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 restricted exposure to the 2013 policy reform

• Pre-policy, full life-cycle data identifies preferences; post-policy behavior identifies repayment friction \(\tau^{\textrm{post}}\)
• Policy change equivalent to imposing a \(67\%\) premium on under-payment

Contribution

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

Agenda

1. Introduction
2. Model framework
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 framework

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 \(r\)
• Borrowing constraints:
  • \(a_{t} \ge \theta^a\,\) unsecured debt allowed and costly, paying \(r^{-}\)
  • Household cannot increase mortgage debt, only repay or maintain balance \(\,d_{t} \ge 0\,\)
• Outstanding mortgage debt demands interest with a positive spread \(r^m > r\)
  • Precautionary motives and impatience only motives to save in \(a\)

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 \(y: y > D^{*}(m_{t-1},\,t) + m_{t-1} (r+s)\)

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:

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

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

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

Data from the Netherlands

Data source: The Eurosystem HFCS - Household Finance and Consumption Survey

• Harmonized survey of households in Euro area. Three waves (2013-14; 2016-17; 2020-21)
• Data on income, consumption, and household balance sheets including mortgages with great detail

A policy reform in 2013 increased the cost of flexibility

• Before 2013, a lot of flexibility in mortgage repayment, large interest-only components at no extra cost
  • The median household in EA amortizes ~10% of yearly income; 2.5% in NL Amortization histograms
• From 2013, mortgage interest deduction in income taxes restricted to conforming part of the mortgage (paid according to standard schedule)
  • Sharp increase in cost of deferred payment for new homebuyers
  • Elimination of income tax rebate \(\equiv\) 2x cost of carrying non-conforming debt

Calibration strategy

1. Externally calibrated parameters from data
   • Income process: life-cycle profiles + stochastic properties
     • Taken from estimates in Netherlands micro data (de Nardi et al. 2021)
     • 2 permanent income types: higher education and others Income process
   • Initial conditions: empirical distributions of wealth, debt, house value
     • Including mortgage interest rates
2. Estimated parameters (4)
   • \(\beta\): patience
   • \(b_0\): bequest motive strength
     
   • \(\tau^{pre}\): pre-2013 cost of under-payment
   • \(\tau^{post}\): post-2013 cost of under-payment
3. Targeted moments (6) (x 2 education types)
   • Early and late stages of the mortgage life cycle (\(\approx\) 5 yrs and 25-30 yrs after origination):
     • Mortgage repayment (% of loan unpaid), pre- and post-policy
     • Liquid wealth / income, pre-policy (post-policy free)

Calibration strategy

Using the policy change to identify effects of repayment rigidity

Policy increased cost of flexibility for new homebuyers from 2013

• Pre-2013 homebuyers have “free” flexibility
  • Observe full mortgage life-cycle
  • Identify preferences governing saving behavior (\(\beta\), \(b_0\), and baseline \(\tau^{pre}\))
• Post-2013 homebuyers face higher cost to delay repayment
  • Assume same preferences
  • Change in early amortization identifies \(\tau^{post}\)

Validation

• Liquid wealth accumulation of post-2013 buyers is not targeted
• Model reproduces observed crowd-out of liquid saving

Initial conditions drawn from empirical distributions

Sample of mortgage holders with ≤ 2 years since origination (purchase time ≈ observation)

• Initial conditions are similar across pre- and post-2013 cohorts
• Estimated separately by education type

Initial conditions at the household level account for cohort differences

• Households are simulated by drawing directly from the joint empirical distribution of:
  • Purchase age (25–40)
  • House value / income \((P_0 / Y_0)\)
  • Loan-to-value \((M_0 / P_0)\)
  • Liquid wealth / income \((A_0 / Y_0)\)
  • Mortgage interest rate
• Accounts for differences and changes in composition of homebuyers
  • e.g. older and wealthier buyers with more liquidity and lower leverage
  • potential reactions to policy at purchase stage

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 post-2013 homebuyers subject to policy is repaying faster

Outstanding loan balance over time since origination

Cohort of post-2013 homebuyers is building less liquid wealth

Liquid wealth over time since origination

Calibration

Income process

Calibration

Externally calibrated parameters

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

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 \(\Rightarrow\) More exposed to shocks
• Matches saving rate differences in the data Saving rate differences: model vs. 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
  • \(\approx\) working an extra year before retirement (given 40y working life)

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

Implying 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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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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Weight of regular mortgage payments on income

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Saving rate measure checks

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Income process

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Data: saving rates in the HFCS

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Data from Euro area countries

Saving rates over the wealth distribution (Q5 = 100)

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Saving rates over the wealth distribution

Mortgaged homeowners vs. others

• Waves 3 and 4:

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Saving rates over the wealth distribution

Mortgaged homeowners vs. others

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Flattening of saving rate differences reproduces pattern in the data

Appendix

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

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Age profiles of debt in the data

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Age profiles of debt in the data

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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