Adaptive Interest Rate Model

The Panoptic Protocol uses a sophisticated adaptive interest rate model based on a PID controller to dynamically adjust borrow rates based on pool utilization. This system ensures equilibrium between liquidity providers and borrowers while remaining responsive to changing market conditions.

Overview

The interest rate model is designed to:

1. Target Optimal Utilization: Drive pool utilization toward a target level
2. Adapt to Market Conditions: Automatically adjust base rates over time
3. Bound Rate Changes: Prevent extreme rate swings through capped adjustments
4. Smooth Responses: Use time-weighted averaging for stable rate progression

Core Parameters

Parameter	Value	Description
TARGET_UTILIZATION	66.67% (2/3)	Optimal pool utilization target
CURVE_STEEPNESS	4	Rate curve multiplier at extremes
MIN_RATE_AT_TARGET	0.1% APY	Floor for target rate
MAX_RATE_AT_TARGET	200% APY	Ceiling for target rate
INITIAL_RATE_AT_TARGET	4% APY	Starting target rate
ADJUSTMENT_SPEED	50/year	Rate at which target rate adapts
IRM_MAX_ELAPSED_TIME	4096 seconds	Maximum time delta for rate updates

WAD Scaling

All rate calculations use WAD (10^18) scaling for precision:

int256 internal constant WAD = 1e18;

Rates are expressed per second, converted from annual rates:

int256 public constant MIN_RATE_AT_TARGET = 0.001 ether / int256(365 days);
int256 public constant MAX_RATE_AT_TARGET = 2.0 ether / int256(365 days);

Interest Rate Functions

interestRate

Returns the current average borrow rate:

function interestRate(
    uint256 utilization,
    MarketState interestRateAccumulator
) external view returns (uint128)

updateInterestRate

Computes the new interest rate and updated rate-at-target for state updates:

function updateInterestRate(
    uint256 utilization,
    MarketState interestRateAccumulator
) external view returns (uint128, uint256)

Rate Calculation Logic

Step 1: Compute Utilization Error

The error measures how far current utilization is from the target:

int256 _utilization = int256(utilization);
int256 errNormFactor = _utilization > TARGET_UTILIZATION
    ? WAD - TARGET_UTILIZATION
    : TARGET_UTILIZATION;
int256 err = Math.wDivToZero(_utilization - TARGET_UTILIZATION, errNormFactor);

The error is normalized:

• When utilization > target: err = (util - target) / (1 - target)
• When utilization < target: err = (util - target) / target

This normalization ensures error ranges from -1 to +1.

Step 2: Determine Rate-at-Target

First Interaction

If no previous rate exists, use the initial value:

if (startRateAtTarget == 0) {
    avgRateAtTarget = INITIAL_RATE_AT_TARGET;
    endRateAtTarget = INITIAL_RATE_AT_TARGET;
}

Subsequent Updates

The rate-at-target adapts based on utilization error over time:

// Speed of rate adjustment
int256 speed = Math.wMulToZero(ADJUSTMENT_SPEED, err);

// Time since last update (capped)
int256 elapsed = Math.min(
    int256(block.timestamp) - int256(previousTime),
    IRM_MAX_ELAPSED_TIME
);

int256 linearAdaptation = speed * elapsed;

The new rate-at-target is computed as:

endRateAtTarget = _newRateAtTarget(startRateAtTarget, linearAdaptation);

Step 3: Apply Exponential Adjustment

The _newRateAtTarget function applies exponential adjustment with bounds:

function _newRateAtTarget(
    int256 startRateAtTarget,
    int256 linearAdaptation
) private pure returns (int256) {
    return Math.bound(
        Math.wMulToZero(startRateAtTarget, Math.wExp(linearAdaptation)),
        MIN_RATE_AT_TARGET,
        MAX_RATE_AT_TARGET
    );
}

Formula: newRate = startRate × e^(linearAdaptation)

Bounded between MIN and MAX to prevent extreme values.

Step 4: Compute Average Rate

The average rate uses trapezoidal integration for accuracy:

int256 midRateAtTarget = _newRateAtTarget(startRateAtTarget, linearAdaptation / 2);
avgRateAtTarget = (startRateAtTarget + endRateAtTarget + 2 * midRateAtTarget) / 4;

This approximation provides better accuracy than simple linear interpolation.

Step 5: Apply Rate Curve

The final rate applies the curve function to the average rate-at-target:

return uint256(_curve(avgRateAtTarget, err));

Rate Curve Function

The curve function adjusts the rate based on utilization error:

function _curve(int256 _rateAtTarget, int256 err) private pure returns (int256) {
    int256 coeff = err < 0
        ? WAD - Math.wDivToZero(WAD, CURVE_STEEPNESS)  // 1 - 1/C = 0.75
        : CURVE_STEEPNESS - WAD;                        // C - 1 = 3
    
    return Math.wMulToZero(Math.wMulToZero(coeff, err) + WAD, _rateAtTarget);
}

Curve Behavior

Utilization	Error	Multiplier	Effect
0%	-1	0.25	Rate = 25% of target
33%	-0.5	0.625	Rate = 62.5% of target
66.67%	0	1	Rate = target rate
83%	0.5	2.5	Rate = 250% of target
100%	1	4	Rate = 400% of target

Visual Representation

Borrow
Rate
  ^
  |                                    /
  |                                  /
4x|                                /
  |                              /
  |                           ./
2x|                        .-´
  |                     .-´
1x|...................-´
  |              .-´
0.25x|         .-´
  |       .´
  +-------+-------+-------+-------+---> Utilization
  0%    33%    66.67%   83%   100%
               Target

Time-Capping Mechanism

The IRM_MAX_ELAPSED_TIME parameter prevents rate drift during periods of inactivity:

int256 elapsed = Math.min(
    int256(block.timestamp) - int256(previousTime),
    IRM_MAX_ELAPSED_TIME  // 4096 seconds ≈ 68 minutes
);

This ensures:

• Long periods without interaction don't cause extreme rate changes
• The model remains responsive but bounded
• Gas optimization (no need for frequent keeper transactions)

Rate Adaptation Examples

Scenario 1: High Utilization

• Current utilization: 90%
• Target utilization: 66.67%
• Error: (90 - 66.67) / (100 - 66.67) = 0.70
• Rate multiplier: ~3.1x

If target rate is 4%, effective rate ≈ 12.4%

Over time, the high utilization error will cause rateAtTarget to increase exponentially, further increasing rates until utilization decreases.

Scenario 2: Low Utilization

• Current utilization: 30%
• Target utilization: 66.67%
• Error: (30 - 66.67) / 66.67 = -0.55
• Rate multiplier: ~0.59x

If target rate is 4%, effective rate ≈ 2.4%

Over time, the negative error will cause rateAtTarget to decrease, lowering rates to attract more borrowers.

Scenario 3: At Target

• Current utilization: 66.67%
• Error: 0
• Rate multiplier: 1x

Rate equals the target rate, and rateAtTarget remains stable.

State Storage

Rate state is stored in the MarketState accumulator:

• 38-bit rateAtTarget: The current target rate (scaled)
• 32-bit epoch: Timestamp (shifted by 2 bits for Y2K38 avoidance)

int256 startRateAtTarget = int256(uint256(interestRateAccumulator.rateAtTarget()));
uint256 previousTime = interestRateAccumulator.marketEpoch() << 2;

Integration with Collateral Tracking

Interest accumulation affects solvency calculations:

// In _getMargin
(uint256 balance0, uint256 interest0) = ct0.assetsAndInterest(user);
(uint256 balance1, uint256 interest1) = ct1.assetsAndInterest(user);

// Interest adds to requirements
tokensRequired = tokensRequired
    .addToRightSlot(uint128(interest0))
    .addToLeftSlot(uint128(interest1));

Accrued interest increases the collateral requirement, ensuring borrowers maintain adequate margin as their debt grows.

Events

When rates are updated, the protocol emits:

event BorrowRateUpdated(
    address indexed collateralToken,
    uint256 avgBorrowRate,
    uint256 rateAtTarget
);

This enables:

• Off-chain rate tracking
• Historical analysis
• Integration with monitoring systems

Design Rationale

Why PID Control?

The PID-style controller provides:

• Proportional response: Immediate reaction to utilization changes
• Integral response: Long-term rate adaptation (via rateAtTarget)
• Bounded behavior: Rate caps prevent extreme values

Why Exponential Adjustment?

Exponential adjustment (e^x) provides:

• Smooth rate transitions
• Percentage-based changes (doubling/halving)
• Natural compounding behavior

Why Trapezoidal Integration?

The trapezoidal method for average rate calculation:

• More accurate than simple averaging
• Accounts for non-linear rate changes over time
• Prevents systematic under/over-estimation

Summary

The adaptive interest rate model:

1. Targets 66.67% utilization through dynamic rate adjustment
2. Adapts over time with exponential rate-at-target changes
3. Bounds rates between 0.1% and 200% at target (0.025% to 800% effective)
4. Uses curve steepness of 4x for 25%-400% rate range around target
5. Caps time deltas to prevent rate drift during inactivity
6. Integrates with solvency by adding accrued interest to requirements

This creates a self-regulating system that naturally balances liquidity supply and demand while remaining resilient to market shocks.