# Autonomous Money Supply Strategy Utilizing Control Theory

Yuval Boneh  Yuval Boneh<EMAIL_ADDRESS>
###### Abstract

Decentralized Finance (DeFi) has reshaped the possibilities of reserve banking
in the form of the Collateralized Debt Position (CDP). Key to the safety of
CDPs is the money supply architecture that enables issued debt to maintain its
value. In traditional markets, and with respect to the United States Dollar
system, interest rates are set by the Federal Reserve in an attempt to
influence the effects of excessive inflation. DeFi enables a more transparent
approach that typically relies on interest rates or other debt recovery
mechanisms being directly informed by asset price. This research investigates
contemporary DeFi money supply and debt management strategies and their
limitations. Furthermore, this paper introduces a time-weighted approach to
interest rate management that implements a Proportional-Integral-Derivative
control system to constantly adapt to market activities and protect the value
of issued currency, while addressing observed limitations.

Keywords: Decentralized Finance, collateralized debt, control system, money
supply, interest rate.

## 1\. Introduction

Blockchain technology has empowered the development of Decentralized Finance
(DeFi), employing programmable smart contracts to enable secure and
transparent financial transactions. DeFi has further enabled a proliferation
of experimental stablecoin architecutres. Stablecoins are cryptocurrencies
designed to trade at par with a reference asset, typically the United States
Dollar (USD). Stablecoins are generally defined by a balance of three key
values: Security, Scalability, and Decentralization. While the nomenclature
may vary, the first principle considerations remain the same, and the general
argument is that a stablecoin design can only optimize for two of the three.
Compromising decentralization and security introduce existential risk into a
protocol, as exhibited by the collapse of Terra’s UST stablecoin ([30]), and
BUSD’s rapid supply fall attributed to regulatory actions ([25]). Meanwhile,
scalability may not be core to a design, but can be achieved over time as the
model develops trust and robustness. To date, the most successful secure and
decentralized architecture concept is that of a Collateralized Debt Position
(CDP) protocol.

CDPs refer to positions created by locking collateral in a smart contract to
generate some kind of debt, typically stablecoins. Stablecoins represent the
system’s debt, so safeguarding their value is cruitial. Established oracle
infrastructures exist to ensure that the value of collateral and debt is known
at all times, however the mechanism by which minted debt assets maintain peg
varies across designs. Liquity’s LUSD stablecoin offers the highest level of
decentralization and security, with rigorous mechanisms designed to ensure the
value of the LUSD token remains pegged. MakerDAO’s DAI stablecoin and Aave’s
GHO stablecoin have interest rates set by governance bodies, which statically
adjust interest rates depending on the need for the stablecoin supplies to
contract or expand. This research provides an analysis of existing CDP
protocols and respective money supply theories and examines potential
shortfalls before proposing a novel design that mitigates identified risks.

## 2\. Industry Review

The purpose of this section is to provide an overview of some of the largest
decentralized CDP protocols in DeFi and assess their money supply strategies
for efficiency. Some mechanisms will be discussed in terms of control theory
to provide a standardized method of communicating assessed drawbacks and
limitations.

###  2.1 MakerDAO

The Maker Protocol, or Multi-Collateral Dai (MCD) system ([31]), accepts
governance-approved collateral assets and allows users to mint the DAI
stablecoin. Opening a CDP with Maker incurs an accruing stability fee, paid in
DAI, to the Maker Buffer. In turn, the Maker Buffer funds the DAI Savings Rate
(DSR) vault; a vault where users can deposit DAI to earn interest. DAI
maintains its price stability by Maker governance adjusting stability fees, as
well as the DSR rate, in order to balance supply and demand mechanics. An
additional Peg Stability Module (PSM) enables 1:1 exchange with a limited
amount of USDC, further tightening the peg by creating rapid arbitrage
opportunities when price shifts sufficiently to make the arbitrage transaction
profitable.

DAI’s stability is contingent on interest rates being modified via governance,
so while they are adaptive, the system response is inherently slow and
inefficient. Maker governance manages the DAI borrow rates of individual
collaterals, as well as the DSR rate, resulting in a cumbersome governance
liability. An April 2024 governance proposal ([21]) included 10 independent
interest rate modifications for the DAI ecosystem based on observations made
by BA Labs. There is a significant amount of research and data accompanying
each proposal, presumably consuming a large quantity of resources, akin to the
US Federal Reserve’s approach to interest rate management. Maker sets the
baseline for CDP technology, deviating from the traditional gold-backed
financial system only insofar as transparency and the use of secure and
decentralized blockchain technology.

In August 2023, Maker drastically increased their DSR savings rate111Source:
https://defillama.com/ (Figure 1), effectively prodiving a conduit between
their large off-chain reserves earning significant interest in a high rates
environment, and the on-chain intrest rate environment.

Figure 1: Maker’s DAI Savings Rate historical interest rates.

This transition rippled throughout the CDP landscape and the instability it
introduced provides an excellent case study for building a more resilient
system.

###  2.2 Liquity V1

The Liquity V1 Whitepaper ([27]) explains their approach to protecting the peg
and the value of their decentralized stablecoin LUSD by way of liquidation and
redemption mechanisms. Of note is the redemption mechanism, which sets a floor
price for LUSD should the price fall below $1. The system allows LUSD holders
to redeem their LUSD for underlying ETH collateral based on the face value of
the redeemed tokens, the current ETH:USD rate, and the current base rate. The
system uses the LUSD to repay debt of the riskiest position (lowest collateral
ratio), and transfers the respective amount of ETH to the redeemer. The
process is fair (the value of ETH transferred from the position is equal to
the amount of LUSD token debt paid) and borrowers do not suffer a net loss
from being redeemed. Redemptions are subject to a redemption fee, which is a
function of the base rate and the redeemed amount of LUSD, with a minimum
redemption fee of 0.5%. Therefore, redemption only becomes profitable when
LUSD falls below $0.995, protecting borrowers from constant redemptions when
LUSD value is maintained at $1.

Liquity offers zero-interest lending, so as broader interest rates rose, the
cost of interest rate arbitrage remained the same while profitability
increased. As a result, LUSD supply shrank dramatically over the last 12
months, for reasons that will be explained. Aave USDC supply on the Ethereum
Mainnet is typically considered the most accessible, lowest risk, stablecoin
strategy, and is thus compared as the risk-adjusted baseline rate for DeFi.
Data can be observed in Figure 2, which demonstrates LUSD circulating
supply222Source: https://defillama.com/ and Aave USDC supply rates333Source:
https://aavescan.com/ over the last 12 months. Strategic users deposited large
amounts of collateral into Liquity to mint LUSD and sell it to the market in
favour of stablecoins that could earn high interest elsewhere. This behaviour
generated sustained selling pressure on LUSD, which led to consistently
profitable redemptions. Consequently, in order to borrow LUSD, users required
increasingly high collateral ratios, which inhibits the capital efficiency of
the protocol. The interest rate arbitrage became inefficient and users exitted
Liquity V1, shrinking its supply.

Figure 2: Effects of Interest Rate Arbitrage on Liquity Scaling.

To be clear, Liquity V1 has always worked exactly as intended. Liquity V1 set
a new standard in decentralized CDP architecture, and was subsequently forked
by many teams. While the Liquity V1 money supply model works exceptionally
well in a low interest rate environment, it can not adapt to a high interest
rate environment. Developing a system that is resilient to to the arbitrage
opportunities around it requires an adaptive cost of borrowing, or interest
rate.

###  2.3 Aave GHO

GHO is a decentralized and collateralized stablecoin native to the Aave
protocol ([18]). GHO has similar supply functionality to Maker in that users
must deposit collateral in order to mint the stablecoin, however the GHO money
market integrated into Aave performs a passively scalable DSR role. Since the
potential supply of GHO is unlimited (ignoring their programmable cap),
conventional lend/borrow interest rate models that reference utilization do
not apply to the CDP borrow rate. Instead, the GHO interest rate is set by
Aave Governance, which "statically adjusts interest rates depending on the
need for the GHO supply to contract or expand" ([19]). As such, Aave’s
stablecoin has the ability to respond to changes in market dynamics by way of
governance voting for adjustments to the global interest rate. This overcomes
Liquity V1’s inability to restore supply dynamics, but introduces other
limitations.

There is a distinct relationship between the cost of borrowing and the price
of the stablecoin. This is a known product of interest rate arbitrage. When
interest rates increased broadly (exemplified on-chain by the DSR increase),
opportunities for profit presented themselves, and the low-interest $GHO
stablecoin faced the same sell pressure as $LUSD. Since Aave employs a global
interest rate in lieu of a redemption mechanism, the $GHO stablecoin lost peg,
falling to a price as low as approximately $0.95. Forum posts as far back as
August of 2023 propose increases in the interest rate in response to market
activities ([17]), with eight additional proposals following in as many months
(Table 1).

Table 1: Sample history of GHO borrow rate adjustments

Date | Adjustment | Forum Post
---|---|---
04 Apr 24 |  from 7.92% to 10%1 | GHO Stewards - Increase GHO Borrow Rate to 10%
13 Mar 24 | from 7.92% to 13% | GHO Stewards + Borrow Rate Update
08 Mar 24 | from 7.48% to 7.92% | Increase GHO Borrow Rate 08-03-2024
29 Feb 24 | from 6.35% to 7.48% | Increase GHO Borrow Rate 29-02-2024
04 Dec 23 | from 5.22% to 6.35% | Increase GHO Borrow Rate 100 bps to 6.35%…
21 Nov 23 | from 4.72% to 5.22% | Increase GHO Borrow Rate to 5.22% on Aave V3
30 Oct 23 | from 3.00% to 4.72% | GHO - Increase Borrow Rate
05 Oct 23 | from 2.50% to 3.00% | Further increase GHO Borrow Rate
23 Aug 23 | from 1.50% to 2.50% | Increase GHO Borrow Rate

  * •

Source: https://governance.aave.com/c/governance/

  * 1

This proposal was submitted as a progressive step towards the previous 13%
proposal.

Historical GHO borrow rates can be observed and compared to other borrow rates
for stablecoins on Aave V3 in Figure 3.

Figure 3:  Historical GHO Borrow Rates ([20]).

The delayed nature of Aave governance’s response is evident as market rates
rise, as well as fall, whereby the interest rate for borrowing GHO is almost
always the cheapest until approximately April 2024, after which it is now the
most expensive, pending further governance proposals to adjust the interst
rate. The effect of such a delayed response is present in GHO’s market price,
which has consistently been traded off in favour of profitable opportunities,
while its market capitalization (and thus circulating supply) has been
steadily increasing444Source: https://www.coingecko.com/ (Figure 4).

Figure 4:  Historical GHO Price and Market Cap over time.

Going forward, for as long as borrowing GHO remains comparably more expensive,
it will face a challenging effort to scale. Adaptive interest rates mitigate
the problems faced by Liquity V1, but reliance on governance is slow,
cumbersome, and the effective control system response is inhibitingly slow.

###  2.4 Liquity V2

Liquity V2 ([29]) aims to address the delay by introducing user-set interest
rates ([28]). Liquity V2 redemption mechanics target individual interest rates
instead of collateral ratios. This means that if users set their interest
rates high enough, they can access the same capital-efficient loans that V1
offered. It can be expected that Liquity interest rates will rise and fall in
tandem with the market’s, within a range of low arbitrage profitability.
However, there remain several drawbacks in the Liquity V2 model.

The first is that in this design, interest rates do not necessarily mitigate
the risk of redemption. [28] describes a redemption scenario in Figure 5, in
which borrowers are ordered by interest rate with their debt and collateral
values expressed as white and black circles. A redemption of four stablecoin
units repays the debt of the two lowest interest rate positions and part of
the 3.9% interest rate position, leaving additional collateral behind for
position owners.

Figure 5:  Liquity V2 redemption scenario ([28]).

It is apparent that maintaining an interest rate that is not the lowest does
not protect users from redemption, as the size of redemption can not be
guaranteed. A smaller redemption would leave the 3.9% interest rate position
unaffected, while an eight unit redemption would affect the 5% interest rate
position as well. The result is that position managers will need to consider
existing positions and potential redemption sizes when managing interest
rates. While it is reasonable that the base rate should be influenced by the
actions of users with highest risk appetite, the redemption mechansim may
continue to adversely impact users whose actions remain aligned with the
protocol.

[32] identifies that Liquity V2 infrastructure allows borrowers to delegate
their interest rates to third parties, smart contracts, or externally owned
accounts. For many users this will provide a solution to the redemption
problem by enabling automation of interest rate management to avoid
redemption. However, this introduces dependencies into the system, and does
not mitigate the requirement to monitor exisitng positions and potential
redemption sizes in order to optimize. The alternative is to pay more interest
than would otherwise be necessary. Additionally, the efficiency of third party
interest rate managers remains to be seen. Regardless, users who want an
optimized interest rate will require third party management, which must always
consider the possibility of redemption, while users who want a fixed interest
rate will have to pay above the base Liquity V2 interest rate.

Finally, the design can be analysed in terms of control theory. When market
rates increase, there is a tendency to take on low interest debt in order to
profit from interest rate arbitrage. This involves selling the $BOLD
stablecoin for others that earn more interest. As the price of $BOLD decreases
below $1, redemptions become profitable, prompting Liquity V2 users to
increase their interest rates, thus restoring interest rate parity. There
exists a relationship between the price of $BOLD and the perceived ’safe’
interest rate, where users are comfortable with their exposure to redemption.
The primary inefficiency of the control system lies in the settling time of
the market response that corrects the interest rate delta. As rates go up,
Liquity V2 will delay in response as users wait until their positions are at
threat of redemption before volunteering to pay more interest. The rolling
threat of redemption will force an overshoot whereby users increase their
interest rates by more than necessarily required in order to avoid having to
continually manage their positions. Similarly, on the way down, users will pay
more interest than they necessarily need to, while they wait for other
positions to buffer their redemption exposure. These effects will not
necessarily affect all Liquity V2 users, however they will affect the system’s
efficiency. The functional tradeoff is that users will always pay more
interest than they necessarily need to, so that the system can offer fixed
interest rate lending, governed by the profitability of redemption.

In order to improve the efficiency and scalability of the CDP construct, focus
should be on a global interest rate solution with minimal response time.
Increasing the system’s ability to respond to market dynamics will reduce the
likelihood and duration of meaningful depeg, eliminating the need for
redemptions, and further improving the system’s efficiency. Features such as
fixed interest rates reduce the core product’s ability to optimize for money
supply and should be considered later as derivative products.

###  2.5 Curve crvUSD

Curve’s crvUSD successfully utilizes an adaptive interterest rate to influence
the cost of borrowing. The equation that governs the interest rate for each
market is derived in their documentation ([24]). The interest rate formula
scales an exponential, in which the exponent is a function of crvUSD price and
a value; sigma. Sigma is a variable that can be configured by the Curve DAO,
such that a lower value makes the interest rates increase and decrease faster
as crvUSD loses and gains value respectively, and vise versa for higher
values.

The strategy is further reinforced by a system called PegKeeper. A PegKeeper
is a contract that helps to stabilize crvUSD price by trading crvUSD with
counterassets. When the price of crvUSD in a pool is above $1, PegKeepers may
take on debt by minting uncollateralized crvUSD and depositing into specific
pools. This increases the balance of crvUSD in the pool, which decreases
crvUSD’s price. If a PegKeeper has taken on uncollateralized debt and
deposited crvUSD into a pool, it may withdraw that crvUSD when the price is
below $1. By withdrawing, the crvUSD token balance will decrease and the price
of crvUSD increases. PegKeepers only provide a small buffer because they
cannot be allowed to mint crvUSD ad infinitum. This is a simple algorithmic
price arbitrage that frontruns the market’s response to interest rate changes.

Curve’s stablecoin system works reasonably well, as can be seen in the data
representing the market capitalization and crvUSD price over the last eight
months555Source: https://www.coingecko.com/ (Figure 6).

Figure 6:  Historical crvUSD Price and Market Cap over time.

Due to limited data, the crvUSD response can only be assessed since August
2023 when MakerDAO’s first large rate hike occurred, so the system’s response
to large systemic changes cannot be assessed because it didn’t face the same
transition as others. However, its ability to respond to consistent changes
since then has been generally promising. The crvUSD stablecoin has remained
within a reasonable range of $1 and the interest rates for borrowing remain
competitive.

Overall, the crvUSD design is considered to be the most advanced money supply
approach of those examined. That said, the crvUSD money supply strategy still
presents interesting limitations. The crvUSD stablecoin relies on debt caps to
protect the DebtFraction parameter (a ratio of the PegKeeper’s debt to the
total outstanding debt) utilized in the interest rate function. As well as
somewhat limiting the system’s scalability, this creates an artificial
utilization signal that informs the interest rate, and is subject to sustained
offset. If the PegKeeper has no counterassets with which to purchase debt and
the market has an appetite to maintain risk, the crvUSD price can remain
offset from the peg. This is mitigated by the sigma variable, however requires
governance intervention, which would introduce the same delay as has been
observed in Aave’s GHO. In its current state, the crvUSD interest rate
strategy has no autonomous way of responding to sustained offset. This can be
achieved with a Proportional-Integral-Derivative (PID) controller, utilizing
control theory.

## 3\. PID Controller Money Supply Strategy

This section introduces a control system for money supply management that
prioritizes the efficiency of the system’s response to market dynamics. The
architecture of the controller itself is closely aligned with a previous
controller architecture ([22]), modified specifically for this application.
Figure 7 provides a general representation of the controller logic.

Figure 7: General representation of controller logic.

In general terms, the controller normalizes an error signal to standardize the
strategy for general use. It then passes the normalized error through a PID
controller that processes independent signals according to Equation 1 to
output a modified controller error signal.

$E_{controller}=E_{P}+E_{I}+E_{D},$ (1)

where $E_{controllerError}$ is the modified error (controller output), and
$E_{P}$, $E_{I}$, and $E_{D}$ are the proportional, integral, and derivative
components of error, respectively.

The controller error then passes through a transfer function, which defines
the relationship between the controller error signal and the interest rate.

###  3.1 Reference and Input

The response mechanisms discussed in the previous section are generally
dependent on price movements. However, there are many potential leading
indicators that may preempt price movements, allowing the system to adjust
before market price changes. This is most applicable for StableSwap pools
([26]), in which the invariant allows for extremely low slippage trades even
when the pool is imbalanced. For a conventional $xy=k$ constant product pool,
imbalances result in impactful slippage, so will reduce the efficiency of the
controller. Research by [23] identifies several strategies that provide
leading indicators to price movement. The purpose of their research is to
develop off-chain infrastructure to alert liquidity providers of potential
depegs, so majority of the leading indicators developed do not suit the
purpose of the money supply strategy, however it does attest to the notion of
responding to pool imbalances before meaningful price changes. The Gini
Coefficient and Shannon’s Entropy, described in the paper as a measure of the
relative balances of a pool’s token, are objective, however are absolute
measures of balance. For the purpose of an autonomous on-chain controller
input signal and reference, a simple directional asset balance assessment
provides a stronger basis.

By assessing the pool balance and targeting a desired weight, a larger weight
indicates a surplus of the stablecoin in the pool (excessive supply), and a
smaller weight indicates a deficit in the pool (excessive demand). The error
signal is defined in Equation 2 as the difference between the current weight
and the reference (target) weight.

$e(w)=w-w_{r},$ (2)

where $w$ is the weight of the stablecoin in the pool, $w_{r}$ is the
reference (target) weight, and $e(w)$ is the weight error. The controller thus
modifies interest rates based on this directional indicator and before the
depeg occurs.

Normalizing the ranges $(0,w_{r}]$ and $(w_{r},1)$ to $(-1,0]$ and $(0,1)$,
respectively, further standardizes the strategy, making it applicable to any
reference weight. The normalized ranges are defined in Function 3.

$E(w)=\begin{cases}\dfrac{e(w)}{w_{r}},&\text{if }e(w)\leq 0\\\\[9.00002pt]
\dfrac{e(w)}{1-w_{r}},&\text{if }e(w)>0,\end{cases}$ (3)

where $E(w)$ is the normalized error signal with range $(-1,1)$. This
standardization makes the strategy applicable to any reference weight and
adaptable for various pool configurations on different exchanges.

A Curve StableSwap pool with amplifiication factor set to 100 has a price
curve represented in Figure 8.

Figure 8: Example StableSwap price curve, A = 100.

With 1,000,000 of token A and 1,000,000 of token B, this pool can facilitate a
swap of 400,000 token A for approximately 398,132 token B, resulting in a pool
balance of approximately 70% token A at a price of approximately $0.995.
Commensurate with Liquity V1’s redemption profitability case, this scenario
forms the base case for the example strategy and tune666Decentralized
Exchanges have varying mechanisms in place for pricing pool imbalances so the
degree of acceptable imbalance as it pertains to the base interest rate should
be tested thoroughly..

###  3.2 Transfer Function

The transfer function is derived theoretically on the basis that at 0% pool
imbalance, the output interest rate should be 0%, and at 100% pool imbalance
(such that the entire liquidity pool has been drained of counterassets, albeit
functionally impossible), the output interst rate should be infinite. It is
reasonable to allow the system to engage in negative interest rates in order
to scale supply for excess demand, however there is no requirement for a
negative interest rate to tend towards infinite. As such, a reasonable
transfer function to reflect the desired behaviour of the interest rate with
respect to pool imbalance is presented in Equation 4.

$r=\alpha\cdot\frac{E_{controller}}{1-E_{controller}},$ (4)

where $r$ is the interest rate, and $\alpha$ is a scaling factor (initially
set to $15e-2$). The interest rate curve can be examined in Figure 9, noting
the plot is a subset of the true pool balance range for demonstration.

(a)

(b)

Figure 9: Transfer function subset: (a) pool weight vs interest rate, (b)
equivalent price vs interest rate.

The transfer function trends appropriately against pool weight, and the
magnitude at 70% weight is refined to 10% with $\alpha=15e-2$. Depending on
the volatility and risk profile of the collateral, as well as the pool
configuration, $\alpha$ should be varied to set an appropriate base rate.

###  3.3 Proportional Component

In general, the purpose of the proportional component is to tune the magnitude
of the initial response, as represented in Equation 5.

$E_{P}=K_{P}\cdot E(w),$ (5)

where $K_{P}$ is the proportional gain factor. The design of the transfer
function is dependent on the range of $E_{controller}$ in that the denominator
offsets the signal from the limit in order to define the asymptote. For a
proportional gain to be applied, the transfer function would be defined as
Equation 6, which would undo the proportional effect of the gain factor and
remove the asymptote effect of the curve design.

$r=\alpha\cdot\frac{E_{controller}}{K_{P}-E_{controller}},$ (6)

For this reason, $K_{P}=1$, and the value of $\alpha$ facilitates a pseudo-
proportional component.

###  3.4 Integral Component

The integral component performs a time series analysis of the pool’s balance
to generate a Time-Weighted Cumulative Error (TWCE) that is scaled by a
tunable gain factor (Equation 7).

$E_{I}=K_{I}\cdot TWCE,$ (7)

where $K_{I}$ is the integral gain factor.

$TWCE$ is achieved by multiplying the normalized error signal each time it is
reported by the time since it was last reported and adding it to the last
TWCE, as summarized in Equation 8.

$TWCE_{i}=\sum_{i=1}^{n}E(w)_{i}\cdot(t_{i}-t_{i-1}),$ (8)

where $E(w)_{i}$ is the $i^{th}$ normalized weight error, $t_{i}$ is the time
at which the $i^{th}$ error is recorded, and $t_{i-1}$ is the previous time at
which error was recorded.

This rolling calculation is mathematically efficient and can operate
effectively ad infinitum. It accumulates when error is positive and dissipates
when error is negative, and does so proportionally to the size and sustained
time of the error. The effect of cumulative error can be observed in Figure
10, with $K_{I}=1.5$, which demonstrates the increase of interest rate leading
to complete liquidation in the absence of market action.

Figure 10:  Interest rate model with $K_{I}=1.5$.

Due to the cumulative nature of the integrator component, the modified weight
error signal transitions from being sensitive to the proportional component,
to being predominantly driven by the integrator component.

####  3.4.1 Phi-Strategy

For a money supply strategy, it is assessed that interest rates should grow or
decay at different rates depending on the class of assets in the system. When
pool balance tends towards a depeg to the downside, premium, low volatility
collateral assets are likely to have a lower Total Collateral Ratio (TCR)
relative to Minimum Collateral Ratio (MCR) and should experience slower
interest rate growth. Conversely, more volatile assets are likely to have a
higher TCR relative to MCR and should experience greater interest rate growth.
Therefore, as $TCR/MCR$ increases, so too should $K_{I}$. While this can be
established by tuning $K_{I}$ for different collateral types, the relationship
between $K_{I}$ and $TCR/MCR$ can be generalized in Equation 9 based on
proportionality.

$K_{I}=\phi\cdot(\frac{TCR}{MCR}-1),$ (9)

where $\phi$ is a scaling factor that governs the aggression of interest rate
growth.

Since $TCR/MCR$ evolves over time, Equations 7 and 8 are adapted to form
Equation 10 to ensure $K_{I}$ is updated with each iteration.

$E_{I}=\sum_{i=1}^{n}K_{I,i}\cdot E(w)_{i}\cdot(t_{i}-t_{i-1}),$ (10)

A dataset can be constructed by sweeping through different collateral ratios
and $\phi$ values to identify the $\phi$ value required to return the system
to a collateral ratio of one, in one year (Figure 11). A base rate of 10%
established by the transfer function means that no integrator value is
required at $TCR/MCR=1.1$, so the simulation sweeps from 1.2, and the
sensitivity to $\phi$ dissipates after $TCR/MCR=1.6$.

Figure 11:  Sweep simulation calculating $\phi$ values that return $TCR/MCR$
conditions to $TCR/MCR=1$ in one year.

Since any precision of the system’s ability to reach a collateralization ratio
of one in one year is invalidated by market response, the value of $\phi$ need
only be within a reasonable order of magnitude, so $\phi$ is set to $4$. The
controller can now be simulated to assess overall effectiveness of returning a
CDP to $TCR/MCR=1$, starting from different points. Simulation outputs are
presented in Figure 12, in which the pool balance deviates from 0.5 to 0.7 in
incremenets of 0.01 and with 12h timesteps. Balance is then maintained at 0.7
for the remainder of the simulation.

(a) Initial $TCR/MCR$ of 1.2.

(b) Initial $TCR/MCR$ of 1.6.

Figure 12: Interest rate controller performance for different $TCR/MCR$
conditions.

In both cases, interest rates initially rise to the same base rate. Once error
is maintained at a pool balance of 70%, the interest rate for the higher
$TCR/MCR$ scenario rises significantly faster, accelerating its return to
$TCR/MCR=1$. Since $\phi$ is set to $4$, the two $TCR/MCR$ cases presented
recover either faster of slower than the one year target. This is expected
from Figure 11, and ignores market responses, while recovering in a reasonable
timeframe. The value of $\phi$ does not necessarily need to be constant,
however it is assessed that the added complexity is not justified.

####  3.4.2 Mitigating Long-Tail Integrator Risks

There is potential for the integrator component to accumulate a significant
amount of directional bias such that it negates the proportional component and
transfer function of the controller. For example if the token weight is under
target for a prolonged period of time, the integrator may accumulate
significant negative error. Should the pool suddenly shift balances to above
target, the proportional component will transition to positive, however the
integral component may suppress the summation and result in a reduced overall
controller error. This can introduce dangerous attack vectors, so the
influence of the integral component must be carefully managed.

This is achieved by programming a limit to the negative accumulation of the
integrator component, so that a sudden shift in pool balances can be safely
accounted for.

This approach addresses the global rate based on pool imbalance as a priority
(pseudo-proportional component), and then increases the burden on riskier
assets over time (integral component), ultimately penalizing the least healthy
positions first. This eliminates the need for redemptions, so users are free
to borrow as they wish and need only focus on their respective position health
and interest rate. $K_{I}$ can be simulated for tuning, or since collateral
ratios are inherently normalized, the Phi-Strategy can be deployed to manage
different collaterals without modification.

This research presents the case of pool balance shifting to $70\%$ and
recovering in isolation in one year, however it should be noted that any
further increase in pool balance will compound interest rate growth and result
in much faster rate growth. Treating $0.995 as an example of meaningful depeg,
one year is the maximum possible time that it can be sustained without market
response. It is expected that as interest rates grow, the market will be
influenced to correct the liquidity imbalance, however the controller is
designed to correct system leverage in isolation, eliminating any dependency
on market activity.

###  3.5 Derivative Component

The derivative component is derived from [22] and performs a time series
analysis of normalized weight error by calculating the gradient over a
specified lookback time (Equation 11).

After defining a period, the timestamp of each update can be compared to a
delayed TWCE. If at least one period has passed, the delayed TWCE and
associated timestamp overwrite the previous TWCE and timestamp, and the
current TWCE and timestamp overwrite the delayed TWCE and timestamp. The
result is two TWCEs that store the cumulative error one period ago, and now,
allowing for the derivative calculation to be completed.

This mitigates the effects of very sudden pool balance changes that would
normally be arbitraged, which would subsequently make interest rates
excessively sensitive to market actions.

$E_{D}=K_{D}\cdot\frac{TWCE_{delayed}-TWCE_{previous}}{t_{delayed}-t_{previous}},$
(11)

where $K_{D}$ is the derivative gain coefficient, $TWCE_{delayed}$ is the most
recent TWCE recorded before the current value, and is only updated after at
least one period, and $TWCE_{previous}$ is the $TWCE$ recorded at least one
period before $TWCE_{delayed}$, and allows for the derivative calculation.
$t_{delayed}$ and $t_{previous}$ are the respective timestamps.

The derivative component allows the system to compensate for short term
(commensurate with the specified lookback time) market actions, to curb rapid
deviations from reference weight, protecting the protocol and users.

$K_{D}$ should be tuned according to the protocol’s risk tolerance under
different market scenarios, with consideration for arbitrage opportunities and
peg support infrastructure.

In essence, the controller modifies the weight error to account for the size
of the error, how long the error has been sustained, and how quickly the error
was incurred.

## 4\. Implementation

This strategy was developed for strict on-chain implementation, with no off-
chain dependencies. The response of the controller can be greatly improved
with off-chain infrastructure, however for the purpose of this research the
trade-off is undesirable.

The arithmetics employed are within the scope of most smart contract
programming languages and calculations can be programmed efficiently in order
to reduce gas costs.

The pseudo-proportional gain and integral gain may be set algorithmically to
achieve generic market responses in the the absense of fine tuning and
simulation, or may be set as discrete constants to achieve a finer response
for independent markets.

Derivative gain architecture may be comparatively expensive so its use should
be considered against the intended network infrastructure. In most cases, it
is anticipated that interest rates should not be highly sensitive to
derivative gain, and its use is not strictly needed.

Contract architecture has been developed and tested separately and compared in
production privately. Some information may be released in future publications.

## 5\. Conclusion

The control system described in this paper enables a reserve currency to
employ passively adaptive money supply theory with autonomous time-based
compensation. There is no requirement for human intervention, and the interest
rate updates as regularly as interactions with the protocol. Exact controller
configurations have been omitted due to the sensitive nature of the
performance of the strategy. As always, security remains the priority, and any
contracts derived from this research should be thoroughly and tested and
independently audited before implementation.

## References

  * [1] Aave “[ARFC] GHO - Increase Borrow Rate” In _Aave Governance_ , 2023 URL: https://governance.aave.com/t/arfc-gho-increase-borrow-rate/15271
  * [2] Aave “GHO Document Hub” In _GHO Document Hub_ , 2024 URL: https://docs.gho.xyz/
  * [3] Aave “Interest and Discount Rate Models” In _GHO Document Hub_ , 2024 URL: https://docs.gho.xyz/concepts/how-gho-works/interest-rate-discount-model
  * [4] Aave “Aave GHO Analytics” In _GHO Analytics_ , 2024 URL: https://aave.tokenlogic.com.au/stablecoin-rates
  * [5] BA Labs “Stability Scope Parameter Changes #12; Under STA Article 3.3” In _MakerDAO Governance_ , 2024 URL: https://forum.makerdao.com/t/stability-scope-parameter-changes-12-under-sta-article-3-3/24132
  * [6] Yuval Boneh “Adaptive money market interest rate strategy utilizing control theory” In _SSRN Electronic Journal_ , 2024 DOI: 10.2139/ssrn.4810469
  * [7] Thomas Cintra and Maxwell Holloway “Detecting Depegs: Towards Safer Passive Liquidity Provision on Curve Finance”, 2023
  * [8] crvUSD “crvUSD Borrow Rate Documentation” In _crvUSD Documentation_ , 2024 URL: https://resources.curve.fi/crvusd/loan-details/#borrow-rate
  * [9] DFS “Notice Regarding Paxos-Issued BUSD”, 2023 URL: https://www.dfs.ny.gov/consumers/alerts/Paxos_and_Binance
  * [10] Michael Egorov “Stableswap-efficient mechanism for stablecoin liquidity” In _Retrieved Feb_ , 2019
  * [11] Robert Lauko “Liquity releases updated whitepaper” In _Liquity_ , 2021 URL: https://www.liquity.org/blog/liquity-releases-updated-whitepaper
  * [12] Sam Lekhak “Liquity v2: why user-set interest rates” In _Liquity Blog_ , 2024 URL: https://www.liquity.org/blog/liquity-v2-why-user-set-interest-rates
  * [13] Liquity “Liquity v2 Whitepaper” In _Liquity Documentation_ , 2024 URL: https://liquity.gitbook.io/v2-whitepaper
  * [14] Jiageng Liu, Igor Makarov and Antoinette Schoar “Anatomy of a run: The terra luna crash” In _SSRN Electronic Journal_ , 2023 DOI: 10.2139/ssrn.4416677
  * [15] Maker “The Maker Protocol: MakerDAO’s Multi-Collateral Dai (MCD) System” In _The Maker Protocol White Paper_ , 2020 URL: https://makerdao.com/whitepaper/White%20Paper%20-The%20Maker%20Protocol_%20MakerDAO%E2%80%99s%20Multi-Collateral%20Dai%20(MCD)%20System-FINAL-%20021720.pdf
  * [16] Bojan Pecek “Liquity v2: why user-set interest rates” In _Liquity Blog_ , 2024 URL: https://www.liquity.org/blog/liquity-v2-enhancing-the-borrowing-experience

## References

  * [17] Aave “[ARFC] GHO - Increase Borrow Rate” In _Aave Governance_ , 2023 URL: https://governance.aave.com/t/arfc-gho-increase-borrow-rate/15271
  * [18] Aave “GHO Document Hub” In _GHO Document Hub_ , 2024 URL: https://docs.gho.xyz/
  * [19] Aave “Interest and Discount Rate Models” In _GHO Document Hub_ , 2024 URL: https://docs.gho.xyz/concepts/how-gho-works/interest-rate-discount-model
  * [20] Aave “Aave GHO Analytics” In _GHO Analytics_ , 2024 URL: https://aave.tokenlogic.com.au/stablecoin-rates
  * [21] BA Labs “Stability Scope Parameter Changes #12; Under STA Article 3.3” In _MakerDAO Governance_ , 2024 URL: https://forum.makerdao.com/t/stability-scope-parameter-changes-12-under-sta-article-3-3/24132
  * [22] Yuval Boneh “Adaptive money market interest rate strategy utilizing control theory” In _SSRN Electronic Journal_ , 2024 DOI: 10.2139/ssrn.4810469
  * [23] Thomas Cintra and Maxwell Holloway “Detecting Depegs: Towards Safer Passive Liquidity Provision on Curve Finance”, 2023
  * [24] crvUSD “crvUSD Borrow Rate Documentation” In _crvUSD Documentation_ , 2024 URL: https://resources.curve.fi/crvusd/loan-details/#borrow-rate
  * [25] DFS “Notice Regarding Paxos-Issued BUSD”, 2023 URL: https://www.dfs.ny.gov/consumers/alerts/Paxos_and_Binance
  * [26] Michael Egorov “Stableswap-efficient mechanism for stablecoin liquidity” In _Retrieved Feb_ , 2019
  * [27] Robert Lauko “Liquity releases updated whitepaper” In _Liquity_ , 2021 URL: https://www.liquity.org/blog/liquity-releases-updated-whitepaper
  * [28] Sam Lekhak “Liquity v2: why user-set interest rates” In _Liquity Blog_ , 2024 URL: https://www.liquity.org/blog/liquity-v2-why-user-set-interest-rates
  * [29] Liquity “Liquity v2 Whitepaper” In _Liquity Documentation_ , 2024 URL: https://liquity.gitbook.io/v2-whitepaper
  * [30] Jiageng Liu, Igor Makarov and Antoinette Schoar “Anatomy of a run: The terra luna crash” In _SSRN Electronic Journal_ , 2023 DOI: 10.2139/ssrn.4416677
  * [31] Maker “The Maker Protocol: MakerDAO’s Multi-Collateral Dai (MCD) System” In _The Maker Protocol White Paper_ , 2020 URL: https://makerdao.com/whitepaper/White%20Paper%20-The%20Maker%20Protocol_%20MakerDAO%E2%80%99s%20Multi-Collateral%20Dai%20(MCD)%20System-FINAL-%20021720.pdf
  * [32] Bojan Pecek “Liquity v2: why user-set interest rates” In _Liquity Blog_ , 2024 URL: https://www.liquity.org/blog/liquity-v2-enhancing-the-borrowing-experience