare loss (%):

• Households would need 2-3% permanent consumption increase to achieve pre-policy welfare
• Caveat: lower borrowing rates could compensate in part for these costs
  • But movements in spreads were low

Forced amortization increases saving rates at the bottom of wealth dist.

Suggesting effects on distribution of total and financial wealth

• Saving rates increase for groups at the bottom wealth groups
• Implications for wealth distribution: \(\downarrow\) total wealth inequality but \(\uparrow\) financial w. ineq., \(\% HtM\)

Conclusion

Appendix

What is an interest-only mortgage?

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Appendix

Data: mortgage amortization in the HFCS

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Appendix

Amortization by wave

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Appendix

Amortization for mortgages before and after 2013

Percentage of obs. where amortization is less than 5% of the regular payment:

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Appendix

Interest rates

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Amortization implied by annuity formula

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Appendix

Weight of regular mortgage payments on income

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Saving rate measure checks

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Appendix

Data: saving rates in the HFCS

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Appendix

Data from Euro area countries

Saving rates over the wealth distribution (Q5 = 100)

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Appendix

Saving rates over the wealth distribution

Mortgaged homeowners vs. others

• Waves 3 and 4:

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Appendix

Saving rates over the wealth distribution

Mortgaged homeowners vs. others

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Saving rates over the life cycle

Saving by homeowners in NL and others

• No substantial difference between post-policy reform mortgages

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Appendix

Saving rates over the wealth distribution

Saving by homeowners in NL and others

• No substantial difference between post-policy reform mortgages

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Appendix

Age profiles of debt in the data

• Life cycle profile of savings and mortgage debt
• Strict subsample of households who:
  • Have never refinanced
  • Live in their first home
    • Roughly identified by age at purchase \(≤35\)
• Interest-only mortgages: those for which amortization is \(<\) 80% implied by annuity formula

Appendix

Age profiles of debt in the data

Appendix

Age profiles of debt in the data

Appendix

Age profiles of debt in the data

Appendix

Model solution details

HH problem and solution

Basic principle uses stochastic gradient descent to find parameters of neural network that solve for the optimal policy function.

• Machine learning techniques allow to compute the gradient \(\nabla_\theta \tilde{V}\left(s_0, \theta ; \hat{\pi}\right)\)
  • Computationally feasible with ML infrastructure, as neural networks are designed to work with problems with many dimension
  • JAX-based solution (implemented by Barrera & Silva, 2024, nndp)
  • Solved using Google Cloud TPU
• Adjust \(\theta\) according to: \[ \Delta \theta = - \alpha \nabla_\theta \tilde{V}\left(s_0, \theta ; \hat{\pi}\right) \]
  • i.e., move in the direction that reduces the loss function (\(-V\)) the fastest
  • \(\alpha\) is the learning rate

Appendix

Bernstein and Koudijs (R&R QJE): mortgage amortization and saving in the Netherlands, 2012-16

Mortgage amortization & wealth accumulation in 2015 by first-time homeowners buying a house in 2012-13 Back to main