---
# System prepended metadata

title: "RAI\_Dollar\_— Technical\_Whitepaper"

---

# RAI Dollar — Technical Whitepaper

*Draft v0.2
16 June 2025*

**Status:** Work‑in‑progress.


---
## Table of Contents

1. Abstract
2. Motivation & Background
3. Borrowing RAI Dollars
      3.1 Borrow Rates
      3.2 Borrow Workflow
      3.3 Redemption Shield
      3.4 Supported Collaterals
4. Price‑Stability Architecture
      4.1 Dynamic Base Rate
      4.2 Controlled Par
      4.3 Summary of Price Stability Dynamics
      4.4 RD/USD Price Bounds
5. Stability Pools & Liquidations
      5.1 Collateral Stability Pools
      5.2 Liquidation Flow
      5.3 Liquidator Compensation
6. Redemptions
      6.1 Redemption Flow
      6.2 Redemption Fee Formula and Decay
      6.3 Redemption Shield Interaction
      6.4 Redemption Example
      6.5 Multi-collateral Redemption 
      6.6 Redemption Collateral Weighting
      6.7 Redemption Collateral Weighting Example
      6.8 Redemption Nuances
7. Collateral Management
      7.1 Automated Per‑Collateral Debt Ceilings
      7.2 Collateral Utilization Multiplier
      7.3 Absolute Supply Caps

8. Tokenomics & Revenue Flow
      8.1 Dynamic Routing of Interest Fees
      8.2 Token Emission Allocation
      8.3 Token Emission Timeline

9. Glossary
10. Appendix: Controller Simulations

---

## 1  Abstract

RAI Dollar (RD) is an immutable, over‑collateralized stablecoin engineered to maintain a **\$1 market price**.  Users can lock their crypto collateral and receive a loan in stable RAI Dollars(RD). The protocol uses a **dynamic borrow rate**, **controlled par** and Liquity‑style **redemptions** to provide peg stability in all market conditions. RAI Dollar loans can be minted from many different collaterals, each with its own risk parameters.

### Collateral Management
As the protocol is immutable, it must have mechanisms to manage collateral safety. To protect from immediate collateral or oracle failure, the protocol can automatically *freeze collaterals*, when their total collateralization ratio level is risky.  The protocol can also automatically *evict "zombie" collaterals*, that no longer back a significant amount of debt.

### Debt Concentration
To keep debt concentraion within safe limits, the protocol charges additional interest rate on over-utilized collaterals. Similarly, under-utitilized collaterals see lower borrow rates.

### Debt Limits

The protocol uses adaptive, per-collateral debt ceilings per-collateral to prevent volatility in debt. In addition, to per-collateral limits, the protocol groups collaterals into clusters, and limits the total % of all RD debt, each cluster can back. This aims to keep the collateral backing RD diversified, reducing risk.


---

## 2  Motivation & Background

Decentralized CDP stablecoins are a desired stable asset due to their lack of ties to off-chain assets. They are also a convenient way for borrowers to get a stable loan against their crypto collateral.

However, CDP stablecoins have a history of peg stability problems in time of high market demand, collateral shocks and volatile risk-free interest rates.

*RAI Dollar solves the major peg problems with current decentralized CDP stablecoins.*


### Liquity 

Liquity V1(LUSD) has encountered two recurring failures:

* **Prolonged under‑pegs** in high risk‑free‑rate environments, leading to mass redemptions and runaway collateral ratios.
* **Sticky over‑peg** after market shocks, as borrowers rush to repay debt and there is no protocol lever to expand supply quickly.

Liquity V2(BOLD) made improvements, but still has shortcomings
* **Rate management is required to  avoid redemptions.**  A borrower must manually adjust rates or outsource rate management to a 3rd party to reduce the risk of redemption against a borrower's trove.
* **Sticky over‑peg** problem in extreme markets most likely persists from Liquity V1.
* **Fee routing split** is set at a fixed 25/75 to stakers/stability pool.
* **Few collaterals** supported.

**RAI Dollar solves all of these issues.**


### RAI vs RAI Dollar
Reflexer's RAI used a floating par, which proved to force market convergence. However, the RAI par and market price are theoretically unbounded.
RAI Dollar continuously attempts to target a market price of $1, as is more attractive to holders and borrowers who desire a $1 stable asset and debt instrument.



---

## 3  Borrowing RAI Dollars

Anyone can permissionlessly get a stable loan against their collateral by opening a trove, locking their collateral and minting RAI Dollars.  RAI Dollars can be minted against several collaterals, including ETH, LSTs, Defi Blue chips and LRTs.
### 3.1 Borrow Rates

The final borrow rate charged to a user for borrowing against collateral *i* has three components, a system-wide base rate, a collateral risk multiplier and a collateral utilization multiplier.


#### Effective Borrow Rate:
$rate_{borrow} = rate_{base} \times k_{i} \times m_{i}$ 

Where:
$rate_{base}$: adaptive base rate managed by RAI Dollar protocol to ensure peg stability
$k_{i}$: a constant, per-collateral multiplier to account for collateral risk
$m_{i}$: an adaptive utilization multiplier to deter a collateral from occupying too much debt, reducing protocol risk

$rate_{borrow}$ will adapt to  current market conditions, stabilizing RAI Dollar to its $1 peg.

#### Borrow Rate Limits
$rate_{base}$ minimum: 0.25%
$rate_{base}$ maximum: 30%


### 3.2 Borrowing Workflow

1. Open trove and deposit collateral.
2. Ensure Individual Collateral Ratio ≥ collateral‑specific Minimum Collateral Ratio.
3. Mint RD loan (>= 2,000 min).
4. If selecting **optional Redemption Shield**(details below), ensure ICR > Shield MCR.



### 3.3 Redemption Shield

#### Background

Redemption is a mechanism where 1 RD can be exchanged for an equal value amount of collateral from trove(s) within the system.   When a trove is redeemed against, its debt is repaid, but the trove's collateral goes to the redeemer. Redemptions are [Liquity V1 style](https://docs.liquity.org/liquity-v1/faq/lusd-redemptions), where the troves with the smallest ICRs are redeemed against first.



Redemption is often an undesirable experience for the trove owner. RAI Dollar introduces *Redemption Shield*, which guarantees a borrower's collateral will not be redeemed against under any circumstance.  Redemption field is an opt-in feature and requires:
1) Paying a higher borrow rate, $rate_{shield}$
 The current borrow rate is multiplied by a constant, $κ$, to yield the final borrow rate paid when shielded from redemptions.

* $rate_{shield} = κ_i rate_{base} * k_i * m_i$
2) Maintaining a higher MCR, **Shield Ratio**

* ${MCR_{i_{shield}}} = 1.3 * MCR_i$


***If a shielded trove's c-ratio drops below the Shield Ratio, it *can* be redeemed against.***

If the user has opted into Redemption Shield, and their $ICR > {MCR_{i_{shield}}}$, their trove will be exempt from redemptions.
### 3.4 Supported Collaterals(wip)


| Asset  | MCR   | Risk Multiplier($k_{i}$) | Optional Shield Multiplier (κ) | Optional Shield Ratio (κ)| Bootstrap Debt‑Cap |
| ------ | ----- | ------------------- | --------------- |--------------- | ------------------ |
| ETH    | 110 % | 1.0                 | 1.8    |180%| \$50 M             |
| wstETH | 120 % | 1.05                 | 1.4     |190%| \$50 M   
| rETH | 120 % | 1.05                 | 1.4     |190%| \$50 M  |
| WBTC   | 130 % | 1.5                 | High            |200%| \$30 M             |
|etc      | …     | …                   | …               || …                  |


---

## 4  Price Stability Architecture


### 4.1 Dynamic Base Rate

#### Overview

RD Dollar automatically adjusts the base rate to help correct peg deviations. Generally, *decreasing* borrow rates increases supply, decreasing the market price. *Increasing* borrow rates decreases supply, increasing the market price.

The protocol uses a DEX-based RD/USD oracle(details below) to determine if RD is trading above or below peg, and corrects accordingly.


#### Behavior
| Market Price | Description  | Correction Needed | Action|
| ----------- |-------|----------|----------|
| **>$1.00** | over-peg | increase supply/reduce demand|**decrease** borrow rate | 
| **<$1.00** | under-peg |decrease supply/increase demand|**increase** borrow rate | 
| **$1.00+-0.003** | at-peg | none| no change| 


#### Visualization: Under-peg scenario

![image](https://hackmd.io/_uploads/rkNwIe57lx.png)

#### Visualization: Details
Step 0: Market is at $1 peg. Base rate is stable at 5%.
Step 6: Market price drops to $0.97. Base rate immediately increases and market starts to respond.
Step 10: Base rate has increased to 6%, increasing demand/decreasing supply. Market price increases.
Step 20: Market price has returned to $1 peg and the base rate has achieved a new equilibrium of 6%.

#### Visualization: Over-peg scenario

![image](https://hackmd.io/_uploads/BJWjDg5Xgg.png)

#### Visualization: Details
Step 0: Market is at $1.00 peg. Base rate is stable at 5%.
Step 6: Market price spikes to $1.03. Base rate immediately decreases and market starts to respond.
Step 10: Base rate has decreased to 4%, decreasing demand and increasing supply. Market price decreases.
Step 20: Market price has returned to $1 peg and the base rate has achieved a new equilibrium of 4.2%.

***Over-peg Note***: Demand for CDP leverage can be inelastic to even very low rates, so lowering rates might *not* return RD to peg. In this case, the par controller(details below) will kick-in, ultimately forcing convergence to $1.00.







#### Base Rate Safety Features

As the base rate adapts to a TWAP of the RD/USD market price, it should be resilient to market spikes and attempted manipulation. There are several features to limit its attack surface and avoid noisy, volatile base rates.

* **Noise Barrier**: $\epsilon=0.003$ around $1.00 where peg deviation is rounded down to 0 and there is no change in $rate_{base}$. Avoids noisy rate response.
* **Noise Ramp**: area beyond noise barrier, where rate response is softened to avoid jumps in rate response. $0.003 < \epsilon < 0.005$
* **Range:** 0.25 % – 30 % APR
* **Rate of Change Bounds:** $rate_{base}$ can change up to +-1 % every 6 hours


### 4.2 Controlled Par

#### Overview

Par is the "face value" of RAI Dollar, the dollar amount of 1 RD for all operations: calculating collateral ratios, redemptions, and liquidations. Almost all other stablecoins have a fixed par of $1. Reflexer's RAI showed that a floating par can influence a stablecoin's  market price.

RD Dollar automatically adjusts the par value, $price_{par}$, of RD to correct prolonged peg deviations. Par is adjusted much more slowly than the borrow rate, responding to deviations of the 24-hr TWAP of $price_{market}$ while rates respond to a shorter 12-hr TWAP.  Rates are the first knob used to correct peg deviations and moving par is the last resort.

*Increasing par* decreases the c-ratio of all troves. This decreases demand for leverage and increases market price. Moreso, it increases demand through redemption, as RD is redeemed at par value. This also *increases the market price*.

*Decreasing par* has the opposite effect, an increase in demand for leverage and a decrease in demand for redemptions. Decreasing par far enought creates a profitable arbitrage(details below). All of these effects *decrease the market price.*

#### Behavior
| $price_{market}$ | Description  | Correction Needed | Action|
| ----------- |-------|----------|----------|
| **>$1.00** | over-peg | increase supply/reduce demand|**decrease** $price_{par}$ | 
| **<$1.00** | under-peg |decrease supply/increase demand|**increase** $price_{par}$ | 
| **$1.00+-0.003** | at-peg | none| no action| 

#### Example #1: $p_{market}$ shock > $1

In this case, the system **lowers** $price_{par}$ until $price_{market}$ responds and lowers to $1. After time $price_{par}$ slowly starts to return to $1.

![image](https://hackmd.io/_uploads/Hy6-3xcmxe.png)


#### Why does $price_{market}$ drop in response to a lower $price_{par}$?

#### 1) Arbitrage
Because of the minimum collateral ratio(CR) of 110% for ETH collateral, there will be an immediate lock+mint+sell arbitrage when $price_{par} < 1/1.10 \approx 0.91$. The total market value of debt will be greater than the value of collateral locked, so arbitragers will lock collateral, sell the debt and "walk away".

As long as $price_{market} >$ $ $1$, the system keeps lowering $price_{par}$.  Market participants understand eventually this will lead to the above abitrage and the market will preemptively go to $1. This threat of future arbitrage invokes market actors to sell before this happens.

This arbitrage is the same mechanism used by Liquity and other CDP protocols to provide an *upper bound* on market price. ***In RD's case moving the controlled par down causes the upper bound to decrease to $1***

#### 2) Higher Collateralization Ratio(CR)

Lowering $price_{par}$ raises all vault CRs, making them safer from liquidation.

This lowers $price_{market}$ by 
a) reducing demand for RD to re-pay debt and avoid liquidation
b) increasing RD supply due to more room for debt to be drawn and sold

#### Example #2: $price_{market}$ shock < $1

Consider a $price_{market}$ below $1.  In this case, the system will raise $price_{par}$ until the market trades at $1.

![image](https://hackmd.io/_uploads/rJHd3gqQle.png)


#### Why does $price_{market}$ rise in response to a higher $price_{par}$?

##### 1) Redemption Arbitrage
RD is directly redeemable at $price_{par}$ value in ETH for a small fee. Therefore, increasing $price_{par}$ will make buy+redeem arbitrarage more attractive, increasing $price_{market}$.

Once this arbitrage makes $price_{market} >$ $ $1$ , $price_{par}$ will start to fall. Redemptions will eventually become unprofitable, and the $price_{market}$ will return to $1.

##### 2) Lower Collateral Ratio

Raising $price_{par}$ lowers all vault collateral ratios, increasing the risk of liquidation and make lending less attractive.

This raises $price_{market}$ due to
a) increasing demand for RD to pay back debt and avoid liquidation
b) decreased supply of RD due to less minting of new debt




#### Par Safety Features

Since $price_{par}$ responds to changes in the RD $price_{market}$ TWAP, the mechanism must be resilient to market shocks and attempted manipulations. It has several features to limit its attack surface and avoid volatile $par$ values.

* **Noise Barrier**: $\epsilon=0.003$ around $1.00 where peg deviation is rounded down to 0 and there is no change in $price_{par}$. Avoids noisy par response.
* **Noise Ramp**: area beyond noise barrier, where par changes are softened. $0.003 < \epsilon < 0.005$
* **Range:** $p_{par} \in$ [$0.85 – $1.15]
* **Maximum Rate of Change:** +-$0.01 every 4 hours.
* **Returns to $1 over time:** $price_{par}$ automatically creeps back towards $1 over time.  This is accomplished by decaying the integral term in par formula with a 48 h half‑life, slowly bringing $price_{par}$ back to $1 after a deviation.



### 4.3 Summary of Price Stability Dynamics

| Scenario | Rate Response | Rate Dynamics| Par Response |Par Dynamics |
| --------------- | ---------------------------- | ----|---------|----------------------- |
| **Under‑peg**   | ↑$rate_{base}$ | ↑SP yield & expensive leverage       | ↑$price_{par}$ |profitable redemptions; cheap de-leverage  |
| **Over‑peg**    | ↓$rate_{base}$ |↓SP yield & cheap leverage          | ↓$price_{par}$ | no redemptions; eventual arbitrage; expensive leverage      |


### 4.4 RD/USD Price Bounds

| Bound        | Bound Type| Value| Dynamics     | 
| --------------- | -------------------|--------- | ----|
| **Market Upper** | hard| $1.10|lock-mint-sell arbitrage; $price_{market} ≤ 1.1\,price_{par}$      Practically, $price_{market} ≤$ $1.10 | 
| **Market Lower** | soft|$1.00 - $fee_{redemption}$|profitable redemptions; fee increases with redemption volume, widening the gap; rate and par respond | 
| **Par Lower** |hard|$0.909|lock-mint-sell arbitrage| 
| **Par Upper** | soft|$1.00 + risk-free rate| reached at parity between redemption profitability and risk-free rates; most likely short-lived due to rising RD rates  | 

---


## 5  Stability Pool and Liquidations

Each collateral **has its own Liquity-style Stability Pool(SP)**. The SP receives a portion of loan fees and also profits from liquidations.

If the SP is empty, liquidated debt and collateral are redistributed to other troves.

### 5.1 Collateral Stability Pools

* Depositors stake RD in the SP for the collateral they wish to secure.
* Stakers receives a pro-rata slice of collateral borrow fees(RD) and liquidations(collateral).
* Stakers will seek SPs with highest yield, given the collateral's risk.

### 5.2 Liquidation Flow

1. Trove(s) can be liquidated when ICR < MCR. Liquidator earns fee.
2. If SP is empty, skip to 6.
3. Corresponding SP burns RD to cover a liquidated trove's debt.
4. SP receives the amount of the trove's collateral that covers the liquidated debt, plus a penalty.
    a. **Most collaterals**: SP receives collateral equal to 105% of liquidated debt
    b. **Volatile collaterals**: SP receives collateral equal to 110% of liquidated debt
5. Trove owner keeps leftover collateral.
6. Remaining debt & collateral are redistributed across troves proportionate to their collateral locked.


### 5.3 Liquidator Compensation

$reward_{liquidation}$ = 200 RD + 0.5% of Trove's collateral



## 6  Redemptions

Redemptions guarantee a **hard price floor** by letting anyone redeem RD for collateral ar $price_{par}$  value, minus a fee. Redemptions realign collateral ratios by preferentially targeting the riskiest troves in each collateral, and weighting redemptions towards collaterals with more risk.

---
### 6.1 Redemption Flow

1. User sends RD to be redeemed to Redemption Aggregator contract
2. Redemption Aggregator splits up redemption amount among all collaterals, weighing riskier collaterals higher in order to reduce protocol risk.
3. Within each collateral, troves are processed in ascending ICR.
4. RD is burned for each collateral $i$.
    * $ΔCollateral_i = ΔRD_i * price_{par} *(1-φ_i) / P_i$
    $φ_i$: redemption fee(details below)
    $P_i$: collateral price
6. Redemption Aggregator sends the basket of collateral to the user.



If insufficient un‑shielded debt is found, the redemption is unsuccessful.

---

### 6.2 Redemption Fee Formula & Decay

Redeemers are charged a fee that is  left in the trove's collateral after redemption.

Let $φ_i(t)$ be the redemption fee charged for collateral $i$ at time $t$.

Then:


$$φ_i(t+Δt) = max(φ_i(t)\,e^{-Δt/τ} +λ\frac{ΔRD}{{RD}_{supply}}, fee_{min})$

where:
$τ = 12h$
$λ = 0.5$
$fee_{min}$: a collateral specific minimum fee

**1) Existing fee is decayed**
The $φ_i(t)\,e^{-Δt/τ}$ term decays the current fee exponentially with twelve‑hour half‑life:
    
**2) Fee is increased proportionate to supply**
The $λ\frac{ΔRD}{{RD}_{supply}}$ term increases the fee more for larger redmptions.
    For example, as $λ = 0.5$ redeeming 10% for supply increases the fee by 5%.
    
Notes:
      **User max fee :** Redeemers supply a $max\_fee$ that is checked against $φ_i$ for atomic protection.
      **Per-collateral $fee_{min}$:** Each collateral has its own minimum fee to account for collateral volatility/oracle divergence.
     **Fee routing:** Collected fees accrue to each SP that burned the debt.


---

### 6.3 Redemption Shield Interaction

* Troves with ICR ≥ `S_i = κ_S·MCR_i` are **skipped**.
* If a shielded trove later drops below `S_i`, it re‑enters the queue automatically.
* Shield owners continue paying their add‑on APR while protected.

---

### 6.4 Collateral Redemption Example

*Input values*

$RD_{amount} = 50,000$
$p_{par}$ = $1.02
$P_{ETH}$ = $3,600
$φ_{ETH}^{decayed} = 0.4\%$
$RD_{supply} = 25 M$

1. **Fee increase**
   $Δφ = 0.5 × 50,000 / 25,000,000 = 0.1 \%$
      $φ_{ETH}^{new} = φ_{ETH}^{decayed} + Δφ$
   $φ_{ETH}^{new} = 0.4\% + 0.1\% = 0.5\%$
      $φ_{ETH}^{new} = 0.5\%$
2. **Total collateral amount**
   $collat_{total} = RD_{amount} × p_{par} / P_{ETH}$
   $collat_{total} = 50,000 × 1.02 / 3,600$
      $collat_{total}≈ 14.17 ETH$
3. **Collateral sent to redeemer**
   $(1-φ_{ETH}^{new}) × 14.17 ≈ 14.10 ETH$
4. **Collateral fee stays in trove**
      $φ_{ETH_{new}} × 14.17 ≈ 0.07 ETH$
   

* Caller ends with **14.10 ETH**
* Trove keeps **0.07 ETH**
* 50k RD is burned.

---
### 6.5 Multi‑Collateral Redemptions

In practice, redemptions in RAI Dollar are split across collaterals.  This is much safer for the protocol than allowing redeemers to pick a specific collateral to redeem their RD for.

The collateral split is not uniform, but favors redemption against riskier collaterals. Redemptions *increase* a collateral's TCR, so favoring riskier collaterals increases the protocol's safety.

### 6.6 Redemption Collateral Weighting
1. Define raw weight for a collateral during redemptions

$weight_i^{raw} = unbacked_i * (MCR_i/TCR_i)^2$

where:
$unbacked_i$ = the amount of collateral $i$ debt that is not in the SP

Multiplying by $(MCR/TCR)^2$ is an improvement over Liquity V2 and prioritizes collaterals that have more debt at risk near the collateral MCR.

2. scale the raw weight

$weight_i^{final}=\frac{weight_i^{raw}}{\sum {weight_i^{raw}}}$

$weight_i^{final}$ is the final proportion of RD that will be redeemed against collateral $i$.


### 6.7 Redemption Collateral Weighting Example

Consider a redemption with 3 collaterals.

$RD_{amount} = 50,000$
$unbacked_{ETH} = 100,000$
$unbacked_{RETH} = 120,000$
$unbacked_{AAVE} = 30,000$
$MCR_{ETH} = 110\%$
$MCR_{RETH} = 120\%$
$MCR_{AAVE} = 150\%$
$TCR_{ETH} = 250\%$
$TCR_{RETH} = 230\%$
$TCR_{AAVE} = 290\%$

**Raw weights**
$weight_{ETH}^{raw} = 100,000 * \frac{110\%}{250\%} = 44,000$
$weight_{RETH}^{raw} = 120,000 * \frac{120\%}{230\%} \approx 62,609$
$weight_{AAVE}^{raw} = 30,000 * \frac{150\%}{290\%} \approx 15517$

Then,

${\sum {weight_i^{raw}}} = 44,000 + 62,609 + 15,517$
$=122,126$

**Scaled weights**

$weight_{ETH}^{final} = \frac{44,000}{122,126} \approx 0.360$

$weight_{RETH}^{final} = \frac{62,609}{122,126} \approx 0.513$

$weight_{AAVE}^{final} = \frac{15,517}{122,126} \approx 0.127$

**Final redemption amounts, per collateral**

$RD_{ETH} = 0.360 * 50,000 = 18,000$

$RD_{RETH} = 0.360 * 50,000 = 25,650$

$RD_{AAVE} = 0.127 * 50,000 = 6,350$







---


## 7  Collateral Risk Management

### 7.1 Per-Collateral Adaptive Debt Ceilings

The maximum debt minted per collateral(debt ceiling) targets a certain % gap, $G$, above the total collateral's debt. The debt ceiling grows and contracts around a current collateral's debt amount.

 During the bootstrap period, the debt ceiling is constant. 

#### Bootstrap Period(first 120 days)


Each collateral *i* has a static debt ceiling, $C_i$ that must remain $G=10\%$ above the collateral debt.

* **Collateral Mint Limit:** $debt_{i} * 1.1 ≤ C_i$


#### Bootstrap Debt Ceilings

| Collateral | Debt Limit(RD) |
| ---- | ------------------------------------------------- | 
| ETH    | 50M |    
| WSETH    | 50M |  
| RETH    | 50M |  
| AAVE    | 25M |  
| .    | . | 


#### Post-Bootstrap Period

Debt ceilings can automatically grow by a factor, $f=1.2$, every 7 days and shrink linearly with a half-life of 60 days.

* **Debt Ceiling:** $debt_{i} ≤ C_{i_{t}}$.
    Debt is now allowed up to the ceiling, but within 10% will trigger bounded ceiling growth.
* **Debt Ceiling Growth:**
    Ceiling can grow up to 20% every 7 days.
    Let:
    $s$ = seconds since last growth
    $s_{7 days}$ = seconds in 7 days
    

    Max Ceiling = $C_{i_{max}} = C_{i} * (1 + \frac{0.2s}{s_{7 days}})$
    
    Then, as debt nears the ceiling:
    
    If $debt_{i} * 1.10> C_{i_{t}}$ then $C_{i_{t+1}} = min(C_{i_{max}}, f C_{i_{t}})$
* **Debt Ceiling Decay:** 
    Ceiling slowly decays back towards 110% of current debt amount. Debt Ceiling will go no lower than 100 times the per-trove debt minimum. 
    
    $C_{min} = 100 * debt_{minimum}$
    $= 100 * 2,000\ RD$
    $= 200,000\ RD$
    
    Let:
    $\alpha = 2^{-1/(86400*60)} \approx 0.999998853923969311$
    $C_{i_{t+s}} = max(\alpha^{s}*C_{i_{t}}, max(C_{min}, 1/G * debt_{i_{t}}))$
    
    
### 7.2 Collateral Over‑Utilization Rate Multiplier

#### 7.2.1 Overview
Ths collateral utilization multiplier, $m_i$, adjusts the final RAI Dollar borrow rate based on a collateral's utilization. The purpose is to nudge the system away from unsafe concentrations of collateral debt. 

The primary tool for avoiding risky concentrations of debt is hard debt limits per collaterals. For example, WSTETH cannot back greater than x% of RAI Dollar debt.

The utilization multiplier is a softer tool, and raises or lowers rates to target ideal concentrations.


#### Effective Borrow Rate

Again, the effective borrow rate for collateral $i$ is:

$rate_{borrow} = rate_{base} * k_{i} * m_{i}$


where:
$rate_{base}$: system-wide dynamic interest rate

$k_{i}$: constant risk multiplier for collateral $i$

$m_{i}$: utilization multiplier for collateral $i$

#### Mechanism Overview

collateral utilization too high $\rightarrow$ borrow rate $\uparrow$

collateral utilization too low $\rightarrow$ borrow rate $\downarrow$

---

#### 7.2.2 Mechanism Details

#### 7.2.2.1 Notation

| Symbol         | Meaning                                                 |
| -------------- | ------------------------------------------------------- |
| **n**          | number of collateral types                              |
| **Dᵢ**         | current debt share of collateral *i*; $Σ Dᵢ = 1$ |
| **Tᵢ**         | target share for collateral *i*; $Σ Tᵢ = 1$              |
| **ρᵢ**         | relative error $ρᵢ = (Dᵢ – Tᵢ) ⁄ Tᵢ$                      |
| **ε**          | dead‑band radius (e.g. 5 %)                  |
| **α** (stick)  | overshoot penalty coefficient                          |
| **κ** ∈ \[0,1] | incentive softness;                             |
|**β = κ α** | incentive  coefficient                        |
|**R** |  base rate                        |
|**kᵢ** |  risk multiplier for collateral *i*                        |
| **mᵢ**         | utilization multiplier for collateral *i*                   |

---

#### 7.2.2.2 Raw multiplier

$mᵢ^{raw} = \begin{cases}
 1 + α\,(ρᵢ-ε) & \text{if } ρᵢ >  ε          \\[6pt]
 1 - β\,( -ρᵢ-ε) & \text{if } ρᵢ < -ε         \\[6pt]
 1               & \text{if } |ρᵢ| ≤ ε
\end{cases}$

*Clamp* `mᵢ^{raw}` to `[M_MIN , M_MAX]` (e.g. 0.5 – 2.0 WAD).

---

#### 7.2.2.3 On-target collaterals

Identify collaterals that are close enough(within $\epsilon$) to their utilization target.. These mutipliers will remain at 1 as their utilization is acceptable.

On-target set 𝔄 ≔ { i ∣ |ρᵢ| ≤ ε } → force `mᵢ = 1`.
Off-target set 𝔑 = complement of 𝔄.

---

#### 7.2.2.4 Off-target collaterals: revenue‑neutral rescale

Rescale the multiplier for all collaterals that are off-target.

Scale factor:

$\text{scale} = \frac{1 - \sum_{i∈𝔄} D_i}{\sum_{j∈𝔑} D_j\,m_j^{raw}}$

Final multipliers:

$mᵢ = \begin{cases} 1                              & i∈𝔄             \\[4pt]
 mᵢ^{raw} · \text{scale} & i∈𝔑
\end{cases}$

This enforces $Σ Dᵢ mᵢ = 1$, aka "revenue neutrality"

---
#### 7.2.2.5 Properties

* **Overall revenue‑neutral:** 
    * ***Changes in collateral distribution due to not alter protocol revenue and thus do not influence RAI Dollar peg***.
    * $Σ Dᵢ mᵢ = 1$
* **Stable Mechanism** 
    * The penalty and incentives coefficients are governed by $k·α < 2$, meeting formal discrete stability requiremens.<Link to proof>
* **Minimize rate changes:**    
    * Rates are unchanged for collaterals that are close to their target.
    * $m = 1$ whenever |ρ| ≤ ε.

---

#### 7.2.2.6 Derivation of the **scale** factor  *(proof)*

We require the debt-weighted sum of all **final** multipliers to remain 1:

$\sum_i D_i\,m_i = 1.$

Split collaterals:
* **𝔄** — on-targets (|ρᵢ| ≤ ε → mᵢ = 1)
* **𝔑** — off-targets (will be scaled)

Explicit sum:

$\sum_{i∈𝔄} D_i·1 \;+\; \text{scale}\,\sum_{j∈𝔑} D_j m_j^{raw} = 1.$

Let
* **A** ≔ $Σ_{𝔄} Dᵢ$     (on-target set debt share)  
* **B** ≔ $Σ_{𝔑} Dⱼ mⱼ^{raw}$ (off-target weighted raw sum)

Then `A + scale·B = 1`, so

$\boxed{\displaystyle\text{scale} = \frac{1 - A}{B}}
      = \frac{1 - \sum_{i∈𝔄} D_i}{\sum_{j∈𝔑} D_j\,m_j^{raw}}.$

After applying this scale we have

$\sum_i D_i m_i=A+(1-A)=1 \square$

Thus revenue neutrality is proven.


### 7.3 Supply Caps

#### Per Collateral Supply Caps



| Cluster |  Supply Cap |
| -------- | -------- | 
| ETH      |  40%     |
| WSETH     |  35%     |
| RETH     |  35%     |
| AAVE     |  25%     |
| LRT     |  20%     |


#### Cluster Supply Caps
| Cluster | Assets | Supply Cap |
| -------- | -------- | -------- |
| ETH Like     | ETH, LSTs     | 45%     |
| BTC Like     | BTC derivatives     | 35%     |
| Blue Chips Like     | AAVE, Link, GNO     | 40%     |
| LRTs     | pufETH     | 25%     |

### 7.4 Collateral Eviction

#### Overview

Collaterals may lose popularity in the market or become deprecated by their communities over time. These collaterals could become volatile or susceptible to manipulation.

As RAI Dollar is immutable, it needs a mechanism to deal with a collateral that is no longer generating a signficant amount of RAI Dollar Debt.

#### Eviction Mechanism

1) Once a collateral's portion of debt has shrank to less than 0.5%, it can be labeled as an *eviction candidate*.
2) After 7 days, if that collateral's portion of debt is still less than 0.5%, it can be evicted.

When a collateral is evicted, the same exact procedures are followed for freezing collaterals.


#### Bootstrap Period

Collaterals cannot be evicted during the initial boot strap period.

### 7.5 RAI Dollar Oracle Architecture **TBD**
Anchor: Balancer RD/USDC/USDT/DAI pool (60 s delayed TWAP)

---

## 8  Tokenomics & Revenue Flow

### 8.1 Dynamic Routing of Interest Fees

Interest fees collected from the protocol will be dynamically allocated among
* Stability Pool depositors
* Liquidity providers
* Market Oracle
* Token stakers


### 8.2 Token Emmisions Allocation


| Entity | Allocation, % | 
| -------- | -------- | 
| Stability Pools     | 50%     | 
| Liquidity Incentives     | 12%     | 
| Team & Advisors     | 20%     | 
| Investors     | 15%     | 
| Foundation multi-sig, reserve     | 3%     | 
| Total    | 100%     | 


### 8.3 Token Emissions Timeline

| Entity | Description| 
| -------- |  -------- | 
| Stability Pools     |  decaying w/ 1-year half-life(1st year: 50%, 2nd year: 25%, 3rd year: 12.5%, etc)     | 
| Liquidity Incentives  | decaying w/   1-year half-life(1st year: 50%, 2nd year: 25%, 3rd year: 12.5%, etc)       | 
| Team & Advisors     |  1-year lockup, then 2.5-year linear stream     | 
| Investors     |  1-year lockup, then 2.5-year linear stream |
| Foundation multi-sig     |  3%     | 


#### Stability Pools

The 50% SP allocation is split proportionately among the collaterals at launch as follows:

| Collateral | Allocation, % | 
| -------- | -------- | 
| ETH    | 5%     | 
| WSTETH     | 5%     | 
| RETH     | 5%     | 
| .     | .     | 
| .     | .     | 


#### Liquidity Incentives

Incentivize liquidity in Balancer RD/DAI/USDC/USDT pool, which is used for RD/USD price oracle.

#### Team & Advisors

Allocations sent to Sablier streams upon launch and will start streaming after 1 year lockup.

### 8.2 Emission Curves and Visualization





![image](https://hackmd.io/_uploads/rJwGfv1Qel.png)


Reference: https://docs.google.com/spreadsheets/d/183B_uRWFIkkUZ8PnGEPc7olgtnlpryRsC1uLGRRpNDQ/edit?gid=0#gid=0

---


## 9  Glossary

| Component                                        | Function                                                                                                                                                                                 |
| ------------------------------------------------ | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| **Trove**                                        | Vault holding one supported collateral; debt denominated in RD.                                                                                                                          |
| **Par**                                          | Internal accounting value of 1 RD, used in redemptions, liquidations and calculating collateralization ratio(c-ratio).                                                                   |
| **Par Controller**                               | Slowly adjusts par(aka redemption value) ($p_{par}$) to correct market deviations from $1 peg.                                                                                         |
| **Base Rate($rate_{base}$)**                   | Base compounding interest rate charged for a loan against collateral.                                                                                                                    |
| **Base Rate Controller**                         | Raises/lowers $rate_{base}$ to correct market deviations from $1 peg.  Works faster than par controller.                                                                                             |
| **Risk Rate ($k_i$)** | Per-collateral rate multiplier to compensates for volatility & tail risk. Fixed at deployment.                                                                                                                             |
| **Utilization Rate ($m_i$)**          | Adaptive rate multiplier that increases with a collateral’s share of total debt, deterring debt concentration in a few collaterals.                                                                                       |
| **Stability Pool**                | Per-collateral Liquity-style pool. Absorbs liquidations for a single collateral; earns liquidation collateral + revenue share. |
| **Stability Pool Share**                         | The proportion of revenue directed to SP stakers. Defaults to 75%, but increases up to 100% during under-peg deviations.|
| **Redemption**                                   | Redemption of RD for collaterals in borrower troves at current par price. Collateral selection is weighted based on current collateral risk, with lowest c-ratio troves processed first. |
| **Redemption Fee**                               | Global Liquity-style fee charged when redeeming RD for collaterals. Weighted by % of total supply being redeemed and decayed with a 6 hour half-life.                                    |
| **Borrow Rate**                                  | Final interest rate charge to borrowers. $rate_{borrow} = rate_{base} \times rate_{risk} \times rate_{util}$ *(floors at 0.25 %, ceilings at 30 %)*.                                                                                                 |
| **Automated Per-Collateral Debt Ceiling**        | Caps & leaks each collateral’s max debt to prevent flash growth.                                                                                                                         |
| **Redemption Shield**                            | Optional per-trove fee add‑on, that blocks redemptions against a trove above a shield C‑ratio.                                                                                           |


---

# Appendix


## Rate Control Formal Definition

The base rate mechanism is a [PI](https://en.wikipedia.org/wiki/Proportional%E2%80%93integral%E2%80%93derivative_controller) controller. It measures the deviation between the current RD/USD price, $price_{market}$, and $1 and controls the value of $rate_{base}$.

$rate_{base} = max(0.25\%, K_pe_{t} + K_i(  \sum_{i=0}^{t-1}e_i \Delta t + e_{t}\Delta t) + u_0)$

| Variable | Formal Name | RD Value |
| -------- | -------- | -------- |
| $rate_{base}$     |  control variable   | $rate_{base}$|
| $u_0$     |  bias term; starting output    | starting $rate_{base}$, 0.5%|
| $e_t$     | current deviation, target-process  | $1 - $price_{market}$ | 
| $r_t$     | target     | $1 |
| $y_t$     |  process output   | $price_{market}$, 8-hr TWAP|        |
| $\sum_{i=0}^{t-1}e_i$   |   integral   | sum of time-weighted price deviations| 
| $K_p$   |   proportional gain   | weight of current deviation |
| $K_i$   |   integral gain   | weight of historical deviations |


## Controlled Peg Formal Definition

Formally, the $price_{par}$ mechanism is a [PI](https://en.wikipedia.org/wiki/Proportional%E2%80%93integral%E2%80%93derivative_controller) controller with integral decay. It measures the error between $p_{market}$ and $1 and controls the value of $p_{par}$


Discrete PI w/ decay:

$u_t = K_pe_{t} + K_i( \alpha^{\Delta t} \sum_{i=0}^{t-1}e_i \Delta t + e_{t}\Delta t) + u_0$

| Variable | Formal Name | System Value |
| -------- | -------- | -------- |
| $u_t$     |  control variable   | $price_{par}$|
| $u_0$     |  bias term, starting $p_{par}$   | $1.00|
| $e_t$     | deviation, target-process  | $1.00 - $price_{market}$ | 
| $r_t$     | target     | $1, constant |
| $y_t$     |  process output   | $price_{market}$, 24-hr TWAP|
| $\alpha$   |     decay coefficent               |    decay coefficent, 14d half-life         |
| $\sum_{i=0}^{t-1}e_i$   |   integral   | integral of deviations | 
| $K_p$   |   proportional gain   | weight of current deviation |
| $K_i$   |   integral gain   | weight of historical deviations |

*End of Draft v0.2*
