The ins and outs of Protocols for Loanable Funds

# The ins and outs of Protocols for Loanable Funds

<h2 class="sm-top-margin">
  Daniel Perez <span class="very minor">Imperial College London</span><br>
</h2>

<h3 class="sm-top-margin">April 22th, Decentralized Finance Spring 2021</h3>

---

# Overview

1. Background
2. PLFs overview
3. PLFs interest rates
4. Collateralization and liquidations
5. Flash loans
6. Summary and open challenges

---

# Background

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# Smart contracts

.pull-left-50[
* Programs deployed on a blockchain
* Usually written in a high-level language and compiled into bytecode
* Interacted with using transactions
* Uses some metering mechanism
]

.pull-right-50[
  ![Smart Contracts](./smart-contracts.png)
  .caption[
  Ethereum Smart Contracts overview]
]

---

# On-chain building blocks

* Facilitate on-chain access to external information
  * Implemented as smart contracts being regularly updated
  * Often require some level of trust
]

* Assets of which the price is pegged to a currency (e.g. USD)
  * Can be implemented in very different ways (e.g. custodial vs non-custodial)
  * Movements above/below target are not uncommon
]

---

# PLFs overview

---

<blockquote>
<i>
The reality of how money is created today differs from the description found in some economics textbooks.
Rather than banks receiving deposits when households save and then lending them out, bank lending creates deposits.
</i>
<br style="margin-bottom: 1em">  
<cite>
Bank of England: <a href="https://www.bankofengland.co.uk/-/media/boe/files/quarterly-bulletin/2014/money-creation-in-the-modern-economy.pdf">https://www.bankofengland.co.uk/-/media/boe/files/quarterly-bulletin/2014/money-creation-in-the-modern-economy.pdf</a></cite>
</blockquote>

---

# Protocols for Loanable Funds (PLF)

* Protocol that intermediates funds between users
* Unlike peer-to-peer lending, funds are pooled
* Requires users to deposit collateral

.text-center[
![Protocols for Loanable Funds overview](./plf.png)
.caption[Protocols for Loanable Funds overview]
]

---

# PLF definitions

* *Market* A smart contract acting as the intermediary of loanable funds for a particular crypto-asset, where users supply and borrow funds.
* *Supply* Funds deposited to a market that can be loaned out to other users and used as collateral against depositors’ own borrow positions.
* *Borrow* Funds loaned out to users of a market.
* *Collateral* Funds available to back a user’s aggregate borrow positions.
* *Locked funds* Funds remaining in the PLF smart contracts, equal to the difference between supplied and borrowed funds.

---

# Agents in the system

* *Supplier* A user who deposits funds to a market.
* *Borrower* A user who borrows funds from a market. Since a borrow position must be collateralized by deposited funds, a borrower must also be a supplier.
* *Liquidator* A user who purchases a borrower’s supply in a market when the borrower’s collateral to borrow ratio falls below some threshold.

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# PLF building blocks

* *Interest rate models* Some function(s) taking liquidity as an argument and returning an interest rate
* *Interest disbursement mechanism* Interest typically accrued per second and paid out per block. Often an interest bearing derivative token used.
* *Collateral* Deposit that can be sold off to recover the debt of a defaulted position
* *Liquidations* The process of selling a borrower’s collateral to recover the debt value upon default
* *Governance mechanism* Decentralized governance typically achieved through an ERC-20 governance token, where token holders' votes are in proportion to their stake

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# PLF use cases

* *Earning interest* Liquidity providers of funds are incentivized by accrued interest
* *Leveraged short position* Borrowing funds of an asset expected to depreciate in value
* *Leveraged long position* Increasing exposure to an asset expected to appreciate in value
* *Liquidity mining* PLFs may distribute governance tokens to their users to incentivize liquidity providers and/or borrowers

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# Interest Rate Models

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# Interest Rates Overview

\$U\_M\$ utilization of deposited funds for market \$m\$, with total loans \$L\$ and gross deposits \$A\$:

The Interest Rate Index \$I\$ and Debt \$D\$ for block \$k\$ are calculated each time an interest rate changes, i.e. as users mint, redeem, borrow, repay or liquidate assets (\$r\$ - per block interest rate, \$t\$ - difference in block height):
<small>
\begin{align}
& \text{Non-compounded} & & \text{Compounded}\\\\
I\_{k,m} &= I\_{k-1,m} (1+rt) & I\_{k,m} &= I\_{k-1,m}(1 + r)^t\\\\
D\_{k,m} &= D\_{k-1,m} (1+rt) & D\_{k,m} &= D\_{k-1,m}(1 + r)^t\\\\
\end{align}
</small>

with a portion of the interest is kept as a reserve (\$\Pi\$), set by reserve factor \$\lambda\$.

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# Linear rates

For loanable funds market \$m\$, borrowing interest rates \$i\_b\$ are given by:

$$
i_{b,m} = \alpha + \beta U_m
$$
where \$\alpha\$ is some constant and \$\beta\$ a slope coefficient on the responsiveness of the borrowing interest rate to the utilization rate.

---

# Non-linear rates

For a loanable funds market \$m\$, the borrowing interest rates \$i\_b\$ follow a non-linear model and are computed as:

$$
    i\_{b,m} = (\alpha \cdot U\_m) + (\beta \cdot U\_m^{32}) + (\gamma \cdot U\_m^{64}).
$$

---

# Kinked rates

Kinked rates can be characterized as:

$$
i_b =
    \begin{cases}
      \alpha + \beta U & \text{if}~U \leq U^{\*}\\\\
      \alpha + \beta U^{\*} + \gamma (U-U^{\*}) & \text{if}~U > U^{\*}
    \end{cases}
$$
  
where \$\alpha\$ denotes a per-block base rate, \$\beta\$ denotes a per-block multiplier, \$U\$ denotes the utilization ratio (with \$U^{\*}\$ denoting the optimal utilization ratio) and \$\gamma\$ denotes a "jump" multiplier.

---

# Borrowing rates for DAI on Compound - Ia

.center-block.max-content[
.caption[21 February - 13 March 2020]
![Borrowing rates on Compound](./compound-observed-borrow.png)
]

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# Borrowing rates for DAI on Compound - Ib

.center-block.max-content[
.caption[21 February - 13 March 2020]
![Borrowing rates vs kink on Compound](./compound-rate-vs-kink.png)
]

---

# DAI Markets

.pull-left-50[
## DAI borrowed and supplied
  ![Total DAI borrowed and supplied](./dai-plf-liquidity.png)
]

.pull-right-50[
  ## Utilization and borrow rates
  ![Utilization and borrow rates](./dai-plfs-illiquidity.png)
]

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# Market risks and liquidations

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# Security Parameters

PLFs have several security parameters, that typically per-market

* *Loan-to-value ratio (LTV)* The ratio of asset that can be borrowed for a given amount of collateral
* *Liquidation threshold* The ratio of asset that can remain borrowed before becoming liquidable
* *Discount factor* The discount at which a liquidator can buy the collateral by repaying the debt of the defaulted position
* *Liquidation ratio* The ratio of the position that can be liquidated at once

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# Conditions for liquidation

* When computing collateral and borrowed funds across markets, amounts are converted to a common currency, e.g. USD or ETH
* A user is liquidable if the *sum of his borrows exceeds the sum of his discounted collateral* across all markets

## Example

Given an ETH market with LTV ratio of 0.5 and liquidation threshold of 0.6

.tight-list[
1. Alice supplies 1 ETH when 1 ETH = 2000 USD
2. Alice borrows 1000 DAI (1 DAI = 1 USD)
3. Price of ETH falls to 1500 USD: Alice is now liquidable
]

---

# Leveraging on PLFs

1. Supply ETH on a PLF.
2. Leverage the deposited ETH to borrow DAI.
3. Sell the purchased DAI for ETH.
4. Repeat steps 1 to 3 as desired.
]

---

# Analyzing Compound

We analyze the different events emitted by Compound smart contracts

.center-block.max-content[
Event     | Description           | State change
----------|-----------------------|-------------
Borrow    | New borrow is created | Borrow
Mint      | cTokens minted for deposit | Supply
RepayBorrow | Borrow is partially/fully repaid | Borrow
LiquidateBorrow | Borrow is liquidated | Supply & Borrow
Redeem | cTokens are used to redeem deposit of asset | Supply

---

# Borrowers and suppliers

Sharp increase when `COMP` rewards started to be distributed

.pull-left-50[
  ![Suppliers and borrowers](./suppliers-borrowers-over-time.png)
  .caption[Number of suppliers and borrowers]
]

.pull-right-50[
  ![Amount of funds supplied, borrowed and locked](./borrow-supply-over-time.png)
  .caption[Amount of funds supplied, borrowed and locked]
]

---

# Distribution of funds

.pull-left-50[
 Top user accounts for 27.4%; top 10 users account for 49%
  ![Distribution of supply](./suppliers-distribution.png)
  .caption[Distribution of supplied funds]
]

.pull-right-50[
 Top user accounts for 37.1%; top 10 users account for 59.9%
  ![Distribution of borrows](./borrowers-distribution.png)
  .caption[Distribution of borrowed funds]
]

---

# Leveraging spirals

* Leveraging spirals is an important reason for concentration in top borrowers/suppliers
* Analysis of the top account
  * Provided in total 55M USD to the protocol
  * Used spirals to supply 342M USD and borrowed 247M
* Analysis of all accounts
  * Over 2,100 accounts (40% of total number borrowers) use leveraging spirals
  * Over 600M USD (~50% of the total supply) is supplied using spirals

---

# Liquidation risk

.pull-left-50[
* `COMP` launch has changed users' behavior
* Before launch, almost all users were at least 25% over their minimum collateral threshold
* After launch, more than 40% of the users were within 5%
]

.pull-right-50[
  ![Collateral locked over time](./supply-borrow-over-time.png)
  .caption[Collateral locked over time, showing how close the amounts are from being liquidated]
]

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# Price fluctuation and liquidation risk

.pull-left-50[
* Users rely on DAI being stable
* Small variations in DAI price can create large liquidations:
  3% price change would have made more than 10M USD liquidable
* This happened last November: over 88M USD liquidated
]

.pull-right-50[
  ![DAI sensitivity](./dai-sensitivity.png)
  .caption[Sensitivity analysis of the liquidable collateral amount given DAI price movement]
]

---

# Liquidations and liquidators

Both liquidated amount and liquidation efficiency has increased with time

.pull-left-50[
  ![Liquidations over time](./liquidation-over-time.png)
  .caption[Liquidations over time]
]

.pull-right-50[
  ![Time to liquidation](./time-to-liquidation.png)
  .caption[Number of blocks elapsed from the time a position can be liquidated to
  actual liquidation]
]

---

# Main observations about liquidations

* Governance token has changed PLF users' behavior significantly
  * A lot of liquidity was attracted but users are taking more and more risks
* Users tend to underestimate the volatility of stable coins
  * Small price deviations can lead to large liquidations
  * Large amounts were liquidated last November
* Liquidators are becoming more efficient
  * More than 70% of the liquidations happen in the block where positions became liquidable

---

# Flash Loans

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# Flash Loans

Flash loans are loans that must be reimbursed within the same transaction.

1. Borrow funds from a PLF
2. Execute arbitrary action with the funds
3. Repay the loan and the fees associated

## Note on Atomicity

* Flash loans require transactions to be atomic: either the transaction fully fails or fully succeed, no partial state update.

* Flash loan providers can check funds are returned using pre/post conditions

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# Flash Loans use-cases

* *Arbitrage* Perform arbitrage using on-chain exchanges and repay the loan and fees using profits from arbitrage

* *Liquidation* Repay debt of a liquidable position, buying collateral at a discount, swap the collateral and repay

* *Collateral swap* Provide flash loan assets as collateral, withdraw collateral in another asset, swap the asset and repay

* *Price oracle manipulation* Use flash loan to artificially change oracle price (e.g. provide a lot of a single asset in a Uniswap pool), perform an action dependent on the price oracle (e.g. deposit price-inflated asset) and use profits to repay

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# Open Challenges and Summary

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# Open Challenges

* *Under-collateralized loans*
  * Current loans require over-collateralization
  * Flash loans allow to borrow without collateral but only for a transaction
  * Under-collateralized loans could allow for other use-cases but are not doable with the current security model
  * Some attempts of under-collateralized loans to white listed smart contracts but support is very limited
* *Composability risks* Flash crash in a stable coin price could lead to huge amount of liquidation
* *Privacy* As implemented, most PLFs are fully transparent, which may or not be desirable

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# Summary

* PLFs allow users to supply and borrow funds
* Borrows must be over-collateralized
* Interest rates incentivize users to keep the utilization ratio close to some desired value
* Liquidations are frequent and most often due to price fluctuations
* Flash loans allow to take non-collateralized loans for a single transaction

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# Relevant literature

* DeFi protocols for loanable funds: Interest rates, liquidity and market efficiency. <small>Lewis Gudgeon, Sam Werner, Daniel Perez, and William J. Knottenbelt. ACM Conference on Advances in Financial Technologies 2020 (AFT'20)</small>
* Liquidations: DeFi on a Knife-edge. <small>Daniel Perez, Sam M. Werner, Jiahua Xu, and Benjamin Livshits. International Conference on Financial Cryptography and Data Security 2021 (FC'21)</small>
* Attacking the DeFi ecosystem with flash loans for fun and profit. <small>Kaihua Qin, Liyi Zhou, Benjamin Livshits, and Arthur Gervais. International Conference on Financial Cryptography and Data Security 2021 (FC'21)</small>
* SoK: Decentralized Finance (DeFi). <small>Sam M. Werner, Daniel Perez, Lewis Gudgeon, Ariah Klages-Mundt, Dominik Harz, and William J. Knottenbelt. arXiv preprint arXiv:2101.08778 (2021)</small>