How do fixed amortization schedules in mortgages affect
homeowners’ saving and the distribution of wealth?

This paper

Life cycle model of homeowners facing uninsurable income risk and a fixed amortization schedule

• Precautionary saving mechanism can explain large effects in empirical literature
• Effects are heterogeneous: younger, poorer homeowners save more; others unaffected

Introduction

Contribution to the literature

• Effects of mortgage amortization on household consumption and saving
  Backman and Khorunzhina (2024), Bernstein and Koudijs (2024), Backman et al. (2024); Larsen et al. (2024); Attanasio et al. (2021)
  • This paper: clarify role of precautionary saving mechanism + long-run effects

Agenda

1. Introduction
2. Model framework
3. Data from Euro area countries
4. Quantitative analysis
5. Conclusion

Model

Overview

Standard incomplete markets model + mortgage debt

• First‐time homebuyer life‐cycle
  • Each agent lives 40 periods, from age 30 (“just bought a house”) to retirement at 70
• Basic features:
  • Two asset types: liquid safe asset (risk‐free) vs. mortgage debt. Housing fixed
  • Idiosyncratic income risk (permanent + transitory)
    
• Key addition – mortgage contract friction
  • Mandatory amortization schedule, transaction cost to reduce payments
  • 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

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

Model

Period problem

\[ \max_{\,c_t,d_t}\; u(c_t)\;+\;\beta\, \mathbb E_t\bigl[V_{t+1}(y_{t+1},a_{t+1},m_{t+1})\bigr] \]

\[ \begin{aligned} a_{t+1} &= (1+r)\bigl[a_t + y_t - (r+s)m_t - d_t - \textcolor{red}{\tau_t}- c_t\bigr] \\[4pt] m_{t+1} &= m_t - d_t &\; \; m_t \ge 0,\; a_t \ge 0 \end{aligned} \]

• Key friction: scheduled repayment \(d_t^{\!*}\), underpaying costs \(\textcolor{red}{\tau_t} \equiv \tau \cdot \max\{0, d_t^* - d_t\}\)

FOC for amortization trades-off marginal value of liquid asset accumulation vs. mortgage repayment

Model

Mechanism: how amortization frictions affect saving

Predictions for consumption and saving under mandatory amortization

• Stronger effects for:
  • Younger: higher expected income growth, lower income, lower wealth (life cycle; down payment)
  • Lower-income: houses, mortgages indivisible
• Little or no effect for wealthier or higher-income homeowners
• Compared to:
  • Flexible repayment scheme (e.g. interest-only mortgages)
  • Renters and others
• Consequence: higher saving rates for constrained mortgaged homeowners
  • Matches stylized facts in Euro area data \(\rightarrow\) life-cycle and income/wealth saving gradients

Data from euro area countries

The Eurosystem HFCS - Household Finance and Consumption Survey

• Harmonized survey of households in Euro area. Three waves (2013-14; 2016-17; 2020-21)
• Compare:
  • Average across Euro area countries (except NL)
  • NL: mostly interest-only mortgages until 2013-14 policy change
• Netherlands policy reform (2013):
  • Until 2013, interest-only mortgages dominated (≈ 60–70% of stock)
  • From 2013, full tax deductibility restricted to fully amortizing annuity 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: \(\textrm{Age} > 70\)

Amortizing mortgages increase saving at the beginning of life cycle

Saving rates over the life cycle (Age 65 = 100)

• Interest-only mortgages show pattern of renters/outright owners Post-2013 policy Life cycle profiles of assets and debt

Amortizing mortgage increase saving only for poorer homeowners

Saving rates over the income distribution (Q5 = 100)

• Again IO mortgages show pattern of renters/outright owners Post-2013 policy
• Age + income heterogeneity \(\rightarrow\) same patterns over the wealth dist Saving rates over wealth dist

Share of income going to amortization declines with age and income

Amortization as % of net income

Quantitative analysis

Income process: inelastic labor supply yields earnings \(Y_{t} = \Gamma_{t}Z_{t}\,\theta_{t}\), as standard: (Carroll & Samwick, 1997)

• \(\Gamma_{t}\) = deterministic life‐cycle profile (grows until age 50, then flat)
  
• \(\ln Z_{t} = \ln Z_{t-1} + \ln \psi_{t}\); \(\ln\psi_{t}\sim N\bigl(-\tfrac12\sigma_{\psi}^{2},\,\sigma_{\psi}^{2}\bigr)\) ; \(\ln \theta_{t}\sim N\bigl(-\tfrac12\sigma_{\theta}^{2},\,\sigma_{\theta}^{2}\bigr)\)

Initial conditions

• A home worth \(P_{0}=5\) (5x annual permanent income)
  
• A 30‐year fixed‐rate mortgage with \(M_{0} \le \theta^{M}P_{0}\) (LTV cap = 80 %)
  
• A small initial liquid buffer: \(A_{0} = 1/12\) of annual income

Bequest motive at retirement (de Nardi, 2004):

\[ B\bigl(a_{T} - m_{T}) = \underline{b} ,\frac{\bigl(a_{T} - m_{T} + \overline{b}\bigr)^{\,1-\gamma}}{1 - \gamma}, \textrm{\; $\underline{b} , \overline{b}$ params} \]

• Mortgage must be fully repaid by retirement \(\Leftrightarrow\) bequest is net wealth \(a_T - m_T\)

Quantitative analysis

Full dynamic household problem

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

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

The household solves the dynamic problem:

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

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

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

Quantitative analysis

Calibration

• Model calibrated for NL (in progress)

• Replicate regime change

  1. Pre-2013: Interest-only free (\(\tau = 0\))
     
  2. Post-2013: High cost of deferred payment (\(\tau = 0.25\))

Model HHs forced to amortize cut consumption until mortgage is repaid

Average age profiles of consumption and saving

Model HHs allowed to optimize backload repayment

Average age profiles of mortgage balance and wealth

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

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

Flattening of saving rate differences reproduces pattern in the data

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
• Sketch of implications for wealth distribution in realistic model

Conclusion

• Mortgage debt repayment is an important part of household saving flows
• Stylized facts from European survey data:
  • Young, low-income homeowners save significantly more than non-mortgaged peers
  • Netherlands interest-only mortgages reveal saving patterns closer to renters
• Can be explained by precautionary saving response of homeowners facing:
• Work in progress with model to:
  • Realistic estimation in the data: income, mortgage costs
  • Endogenize initial contract choice
  • Quantify implications for the wealth distribution

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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Appendix

Amortization implied by annuity formula

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Appendix

Weight of regular mortgage payments on income

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Appendix

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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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

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