# Phase transitions in debt recycling

Sabrina Aufiero  Preben Forer  Pierpaolo Vivo  Fabio Caccioli  Silvia
Bartolucci

###### Abstract

Debt recycling is an aggressive equity extraction strategy that potentially
permits faster repayment of a mortgage. While equity progressively builds up
as the mortgage is repaid monthly, mortgage holders may obtain another loan
they could use to invest on a risky asset. The wealth produced by a successful
investment is then used to repay the mortgage faster. The strategy is riskier
than a standard mortgage-repayment plan since fluctuations in the house market
and investment’s volatility may also lead to a fast default, as both the
mortgage and the liquidity loan are secured against the same good. The general
conditions of the mortgage holder and the outside market under which debt
recycling may be recommended or discouraged have not been fully investigated.
In this paper, in order to evaluate the effectiveness of traditional monthly
mortgage repayment versus debt recycling strategies, we build a dynamical
model of debt recycling and study the time evolution of equity and mortgage
balance as a function of loan-to-value ratio, house market performance, and
return of the risky investment. We find that the model has a rich behavior as
a function of its main parameters, showing strongly and weakly successful
phases – where the mortgage is eventually repaid faster and slower than the
standard monthly repayment strategy, respectively – a default phase where the
equity locked in the house vanishes before the mortgage is repaid, signalling
a failure of the debt recycling strategy, and a permanent re-mortgaging phase
– where further investment funds from the lender are continuously secured, but
the mortgage is never fully repaid. The strategy’s effectiveness is found to
be highly sensitive to the initial mortgage-to-equity ratio, the monthly
amount of scheduled repayments, and the economic parameters at the outset. The
analytical results are corroborated with numerical simulations with excellent
agreement.

###### keywords:

Debt recycling, equity release, household finance, loan-to-value ratio,
mortgage affordability.

[1]organization=Department of Computer Science, University College London,
addressline=66-72 Gower Street, city=London, postcode=WC1E 6BT, country=UK
[2]organization=Department of Mathematics, King’s College London,
addressline=Strand, city=London, postcode=WC2R 2LS, country=UK
[3]organization=Systemic Risk Centre, London School of Economics and Political
Sciences, addressline=, city=London, postcode=WC2A 2AE, country=UK

##  1 Introduction

In the contemporary financial landscape, the issuance of mortgages and the
strategies for debt repayment are core elements that support the stability and
growth of both individual finances and the global economy. The decision to
take on a mortgage represents one of the most significant financial
commitments for individuals and families, entailing long-term implications for
their economic well-being [1]. Consequently, the feasibility of debt
repayments emerges as a critical area of study, offering essential insights
into the sustainability of financial practices and the avoidance of over-
leveraging, which can lead to distressing outcomes such as defaults and
foreclosures.From an economic standpoint, mortgages play a central role in the
housing market, influencing not only the dynamics of supply and demand, but
also the overall economic cycle. The conditions under which mortgages are
issued – including the assessment of borrowers’ ability to repay – can
significantly affect the health of the financial system. Lax lending standards
can lead to increased default rates, with cascading effects on the housing
market, financial institutions, and the broader economy, as vividly
demonstrated by the 2008 financial crisis [2].The study of debt repayment
feasibility and mortgage issuance is thus imperative for crafting policies and
practices that promote financial stability and responsible lending. It
involves analyzing various factors, including interest rates, income
stability, debt-to-income ratios, and economic conditions, to assess the risk
profiles of borrowers and the sustainability of debt obligations. Moreover,
this analysis is crucial for the development of innovative financial products
and strategies that can enhance access to housing, while managing the risks
associated with borrowing. Traditional mortgage schemes, while providing a
structured path towards home ownership, inherently come with a significant
limitation: the equity accumulated in the property remains largely
inaccessible and non-productive until the mortgage is fully repaid. This
standard feature of conventional mortgages may result in a sub-optimal use of
homeowners’ financial resources, as the equity tied up in the property does
not yield any financial benefits or returns during the loan’s tenure.
Essentially, while the equity builds up in time, it remains dormant and does
not contribute to the mortgage holder’s broader financial well-being and/or
investment capacity [3].This issue has prompted financial experts and
institutions to design and propose alternative mortgage and debt management
strategies. These innovative approaches aim to address the inefficiencies of
traditional models by providing mechanisms through which homeowners can
leverage their accumulated equity more effectively. The goals of these
strategies are multifaceted and include accelerating the repayment process,
enhancing the utility of locked equity, and potentially reducing the rates of
default by improving the financial flexibility and resilience of borrowers
against external stresses.

In this paper, we will focus primarily on debt recycling, an aggressive
strategy of equity extraction currently available only to Australian mortgage
holders. Debt recycling leverages home equity building up in time to finance
(risky) investments, effectively converting non-deductible mortgage debt into
tax-efficient investment debt. This strategy offers a stark departure from
traditional and more conservative financial strategies, which typically
prioritize mortgage repayment before any investment activity is undertaken.

The main idea is fairly simple: while the mortgage is being repaid, the home
equity (essentially, the value of the fraction of the house owned thus far) is
used to back a second loan, which in turn is invested into a (risky) asset.
This way, the non-deductible Principal Place of Residence (PPOR) mortgage debt
is converted into tax-efficient investment debt111The key to the success of
debt recycling lies in the tax-deductibility of the interest on the loan used
for investment purposes. The Australian Taxation Office (ATO) [5] states when
the interest on a loan is considered tax deductible: “[…] If you borrow money
to buy shares or related investments from which you earn dividends or other
assessable income, you can claim a deduction for the interest you pay. Only
interest expenses you incur for an income-producing purpose are deductible.”,
with the extra income hopefully generated by the investment becoming now
available to repay the home mortgage faster. Recycling borrowed money
therefore converts “unproductive” mortgage debt into additional cash flow that
enjoys tax benefits, at the same time helping ease the financial burden on the
house while it is being repaid. Other forms of less risky equity release are
available in various markets, for example financial strategies to allow
elderly citizens to extract liquid income from their property without having
to relocate [3] or debt swap strategies [4], which are reviewed in the next
section.

The peculiar and rather extreme feature of the Australian debt recycling
scheme, though, precisely lies in the interplay between _two_ (or more) loans
that are backed by the _same_ good (the Principal Place of Residence, whose
full ownership is being secured). While a good investment coupled with a
booming house market can lead to increased locked-in equity being available,
and in turn to a faster repayment of the mortgage, a bad investment or a
depreciation of one’s own house due to an economic downturn may increase the
financial strain on the loans’ holder to the point that they are no longer
able to fulfill their obligations towards the lender, with potentially severe
consequences.

Let us consider the case of a home valued at $800,000 with a remaining
mortgage of $500,000. The owner may access $100,000 from their equity via a
line of credit. The expected annual return from the $100,000 investment would
be 7%. The owner can then use the $7,000 annual return to make additional
payments on their home mortgage. Over time, these extra payments not only
reduce the principal faster, but also decrease the amount of interest they
would pay over the life of the mortgage. Moreover, the interest they pay on
the $100,000 line of credit, which could be around 4% annually, is tax-
deductible because it is tied to investment purposes. This deduction can
offset the taxable income, providing further financial benefit. As their
mortgage balance decreases and their home equity increases due to both
repayments and potential appreciation of their home’s value, they could
consider drawing additional funds for further investments. They might also re-
invest dividends received, compounding their returns over time.

The precise conditions of the borrower and the general economic climate that
may lead to one outcome or the other are not described yet, and the literature
on the topic is quite scarce, due to the novelty of the scheme, which
flourished only recently due to rising home equity tied to booming real estate
markets [6]. Financial advisors and planners began advocating for more
sophisticated wealth-building strategies, including debt recycling, as part of
a broader approach to financial management.

In this paper, we provide a first dynamical model in discrete time of the
joint evolution of home equity and mortgage, assuming that the mortgage holder
follows a regular repayment schedule on top of converting a fraction of the
equity to-date into a risky investment, whose performance is fluctuating
randomly in time. The house market is also fluctuating randomly, leading to
larger or smaller equity available to the investment. The average behavior in
time of the equity and the remaining mortgage can be computed analytically as
a function of the main parameters. We study the first _hitting time_ , namely
the first time at which either the remaining mortgage hits zero (signaling
that the house has been repaid in full), or the equity locked in the house
hits zero (signaling that the house is fully mortgaged and the debt recycling
strategy has failed). In the former case, we also monitor whether the full
repayment (and consequent full ownership of the house) has materialized before
or after what would have naturally happened if only the traditional monthly
repayment strategy were adopted.

We find that the complex interplay between house market and investment
performance leads to a rich phase diagram, with regions of (strong or weak)
success for the debt recycling strategy sharply separated from regions where
the strategy fails, or where _permanent re-mortgaging_ may occur. In this
scenario, securing additional investment funds from the lender creates a cycle
where, despite ongoing investment losses, equity consistently exceeds the
mortgage. This dynamic could potentially lead to an endless cycle of re-
mortgaging. Although our model accommodates this theoretical possibility, it
is unlikely to materialize in real-life settings. Our work therefore paves the
way for a firmer quantitative understanding of the personal and environmental
circumstances that make debt recycling a viable or non-viable strategy to
repay one’s mortgage faster.

The plan of the paper is as follows. In Section 2, we briefly review the
existing literature on mortgages and risks of default, and strategies for
equity release. In Section 3, we set up the model and discuss the relevant
variables and dynamical equations. In Section 4 we briefly summarize our
results for the phase diagram and the main findings from analytical
considerations and numerical simulations. In Section 5 we describe the
mathematical methods we employ to lead to the analytical solution of the
model. In Section 6 we report an extended discussion of results, including
parameter dependence and different scenarios. Finally, in Section 7 we offer
some concluding remarks and outlook for future research. The Appendix is
devoted to technical details.

##  2 Related Works

In this section we analyse the main literature related to (i) modelling and
data analysis of mortgage defaults and (ii) assessment of debt recyclying and
other equity extraction strategies.

###  2.1 Models and Analyses of Mortgage Default

Credit default models are essential tools for understanding the dynamics that
influence homeowners’ decisions to default on mortgages. These analyses
incorporate a variety of risk factors and economic variables to predict
default behaviors under different circumstances.

The life-cycle model illustrated in [7] shows that households face income and
house-price risk when buying homes with mortgages. Combining recourse
mortgages and loan-to-valie (LTV) limits effectively reduces default rates,
and increases housing demand. The effect of monthly payment size on mortgage
default is studied by Fuster et al. in [8]. They find that payment reductions
substantially lower mortgage defaults, even for borrowers that are deeply
underwater: for instance, cutting the required payment in half reduces the
delinquency hazard by about $55\%$.

Campbell et al. in [9] solve a dynamic model of households’ mortgage decisions
incorporating labor income, house price, inflation, and interest rate risk.
The model quantifies the effects of adjustable versus fixed mortgage rates,
LTV ratios, and mortgage affordability measures on mortgage premia and
default. The model highlights the fact that the default decision depends not
only on the extent to which a borrower has negative home equity, but also on
the extent to which borrowers are constrained by low current resources. In the
model, constraints shift the threshold at which a borrower optimally decides
to exercise the irreversible option to default.

The impact of rising income inequality on the overall level of debt in the
system as well as the default rates was studied by Kösem in [10]. By
developing an equilibrium-based model of the mortgage markets, they show that
rising income inequality actually leads to lower amount of mortgage debt, but
higher mortgage default rates. This is due to a higher share of borrowers
forced into risky low-value loans.

An empirical study into the effect of personal financial health on the
probability of delinquency is done by Mocetti and Viviano in [11]. Using
datasets from Italian tax returns office and the Bank of Italy, they examined
how banks’ selection policies and income shocks affect the rate of default. In
particular, they show that the trend in default rates could be explained by
the tightening of selection policies post 2008, and that individual financial
health shocks have an out-sized impact on default rates. An income drop of 10%
caused default rates to go up by 5%, while job losses double the likelihood of
default.

Another study empirically looking at the link between LTV ratios and the risk
of defaults was done by González et al. [12]. Using regression models on data
from mortgage loan portfolios in Spain between 2005-2008, they found a non-
linear relationship between the two. Indeed, higher LTV ratios were associated
with higher default rates, with a sharp increase seen for values greater than
80%.

The study in [13] shows that real estate properties can be sold for more than
their collateral values, leading to a significant bias in the LTV ratio that
understates credit risk, especially with longer mortgage terms, higher LTV
ratios, and aggressive lending products. This bias suggests that many
mortgages, particularly those from the housing bubble peak, were already risky
at the outset, questioning current underwriting and risk control practices.

The relationship between an individual’s mortgage repayment capacity and its
impact on mortgage default risk was studied by O’Toole et al. [14]. Using data
from 2004-2007 Ireland, they used a household’s debt service ratio (i.e., the
portion of net income that goes towards mortgage repayments), as a measure of
a household’s repayment capacity. They found that a deterioration of the debt-
service ratio led to an increase in default risk, regardless of the initial
debt load. Additionally, in times of crisis this is worsened due to the
presence of negative equity and liquidity constraints, essentially removing
the household’s buffer.

###  2.2 Equity Release

A popular home-loan investment strategy in Europe and U.S. is the so-called
equity release or reverse mortgage strategy. Equity release, often offered and
considered by individuals aged 55 or above, represents a financial strategy
enabling homeowners to access their property’s equity without relocating. As
eligibility criteria for older homeowners has become increasingly strict for
traditional mortgages, using equity release to pay off a mortgage early has
emerged as a popular way for homeowners in retirement to clear their existing
mortgage debt [3, 15, 16]. This mechanism permits the extraction of funds
either as a lump sum or in increments, without necessitating immediate
repayments. The schemes embed a No Negative Equity Guarantee (NNEG),
guaranteeing that the amount owed to the bank cannot be higher than the value
of one’s house when sold. However, the compounded interest over time can
significantly inflate the owed amount, potentially impacting the estate’s
value for inheritors.

In [17], a quantitative model is developed to study the differences between
equity release schemes from the perspective of a retired client: the reverse
mortgages are the best option for prospective clients due to the higher lump
sums paid out, as well as the protection against falling house prices provided
via the NNEG.

The NNEG has been found by Sharma et al. [18] to be one of the reasons for the
disparity in use patterns of equity release schemes across the UK. They found
that equity release schemes should only be used in areas of high house price
due to the regional fluctuations of the cost of the NNEG that are not
accounted for by market providers. Ignoring these regional fluctuations
outside of areas of high house price growth leads to pre-mortgage equity
release schemes being unprofitable to providers. The question of profitability
of equity release schemes has also been studied by Hosty et al. [19]. They
investigate whether or not the competitiveness of the equity release market
has pushed the sector towards unsustainable prices. To do so, they model the
pricing of such products through ‘average’ pricing assumptions that are meant
to provide a benchmark to industry members. Another way of modelling the NNEG
price was developed by Lie et al. in [20]. They propose a hedging strategy for
the provider to use in order to hedge the house inflation risk. Using Monte
Carlo simulations they could then estimate the hedging cost which allows them
to price the NNEG. An overview and comparison of the ways of modelling the
NNEG, and thus pricing equity release programs, was done by Tunaru in [21]:
there is no best way of modelling the NNEG itself, and many different
approaches have different benefits and drawbacks. The key to making each
approach work is indeed accurate modelling of house prices, as well as testing
the models for a variety of different scenarios.

A qualitative study has also been undertaken by the Financial Conduct
Authority of the United Kingdom, aiming to understand the motivations and
thought process of people who purchase reverse mortgages and equity release
schemes [22]. They found that negative outcomes for the client were driven by
the following key factors: the presence of a strong time constraint, being
vulnerable to outside pressure, being financially unsophisticated, not seeking
professional advice, and failing to explore other options.

More specifically, within strategies of debt conversion we also recall the
debt swap strategy [4]. The homeowner disposes of their non-registered
investments and uses the proceeds from the sale to pay off the mortgage (the
non-deductible debt). They subsequently re-borrow the same amount (secured by
the home) and use the proceeds to purchase a non-registered portfolio of
income-producing assets. At this point, since they are then directly using the
borrowed money for the purpose of earning income, the interest paid on the re-
borrowing may be deductible for tax purposes.

Concerning the more aggressive debt recycling strategy, there is very little
literature available. Most of the available information explain the risks and
benefits associated with the strategy in a qualitative way. There are
seemingly no quantitative analyses of the topic, as opposed to the extensive
economic models developed for reverse mortgages and the NNEG discussed
earlier.

##  3 Model Setup

In this section, we introduce a discrete-time model for the evolution of
mortgage and equity values via a debt-recyclying strategy incorporating the
following update mechanisms: (i) monthly loan repayments, accounting for the
principal and interest components of each monthly mortgage payment arranged
with the lender, (ii) fluctuations of the house market that affect the
property value, (iii) dynamics of equity investment, as a fraction of their
equity is invested via different routes.

At time $t=0,\ldots,T$ the house value $H_{t}$ is simply the sum of the equity
owned $E_{t}$ and the mortgage $M_{t}$,

$H_{t}=E_{t}+M_{t}\ .$ (1)

We assume that the usable equity at time $t$ – the amount that that can be
used to back an investment – is a fraction of the equity owned

$U_{t}=\ell E_{t}\ ,$ (2)

where $\ell\in[0,1]$ is the so-called loan-to-value ratio (LTV). The LTV ratio
– calculated by dividing the amount of the loan by the appraised value or
purchase price of the property – is used by financial institutions and other
types of lenders to assess the lending risk before approving a mortgage [1].
Typically, loan assessments with high ($>80\%$) LTV ratios are considered
higher-risk. In this scenario, mortgage and equity values can vary depending
on the (i) house market dynamics, and (ii) the performance of the investment.
At each time step, a fraction $I_{t}$ of the investment $U_{t}$ can be either
earned or lost depending on the inherent risk associated with the investment

$I_{t}=\sigma_{t}\mu U_{t}\ ,$ (3)

where the risk factor $\mu\in[0,1]$ directly correlates to the investment’s
risk level. If the investment is successful, the wealth gained can be used to
further cover part of the mortgage, leading in turn to an increase in equity
owned. A higher risk factor $\mu$ signifies a riskier investment. The sequence
of random variables ${\bf\sigma}=\\{\sigma_{0},\dots,\sigma_{T}\\}$ with
$\sigma_{t}=\pm 1$ models the investments’ positive or negative trend. The
house market fluctuates as well, leading to an increase or decrease in the
house value in time, therefore directly impacting the equity owned as well. In
particular, at each time step a fraction $H_{t}r_{t}$ of the house value may
be either lost or gained, depending on the house market trend. The random
variable $r_{t}$ represents the percentage price change of the house market at
time $t$.

So, at a generic time step $t$, $E_{t}$ and $M_{t}$ are assumed to evolve
according to the following set of coupled equations:

$\displaystyle E_{t}$ $\displaystyle=E_{t-1}+\pi_{t}+I_{t-1}+H_{t-1}r_{t}\ ,$
(4) $\displaystyle M_{t}$ $\displaystyle=M_{t-1}-\pi_{t}-I_{t-1}\ .$ (5)

The meaning of the evolution equations (4),(5) is as follows. The value of the
equity increases at each step by an amount $\pi_{t}$ repaid regularly against
the mortgage, which in turn decreases by the same amount. The equity may also
increase (decrease) if the house market is growing (shrinking), with $r_{t}>0$
($r_{t}<0$). A positive (negative) return on the investment $I_{t}$ leads
respectively to an increase (decrease) in equity owned that can be re-
invested, and a decrease (increase) in exposure towards the lender via the
mortgage term.

Via simple manipulations, we can rewrite the process as follows:

$\displaystyle E_{t}$ $\displaystyle=\alpha_{t}E_{t-1}+\pi_{t}+r_{t}M_{t-1}\
,$ (6) $\displaystyle M_{t}$ $\displaystyle=M_{t-1}-\pi_{t}+\delta_{t}E_{t-1}\
,$ (7)

where

$\displaystyle\alpha_{t}$ $\displaystyle=1+\ell\mu\sigma_{t-1}+r_{t}\ ,$ (8)
$\displaystyle\delta_{t}$ $\displaystyle=-\ell\mu\sigma_{t-1}\ .$ (9)

In matrix form

$\begin{pmatrix}E_{t}\\\ M_{t}\end{pmatrix}=\begin{pmatrix}\alpha_{t}&r_{t}\\\
\delta_{t}&1\end{pmatrix}\begin{pmatrix}E_{t-1}\\\
M_{t-1}\end{pmatrix}+\begin{pmatrix}\pi_{t}\\\ -\pi_{t}\end{pmatrix}\ .$ (10)

The process in Eq. (10) ends at the first _hitting time_
$t^{\star}=\min(t_{E},t_{M})$ (unless it continues indefinitely). The time
$t_{E}$ characterizes the first time at which $E_{t_{E}}=0$, signaling that
the homeowner’s equity (seen as value of the fraction of the house owned) has
been massively depleted: the house is now entirely mortgaged, and the owner
can only rely on monthly repayments to regain ownership of the house. In this
extreme scenario, the debt recycling strategy has clearly failed. The time
$t_{M}$ characterizes instead the first time at which $M_{t_{M}}=0$, showing
that the borrower has managed to pay off their mortgage entirely, therefore
now fully owning their house. In this scenario, we deem the strategy weakly
successful if the time taken to acquire the house through debt recycling,
$t^{\star}$, exceeds the time it would have taken to repay the house by
standard monthly repayments, and as strongly successful if the time required
to own the house with the strategy is less than it would have been without
recycling.

The question is therefore how to characterize $t^{\star}$, the _first hitting
time_ of the stochastic process onto the absorbing boundaries $E=0$ or $M=0$,
and consequently which boundary is reached first.

In the following section, we are going to define the probability distributions
and parameter assumptions of our model. Later, we are going to summarize the
results for the hitting time of the average processes $\langle E_{t}\rangle$
and $\langle M_{t}\rangle$, where the averages are taken over the realizations
of the random variables involved.

###  3.1 Assumptions

  * 1.

We assume that the initial values of both equity and mortgage, denoted as
$E_{0}$ and $M_{0}$ respectively, are fixed. Consequently, the average values
are $\langle E_{0}\rangle=E_{0}$ and $\langle M_{0}\rangle=M_{0}$.

  * 2.

The return of the investment $I_{t}$, defined in Eq. (3), depends on the
random variable $\sigma_{t}$. We assume that $\sigma_{t}$ at different times
are independent Bernoulli variables

$p(\sigma)=p\delta_{\sigma,+1}+(1-p)\delta_{\sigma,-1}\ ,$ (11)

accounting for the possibility of investment gain $(+1)$ or loss $(-1)$ with
respective probabilities $p$ and $1-p$. Consequently, the average investment
outcome is:

$\langle\sigma\rangle=2p-1\ ,$ (12)

where the parameter $p\in[0,1]$ denotes the probability of earning from the
investment. A higher value of $p$ implies a greater likelihood of generating
wealth from the investment, which can be funneled into the mortgage repayment.

  * 3.

We assume that the fluctuations in the house market value, introduced in Eq.
(4) with the term $H_{t-1}r_{t}$, are encoded in a normal distribution $p(r)$

$p(r)\sim\mathcal{N}(s,\phi^{2})$ (13)

with average222The choice to let $s$ vary within the range $[-4\%,+4\%]$ is
based on data regarding the percentage change in the Residential Property
Price Index (RPPI). The RPPI aggregates the Established House Price Index
(HPI) and the Attached Dwelling Price Index (ADPI) to measure changes in
residential dwelling prices across Australia’s eight Greater Capital City
Statistical Areas (Sydney, Melbourne, Brisbane, Adelaide, Perth, Hobart,
Darwin, and Canberra). This index, published by the Australian Bureau of
Statistics (ABS) assesses the residential property price inflation, covering
all residential properties, regardless of ownership or the occupants’ tenure.
$s\in[-4\%,+4\%]$ and variance $\phi^{2}$. A higher $s$ value indicates a
positive market trend, leading to an increase in house value; instead, a
negative $s$ value implies poor market performance, resulting in a loss in
house value.

  * 4.

The scheduled repayment to the bank $\pi_{t}$, introduced in Eq. (4), is
assumed to be non-negative ($\pi_{t}>0$) at every time step $t$, indicating
consistent payments to the bank. However, our model allows for the rare event
of a few installments of the repayment being skipped (with probability $q$).
Otherwise, with probability $1-q$ each installments has value $\pi^{\star}$.

$p(\pi)=q\delta(\pi)+(1-q)\delta(\pi-\pi^{\star})\ ,$ (14)

implying an average installment of

$\langle\pi\rangle=(1-q)\pi^{\star}\ .$ (15)

In summary, the parameters of the models are:

  * 1.

$\ell\in[0,1]$: LTV ratio;

  * 2.

$\mu\in[0,1]$: risk factor of the investment;

  * 3.

$p\in[0,1]$: probability that the investment produces a positive income
(gain);

  * 4.

$s\in[-4\%,+4\%]$: average of the fluctuations in the house market value.

  * 5.

$q\ll 1$: probability of skipping an installment of the repayment to the
lender;

  * 6.

$\pi^{*}$: value of the monthly installment;

  * 7.

$E_{0},M_{0}$: initial conditions of equity and mortgage respectively.

Given the availability of official housing market data, which are typically
reported on a quarterly basis, we define our model’s time steps as quarters,
equating to 3-month periods.

##  4 Summary of Results

In this section, we are going to present a concise summary of our findings. We
then provide the methodology in Section 5, and a more detailed discussion of
the results in Section 6.

In Section 5, we will find that the evolution in time of the equity $E$ and
mortgage $M$ is described by a $2\times 2$ matrix. Since we are interested in
the average processes $\langle E_{t}\rangle$ and $\langle M_{t}\rangle$ –
where the averages are taken over the realizations of the random variables
involved – we focus on the average of the matrix that describes the evolution.
We find that the relevant variables to describe our processes are the
combinations $\lambda_{1}=s+1$ and $\lambda_{2}=\ell\mu(2p-1)+1$, which
correspond to the eigenvalues of the average matrix.

To further fix the setup, we now consider $\ell=0.5$ and $\mu=0.5$,
corresponding to a scenario where the LTV ratio is below the suggested maximum
value of $\sim 80\%$, and the fraction earned or lost is half of the
investment.

In Fig. 1, we present a schematic summary of the space of possible values
taken up by the eigenvalues. The markers indicate the four possible
combinations of their magnitudes – reflecting all potential outcomes, given
that $\lambda_{1}$ and $\lambda_{2}$ cannot be negative, regardless of
parameter choices. The red lines represent the transitions between different
zones. Alterations in the parameters $\ell$ and $\mu$ affect only the range
within which $\lambda_{2}$ varies, but the transition thresholds are
consistent: at $s=0\%$, where $\lambda_{1}$ exceeds $1$, and at $p=0.5$, where
$\lambda_{2}$ exceeds $1$.

Figure 1: $\lambda$-space for $\ell=0.5$, $\mu=0.5$.
Legend:
I quadrant ($\bullet$): $\lambda_{1,2}>1$
II quadrant ($\blacksquare$): $\lambda_{1}>1;0<\lambda_{2}<1$
III quadrant ($\times$): $0<\lambda_{1,2}<1$
IV quadrant ($\bigstar$): $\lambda_{2}>1;0<\lambda_{1}<1$

The four regions depicted in the $\lambda$-space in Fig. 1 correspond to the
four dynamical scenarios summarized in Table 1, which in turn end up in three
possible outcomes of the debt recycling strategy. The strategy is deemed
successful when the average process reaches the $M=0$ boundary first, a result
achievable with all four potential combinations of eigenvalue magnitudes.
Whether the strategy is weakly successful – if the time $t^{\star}$ needed to
acquire the house via debt recycling exceeds the time needed without the
strategy – or strongly successful – if $t^{\star}$ is smaller than the time
needed without debt recycling – depends on the specific scenario and the
values of the parameters involved. The strategy defaults (fails) when the
process hits the $E=0$ boundary first, an occurrence seen in the III and IV
quadrants of the $\lambda$-space. Lastly, the strategy may fall into a state
of permanent re-mortgaging where neither process ever hits the absorbing
boundary: as the house value increases, securing additional bank funding for
investment creates a virtuous cycle of equity extraction. Despite sustained
investment losses, equity remains above the mortgage, a situation observed in
the II quadrant. In practice, this theoretical outcome will eventually be
halted by the lender’s decision to discontinue funding at some point. We will
now discuss the possible dynamical scenarios in details.

  * 1.

I quadrant ($\bullet$): $\lambda_{1,2}>1$; Strong Success.

The parameters significantly favor the mortgage holder due to a positive trend
in the housing market ($s>0$) combined with high likelihood of the investment
yielding a positive return ($p>0.5$). The average equity displays an
increasing exponential trend, whereas the average mortgage rapidly decreases,
reaching the absorbing boundary at zero within a few time steps. Consequently,
the strategy proves to be strongly successful.

  * 2.

II quadrant ($\blacksquare$): $\lambda_{1}>1$, $0<\lambda_{2}<1$; Weak Success
or Permanent Re-Mortgaging.

The housing market shows a positive trend ($s>0$), this time paired with a low
probability of the investment yielding a positive return ($p<0.5$). This
configuration leads to two possible outcomes for the strategy, determined by
the value of the product $\ell\mu$: either the average equity increases while
the average mortgage decreases and eventually hits zero, marking the strategy
as weakly successful, or neither process ever hits the absorbing boundary,
displaying either exponential growth or stabilizing at a constant value,
determined by the magnitude of $s$. This dynamics happens because, in
scenarios where the house keeps increasing in value ($s>0$), obtaining
additional funds from the bank for investment initiates a cycle where, despite
persistent investment losses, equity consistently exceeds the mortgage at each
time step $t$. This cycle leads to further lending, potentially resulting in a
never-ending sequence of re-mortgaging. While our model theoretically allows
for this scenario, this is unlikely to occur in practice, indicating the need
for external mechanisms to suppress it (typically, a time cap in the lender’s
willingness to provide credit).

  * 3.

III quadrant ($\times$): $0<\lambda_{1,2}<1$; Success or Default.

In this region, the values of the parameters should discourage the mortgage
holder from undertaking a debt recycling strategy due to a negative trend in
the housing market ($s<0$), and a low probability of the investment yielding a
positive return ($p<0.5$). While the average mortgage consistently decreases,
the average equity reaches an absolute minimum in the early stages of the
process before following an increasing trend. For a given $p$, but with
increasing $s$, the average equity curves move upwards, thereby creating the
possibility of an early default if the minimum of the equity is negative, but
also the possibility of a success (both strong and weak) if the minimum is
positive. In the majority of scenarios, though, due to unfavorable parameters,
the time required for the strategy to be successful is considerably high,
suggesting that a simple monthly repayment without recycling is the better
strategy. In Fig. 2 the equity for fixed $p=0.4$ and increasing $s$ is shown.

Figure 2: Average equity dynamics for increasing $s$ and fixed $p$, in the III
quadrant ($\times$) of the $\lambda$-space.

  * 4.

IV quadrant ($\bigstar$): $\lambda_{2}>1$, $0<\lambda_{1}<1$; Success or
Default.

There is a negative trend in the housing market ($s<0$), but a high
probability of the investment yielding a positive return ($p>0.5$). The
outcome of the strategy can either be success (both strong and weak), with the
average equity showing an increasing exponential trend and the average
mortgage steadily decreasing, or default if the trends are respectively
decreasing and increasing. When the strategy is successful, it is almost
always strongly successful, as a $s<0$ value implies that, with each time
step, the total value of the house decreases, thereby also lowering the paid
equity threshold required to achieve ownership of the house.

In Table 1, we summarize the various outcomes of the strategy in different
regions of the phase diagram, and the typical process dynamics in each case.

We can now construct a phase diagram in the $(p,s)$ plane, keeping the other
parameters fixed.

In Fig. 3 we present the phase diagram of the debt recycling strategy with the
following parameters: average initial equity $\langle E_{0}\rangle=\$30,000$,
i.e., the amount that the homeowner paid at the beginning of the mortgage
agreement, or equivalently the value of the portion of the house that the
borrower owns totally from the beginning, free from any liabilities or
mortgages; average initial mortgage $\langle M_{0}\rangle=\$300,000$, i.e.,
the part of the property’s purchase price that is financed through the loan;
value of the installment of the repayment to the bank $\pi^{*}=\$3,000$;
$q=1\%$ chance of skipping an installment; LTV ratio $\ell=0.5$; risk factor
of the investment $\mu=0.5$.

Within the parameter space, we analyze the average equity and mortgage,
coloring each point of the phase diagram in orange, purple, or gray to
represent the three potential outcomes of the strategy detailed in Table 1.
Orange indicates a successful strategy (both weakly and strongly), where the
mortgage holder is generally able to fully repay the mortgage and achieve
complete ownership of the house. Purple signifies a default scenario, where
the house value is entirely mortgaged, suggesting that a standard monthly
repayment strategy (without recycling) should have rather been undertaken.
Gray is used for scenarios leading to permanent re-mortgaging.

The color intensity directly correlates with how fast the final outcome is
achieved: darker orange (purple) indicates faster acquisition (loss) of house
ownership. For visual clarity, the maximum intensity of orange is limited to
$t=400$ quarters, or $100$ years, with points depicting weakly successful debt
recycling beyond $100$ years shown in beige. The beige region is, obviously,
unphysical.

Markers on the diagram align with the legend in Fig. 1 or the first column of
Table 1, showing the possible combinations of the magnitudes of eigenvalues
$\lambda_{1}$ and $\lambda_{2}$.

Lime lines represent contour lines, which are the locus of points on the plane
$(p,s)$ that have the same first hitting time $t^{\star}$. We choose to
illustrate the points of the plane where the process concludes at 5 and 15
years, aligning with the guideline that debt recycling is best suited for
individuals who can sustain the cycle for a minimum of 5 years. This approach
is shared by many financial advisory firms and mortgage consultancy companies
that provide guidance on loans and accounting services.

| Strategy Outcome | Process Dynamics
---|---|---
| $\bullet$
---
$\lambda_{1,2}>1$
|  Strong Success The average equity exhibits an increasing exponential trend;
the average mortgage shows a decreasing exponential pattern, until it is
eventually extinguished.
---
|
---
| $\blacksquare$
---
$\lambda_{1}>1$
$0<\lambda_{2}<1$
|  Weak Success (low $\ell\mu$) The average equity exhibits an increasing
trend; the average mortgage shows a decreasing pattern, until it is eventually
extinguished.
---
Permanent Re-Mortgaging
(high $\ell\mu$) Neither process ever hits the absorbing boundary: they either
display increasing behavior or converge to a constant.
|
---
| $\times$
---
$0<\lambda_{1,2}<1$
|  Success (high $s$) The average equity displays a positive minimum in the
initial phase, then follows an increasing trend; the average mortgage shows a
decreasing pattern, until it is eventually extinguished.
---
Default (low $s$) The average equity rapidly reaches a negative minimum in the
initial phase; the average mortgage shows a decreasing pattern, but slower.
The strategy defaults.
|
---
| $\bigstar$
---
$\lambda_{2}>1$
$0<\lambda_{1}<1$
|  Success The average equity exhibits an increasing exponential trend; the
average mortgage shows a decreasing exponential pattern, until eventually it
is extinguished.
---
Default The average equity exhibits a decreasing exponential trend, hitting
the absorbing boundary early on; the average mortgage shows an increasing
exponential pattern. The strategy defaults.
|
---

Table 1: Average equity and mortgage dynamics, based on the value of the
eigenvalues $\lambda_{1}$ and $\lambda_{2}$. Figure 3: Here and in the
following phase diagrams, the plot is constructed as follows: area in orange
represents where debt recycling is (either weakly or strongly) successful;
area in purple represents regions in the plane where debt recycling fails;
area in gray denotes the zone of permanent re-mortgaging. The intensity of the
orange (purple) directly correlates with the speed at which the strategy
succeeds (fails). The colorbar is measured in quarters. Lime lines are contour
lines, and the markers on the plot represent the values of the eigenvalues of
the process at that point, with white lines marking the transition between one
area and another. The markers follow the legend:
($\bullet$): $\lambda_{1,2}>1$
($\blacksquare$): $\lambda_{1}>1;0<\lambda_{2}<1$
($\times$): $0<\lambda_{1,2}<1$
($\bigstar$): $\lambda_{2}>1;0<\lambda_{1}<1$

In Fig. 4, we further distinguish between areas of strong success (in dark
orange), where the time to achieve house ownership through debt recycling is
shorter than without any strategy, and areas of weak success (in light
orange), where the strategy requires more time than the conventional monthly
repayment approach, suggesting the latter as the more advisable option. The
cyan line is also a contour line, similar to the ones in lime shown in the
previous Fig. 3. It is still calculated as the locus of points on the plane
where the first hitting time takes on the same value $t_{\text{ no
recycling}}$, the time required to achieve house ownership without any debt
recycling strategy.

Figure 4: Another representation of the phase diagram in Fig. 3, with focus on
the strong (dark orange) and weak (light orange) success of the strategy. In
cyan, the contour line of $t^{*}=t_{\text{ no recycling}}=\langle
M_{0}\rangle/\pi^{*}=25$ years for our choice of parameters $\langle
M_{0}\rangle=\$300,000$ and $\pi^{*}=\$3,000$.

Strong success is consistently observed in the I quadrant and weak success in
the II quadrant. In the majority of standard scenarios, where parameters are
within typical ranges, the III quadrant typically exhibits weak success, and
the IV quadrant strong success, with $p=0.5$, serving as the separating
threshold between the two. In rare cases involving extreme parameter values,
specifically when the total value of the house is exceptionally high (worth
millions) or the initial mortgage-to-equity ratio is significantly skewed
towards a high level of debt with very little initial equity (over-
collateralized situations), strong success can manifest in the III quadrant or
weak success in the IV quadrant.

The outcome of the debt recycling strategy is primarily influenced by the
mortgage-to-equity ratio and the monthly repayment schedule. Raising the LTV
ratio, the risk parameter, or both can accelerate the process and shift
outcomes from default to success in the IV quadrant, where the housing market
has a negative trend. This shift is more pronounced when the total house value
is lower, showing that the strategy’s effectiveness is also significantly
impacted by risk levels.

For a more detailed and extended discussion of the results, please refer to
Section 6. In the next section, we are going to describe the mathematical
methods used to solve the joint stochastic process $(E_{t},M_{t})$ defined in
Eq. (10).

##  5 Methods

In this section, we derive the solution for the average behaviour of the
equity and mortgage process. Introducing the $2\times 2$ matrix $A_{t}$

$A_{t}=\begin{pmatrix}\alpha_{t}&r_{t}\\\
\delta_{t}&1\end{pmatrix}=\begin{pmatrix}1+\ell\mu\sigma_{t-1}+r_{t}&r_{t}\\\
-\ell\mu\sigma_{t-1}&1\end{pmatrix}\ ,$ (16)

the coupled dynamical process (10) can be iterated to yield

$\displaystyle\begin{pmatrix}E_{t}\\\ M_{t}\end{pmatrix}$
$\displaystyle=A_{t}A_{t-1}\cdots A_{1}\begin{pmatrix}E_{0}\\\
M_{0}\end{pmatrix}+A_{t}A_{t-1}\cdots A_{2}\begin{pmatrix}\pi_{1}\\\
-\pi_{1}\end{pmatrix}+A_{t}A_{t-1}\cdots A_{3}\begin{pmatrix}\pi_{2}\\\
-\pi_{2}\end{pmatrix}$ $\displaystyle+\ldots+\begin{pmatrix}\pi_{t}\\\
-\pi_{t}\end{pmatrix}\ ,$ (17)

where $E_{0}$ and $M_{0}$ are the initial conditions of equity and mortgage
respectively, and $\pi_{t}$ is the scheduled repayment to the bank at the time
$t$. Isolating $E_{t}$ and $M_{t}$, and taking the average over the disorder,
we may use independence at different times, and the fact that the matrix
entries are identically distributed to deduce the following for product of
operators

$\Big{\langle}\prod_{t^{\prime}=k}^{t}A_{t^{\prime}}\Big{\rangle}=\prod_{t^{\prime}=k}^{t}\langle
A_{t^{\prime}}\rangle=\bar{A}^{t-k}\ ,$ (18)

where $\bar{A}$ is the average matrix

$\bar{A}=\begin{pmatrix}\langle 1+\ell\mu\sigma+r\rangle&\langle r\rangle\\\
\langle-\ell\mu\sigma\rangle&1\end{pmatrix}=\begin{pmatrix}1+\ell\mu(2p-1)+s&s\\\
-\ell\mu(2p-1)&1\end{pmatrix}\ ,$ (19)

and $\bar{A}^{t-k}$ stands for matrix power. The average matrix depends on
four main parameters: $\ell$ (LTV ratio), $\mu$ (risk factor), $p$
(probability of gaining from the investment), and $s$ (average value of the
market’s fluctuations). Therefore from Eq. 5 we obtain

$\langle E_{t}\rangle=(\bar{A}^{t})_{11}\langle
E_{0}\rangle+(\bar{A}^{t})_{12}\langle M_{0}\rangle+\\\
(\bar{A}^{t-1})_{11}\langle\pi\rangle+(\bar{A}^{t-1})_{12}\langle-\pi\rangle+(\bar{A}^{t-2})_{11}\langle\pi\rangle+(\bar{A}^{t-2})_{12}\langle-\pi\rangle+\ldots+\langle\pi\rangle\
.$ (20)

Now, the matrix $\bar{A}$ can be decomposed as follows

$\bar{A}=U\Lambda U^{-1}\ ,$ (21)

where

$\Lambda=\begin{pmatrix}s+1&0\\\ 0&\ell\mu(2p-1)+1\end{pmatrix}$ (22)

is the diagonal matrix of (distinct) eigenvalues, and

$\displaystyle U$ $\displaystyle=\begin{pmatrix}-\frac{s}{\ell\mu(2p-1)}&-1\\\
1&1\end{pmatrix}$ (23) $\displaystyle U^{-1}$
$\displaystyle=\begin{pmatrix}\frac{l\mu(2p-1)}{l\mu(2p-1)-s}&\frac{l\mu(2p-1)}{l\mu(2p-1)-s}\\\
-\frac{l\mu(2p-1)}{l\mu(2p-1)-s}&-\frac{s}{l\mu(2p-1)-s}\\\ \end{pmatrix}\ $
(24)

are the matrix of the eigenvectors and its inverse, respectively. It should be
noted that $\lambda_{1}=s+1>0$ is always true, since $s$ represents the
percentage change in market fluctuations, and $\lambda_{2}=\ell\mu(2p-1)+1>0$
is always true as well, as $\ell$, $\mu$, and $p$ are defined to be within the
range $[0,1]$. Eq. 20 can therefore be rewritten as

$\frac{\langle E_{t}\rangle}{\langle
E_{0}\rangle}=(U\Lambda^{t}U^{-1})_{11}+c_{1}(U\Lambda^{t}U^{-1})_{12}+c_{2}\sum_{k=1}^{t}\left[(U\Lambda^{t-k}U^{-1})_{11}-(U\Lambda^{t-k}U^{-1})_{12}\right]\
,$ (25)

with

$c_{1}=\frac{\langle M_{0}\rangle}{\langle E_{0}\rangle};\qquad
c_{2}=\frac{\langle\pi\rangle}{\langle E_{0}\rangle}\ .$ (26)

Following the same procedure, we can find the evolution equation for the
average mortgage value

$\langle M_{t}\rangle=(\bar{A}^{t})_{21}\langle
E_{0}\rangle+(\bar{A}^{t})_{22}\langle M_{0}\rangle+\\\
(\bar{A}^{t-1})_{21}\langle\pi\rangle+(\bar{A}^{t-1})_{22}\langle-\pi\rangle+(\bar{A}^{t-2})_{21}\langle\pi\rangle+(\bar{A}^{t-2})_{22}\langle-\pi\rangle+\ldots-\langle\pi\rangle\
,$ (27)

or, equivalently:

$\frac{\langle M_{t}\rangle}{\langle
M_{0}\rangle}=c_{3}(U\Lambda^{t}U^{-1})_{21}+(U\Lambda^{t}U^{-1})_{22}+c_{4}\sum_{k=1}^{t}\left[(U\Lambda^{t-k}U^{-1})_{21}-(U\Lambda^{t-k}U^{-1})_{22}\right]\
,$ (28)

with

$c_{3}=\frac{1}{c_{1}}=\frac{\langle E_{0}\rangle}{\langle
M_{0}\rangle};\qquad c_{4}=\frac{\langle\pi\rangle}{\langle M_{0}\rangle}\ .$
(29)

We now wish to extract from Eq. 25 and Eq. 28 the first time $t^{\star}$ at
which either of these processes hits zero. After some lengthy algebra (see A
for details) we manage to extract the $t$-dependence exactly as

$\displaystyle\frac{\langle E_{t}\rangle}{\langle E_{0}\rangle}$
$\displaystyle=\mathcal{A}\lambda_{1}^{t}+\mathcal{B}\lambda_{2}^{t}+\mathcal{C}\
,$ (30) $\displaystyle\frac{\langle M_{t}\rangle}{\langle M_{0}\rangle}$
$\displaystyle=\mathcal{D}\lambda_{1}^{t}+\mathcal{E}\lambda_{2}^{t}+\mathcal{F}\
,$ (31)

where the constants are given explicitly as

$\displaystyle\mathcal{A}$
$\displaystyle=-\frac{s(c_{1}+1)}{\ell\mu(2p-1)-s}=\frac{(c_{1}+1)(\lambda_{1}-1)}{\lambda_{1}-\lambda_{2}}\
,$ (32) $\displaystyle\mathcal{B}$
$\displaystyle=\frac{-c_{2}s+\ell\mu(2p-1)(c_{1}s+c_{2}+\ell\mu(2p-1))}{\ell\mu(2p-1)(\ell\mu(2p-1)-s)}=$
$\displaystyle=\frac{c_{2}\left(\lambda_{1}-1\right)-\left(\lambda_{2}-1\right)\left(c_{1}\left(\lambda_{1}-1\right)+c_{2}+\lambda_{2}-1\right)}{\left(\lambda_{1}-\lambda_{2}\right)\left(\lambda_{2}-1\right)}\
,$ (33) $\displaystyle\mathcal{C}$
$\displaystyle=-\frac{c_{2}}{\ell\mu(2p-1)}=-\frac{c_{2}}{\lambda_{2}-1}\ ,$
(34) $\displaystyle\mathcal{D}$
$\displaystyle=\frac{(c_{3}+1)\ell\mu(2p-1)}{\ell\mu(2p-1)-s}=-\frac{\left(c_{3}+1\right)\left(\lambda_{2}-1\right)}{\lambda_{1}-\lambda_{2}}\
,$ (35) $\displaystyle\mathcal{E}$
$\displaystyle=\frac{c_{4}s-\ell\mu(2p-1)(c_{3}\ell\mu(2p-1)+c_{4}+s)}{\ell\mu(2p-1)(\ell\mu(2p-1)-s)}=$
$\displaystyle=\frac{-c_{4}\left(\lambda_{1}-1\right)+\left(\lambda_{2}-1\right)\left(c_{3}\left(\lambda_{2}-1\right)+c_{4}+\lambda_{1}-1\right)}{\left(\lambda_{1}-\lambda_{2}\right)\left(\lambda_{2}-1\right)}\
,$ (36) $\displaystyle\mathcal{F}$
$\displaystyle=\frac{c_{4}}{\ell\mu(2p-1)}=\frac{c_{4}}{\lambda_{2}-1}\ ,$
(37)

while the eigenvalues $\lambda_{1}$ and $\lambda_{2}$ are given in Eq. (22).
It should be noted that the parameters $\ell$ and $\mu$ never occur
independently of each other: it is the product $\ell\mu\in[0,1]$ that acts as
the critical factor influencing the amplitude of the fluctuations in $E_{t}$
and $M_{t}$. In the next section we present a more detailed and extended
discussion of our results.

##  6 Results

We recall that the parameters involved in the simulations are the following:

  * 1.

$\ell\in[0,1]$: LTV ratio;

  * 2.

$\mu\in[0,1]$: risk factor of the investment;

  * 3.

$p\in[0,1]$: probability that the investment generates a gain;

  * 4.

$s\in[-4\%,+4\%]$: average value of the market’s fluctuations;

  * 5.

$q\ll 1$: probability of skipping an installment of the repayment to the
lender;

  * 6.

$\pi^{*}$: value of the regular installment;

  * 7.

$E_{0},M_{0}$: initial conditions of equity and mortgage respectively.

We have derived the evolution equation for the average equity process $\langle
E_{t}\rangle$ in Eq. (30), and for the average mortgage process $\langle
M_{t}\rangle$ in Eq. (31). We are interested in the time
$t^{\star}=\min(t_{E},t_{M})$, where $t_{E}$ is the first time at which
$\langle E_{t}\rangle=0$, while $t_{M}$ is the first time at which $\langle
M_{t}\rangle=0$. If $t^{\star}=t_{E}$, the homeowner’s equity (or value of the
fraction of the house owned) has been massively depleted. In this scenario,
the borrower would need to rely uniquely on monthly repayments to regain
ownership, indicating that the strategy has not succeeded. Instead, $t_{M}$
signifies that the homeowner has managed to pay off the entire mortgage,
achieving full ownership of the house. If $t^{\star}=t_{M}$ then the strategy
has been successful. In this case we distinguish between _strong_ success,
where the time to achieve house ownership through debt recycling is shorter
than without any strategy, and _weak_ success, where the strategy requires
more time than the conventional monthly repayment approach, suggesting that
the latter would still be the more advisable option – even if eventually full
home ownership is reached anyway.

The question is therefore which of the two scenarios occurs first. To shed
some light on the question, we first conducted a few numerical simulations of
the joint dynamical process $(E_{t},M_{t})$ described by Eq. (10) for
different values of the model parameters. Next, we present a more detailed
investigation of phase diagrams for this problem.

###  6.1 Simulations of trajectories for mortgage balance and equity

In Fig. 5 we display the time evolution of the equity $E_{t}$ (in units of the
initial equity $E_{0}=\$30,000$) [left panel] and the mortgage balance $M_{t}$
(in units of the initial mortgage $M_{0}=\$300,000$) [right panel] in a
scenario where the parameters are chosen to be firmly in favor of the
homeowner being able to successfully repay their mortgage: $\ell=0.5$,
$\mu=0.5$, $p=0.8$, $s=2\%$, $q=1\%$, and $\pi^{*}=3,000\$$. In particular, we
have a high probability $p=0.8$ that the asset being invested on will generate
a positive income, and we have a positive trend $s=2\%$ of the house market
leading to the possibility to extract more equity.

In line with our expectations, most of the equity trajectories as well as the
average process (solid green line) are steadily growing in time, while the
mortgage balance curves and their average (solid red line) are steadily
decreasing in time, until at time $t^{\star}=11.4$ the mortgage holder has (on
average) successfully managed to acquire full ownership of the house. From
Fig. 4, we know that $(p,s)=(0.8,2\%)$ corresponds to a point in the phase
diagram indicating strong success of the strategy. Indeed, the time required
to acquire the house without any strategy, calculated as
$t=M_{0}/\langle\pi\rangle\sim 101$ quarters ($\sim 25$ years), is
significantly higher than $t^{*}$ (equal to nearly $3$ years).

(a) Equity over time. The average process, Eq. (30), is illustrated by the
solid green line; some simulations of the process, Eq. (10), are shown in
black; the standard deviation band, calculated using $N=10^{5}$ simulations,
is displayed in gray.

(b) Mortgage over time. The average process, Eq. (31), is illustrated by the
solid red line; some simulations of the process, Eq. (10), are shown in black;
the standard deviation band, calculated using $N=10^{5}$ simulations, is
represented in gray. The vertical line at $t^{\star}=11.4$ signals the moment
the average mortgage balance hits the absorbing wall at zero.

Figure 5: Example of time evolution of the equity and mortgage balance
processes with the debt recycling strategy: the equity increases steadily,
while the mortgage balance decreases steadily until it hits the absorbing
boundary within $3$ years, marking the strategy as strongly successful.
Referring to Tab. 1, we are in the scenario $(\bullet):\lambda_{1,2}>1$.

In Fig. 6, we instead present a scenario where the parameters are unfavorable
to the mortgage holder: here, we have a low probability $p=0.2$ that the
investment will result in a positive gain and an unfavorable housing market
with $s=-3\%$. All other parameters remain unchanged. Both equity and mortgage
balance exhibit a decreasing trend (due to the negative value of $s$), but the
equity curves (as well as the average process) hit the $E=0$ absorbing wall in
just over $3$ time steps, indicating a default situation where the house value
is entirely eroded by the mortgage.

(a) Equity over time. The average process, Eq. (30), is illustrated by the
solid green line; some simulations of the process, Eq. (10), are shown in
black; the standard deviation band, calculated using $N=10^{5}$ simulations,
is displayed in gray. The vertical line at $t^{*}=3.2$ signals the moment the
average equity hits the barrier at zero.

(b) Mortgage over time. The average process, Eq. (31), is illustrated by the
solid red line; some simulations of the process, Eq. (10), are shown in black;
the standard deviation band, calculated using $N=10^{5}$ simulations, is
displayed in gray.

Figure 6: Example of time evolution of the equity and mortgage balance
processes with the debt recycling strategy: they both decrease, but the equity
does it faster and hits the absorbing boundary within $9$ months, marking the
strategy as failing. Referring to Tab. 1, we are in the scenario
$(\times):0<\lambda_{1,2}<1$. Note that the equity trajectory in the left
panel is limited to the initial few time steps, as we aim to emphasize the
first hitting time. As a result, the increasing trend potentially observed
_after_ the hitting time (see scenario $(\times)$ in Tab. 1) is not shown.

###  6.2 Phase Diagrams and dependence on parameters

In this section, we present more phase diagrams of the debt recycling process,
for different initial conditions. Starting from individual trajectories of the
average process, which depend on the parameters ($\ell$, $\mu$, $p$, $s$) as
seen in Sec. 6.1, we explore the parameter space $(p,s)$ and build the phase
diagrams, dependent on the parameters $\ell$ and $\mu$. Every point in the
$(p,s)$ space is colored in orange if the average process trajectories
correspond to a successful strategy outcome (see example in Fig. 5), in purple
if they correspond to a strategy failure (see example in Fig. 6), in gray if
they lead to permanent re-mortgaging. These processes follow an underlying
dynamic that is determined by the magnitudes of the eigenvalues $\lambda_{1}$
and $\lambda_{2}$ at that point in parameter space (i.e. the markers on the
phase diagram relate to the dynamical situation at that point). Based on the
combination of the parameters $\lambda_{1}=s+1$ and
$\lambda_{2}=\ell\mu(2p-1)+1$, the dynamics of the processes is described by
one of the cases showed in Tab. 1. For each combination of symbols and colors
within the parameter space, the dynamics of $\langle E\rangle$ and $\langle
M\rangle$ is the same. The results are of course consistent for different
initial conditions.

The strategy is always strongly successful in the I quadrant, as suggested by
a high chance of investment’s success ($p>0.5$) and a growing housing market
($s>0$). The equity exhibits an increasing exponential trend, while the
mortgage balance shows a decreasing exponential trend, reaching the $M=0$
barrier early in the process. The lime-colored flow lines in the phase diagram
presented in Fig. 3 illustrate that progressing towards the upper right corner
of the I quadrant shortens the timeframe for acquiring house ownership. In the
II quadrant, the strategy can be weakly successful with lower values of the
product $\ell\mu$, or it may lead to permanent re-mortgaging with higher
values of $\ell\mu$, in which case neither process ever hits the absorbing
boundary. The III and IV quadrants are where default can potentially occur.
The other possible outcome is success (both strong and weak).

As stated in Eqs. (26) and (29) the parameters leading the evolution of the
equity and the mortgage balance are:

$c_{1}=\frac{\langle M_{0}\rangle}{\langle E_{0}\rangle};\qquad
c_{2}=\frac{\langle\pi\rangle}{\langle E_{0}\rangle};\qquad
c_{3}=\frac{1}{c_{1}}=\frac{\langle E_{0}\rangle}{\langle M_{0}\rangle};\qquad
c_{4}=\frac{\langle\pi\rangle}{\langle M_{0}\rangle}\ .$ (38)

In Sec. 4 we presented a phase diagram with the following parameters. Average
initial equity $\langle E_{0}\rangle=\$30,000$; average initial mortgage
$\langle M_{0}\rangle=\$300,000$; value of the installment of the repayment to
the bank $\pi^{*}=\$3,000$; $q=1\%$ chance of skipping an installment;
$\ell=0.5$; $\mu=0.5$. Indeed, under unfavorable investment parameters ($p\sim
0$ and $s\sim-4\%$), the house is entirely mortgaged, and the strategy
defaults; with favorable parameters ($p\sim 1$ and $s\sim+4\%$), the strategy
is strongly successful. We now analyze and present in Fig. 7 how the phase
diagram changes for a wider range of initial conditions (average equity
$\langle E_{0}\rangle$, average mortgage $\langle M_{0}\rangle$, average
installment of the repayment to the bank $\langle\pi\rangle$), but keeping the
same value of the product of the LTV ratio and risk factor $\ell\mu$333The
official data available on the Australian Bureau of Statistics (ABS), updated
to December 2023, states that the mean price of residential dwellings is
$\$933,800$. We chose to reference Australian data because this strategy is
already in use there.:

  * 1.

Case $1a$: $\langle E_{0}\rangle=\$300,000$ and $\langle
M_{0}\rangle=\$300,000$ for houses of low value, with fixed
$\langle\pi\rangle=\$3,000$.

  * 2.

Case $1b$: $\langle E_{0}\rangle=\$800,000$ and $\langle
M_{0}\rangle=\$800,000$ for houses of high value, with fixed
$\langle\pi\rangle=\$3,000$.

  * 3.

Case $2a$: $\langle E_{0}\rangle=\$90,000$ and $\langle
M_{0}\rangle=\$900,000$ for houses of medium value, with fixed
$\langle\pi\rangle=\$3,000$.

  * 4.

Case $2b$: $\langle E_{0}\rangle=\$150,000$ and $\langle
M_{0}\rangle=\$1,500,000$ for houses of high value, with fixed
$\langle\pi\rangle=\$3,000$.

  * 5.

Case $3a$: $\langle E_{0}\rangle=\$90,000$ and $\langle
M_{0}\rangle=\$900,000$ for houses of medium value, with fixed
$\langle\pi\rangle=\$10,000$.

The comparison between Case $1a$ and $1b$ shows the dependence on the initial
conditions $\langle E_{0}\rangle$ and $\langle M_{0}\rangle$ separately, with
fixed ratio $c_{1}=1$; the comparison between Case $2a$ and $2b$ shows the
dependence on the initial conditions $\langle E_{0}\rangle$ and $\langle
M_{0}\rangle$ separately, with fixed ratio $c_{1}=10$; the comparison between
Case $2a$ and $3a$ shows the dependence on the value of repayment installments
$\langle\pi\rangle$.

Comparing Case $1a$ with Case $1b$, and Case $2a$ with Case $2b$, where the
difference is in the initial conditions of equity and mortgage, but with the
mortgage-to-equity ratio $c_{1}$ kept constant, we find that the areas in the
phase diagram for the three different possible outcomes of the strategy remain
largely the same, with the purple area slightly bigger when the overall house
value is higher. What changes is the speed at which the processes come to a
definite outcome. In the orange area, where the strategy is successful, an
increase in the total value of the house $H$ ($H=E+M$) corresponds to a
slightly longer time required to fully repay it, converting its entire value
into equity. Meanwhile, again in the case of a higher house value, the time to
reach failure in the area where the strategy fails (in purple) is shorter.
Where the strategy is successful, a higher house value means there is more
capital to be managed and repaid. This results in a longer time period needed
to transform all mortgage into equity, even though the strategy ultimately
succeeds. In contrast, in scenarios where the strategy fails, the financial
strain of maintaining large mortgage payments against a failing investment
strategy becomes unsustainable more quickly, leading to a faster default.
These observations imply that while higher property values can enhance the
potential gains from successful debt recycling, they also increase the risks
and consequences of failure, requiring a more extended commitment period.

Comparing Case $1a$ and $1b$ with Case $2a$ and $2b$, which differ in the
initial mortgage-to-equity ratio $c_{1}$, we notice that a higher $c_{1}$
ratio, indicating a higher percentage of the mortgage yet to be repaid
relative to the already owned equity, results in an expanded default outcome
space, extending well into the IV quadrant. This expansion suggests that even
with a good chance of investment gains ($p>0.5$), if the average market value
trend is negative ($s<0$), the strategy on average still defaults.

Comparing Case $2a$ with Case $3a$, where the difference lies in the scheduled
monthly repayment to the bank, we see that the phase diagram shifts, enlarging
the success zone as the value of the scheduled repayment increases.

Therefore, the most influential parameters are the mortgage-to-equity initial
ratio, rather than the individual values of mortgage and equity themselves
(meaning the strategy’s outcome does not depend strongly on whether we are
dealing with a low or high-value house) and the monthly repayments.

(a) Case $1a$: $\langle E_{0}\rangle=\$300,000$; $\langle
M_{0}\rangle=\$300,000$;
$\langle\pi\rangle=\$3,000$

(b) Case $1b$: $\langle E_{0}\rangle=\$800,000$; $\langle
M_{0}\rangle=\$800,000$;
$\langle\pi\rangle=\$3,000$

(c) Case $2a$: $\langle E_{0}\rangle=\$90,000$; $\langle
M_{0}\rangle=\$900,000$;
$\langle\pi\rangle=\$3,000$

(d) Case $2b$: $\langle E_{0}\rangle=\$150,000$; $\langle
M_{0}\rangle=\$1,500,000$;
$\langle\pi\rangle=\$3,000$

(e) Case $3a$: $\langle E_{0}\rangle=\$90,000$; $\langle
M_{0}\rangle=\$900,000$;
$\langle\pi\rangle=\$10,000$

Figure 7: Phase diagrams for different initial conditions. Area in orange
represents where debt recycling is (either weakly or strongly) successful;
area in purple represents regions in the plane where debt recycling fails;
area in gray denotes the zone of permanent re-mortgaging. The intensity of the
orange (purple) directly correlates with the speed at which the strategy
succeeds (fails). The colorbar is measured in quarters. Lime lines are contour
lines, and the markers on the plot represent the values of the eigenvalues of
the process at that point, with the white lines marking the transition between
one area and another. The markers follow the same legend described in Fig. 3.

Now, we analyze the phase diagrams for different values of $\ell$ and $\mu$
(see Fig.8) . As showed in Sec. 5, the product $\ell\mu\in[0,1]$ is the
critical factor influencing the amplitude of the fluctuations of $E_{t}$ and
$M_{t}$.

For the following phase diagrams (see Fig. 8) we fix the values of the initial
conditions to a house of medium value with $\langle E_{0}\rangle=\$90,000$,
$\langle M_{0}\rangle=\$900,000$, $\langle\pi\rangle=\$3,000$, and we explore
the following combinations:

  * 1.

Case $4a$: $\ell=0.1$ and $\mu=0.1$ $\rightarrow\ell\mu=0.01$.

  * 2.

Case $4b$: $\ell=0.9$ and $\mu=0.9$ $\rightarrow\ell\mu=0.81$

  * 3.

Case $5a$: $\ell=0.5$ and $\mu=0.1$ $\rightarrow\ell\mu=0.05$.

  * 4.

Case $5b$: $\ell=0.5$ and $\mu=0.9$ $\rightarrow\ell\mu=0.45$.

The comparison between Case $4a$ and $4b$ shows the dependence on the
parameter $\ell\mu$ combined; the comparison between Case $5a$ and $5b$ shows
instead the dependence on the risk factor only when we decide to keep the
other one (LTV ratio) constant.

Comparing Case $4a$ with Case $4b$, we observe that increasing the value of
the product $\ell\mu$ yields two effects: it reduces the default zone in the
IV quadrant and accelerates the strategy outcome. With higher $\ell\mu$, the
system becomes more sensitive to changes in parameters $p$ and $s$. Small
variations in these parameters can lead to significant changes in the outcome,
as indicated by the steeper gradients in Case $4b$. While higher $\ell\mu$
values reduce the risk of default and accelerate success, they also imply that
rapid changes of the outcome of the strategy are more likely. Additionally,
one may note that only low values of $\ell\mu$ may lead to a weakly successful
strategy in the II quadrant, as outlined in Tab. 1.

The straightforward comparison between Case $5a$ and Case $5b$, where only the
risk factor $\mu$ increases, leads to qualitatively similar conclusions. We
can either increase the LTV ratio parameter, the risk parameter, or both
simultaneously for an amplified effect: the result is a faster process, and in
the IV quadrant, default is shifted back in favor of success. This effect is
amplified as the total value of the house decreases.

(a) Case $4a$: $\ell=0.1$; $\mu=0.1$

(b) Case $4b$: $\ell=0.9$; $\mu=0.9$

(c) Case $5a$: $\ell=0.5$; $\mu=0.1$

(d) Case $5b$: $\ell=0.5$; $\mu=0.9$

Figure 8: Phase diagrams for different values of LTV ratio $\ell$ and risk
factor $\mu$. Area in orange represents where debt recycling is (either weakly
or strongly) successful; area in purple represents regions in the plane where
debt recycling fails; area in gray denotes the zone of permanent re-
mortgaging. The intensity of the orange (purple) directly correlates with the
speed at which the strategy succeeds (fails). The colorbar is measured in
quarters. Lime lines are contour lines, and the markers on the plot represent
the values of the eigenvalues of the process at that point, with the white
lines delineating the transition between one area and another. The markers
follow the same legend described in Fig. 3.

####  6.2.1 Velocity of the outcome

The phase diagrams are color-coded to indicate whether debt recycling
represents a strong/weak winning or losing strategy, with the intensity of the
color reflecting the speed at which the average equity/mortgage process
reaches the absorbing boundary first. In this section we present slice plots,
an analysis of the phase diagrams by fixing the variable $p$ ($s$) on the
$x-$axis ($y-$axis) and examining the variations in first hitting time along
the other dimension.

We show in Fig. 9 the plot of the first hitting time for $\ell=0.5$,
$\mu=0.5$, and $p=0.4$, the same parameters chosen for the plot in Fig. 2.

Figure 9: First hitting time varying $s$, with $\ell=0.5$, $\mu=0.5$, $p=0.4$.

Taking Fig. 3 as a reference, we travel along the vertical line at $p=0.4$
from bottom to top, increasing the value of $s$. At first, we are deeply into
the default (purple) region, but increasing $s$ the time it takes for the
average equity to hit zero increases. This is in clear agreement with the
housing market becoming progressively more favorable, as also evidenced by the
first hitting root of the average equity curves in Fig. 2. At $s=-1.28\%$, we
cross the purple-orange boundary in the phase diagram in Fig. 3, meaning that
the strategy turns into a (weakly) successful one (light orange). The time
required to gain ownership of the house indeed turns out to be longer than
without any strategy – just paying back the lender on a monthly basis (cyan
line). The time required to repay the debt without applying the strategy is
computed assuming a constant repayment amount $\pi^{*}$ to the bank at each
time step. With an initial mortgage $M_{0}$ of $300,000 and $\pi^{*}$ set at
$3,000, the required repayment period is $100$ time steps (or $25$ years). For
visual clarity, in the phase diagram we have limited the maximum color
intensity to $t=400$ quarters; instances where debt recycling is effective
beyond $100$ years are here represented at $t=400$, as well. At $s=0\%$ we
further cross the orange-gray boundary and enter the phase of permanent re-
mortgaging.

To further validate the findings presented in Section 6.2, we show in Fig. 10
a comparison of the process speed between Cases $1a$ and $1b$, reproduced here
for ease of reference:

  * 1.

Case $1a$: $\langle E_{0}\rangle=\$300,000$ and $\langle
M_{0}\rangle=\$300,000$ for houses of low value, with fixed
$\langle\pi\rangle=\$3,000$.

  * 2.

Case $1b$: $\langle E_{0}\rangle=\$800,000$ and $\langle
M_{0}\rangle=\$800,000$ for houses of high value, with fixed
$\langle\pi\rangle=\$3,000$.

The mortgage-to-equity ratio $c_{1}$ is held constant but we have different
initial conditions of equity and mortgage. Setting $p=0.49$ in Fig. 7(a) and
7(b) and increasing $s$, we confirm that the purple region (strategy failing)
is stretched out for higher-value houses, with a purple-to-orange threshold
moving up from $s=-2.28\%$ to $s=-0.67\%$). It is noteworthy that in both
cases, the strategy past the threshold is only weakly successful,
corroborating the result found in Section 4 that $p=0.5$ is the threshold
between weak and strong success.

(a) First hitting time of Case $1a$, with $p=0.49$.

(b) First hitting time of Case $1b$, with $p=0.49$.

Figure 10: First hitting time varying $\langle E_{0}\rangle$ and $\langle
M_{0}\rangle$.

In Fig. 11, we present a comparison of the process speed between Cases $4a$
and $4b$ (see Fig. 8(a) and 8(b) for reference), where $p=0.6$. As expected,
an increase in the product $\ell\mu$ results in a reduction of the default
region (transitioning from a threshold of $s=-0.51\%$ to $s=-1.99\%$) and a
noticeable acceleration of the outcome. In this scenario, the strategy is
strongly successful beyond the threshold, as gaining ownership via regular
monthly repayments would take much longer (see the cyan constant line).

(a) First hitting time of Case $4a$, with $p=0.6$.

(b) First hitting time of Case $4b$, with $p=0.6$.

Figure 11: First hitting time varying $\ell\mu$.

Finally, to show that $p=0.5$ is the threshold between weak and strong success
in the majority of cases, we show in Fig. 12 a phase diagram for Case $3a$,
focusing just on the success area. This case outlines a realistic and not too
risky situation for a mid-value house: $\langle E_{0}\rangle=\$90,000$;
$\langle M_{0}\rangle=\$900,000$; $\langle\pi\rangle=\$10,000$; $\ell=0.5$;
$\mu=0.5$; $s=-1\%$.

Figure 12: First hitting time of Case $3a$, with $s=-1\%$.

Below, in Fig. 13, we present two edge-case scenarios: in the Left Panel, a
case where weak success occurs even for $p>0.5$, and in the Right Panel, a
scenario where strong success is achieved for $p<0.5$.

(a) $\langle E_{0}\rangle=\$15,000$; $\langle M_{0}\rangle=\$150,000$.
In light orange, weak success; in dark orange, strong success. In cyan, the
contour line $t^{*}=t_{\text{ no recycling}}$.

(b) $\langle E_{0}\rangle=\$150,000$; $\langle M_{0}\rangle=\$1,500,000$.
In dark orange, strong success.

Figure 13: Scenarios where $p=0.5$ is not the threshold between strong and
weak success.

The Left Panel is defined by parameters where the equity and mortgage values
are relatively low: $\langle E_{0}\rangle=\$15,000$; $\langle
M_{0}\rangle=\$150,000$; $\pi^{*}=\$3,000$; $q=1\%$; $\ell=0.1$; $\mu=0.1$. In
contrast, the Right Panel features significantly higher values: $\langle
E_{0}\rangle=\$150,000$; $\langle M_{0}\rangle=\$1,500,000$;
$\pi^{*}=\$3,000$; $q=1\%$; $\ell=0.1$; $\mu=0.1$. Both scenarios represent
over-collateralization of the house, as evidenced by the consistent mortgage-
to-equity ratio $c_{1}=10$ across both panels, and the product $\ell\mu$
remains low in each. The key distinction between these cases lies in the total
value of the house, which is on the order of $10^{5}$ in the Left Panel and
$10^{6}$ in the Right Panel, reflecting the scale of financial involvement and
potential risk exposure. In the Right Panel, monthly repayments should be
properly rescaled (e.g., increased) to account and hedge for the increased
risk of the debt recycling strategy – in which case the two phase diagram
would become more similar. The scale of the problem, therefore, is also an
important parameter to consider to assess the success of the strategy, and a
re-evaluation of all parameters’ involved in the initial mortgage issuance
(e.g., monthly repayment schedule) should be carried out.

##  7 Conclusions and Outlook

We have considered a simple dynamical model for the joint time evolution of
equity $E_{t}$ and mortgage balance $M_{t}$ in the case where an aggressive
“debt recycling” strategy is employed. Debt recycling consists in a mortgage
holder investing in an income-producing asset backed by their equity (the
current market value of the fraction of the house they have already repaid),
with the aim to hopefully pay off their home loan quicker. When interest rates
rise, the immediate benefit of paying down a mortgage becomes evident due to
the higher cost of borrowing. However, debt recycling emerges as a powerful
strategy, leveraging higher interest rates to maximize tax deductions
associated with mortgage interest payments. While debt recycling offers
significant advantages, especially in higher interest environments, its
success strongly depends on a proper execution and continuous management. The
strategy can outperform regular mortgage repayment when executed with
precision and supported by favorable house market and investment conditions.
However, the risk associated with investments must be managed to prevent
potential setbacks from market volatility. The choice between paying down a
mortgage directly or engaging in debt recycling should be informed by an
understanding of both financial strategies and personal financial goals,
however this hoped-for level of awareness is hindered by the current lack of
workable models for the joint evolution of equity and mortgage burden.

Our paper aims at filling this gap, by incorporating a risk factor $\mu$ of
the investment in our model, as well as a stochastic trajectory $\bm{\sigma}$
of positive or negative returns $\sigma_{t}\in\\{\pm 1\\}$ of the investment
performance. The mortgage is assumed to decrease in time due to regular
repayments, but also due to a larger influx of cash coming from a rewarding
investment. The flip side is of course that the mortgage holder’s debt or
exposure can actually increase in case of a bad investment, or due to
shrinking available equity in a contracting house market. These interacting
effects are modeled by the set of coupled dynamical equations in (10). Under
reasonable assumptions about the parameter ranges, we found that the average
processes can be characterized analytically in terms of the eigenvalues
$\lambda_{1}=s+1$ and $\lambda_{2}=\ell\mu(2p-1)+1$ of the average matrix
given in Eq. (19). The eigenvalues depend on four parameters: $\ell$ (LTV
ratio), $\mu$ (risk factor), $p$ (probability of gaining from the investment),
and $s$ (average value of the market’s fluctuations).

Depending on these combinations of parameters, we find three outcomes for the
strategy: successful, unsuccessful, permanent re-mortgaging. The strategy is
successful when the average mortgage process $\langle M_{t}\rangle$ hits the
absorbing boundary $\langle M_{t}\rangle=0$ first. In this scenario, we
distinguish between strong success, where the time to achieve house ownership
through debt recycling is shorter than without any strategy, and weak success,
where the strategy requires more time than the conventional monthly repayment
approach, suggesting the latter as the more advisable option. In either case,
though, the house is eventually secured. Conversely, the strategy is deemed
unsuccessful when it is the average equity process $\langle E_{t}\rangle$ that
hits the absorbing boundary $\langle E_{t}\rangle=0$ first. In this case the
homeowner owns far less equity (seen as portion of ownership of the house)
than they would have had they not engaged in debt recycling, meaning that the
value of the house is now entirely mortgaged, and only monthly repayments
could gain the house back. The strategy may also lead to a state of permanent
re-mortgaging where neither process ever hits the absorbing boundary: as the
house value increases, it is theoretically possible to keep securing
additional bank funding for investment that feeds a cycle of equity
extraction. Without regulatory intervention to cap bank lending, individuals
could in principle exploit this strategy indefinitely, potentially leading to
financial malpractice.

The debt recycling strategy is overall strongly successful with favorable
parameters ($p>0.5$, $s>0$); however, it leads to a state of permanent re-
mortgaging when $p<0.5$, $s>0$. With $s<0$, there is a potential for either
default or success. The phase transition between default and success marks a
critical region within the parameter space: small variations in $s$ or $p$ can
lead to drastically different outcomes. In this region, the time it takes for
the strategy to lead to a final outcome experiences a discontinuous jump,
resembling a first-order transition.

The outcome of the strategy is sensitive to various parameters with different
effects: a higher percentage of the mortgage yet to be repaid relative to the
already owned equity increases the likelihood of default, expanding the region
where the strategy is expected to fail, especially at high $p$ (probability of
successful investment). However, increasing the scheduled monthly repayment to
the bank expands the region where the strategy is expected to be successful,
even for $p<0.5$. The factors $\ell$ (LTV ratio) and $\mu$ (risk factor) have
the same influence on the outcome of the strategy: the higher their values,
the more they expand the region where the strategy is expected to be
successful (for $p>0.5$ and $s<0$).

Future research could focus on the following issues:

  1. 1.

Microscopic modeling of taxation issues. In our current model, we have ignored
the interplay between deductible and non-deductible debt, and subtleties
related to different taxation regimes for locked-in equity and investment-
generated wealth. It would be interesting to incorporate this further
complexity layer in our model. Also, our model does not currently allow for a
finer sub-splitting of the primary debt (mortgage) and the locked-in equity.

  2. 2.

Determining the optimal parameters for different phases of the mortgage’s
life. For example, how should the monthly repayment amounts or optimal
investment strategies be scaled to ensure that the mortgage is repaid on
average in 20 years?

  3. 3.

A potential diversification of the asset portfolio is another research avenue
worth pursuing. We have currently limited our setting to a single risky
investment for simplicity, but extending to the setting to multiple assets
being traded is potentially very interesting.

  4. 4.

It will be also interesting to include potential correlations between the
house market and the return/volatility performance of the risky asset, which
are considered for simplicity independent of each other at the moment.

  5. 5.

While we have studied analytically only the average processes, it would be
interesting to characterize fluctuations of the $(E_{t},M_{t})$ processes, as
well as to characterize the first hitting time $t^{\star}$ analytically
without any reference to the average process.

  6. 6.

It will be also interesting to add some mechanism to the model to suppress the
permanent re-mortgaging phase, for example a time cap to the propensity of the
lender to provide a line of credit.

  7. 7.

In traditional banking, mortgages serve as collateral where the financed
property secures the loan. In crypto-backed lending, cryptocurrencies act as
collateral, allowing borrowers to access liquidity without selling their
assets. This enables investment in additional assets, leveraging potential
returns. However, the volatile nature of cryptocurrency markets can
significantly affect the leverage or loan value, with interest rates also
varying based on the lending platform and the collateral used [23]. For
instance, an investor might use Bitcoin to secure a loan in USDC through a
decentralized finance protocol, then reinvest these funds in yield-generating
activities. Given the liquidity and potential high returns of cryptocurrency
investments, this could potentially accelerate the benefits of debt recycling.
Combining debt recycling with the peculiar features of crypto markets is
another exciting avenue for further research.

## Acknowledgments

P.V. acknowledges support from UKRI Future Leaders Fellowship Scheme (No.
MR/S03174X/1). For the purpose of open access, the authors have applied a
creative commons attribution (CC BY) licence to any author-accepted manuscript
version arising.

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##  Appendix A Extended Derivation of Main Equations for the Average Process

We start from Eq. (25):

$\frac{\langle E_{t}\rangle}{\langle
E_{0}\rangle}=(U\Lambda^{t}U^{-1})_{11}+c_{1}(U\Lambda^{t}U^{-1})_{12}+c_{2}\sum_{k=1}^{t}\left[(U\Lambda^{t-k}U^{-1})_{11}-(U\Lambda^{t-k}U^{-1})_{12}\right]\
,$ (39)

with

$c_{1}=\frac{\langle M_{0}\rangle}{\langle E_{0}\rangle},\qquad
c_{2}=\frac{\langle\pi\rangle}{\langle E_{0}\rangle}\ .$ (40)

Writing the matrix entries explicitly, we have

$\displaystyle\frac{\langle E_{t}\rangle}{\langle E_{0}\rangle}$
$\displaystyle=\sum_{l=1}^{2}U_{1l}\lambda_{l}^{t}(U^{-1})_{l1}+c_{1}\sum_{l=1}^{2}U_{1l}\lambda_{l}^{t}(U^{-1})_{l2}$
$\displaystyle+c_{2}\sum_{k=1}^{t}\sum_{l=1}^{2}\left[U_{1l}\lambda_{l}^{t-k}(U^{-1})_{l1}-U_{1l}\lambda_{l}^{t-k}(U^{-1})_{l2}\right]=$
$\displaystyle=\sum_{l=1}^{2}\phi_{l}\lambda_{l}^{t}+c_{1}\sum_{l=1}^{2}\psi_{l}\lambda_{l}^{t}+c_{2}\sum_{l=1}^{2}\phi_{l}\sum_{k=1}^{t}\lambda_{l}^{t-k}-c_{2}\sum_{l=1}^{2}\psi_{l}\sum_{k=1}^{t}\lambda_{l}^{t-k}=$
$\displaystyle=\sum_{l=1}^{2}\phi_{l}\lambda_{l}^{t}+c_{2}\sum_{l=1}^{2}\phi_{l}\sum_{k=1}^{t}\lambda_{l}^{t-k}+c_{1}\sum_{l=1}^{2}\psi_{l}\lambda_{l}^{t}-c_{2}\sum_{l=1}^{2}\psi_{l}\sum_{k=1}^{t}\lambda_{l}^{t-k}$
(41)

where we have defined for $l=1,2$

$\displaystyle\phi_{l}$ $\displaystyle\equiv U_{1l}(U^{-1})_{l1}$ (42)
$\displaystyle\psi_{l}$ $\displaystyle\equiv U_{1l}(U^{-1})_{l2}\ ,$ (43)

with the $2\times 2$ matrices $U$ and $U^{-1}$ given in Eqs. (23) and (24).

Using the geometric sum

$\sum_{k=1}^{t}\lambda_{l}^{t-k}=\frac{1-\lambda_{l}^{t}}{1-\lambda_{l}}$ (44)

we can write

$\displaystyle\frac{\langle E_{t}\rangle}{\langle
E_{0}\rangle}=\sum_{l=1}^{2}\phi_{l}\lambda_{l}^{t}+c_{2}\sum_{l=1}^{2}\phi_{l}\frac{1-\lambda_{l}^{t}}{1-\lambda_{l}}+c_{1}\sum_{l=1}^{2}\psi_{l}\lambda_{l}^{t}-c_{2}\sum_{l=1}^{2}\psi_{l}\frac{1-\lambda_{l}^{t}}{1-\lambda_{l}}=$
$\displaystyle=\phi_{1}\lambda_{1}^{t}+\phi_{2}\lambda_{2}^{t}+c_{1}\psi_{1}\lambda_{1}^{t}+c_{1}\psi_{2}\lambda_{2}^{t}+c_{2}\left[\phi_{1}\frac{1-\lambda_{1}^{t}}{1-\lambda_{1}}+\phi_{2}\frac{1-\lambda_{2}^{t}}{1-\lambda_{2}}\right.$
$\displaystyle\left.-\psi_{1}\frac{1-\lambda_{1}^{t}}{1-\lambda_{1}}-\psi_{2}\frac{1-\lambda_{2}^{t}}{1-\lambda_{2}}\right]\
.$ (45)

Collecting now the coefficients of $\lambda_{1}^{t}$ and $\lambda_{2}^{t}$, we
obtain the main result in Eq. (30) in the main text, with

$\displaystyle\mathcal{A}$
$\displaystyle\equiv\phi_{1}+c_{1}\psi_{1}-\frac{c_{2}\phi_{1}}{1-\lambda_{1}}+\frac{c_{2}\psi_{1}}{1-\lambda_{1}}$
(46) $\displaystyle\mathcal{B}$
$\displaystyle\equiv\phi_{2}+c_{1}\psi_{2}-\frac{c_{2}\phi_{2}}{1-\lambda_{2}}+\frac{c_{2}\psi_{2}}{1-\lambda_{2}}$
(47) $\displaystyle\mathcal{C}$
$\displaystyle\equiv\frac{c_{2}\phi_{1}}{1-\lambda_{1}}+\frac{c_{2}\phi_{2}}{1-\lambda_{2}}-\frac{c_{2}\psi_{1}}{1-\lambda_{1}}-\frac{c_{2}\psi_{2}}{1-\lambda_{2}}\
,$ (48)

which reduce to the expressions in Eqs. (32),(33), (34) once the values are
substituted in. The derivation for the average mortgage process is entirely
analogous.