DeFi Protocols for Loanable Funds: Interest Rates, Liquidity and Market Efficiency
Lewis Gudgeon, Sam M. Werner, Daniel Perez, William J. Knottenbelt
DDeFi Protocols for Loanable Funds:Interest Rates, Liquidity and Market Efficiency
Lewis Gudgeon Sam M. Werner Daniel Perez William J. Knottenbelt
Imperial College London
ABSTRACT
We coin the term
Protocols for Loanable Funds (PLFs) to refer to pro-tocols which establish distributed ledger-based markets for loanablefunds. PLFs are emerging as one of the main applications within De-centralized Finance (DeFi), and use smart contract code to facilitatethe intermediation of loanable funds. In doing so, these protocolsallow agents to borrow and save programmatically. Within theseprotocols, interest rate mechanisms seek to equilibrate the supplyand demand for funds. In this paper, we review the methodologiesused to set interest rates on three prominent DeFi PLFs, namelyCompound, Aave and dYdX. We provide an empirical examinationof how these interest rate rules have behaved since their inceptionin response to differing degrees of liquidity. We then investigatethe market efficiency and inter-connectedness between multipleprotocols, examining first whether Uncovered Interest Parity holdswithin a particular protocol and second whether the interest ratesfor a particular token market show dependence across protocols,developing a Vector Error Correction Model for the dynamics.
A recent development within financial architecture based on de-centralized ledgers, or DeFi for short, is the emergence of protocolswhich facilitate programmatic borrowing and saving. Such pro-tocols represent a significant advancement for DeFi due to theimportance of these operations to an economy. Markets for loan-able funds, a matching market for savers and would-be borrowers,in principle enable agents to engage in intertemporal consumptionsmoothing, whereby agents choose their present and future con-sumption to maximize their overall welfare [10]. That is, accessto loans enables a borrower to consume more today than theirincome would permit, paying back the loan when their income ishigher. On the other hand, savers, for whom income is higher thantheir present consumption, are able to deposit their funds and earninterest on them [22, 25].Here, we term protocols that intermediate funds between usersas Protocols for Loanable Funds (PLFs). In doing so, we note suchprotocols are not directly acting as a fully-fledged replacement forbanks, not least because traditional banks are not intermediariesof loanable funds: rather, they provide financing through moneycreation [14] (see Section 3). Further, at present PLFs only offer secured lending, where agents can only borrow an amount pro-vided they can front at least this amount as collateral. This reflectsthe trustless setting within which PLFs operate: absent the typi-cal repurcussions of reneging on debt commitments in traditionalfinance since in DeFi agents could simply default on their loanswithout recourse. Therefore, at present the extent to which PLFs The enforcement of strong-identities, a mapping of on-chain to real world identities,would plausibly alter this tradeoff. facilitate ‘true’ borrowing—where an agent gets into a position ofnet debt—is limited.In PLFs, interest rates reflect the prevailing price of funds result-ing from supply and demand. The mechanism used to set these ratesis therefore a crucial aspect of protocol design: it provides the pre-conditions under which the process of tatonnement —or reachingthe equilibrium—occurs [28]. In traditional finance, interest ratesare primarily set by central banks—via a base rate—and functionas a key lever in the management of credit in economies [4, 21].The lowering of the base rate makes it relatively cheaper to borrow,while discouraging saving. In the context of PLFs, the interest ratesetting mechanism is decided upon at the protocol level, commonlyvia a governance process.In this paper, we seek to gain insights into how currently-deployedPLFs operate, setting out the interest rate models they employ. More-over we seek to characterize the periods of illiquidity—roughly,where most of the funds within a PLF are loaned out and unavail-able for withdrawal by their depositors—that these protocols haveexperienced. We then seek to understand how efficient these proto-cols are at present, investigating whether the no-arbitrage conditionof Uncovered Interest Parity (UIP) holds within a particular protocol.The efficiency of the markets serves to provide indication of thelevel of financial maturity, as well as the responsiveness of agents toeconomic incentives. Finally, we look at the interrelation of interestrate markets across protocols, developing a Vector Error CorrectionModel (VECM) for the dynamics between Compound [9], dYdX [8]and Aave [1] in the markets for the stablecoins DAI and USDC.
Contributions
This paper makes the following contributions: • We provide a taxonomy of the interest rate models currentlyemployed by PLFs, resulting in three categories: linear, non-linear and kinked rates. • We collect and analyze data on interest rates, utilization andthe total funds borrowed and supplied on three of the largestPLFs. We have made the dataset publicly available. • We present the first liquidity study of the markets for DAI,ETH and USDC across these PLFs, finding that periods ofilliquidity are common, often shared between protocols andthat liquidity reserves can be very unbalanced, with in somecases as few as three accounts controlling c. 50% of the totalliquidity. We also find that realized interest rates tend tocluster around the kink of a kinked interest rate model. • Investigating the largest PLF, Compound, we find that theno arbitrage condition of Uncovered Interest Parity typicallydoes not hold, suggesting that at markets associated withthese protocols may be relatively inefficient and agents maynot be optimally reacting to interest rate incentives. a r X i v : . [ q -f i n . GN ] J un udgeon et al. • We examine the market dependence between PLFs and findthat the borrowing interest rates exhibit some interdepen-dence, with Compound appearing to influence borrowingrates on other, smaller PLFs.The remainder of this paper is organized as follows. Section 2presents relevant background material, while Section 3 outlines thegeneral design of PLFs. Section 4 presents a taxonomy of differentinterest rate models. Sections 5, 6 and 7 provide an analysis onmarket liquidity, efficiency and dependence, respectively. Section 8discusses related work, before Section 9 concludes.
The Ethereum [30] blockchain allows its users to run smart con-tracts , programs designed to run on its distributed infrastructure.Smart contracts and the interactions between them are fundamen-tal building blocks of DeFi. They are almost feature-equivalentto programs written in any Turing-complete language but have afew particularities. For instance, smart contracts must be strictlydeterministic. For this reason, they can only communicate withthe outside world through transactions executed on the Ethereumblockchain. On the other hand, smart contracts can easily interactwith other smart contracts, allowing complex interactions betweendifferent parties as long as these interactions happen directly onchain. Another particularity of Ethereum smart contracts is theiratomicity: they can only be executed within a transaction. If anerror happens during the execution, the transaction is reverted. Insuch an event, any change of state that occurred in this contract orany other interaction with other contracts will be reverted and nochange of state will happen.
DeFi refers to a financial system which relies for its security andintegrity on distributed ledger technology. Applications of suchtechnology include lending, decentralized exchange, derivativesand payments. At the time of writing on 9 June 2020, DeFi has atotal value locked of over 1bn USD, with most applications deployedon the Ethereum blockchain [23]. Unlike regular finance where theidentity of all participants is known and correct behavior can beenforced via regulation, DeFi actors are pseudonymous and DeFisystems need other means to prevent users from misbehaving. Inthe absence of traditional credit-rating mechanisms, the systemrules are typically “enforced” by incentivizing actors to behaveaccording to the rules of the system [13].
PLFs intermediate markets for loanable funds, with suppliers offunds earning interest. As mentioned above, protocols need to pro-tect against borrowers defaulting on their debt obligations. Whereloans need to be valid for more than a single transaction, thisprotection is currently achieved by requiring borrowers to over-collateralize their loans, allowing the lender to redeem the pledgedcollateral should a borrower default on a position . Where the loan Therefore loans of this type on DeFi lending protocols are instances of secured loans,where an agent can only borrow against collateral they already own; they cannot enterinto ‘net debt’. We address this further in Section 3. needs to be valid only for a single transaction, flash loans enableagents to borrow without collateral, whereby the loaned amount isprotected by the atomicity afforded by smart contracts: if the loanis not repaid with interest, the whole transaction is reversed [1].In the context of lending protocols, a borrower defaults on a loanwhen the value of the locked collateral drops below some fixedliquidation threshold. The over-collateralization and liquidationthresholds vary between asset markets across different protocols.In an event of default, the lending protocol seizes and liquidatesthe locked collateral at a discount to cover the underlying debt.Additionally, a penalty fee is charged against the debt, prior topaying out the remaining collateral to the borrower.
In order for a cryptoasset to be a viable medium of exchange andstore of value, price stability needs to be guaranteed.
Stablecoins are cryptoassets which possess a price stabilization mechanism tomaintain some target peg. Here we briefly outline two of the mostwidely used stabilization mechanisms [20]:
Fiat-collateralized.
Each unit of stablecoin is pegged to somefixed amount of fiat currency (typically USD). This is gen-erally realized via a network of banks maintaining the fiatcollateral and is therefore not decentralized. Stablecoins suchas USDT [16] and USDC [6] belong in this category.
Cryptoasset-collateralized.
Each unit of stablecoin is backedby an amount of some other cryptoasset. A stabilizationmechanism is needed to protect against the volatility of thecollateral. Perhaps the most prominent of such stablecoinsis DAI [17]. In order to borrow newly minted units of DAI,where one DAI is pegged to 1 USD, a user has to pledgean over-collateralized amount of cryptocurrency (e.g. ETH),which becomes locked up in a smart contract. In case theprice of DAI deviates from its peg, arbitrageurs are incen-tivized to buy or sell DAI should the price drop below or riseabove 1 USD, respectively. A borrower of DAI has to ensureto keep the associated collateralization ratio above some liq-uidation threshold, as otherwise the borrow position will beliquidated at a discount and a penalty fee will be chargedagainst the debt.
PLFs facilitate the matching of would-be borrowers and lenders,with the interest rate set programmatically. Importantly, unlikepeer-to-peer lending, funds are pooled, such that a lender may lendto a number of borrowers and vice versa. In so doing, an openlending protocol provides a market for loanable funds, where therole that an intermediary would play in traditional finance has beenreplaced by a set of smart contracts.It should be stated that by creating markets for loanable funds—as protocols for loanable funds—such protocols are not functionallyequivalent to banks. The construal of banks as primarily intermedi-aries of loanable funds, as in some economic theory, has been de-bunked (see e.g. [14]). Rather than accepting deposits of pre-existingfunds from savers and then lending these funds out to borrowers,banks primarily provide financing through money creation , creating eFi Protocols for Loanable Funds:Interest Rates, Liquidity and Market Efficiency new money at the point of making a loan and constrained by theirprofitability and solvency requirements [14]. Therefore since banksare not primarily ILFs, PLFs are not functional replacements.
The introduction of PLFs significantly extends the existing trad-ing capabilities in DeFi, offering several use cases for DeFi actors.Predominantly, PLFs empower decentralized margin trading byfacilitating short sells and leveraged longs . In a short sell, a tradersells the borrowed funds, seeking to make a profit by repurchasingthe borrow position at a lower price. Similarly, in a leveraged long atrader buys some other asset with the borrowed funds and profits incase the purchased asset appreciates in value. As a consequence ofmargin trading, suppliers of loanable funds are able to earn interest.A further use case of PLFs lies in borrowers being able to leveragetheir funds as collateral, while maintaining the right to repurchasethe collateralized token, thereby not giving up direct ownership.
Suppliers of loanable funds receive in-terest over time, while borrowers have to pay interest. A key differ-entiating factor across lending protocols is the chosen interest ratemodel, which is generally some linear or non-linear function ofthe available liquidity in a market. As loans on protocols for loan-able funds have unlimited maturities, variable interest rates mayfluctuate from the opening of a borrow position. By using variablerate models, lending protocols are able to dynamically adjust theinterest rate depending on the ratio of funds borrowed to supplied,which can prove particularly useful during periods of low liquidityby incentivizing borrowers to repay their loans.
Additionally, lending protocols employ a re-serve factor , specifying the amount of a borrower’s accrued interestto be deducted and set aside for periods of illiquidity. Hence, theinterest earned by lenders is a function of the interest paid byborrowers less the reserve factor.
Interest is typically ac-crued per second and paid out on a per block basis. Since the re-peated payment to lenders of the accrued interest (denoted in thesupplied token) would incur undesired transaction costs, accruedinterest is often payed out through the use of interest-bearing de-rivative tokens , which are ERC-20 tokens that are minted upon thedeposit of funds and burned when redeemed. Each market has suchan associated derivative token, which appreciates with respect tothe underlying asset at the same rate as interest is compounded,thereby accruing interest for the token holder. Even though loansare made with indefinite maturity, a loan is liquidated should thevalue of the borrowed asset’s underlying collateral fall below afixed liquidation threshold. In the case of an undercollateralizedborrow position, so-called liquidators can purchase the collateralat a discount and a penalty fee is imposed upon the borrower.
A critical component of lendingprotocols is decentralized governance. Lending protocols tend toachieve decentralized governance through the use of ERC-20 gover-nance tokens specific to the lending protocol, whereby token hold-ers’ votes are weighted proportionally to their stake. Token holders are thereby empowered to propose new features and changes tothe existing protocol.
In this section, we outline the main classes of interest rate modelsemployed by PLFS. We also describe an approach that has beentaken to enable these variable rate models to offer more interestrate stability. We illustrate each case with an example taken froman implemented protocol.
Definitions.
For a market m , total loans L and gross deposits A ,we define the utilization of deposited funds in that market as U m = LA (1)The Interest Rate Index I for block k is calculated each time aninterest rate changes, i.e. as users mint, redeem, borrow, repay orliquidate assets. It is given by: I k , m = I k − , m ( + rt ) (2)where r denotes the per block interest rate and t denotes the differ-ence in block height. Therefore debt D in a market is given by D k , m = D k − , m ( + rt ) (3)where a portion of the interest is kept as a reserve ( Π ), set by reservefactor λ : Π m = Π k − , m + D k − , m ( rtλ ) (4)We now turn to the classification of the extant interest rates intothree main models. The first model we present is one in which interest rates are set asa linear function of utilization. With a linear interest rate model,interest rates are determined algorithmically as the equilibriumvalue in a loanable market m , where the borrowing interest rates i b are given by (e.g., following Compound [9]): i b , m = α + βU m (5)where α is some constant and β a slope coefficient on the re-sponsiveness of the borrowing interest rate to the utilization rate.Saving interest rates i s are given by: i s , m = ( α + βU m ) U m (6)where in essence the interest rate i b , m is scaled by the utilizationto arrive at an interest rate for saving that is lower than that of therate paid by borrowers. This serves to ensure that the interest ratespread ( i b , m − i s , m ) is positive. Some portion of this spread can bekept for reserves. Interest rates may also be set non-linearly, and here we present thenon-linear continuous model employed by dYdX [8]. For a loanablefunds market m , the borrowing interest rates i b follow a non-linearmodel and are computed as: i b , m = ( α · U m ) + ( β · U m ) + ( γ · U m ) . (7)The saving interest rates i s with reserve factor λ are given by: udgeon et al. Protocol Interest RateModel StableInterest Rate VariableInterest Rate GovernanceToken Interest-bearingDerivative Token AdditionalFunctionalities
Compound Kinked ✗ ✓ ✓ ✓ –Aave Kinked ✓ ✓ ✓ ✓
Swap rates, flash loansdYdX Non-linear ✗ ✓ ✗ ✗
Decentralized Exchange
Table 1: Comparison of different protocols for loanable funds. i s , m = ( − λ ) · i b , m · U m (8)In comparison to the linear rate model, a non-linear model allowsfor the interest rate to increase at an increasing rate as the protocolbecomes more heavily utilized, creating an non-linearly increasingincentive for suppliers to supply to the protocol and for borrowersto repay their borrows. In the final interest rate model, interest rates exhibit some form ofkink: they sharply change at some defined threshold. Such interestrates are employed by a number of protocols, including [1, 9].Mathematically, kinked interest rates can be characterized asfollows. i b = (cid:40) α + βU if U ≤ U ∗ α + βU ∗ + γ ( U − U ∗ ) if U > U ∗ (9)where α denotes a per-block base rate, β denotes a per-blockmultiplier, U denotes the utilization ratio (with U ∗ denoting theoptimal utilization ratio) and γ denotes a ‘jump’ multiplier.In the case of Compound, the associated saving rates are givenby Equation (10). i s = U ( i b ( − λ )) (10)where λ is a reserve factor.Such models share the property of sharply changing the incen-tives for borrowers and savers beyond some utilization threshold,as with the non-linear model. However, they also introduce a pointof sharp change in the interest rate, beyond which the interestrates starts to sharply rise, in contrast to non-linear models with nokink. Therefore it might be expected that this kink would becomea Schelling point of convergence among agents [26]. Some platforms, such as Aave, allow the borrower to choose betweena variable and a stable interest rate. However, it is important to notethat the “stable” interest rate is not entirely stable, as it can be revisedin the event that it significantly deviates from the market average.Examining Aave’s implementation in detail, we first present theirinstantiation of a kinked interest rate model before showing howthe stable rate is derived.The variable interest rate is based on several parameters definedby the system. Given the utilization rate U of a particular asset, theparameter U optimal is the optimal utilization. In practice, this valuewas set to 0 . . R slope1 is used when U < U optimal and R slope2 when U ≥ U optimal . Finally, given a base variable borrowrate i b , m , v , the variable borrow interest rate i b for market m iscomputed as follows: i b , m , v = i b , m , v + UU optimal · R slope1 if U < U optimal i b , m , v + R slope1 + U − U optimal − U optimal · R slope2 if U ≥ U optimal (11)To compute the stable rate, Aave computes the lending protocol-wide market rate m r as the arithmetic mean of the total borrowedfunds weighted by the borrow rate i b , m for given platform p asfollows: m r = (cid:205) np = i b , m , p · B m , p (cid:205) np = B m , p (12)where B m , p denotes the total amount of borrowed funds formarket m on lending protocol p . Hence, using the m r as the baserate, the stable borrowing rate i b , s for a market m is given by: i b , m , s = m r + UU optimal · R slope1 if U < U optimal m r + R slope1 + U − U optimal − U optimal · R slope2 if U ≥ U optimal (13)In case the stable rate deviates too much from the market rate, itwill be revised. The stable borrow rate i b , m , s for user z is revisedupwards to the most recent stable borrow rate for the respectivemarket when i b , m , s , z < B m , v · i b , m , v + B m , s · i b , m , s B m , v + B m , s (14)If (14) holds, a borrower of funds would be able to earn interestfrom a borrow position. On the contrary, should the stable rate of aborrow position exceed the latest stable rate it would be adjusteddownwards should i b , m , s , z > i b , m , s · ( + ∆ i b , m , s , t ) (15)where ∆ i b , m , s , t denotes the change in the stable rate for a spec-ified adjustment window t . Note that unlike for variable interestrate denominated loans, stable rate loans have a definite maturity. We have reviewed the three main interest rate models for variableinterest rates, and explained a mechanism which seeks to bringstability to these rates. An emergent key feature of these modelsis the incentive they provide to borrowers and savers at times of eFi Protocols for Loanable Funds:Interest Rates, Liquidity and Market Efficiency high utilization. In the next section, this behavior at high utilizationbecomes a central object of concern.
In this section we provide an analysis of liquidity and interest ratesfor loanable funds markets on Compound, dYdX and Aave.
The total amount of locked loanable funds for the largest marketsacross Compound, Aave and dYdX are given in Table 2.Currency Total Amount Locked(median in millions of USD)Compound Aave dYdX(W)ETH 76.58 4.80 19.41USDC 31.54 4.12 6.58DAI 24.82 0.95 4.64SAI 36.94 - -USDT - 3.92 -BAT 0.95 0.08 -LEND - 3.60 -LINK - 12.21 -
Table 2: Median of total supply of loanable funds in USDfor the largest markets on Compound, Aave and dYdX, sinceeach market’s inception until 7 May 2020.
It can be seen that ETH, USDC and DAI account for the majorityof loanable funds on all three PLFs. Hence we focus on thesemarkets for an in-depth analysis. From Figure 1 it becomes apparentthat these three markets are very similar in terms of their averageborrow and utilization rates, particularly for DAI and ETH.
The available liquidity for loanable funds for anasset is given by the difference between the total supply and totalborrows in the respective market. High liquidity allows actors toborrow funds at lower rates, while guaranteeing suppliers of fundsthat funds can be withdrawn at any point in time. On the onehand, regarding the liquidity for ETH (see Figure 2) all three PLFsmaintain high liquidity over time, largely due to the total borrowsremaining relatively stable. On the other hand, the markets for DAIand USDC (see Figures 3 and 4) frequently exhibit periods of muchlower liquidity, with utilization exceeding 80% and 90% respectively.Moreover, it appears that such periods of low liquidity are to someextent shared across protocols, in particular for the smaller PLFsdYdX and Aave for the period January to mid-March 2020.On Thursday March 12, 2020—
Black Thursday [24]—the totalamount of locked funds across all DeFi protocols dropped from897.2m USD to 559.42m USD. For DAI, it can be seen how onBlack Thursday even the largest PLF, Compound, was exposedto prolonged periods of low liquidity, before attracting increasedliquidity again at the same time as dYdX and Aave. However, after As single-collateral DAI (SAI) has been replaced by multi-collateral DAI (DAI), wesolely focus on the latter for this analysis. Source: https://defipulse.com. Accessed: 05-06-2020. D A I E T H U S D C W B T C B A T Z R X R E P L E N D L I N K U S D T K N C S N X T U S D M A N A M K R B U S D S A I A v g . U t ili z a t i o n A v g . B o rr o w R a t e AavedYdXCompound
Figure 1: Average utilization and borrow interest rates forall markets on Aave, Compound and dYdX.Figure 2: Total funds borrowed and supplied (i.e. liquidity)for ETH markets on dYdX, Compound and Aave. mid-April, the market for DAI on Compound re-experienced lowliquidity.
On PLFs agents are incentivized to provide liq-uidity via the employed interest rate model, as high interest rateswould make borrowing more cost prohibitive in periods of lowliquidity. However, if borrowers are not incentivized to repay theirloans by sufficiently high interest rates at times of full utilization,insufficient liquidity may materialize. In the event of such illiquiditymaterializing, suppliers of funds would be unable to withdraw them,being forced to hold on to and continue to earn interest throughtheir cTokens.Out of the three PLFs, only Aave enforces a utilization ceilingat 100%, while Compound and dYdX permit borrows even beyond full utilization. When examining the market for DAI in Figure 5, it udgeon et al.
Figure 3: Total funds borrowed and supplied (i.e. liquidity)for DAI markets on dYdX, Compound and Aave. Periodswhere utilization was between 80% and 90% are highlightedin salmon, while utilization higher than 90% is shaded in red.Figure 4: Total funds borrowed and supplied (i.e. liquidity)for USDC markets on dYdX, Compound and Aave. Periodswhere utilization was between 80% and 90% are highlightedin salmon, while utilization higher than 90% is shaded in red. can be seen how utilization of funds has in the past been multipletimes at and even above 100% on Compound and dYdX.It can be seen that Aave has experienced periods of near-illiquidity,while Compound and dYdX have experienced periods of full illiquid-ity for DAI, i.e. all supplied funds were loaned out. When comparingthe DAI borrow rates during periods of full utilization (red) in Fig-ure 5, notable differences can be made out between the differentinterest rate regimes. On dYdX, the borrow rate hits the by the
Figure 5: Utilization and borrow rates for DAI on Aave (top),dYdX (middle) and Compound (bottom). Time periods inwhich utilization equaled or exceeded 100% are highlightedin red. model imposed interest rate ceiling of 50%, while on Compound,the rate does not exceed 25% even at full utilization, which can beexplained by the linear nature of Compound’s interest rates. De-spite Aave never reaching full utilization for DAI, due to an optimalutilization target of 80% during the measurement period, borrowrates on Aave exceed rates on Compound during periods of highutilization. This suggests that holding on to loans during periods ofilliquidity is notably cheaper on Compound than on dYdX or Aave.
Periods of low liquidity have several impli-cations for market participants. On one side, high utilization implieslucrative interest rates for suppliers of funds, thereby attractingnew liquidity. On the other hand, suppliers are faced with the riskof being unable to redeem their funds, for example, in the case of a‘bank run’.In order to better assess the risk of a market becoming fullyilliquid, we examine the cumulative percentage of locked funds forthe number of Ethereum accounts on Compound in Figure 6. Notethat as a similar pattern was found for Aave and dYdX, we decidedto solely focus on Compound. The distribution of funds acrossaccounts is very similar for DAI, ETH and USDC in that a verysmall set of accounts controls the majority of all supplied funds. Forinstance, 50.3% of total locked DAI is controlled by only 3 accounts.Similarly, for ETH and USDC, the same number of accounts control60.0% and 47.3%, respectively. Hence, for all three markets, even intimes of high liquidity, a small number of suppliers of funds are ina position to to drastically reduce liquidity, or possibly even causefull illiquidity.
Case Study : DAI on Compound
In the context of liquidity, we present a case study of interest ratebehavior in the market for DAI on Compound, focusing on the eFi Protocols for Loanable Funds:Interest Rates, Liquidity and Market Efficiency
Number of accounts0%20%40%60%80%100% C u m u l a t i v e p e r c e n t a g e o f l o c k e d f un d s (a) DAI Number of accounts0%20%40%60%80%100% C u m u l a t i v e p e r c e n t a g e o f l o c k e d f un d s (b) ETH Number of accounts0%20%40%60%80%100% C u m u l a t i v e p e r c e n t a g e o f l o c k e d f un d s (c) USDC Figure 6: Cumulative percentage of locked funds on Com-pound for DAI, ETH and USDC on 2020-06-04. period of 21 February to 21 April 2020 and its interest-bearingtoken cDAI. It could be seen in Figure 5 that for the aforementionedperiod, this market was exposed to a range of different utilizationlevels, experiencing periods of relatively high liquidity but alsoilliquidity. Hence, we investigate market participants’ behavior—given by the interest rates that are actually observed—for differentinterest rate regimes during the period of interest.
Interest rate models for the cDAI contract.
To illustrate kinkedrates, we present the case of the DAI interest rate in CompoundFinance. The cDAI token is an example of an interest-bearing de-rivative token based on a linear kinked interest rate model. Sincethe 17th December 2019, the borrowing rates ( i b ) have operatedwith Equation (9). However, the precise parameter values used bythe model have been revised multiple times. We include a list ofthese modifications in Table 3 in the appendix. Mar2020 Apr24 02 09 16 23 30 06 13 201.501.752.002.252.502.753.003.253.50 c D a i Figure 7: Three interest rate regimes in Compound.
Interest rate behavior.
We consider in detail how since 17 Decem-ber 2019 agents have optimized their selection of borrowing andsaving amounts given an interest rate schedule. Here we focus ona subset of three periods, namely: •
21 February - 13 March 2020 •
14 March - 5 April 2020 • α is reduced via aparameter change by 49.04%. Despite this change, we continueto observe a clustering of the realized interest rates at the kink,although there does appear to be some effect of reducing the typicalutilization ratio to below the kink.Figure 8c shows how the system behaves once the base rate α is set to zero, while the multiplier β is increased by nearly 1000%.Again, we observe a similar pattern: most of the realized interestrates appear to be at the kink. However, if not at the kink, nowtypically utilization is above 90%. We saw that, especially for DAI, there were several periods ofilliquidity and that they were often shared across the three protocols.We also showed that the locked funds were very concentrated andthat a very small amount of accounts had the potential to makethe markets illiquid. Finally, we analyzed the interest rate behaviorof DAI on Compound and showed that during all the observationperiods, the interest rates appeared clustered around the kink ofthe interest rate function. udgeon et al. S u p p l y e B o rr o w s e I n t e r e s t r a t e (a) 21 February - 13 March S u p p l y e B o rr o w s e I n t e r e s t r a t e (b) 14 March - 5 April S u p p l y e B o rr o w s e I n t e r e s t r a t e (c) 6 April - 21 April Figure 8: Borrowing rates surface for DAI.
In this section we consider the capital market efficiency of DeFilending protocols. Loosely, a capital market is said to be efficient ifin the process of determining prices, it fully and correctly reflectsall relevant information [19]. More precisely:
Figure 9: Borrowing rate distribution around kink, 21 Febru-ary - 13 March.
Definition 6.1 (Market efficiency).
A market is efficient with re-spect to some information set ϕ if prices would be unaffected byrevealing that information to all market participants [19].A notable consequence of Definition 6.1 is that such efficiencyimplies it the impossibility of making economic profits on the basisof the information set ϕ . The market efficiency of PLFs is a questionof central interest because it provides a mechanism to assess thematurity of the markets and to understand the responsiveness ofagents to changes in the information set ϕ . Moreover, since a coremechanism common to many PLFs is the use of high interest ratesat times of high utilization—to encourage saving and discourageborrowing, incentivizing agents to behave in a certain way—theextent to which PLFS are capital efficient will inform how reliablethis mechanism is, at present, in incentivizing agents to act in theintended way. If agents do not in fact respond to high interest ratesby reducing their borrowing requirements and increasing theirsupply of funds to a PLF, illiquidity resulting from high utilizationrates on a given protocol may be expected to result. Such illiquidityevents, where agents cannot withdraw their funds, can be expectedto cause panic in financial markets. Therefore from the point ofview of financial risk, the efficiency of markets is of central interest.Thus in this section we consider whether PLFs are efficientwithin a given protocol, considering Compound [9] within a frame-work which is standard in assessing the efficiency of markets inthe context of foreign exchange: Uncovered Interest Parity. First, we set out Uncovered Interest Parity (UIP) as it would nor-mally appear in the context of foreign exchange between two coun-tries: domestic and foreign . An investor has the choice of whetherto hold domestic or foreign assets. UIP is a theoretical no-arbitragecondition, which states that in equilibrium, if the condition holds,a risk-neutral investor should be indifferent between holding thedomestic or foreign assets because the exchange rate is expected toadjust such that returns are equivalent. eFi Protocols for Loanable Funds:Interest Rates, Liquidity and Market Efficiency I n t e r e s t R a t e Borrow RateInterest Rate at Kink (a) 21 February - 13 March I n t e r e s t R a t e Borrow RateInterest Rate at Kink (b) 14 March - 5 April I n t e r e s t R a t e Borrow RateInterest Rate at Kink (c) 6 April - 21 April
Figure 10: Borrowing rate (hourly mean) distributionaround kink for DAI.
For example, consider UIP holding between GBP and USD. Aninvestor starting with Âč1m at t = • receive a return of (e.g.) 3% in GBP at t =
1, resulting inÂč1.03m • buy $1.23m USD at t = + ι i = E t [ S t + k ] S t ( + ι j ) (16)where E t [ S t + k ] denotes the expectation in period t of the ex-change rate between assets i and j at time t + k , k is an arbitrarynumber of periods into the future, S t is the current spot exchangerate between assets i and j , ι i is the interest rate payable on asset i and ι j is the interest rate payable on asset j . If Equation (16) holds,then investors cannot make risk free profit. Here, analogously, we perform a pairwise analysis of all possiblepairs of tokens available within a protocol, seeking to establishwhether UIP holds for that pair. For UIP to hold it must be the casethat a risk-neutral investor would be indifferent between saving(or borrowing) either of the tokens within the pair, because theexchange rate between any token pair adjusts such that no risk-freeprofit can be made. As it is the largest PLF [23], we consider to whatextent the condition holds within Compound [9].
To develop our empirical specification, we assume that agents haverational expectations: S t + k = E t [ S t + k ] + ϵ t + k (17)where ϵ denotes a random error. We test whether UIP obtains withthe following empirical specification. S t + − S t = α + β ( ι i − ι j ) + ϵ (18) H Strict form UIP: α = β = α perhaps reflect-ing a risk premium [3]. H’ Weak form UIP: β = udgeon et al. For both borrowing and saving rate regressions, we use heteroskedas-ticity and autocorrelation robust standard errors, which we reportin brackets in the results. We find that for the 28 market pairs, theevidence suggests that in 8 cases UIP in both weak and strong formholds: we are unable to reject the null hypotheses of 19 and 20 atthe 1% significance level. However, in all but one of the cases, thestandard errors are large, such that it would be difficult to reject anynull hypothesis. The one case with statistical evidence consistentfor the UIP is the ETH/REP pair, where the β coefficient is statisti-cally indistinguishable from 1 and the standard error is relativelysmall. The results are reported in table 6 in the appendix. Overall,for daily data on borrowing rates we find very weak evidence thatin some cases UIP may hold, but that in most cases it does not andwe are able to reject both H and H’ at the 1% level. Looking now at saving rates, first, similar to the borrowing case,we find that in 7/28 cases (the same market pairs as for borrowingrates, aside from ETH/BAT), UIP in both weak and strong formsseems to hold (that is, we are unable to reject either H or H’ ).However, again the standard errors are typically large, such that itwould be difficult to reject any hypothesis. The results are reportedin table 7 in the appendix. Overall, for daily data on saving rateswe again find only very weak evidence that in some cases UIP mayhold. Looking at daily frequency data for borrowing and saving, wefind only very weak evidence that UIP holds in some cases. Thistherefore suggests that overall the markets within the CompoundPLF may not be fully capital efficient at present, and it seems primafacie plausible that these results are not only idiosyncratically trueof Compound. The finding that this PLF is not capital efficientat the daily frequency is not surprising - there is considerable ofevidence that UIP does not hold even in traditional foreign exchangemarkets [7]. However, we submit that in the context of a PLF, tothe extent that there is market inefficiency, agents may not be fullyresponding to these incentives.
We now consider the extent of inter-connectedness between pro-tocols by considering how changes in an interest rate for a giventoken on one PLF are related to changes in the interest rate for thetoken on another PLF.For example, consider the borrowing rate for DAI, i b , Dai . A pri-ori, we would expect that if i b , Dai is higher on one PLF than others,agents would be incentivized to borrow from those PLFs with alower borrowing rate, deleveraging on one PLF and leveraging onothers. But this influx of borrowers for the token on other PLFswould, in turn, increase the borrowing rates on those protocols.In this section, taking the stablecoins DAI and USDC, we investi-gate whether there is evidence of such dynamics, and find that suchbehavior is indeed observable. Moreover, we quantify the speed ofadjustment to new equilibria values, and in so doing measure inone way the responsiveness of agents to their incentives in PLFs. . . . . B o rr o w i n t e r e s t r a t e Figure 11: Daily borrowing interest rates on DAI across pro-tocols.
We model both the short and long run dynamics between borrowingrates for DAI and USDC by using a Vector Error Correction Model(VECM).Where time series are non-stationary (e.g. a random walk), therequired criteria for a regression to produce be the Best LinearUnbiased Estimator (BLUE) are not satisfied, because the variablesare not covariance stationary. However, if there exists a linearcombination of non-stationary time series, where this combinationis itself stationary, the series are said to be cointegrated . VECMspermit the modelling of the stationary relationships between suchtime series, and allow estimation of both the long-run and short-runadjustment dynamics. A VECM model is as follows. ∆y t = v + Πy t − + p − (cid:213) i = Γ i ∆y t − i + ϵ i (21)where ∆ denotes a single time step, y t is a vector of K variables, v is a vector of K × Π = (cid:205) j = pj = A j − I k ( I k denotesan indicator vector), where A j is a matrix of K × K parametersfrom a vector autoregression (VAR) , Γ i = − (cid:205) j = pj = i + A j and ϵ is a K × Π has reduced rank0 < r < K it can further be expressed as Π = αβ ′ [27]. In terms ofinterpretation, α provides the adjustment coefficients, β providesthe parameters of the cointegrating (i.e. long-run) equations. Separately, we focus on the borrowing rates for DAI and USDCseparately, considering Compound, Aave and dYdX. We present theborrowing rates for DAI in Figure 11 and for USDC in Figure 12.
DAI Results.
First, we consider the markets for DAI. We do notfind evidence of a cointegrating relationship between DAI on dYdXand either on Compound or Aave via a Johansen test [27], so we Covariance stationary means that the mean and autocovariances are finite and timeinvariant. A VAR(p) can be expressed as y t = v + A y t − + A y t − + ... + A p y t − p + ϵ eFi Protocols for Loanable Funds:Interest Rates, Liquidity and Market Efficiency . . . . B o rr o w i n t e r e s t r a t e Figure 12: Daily borrowing interest rates on USDC acrossprotocols. vec1, c_dai, a_dai
Steps from shock
Graphs by irfname, impulse variable, and response variable
Figure 13: Impulse Response Function: impact of a shock toCompound’s DAI borrow rate on Aave’s DAI borrow rate. remove dYdX from the analysis. We find the optimum lag length tobe 4. The results are presented in full in Table 4 in the appendix.In terms of short adjustment coefficients, we find a statisticallysignificant coefficient on Aave DAI of 0 .
3, such that when theborrowing rate on Compound is high, Aave’s borrow rate quicklyadjusts by increasing to match it, at a speed of 0 .
3. Interestingly,we do not find evidence of the Compound DAI rate adjusting tochanges in the Aave DAI rate, suggesting that Compound’s interestrate changes drive changes in Aave’s borrowing rates, which maysuggest that Compound has market power. This is perhaps to beexpected: as we show in Fig. 3, Compound has the largest borrowand supply volumes for DAI compared to the other two PLFs andthus will plausibly shape interest rates across protocols. In terms oflong-run relationship between Aave DAI and Compound DAI, wefind that the series share a long run relationship, with each seriesmoving with the same sign.We present the impact of a shock to Compound’s DAI borrowrate on Aave’s in Figure 13. It can be seen that a positive shock tothe borrowing rate results in a permanent increase in the borrowingrate on Aave. vec1, c_usdc, a_usdc vec1, c_usdc, d_usdc
Steps from shock
Graphs by irfname, impulse variable, and response variable
Figure 14: Impulse Response Function: impact of a shock toCompound’s USDC borrow rate on Aave and dYdX’s USDCborrow rates.
USDC Results.
For USDC the results are reported in Table 5. ForUSDC, we find that between the series there are 2 cointegratingrelationships [27] and therefore model all three series. In this case,after iteratively tuning the model with postestimation results, wefind the optimum lag length to be 3.It appears that again, Compound may have market power, withthe borrowing rates on Aave and dYdX adjusting to match theCompound interest rate level. Aave appears to adjust with a fasterspeed of 0 . . We performed extensive robustness checks on the fitted VECMmodels. Since our ability to draw sound inference on the adjust-ment parameters depends on the cointegrating equations beingstationary, we plot the cointegrating equations over time (see Fig-ures 15, 16 and 17 in the appendix.) We argue that the cointegratingequations appear without significant trends and therefore are sta-tionary. Furthermore, we check that we have correctly specified thenumber of cointegrating equations in Figures 18 and 19. We find noevidence that any of the eigenvalues are close to the unit circle, andtherefore no evidence that the model is misspecified (see [27] fordetails on this test.) Additionally, we test for serial correlation inthe residuals of the regressions and find no evidence of this. A testfor the normality of the errors in our models does suggest that theerrors are non-normally distributed, which may affect our standarderrors but should not result in parameter bias. Jointly this panelof robustness tests gives us confidence that the VECM models arereasonably well specified. udgeon et al.
Overall we find evidence of cointegrating relationships betweenmarkets for DAI and USDC. In turn, this suggests that to someextent interest rate changes in one protocol are associated withinterest rate changes in others, perhaps in turn providing evidenceof agents being incentivized to change protocol by the rates theyobserve. Moreover, we also find some evidence of Compound havingmarket power.
In this section, we present related work about interest rates in bothtraditional finance as well as in DeFi protocols. This paper being,to the best of our knowledge, the first academic work to analyzein-depth PLFs, we include some non-academic work covering someaspects of PLFs interest rates.In [15], the author describes how the interest rate models workin PLFs. The author first provides a definition of the utilization ratioof a PLF, then describes linear and polynomial interest rate modelsand finally presents how these different models are used by threemajor PLFs, namely, Compound, dYdX and DDEX [29].The author of [2] analyzes Compound to show the risks inherentto decentralized lending. In particular, they focus on the risks asso-ciated with illiquidity and bank runs. The authors analyze the SAImarket on compound and find that there were several periods ofnear-illiquidity and actual-illiquidity. They present instances wherethe illiquidity is created because of large loans in a short periodof time and others where it is created by the lenders withdrawinglarge amount of funds they had locked. In particular, they show thaton five occasions, a single transaction was sufficient to withdrawmore than a quarter of the available liquidity, and in the worstcase a single transaction drained more than 95% of the availableliquidity.The author of [12] focuses on how Black Thursday [24] in March2020 affected the Aave market. They first show that the amountof money borrowed through flash loans went up by more than10,000% in only a few hours because users were leveraging theseto liquidate their collateralized debt positions [17, 18]. The authoralso highlights the fact that the amount of borrows liquidated onAave during Black Thursday was more than 100 times higher thanthe typical amount liquidated, reaching a total of more than $550kUSD in a single day. Finally, the author show that during the BlackThursday crisis, some design flaws of MakerDAO’s protocol [17]caused Maker to loose a total of more than $4m USD worth ofcollateral.In [14], the authors elucidate the difference between the inter-mediation and financing roles of traditional banks, and show thatwhen modelling banks with financing models as opposed to inter-mediation models, identical shocks have much greater effects onthe real economy.Finally, Brody et al. [5] present a work about interest rates in thecontext of cryptocurrencies but centered on a different problem.The focus of their work lies on how cryptocurrencies could set aninterest rate for their holders, such that that they accumulate theseinterest rates in a continuous manner.
In this paper, we coin the phrase Protocol for Loanable Funds, todescribe DeFi equivalents of Intermediaries for Loanable Funds intraditional finance, providing a classification framework for theextant interest rate models. We analyze three of the largest PLFs interms of market liquidity, efficiency and dependence.In terms of market liquidity we find we find that individuallyPLFs often operate at times of high utilization, and moreover, of-ten these moments of high utilization are shared across protocols.Moreover, we find that token holdings can be concentrated to a verysmall set of accounts, such that at any time were a small number ofsuppliers to withdraw their funds, perhaps in concert, they couldsignificantly reduce the liquidity available on markets and perhapsmake such markets illiquid.In terms of market efficiency, we consider whether uncoveredinterest parity holds. On the whole, we find that it does not, sug-gesting that token markets are at present relatively inefficient. Thisalso suggests that at present agents may not be fully responsive tointerest rate incentives.In terms of market dependence we find that the borrowing rateson these protocols influence each other, an in particular that Com-pound appears to have some market power to set the prevailingborrowing rate for Aave and dYdX.
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Testing uncovered interest parity at shortand long horizons during the post-Bretton Woods era . Technical Report. NationalBureau of Economic Research.[8] dYdX. 2019. dYdX. https://dydx.exchange/[9] Compound Finance. 2019. Compound Finance. https://compound.finance/[10] Irving Fisher. 1930.
Theory of interest: as determined by impatience to spend incomeand opportunity to invest it . Augustusm Kelly Publishers, Clifton.[11] Emilio Frangella. 2020. Aave Borrowing Rates Upgraded. https://medium.com/aave/aave-borrowing-rates-upgraded-f6c8b27973a7[12] Emilio Frangella. 2020. Crypto Black Thursday: The Good, the Bad, and theUgly. https://medium.com/aave/crypto-black-thursday-the-good-the-bad-and-the-ugly-7f2acebf2b83[13] Lewis Gudgeon, Daniel Perez, Dominik Harz, Arthur Gervais, and BenjaminLivshits. 2020. The Decentralized Financial Crisis: Attacking DeFi. arXiv preprintarXiv:2002.08099 (2020).[14] Zoltan Jakab and Michael Kumhof. 2015. Banks are Not Intermediaries ofLoanable Funds And Why this Matters.
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529 (2015).https://doi.org/10.2139/ssrn.2612050[15] Tian Li. 2019. How Lending Pool Interest Rates actually work.https://medium.com/hydro-protocol/how-lending-pool-interest-rates-actually-work-375794e71716[16] Tether Limited. 2016. Tether: Fiat currencies on the Bitcoin blockchain. https://tether.to/wp-content/uploads/2016/06/TetherWhitePaper.pdf Accessed: 08-06-2020.[17] Maker. [n.d.]. The Maker Protocol: MakerDAO’s Multi-Collateral Dai (MCD)System. https://makerdao.com/en/whitepaper/ Accessed: 08-06-2020.[18] MakerDAO. 2019. MakerDAO. https://makerdao.com/en/[19] Burton G. Malkiel. 1989.
Efficient Market Hypothesis . Palgrave Macmillan UK,London, 127–134. https://doi.org/10.1007/978-1-349-20213-3_13[20] Amani Moin, Emin Gün Sirer, and Kevin Sekniqi. 2019. A Classification Frame-work for Stablecoin Designs. arXiv preprint arXiv:1910.10098 (2019). eFi Protocols for Loanable Funds:Interest Rates, Liquidity and Market Efficiency
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95, 4 (1987), 758–774.[29] Scott Winges. 2019. DDEX FAQ: Margin Trading. https://medium.com/hydro-protocol/ddex-faq-margin-trading-bd4b32beb9f[30] Gavin Wood et al. 2014. Ethereum: A secure decentralised generalised transactionledger.
Ethereum project yellow paper
A APPENDIX
ParametersDate α β γ U ∗
17 Dec ’19 19637062989 264248265 570776255707 9e178 Jan ’20 29174130900 264248265 570776255707 9e1726 Jan ’20 37372598273 264248265 570776255707 9e174 Feb ’20 41997859121 264248265 570776255707 9e179 Feb ’20 36209575847 705029680 570776255707 9e1721 Feb ’20 38532925389 264248265 570776255707 9e1714 Mar ’20 19637062989 264248265 570776255707 9e176 Apr ’20 0 2900146648 570776255707 9e1721 Apr ’20 0 264248265 570776255707 9e1727 Apr ’20 0 10569930661 570776255707 9e17
Table 3: Interest rate model and parameter changes for thecDAI contract since 17th December 2019 (prior to this datean earlier variation of the interest rate model —‘Jump RateModel’—was in force since 23rd November 2019; we omit thisperiod for expositional clarity.). - . - . - . . . P r ed i c t ed c o i n t eg r a t ed equa t i on Figure 15: DAI cointegrating equation udgeon et al.
Table 4: Vector Error Correction Model Results - DAI. (1)D_c_daiL._ce1 -0.0332(-0.74)LD.Compound Dai -0.267 ∗∗ (-2.84)L2D.Compound Dai -0.198 ∗ (-2.07)L3D.Compound Dai -0.406 ∗∗∗ (-4.47)LD.Aave Dai -0.0861(-1.24)L2D.Aave Dai -0.0763(-1.23)L3D.Aave Dai -0.0718(-1.22)Constant -0.000956(-0.60)D_a_daiL._ce1 0.300 ∗∗∗ (4.44)LD.Compound Dai -0.247(-1.75)L2D.Compound Dai 0.0405(0.28)L3D.Compound Dai 0.116(0.85)LD.Aave Dai 0.0952(0.91)L2D.Aave Dai -0.0843(-0.90)L3D.Aave Dai 0.0442(0.50)Constant -0.000106(-0.04)Long-run (_ce1)Compound Dai 1Aave Dai -1.4010 ∗∗∗ (-5.64)Constant 0.0402Observations 117 t statistics in parentheses ∗ p < . , ∗∗ p < . , ∗∗∗ p < . Table 5: Vector Error Correction Model Results - USDC(abridged). (1)D_c_usdcL._ce1 0.0146(0.83)L._ce2 0.0271(1.89)Constant -0.000189(-0.89)D_a_usdcL._ce1 0.607 ∗∗∗ (3.42)L._ce2 -0.720 ∗∗∗ (-4.97)Constant -0.00000564(-0.00)D_d_usdcL._ce1 0.115 ∗∗ (2.75)L._ce2 0.0200(0.59)Constant 0.0000538(0.11)Long-run (_ce1)Compound USDC 1Aave USDC 5.55e-17(-5.64)DyDx USDC -1.353 ∗∗∗ (-7.77)Constant 0.0066Long-run (_ce2)Compound USDC -2.78e-17Aave USDC 1DyDx USDC -1.3547 ∗∗∗ (-7.95)Constant 0.0028Observations 119 t statistics in parentheses ∗ p < . , ∗∗ p < . , ∗∗∗ p < . - - . . I m ag i na r y -1 -.5 0 .5 1Real The VECM specification imposes 1 unit modulus
Roots of the companion matrix
Figure 19: USDC cointegrating equations misspecificationtest. eFi Protocols for Loanable Funds:Interest Rates, Liquidity and Market Efficiency
Table 6: Table of UIP results for daily frequency data, using borrowing rates. Using Newey-West heteroscedasticity and auto-correlation robust standard errors (reported in parentheses.)
Pair N.obs α β
R-squared α p-value β p-value Strict form (19) p-value Weak form (20) p-value( α =
0) ( β =
0) ( α = , β =
1) ( β = udgeon et al. Table 7: Table of UIP results for daily frequency data, using saving rates. Using Newey-West heteroscedasticity and autocorre-lation robust standard errors (reported in parentheses.)
Pair N.obs α β
R-squared α p-value β p-value Strict form (19) p-value Weak form (20) p-value( α =
0) ( β =
0) ( α = , β =
1) ( β = eFi Protocols for Loanable Funds:Interest Rates, Liquidity and Market Efficiency - . . P r ed i c t ed c o i n t eg r a t ed equa t i on Figure 16: USDC cointegrating equation 1 - . - . . . P r ed i c t ed c o i n t eg r a t ed equa t i on Figure 17: USDC cointegrating equation 2. - - . . I m ag i na r y -1 -.5 0 .5 1Real The VECM specification imposes 1 unit modulus