Premium Economics

A sustainable and scalable premium model that auto-adjusts to protocol risks per market per asset.

1. Total Staked LST v/s Coverage Amount Requested - A Bonding Curve

Variables:

• B: Total staked LST value in the AVS pool (y-axis)

• C: Coverage amount requested by a protocol (LST value) (This acts like an input similar to buying tokens on an AMM)

• R: Risk score assigned to the protocol by the AVS risk assessment engine $(0 <= R <= 1)$

• K: Overall risk factor of the AVS pool (set by AVS governance) $(0 <= K <= 1)$

• F: Factor representing the influence of risk score on pool size (set by AVS governance) - Higher F values lead to a steeper curve for higher-risk protocols.

• S: Scaling factor (positive constant) - Adjusts the overall curve's position on the graph.

The Formula:

$$B = S * (C ^ {(1 + F*R)}) \over (1 + K*R)$$

Explanation:

• This formula resembles a power function with an exponent influenced by the risk score (R) and overall pool risk (K).

• $C ^ {1 + F*R}$: This term represents the purchased coverage (C) raised to a power influenced by the risk score (R) and factor F. Higher risk protocols (higher R) see a steeper increase in required pool size (B) due to the exponent.

• $(1 + K*R)$: This term acts like a denominator, slowing down the growth of the required pool size (B) as the total coverage (represented by cumulative C values) increases. This reflects the benefit of spreading risk across multiple protocols.

• $S$: The scaling factor adjusts the overall curve's position on the y-axis. A higher S value would raise the entire curve, requiring a larger pool size for all coverage levels.

The Graph:

The graph will depict a curve where the y-axis represents the total staked LST value (B) in the AVS pool and the x-axis represents the cumulative coverage amount (C) requested by various protocols.

• The curve will start at the origin (0, 0), indicating no coverage requires no staked LST.

• The curve will have a positive slope, signifying that as the total requested coverage (C) increases, the required pool size (B) also increases.

• The curve will be steeper for higher-risk protocols (higher R) due to the exponent term (C ^ (1 + FR)). This visually represents the need for a larger pool to cover potential claims from high-risk protocols.

• The curve will flatten out slightly as the total coverage increases (due to the denominator term (1 + KR)), reflecting the risk mitigation from having a diverse pool of insured protocols.

Benefits of this Bonding Curve Approach:

• Visually Represents Risk: The curve's slope offers a visual representation of how risk score (R) affects the required pool size (B).

• Incentivizes Lower-Risk Protocols: The flatter curve for lower-risk protocols can incentivize them to participate by requiring a proportionally smaller pool contribution.

• Dynamic Pool Size: The pool size (B) automatically adjusts based on the requested coverage (C) and risk profile of participating protocols.

Example:

Let's consider the following scenario:

• AVS Parameters:

  • K (Overall Risk Factor) = 0.2 (Base premium is 20% of coverage)

  • F (Risk Score Factor) = 0.5 (Risk score has a moderate influence)

  • S (Scaling Factor) = 1 (For simplicity, we'll set the scaling factor to 1, keeping the curve centered)

• DeFi Protocols:

  • Protocol A: Requests 100,000 LST in coverage (Low Risk, R = 0.3)

  • Protocol B: Requests an additional 900,000 LST in coverage (High Risk, R = 0.8) (Total coverage becomes 1,000,000 LST)

Calculating Staked LST Value (B):

1. Protocol A:

   • B = 1 * (100,000 ^ (1 + 0.5 * 0.3)) / (1 + 0.2 * 0.3)

   • B ≈ 134,217 LST (This is the minimum staked LST value needed for Protocol A's coverage)

2. Protocol B (Total Coverage):

   • B = 1 * (1,000,000 ^ (1 + 0.5 * 0.8)) / (1 + 0.2 * 0.8)

   • B ≈ 10,240,000 LST (This is the total staked LST value needed after both protocols join)

Graph Representation:

Imagine a graph with "Cumulative Coverage Amount (LST)" on the x-axis and "Total Staked LST Value" on the y-axis. The curve will start at the origin (0, 0) and rise rapidly as the coverage amount (x-axis) increases. This rapid rise is due to the exponential term (C ^ (1 + FR)) in the formula.

• The curve will be steeper for Protocol B (higher risk score, R = 0.8) compared to Protocol A (lower risk score, R = 0.3). This steeper slope signifies the significantly larger pool size required to cover potential losses from a high-risk protocol.

• As the total coverage increases (adding Protocol B's coverage), the curve continues to rise, but at a slightly slower rate due to the denominator term (1 + KR). This reflects the benefit of spreading risk across multiple protocols.

Benefits of Exponential Bonding Curve:

• Prioritizes Capital Adequacy: The steeper curve for high-coverage and high-risk scenarios ensures a faster increase in the pool size, promoting sufficient capital reserves for potential claims.

• Discourages Excessive Coverage: The steeper curve acts as a natural disincentive for protocols seeking unreasonably large insurance amounts, as the required pool size grows rapidly.

Exponential Bonding Curve v/s logarithmic Bonding Curve.

• Exponential Curve: As the total coverage purchased (x-axis) increases, the required pool size (y-axis) increases at an accelerating rate. This aligns well with the insurance AVS model where higher coverage demands necessitate a more significant increase in the pool size to ensure sufficient funds for potential claims.

• Logarithmic Curve: As the total coverage purchased (x-axis) increases, the required pool size (y-axis) increases at a decelerating rate. This means that even with substantial coverage demands, the pool size wouldn't grow proportionally, potentially leading to insufficient funds in case of large claims.

Example: Logarithmic Curve Disadvantage

Imagine two DeFi protocols:

• Protocol A: Requests 100,000 LST in coverage.

• Protocol B: Requests an additional 900,000 LST in coverage (total coverage becomes 1,000,000 LST).

With a logarithmic bonding curve, the pool size increase for Protocol B (adding 900,000 LST in coverage) might be relatively small compared to the initial pool size needed for Protocol A (100,000 LST coverage). This could lead to a situation where the pool isn't adequately capitalized to cover significant claims from Protocol B, jeopardizing the AVS's ability to fulfill its insurance obligations.

Logarithmic Curve Example Graph:

Imagine a graph similar to the one described earlier for the exponential curve. However, in this case, the curve would have a much gentler slope, especially as the total coverage (x-axis) increases. This signifies a slower rise in the required pool size (y-axis), potentially creating capital shortfalls for the AVS.

Benefits of Exponential Curve:

• Prioritizes Capital Adequacy: The exponential curve ensures a faster increase in the pool size with higher coverage demands, promoting capital adequacy for the AVS.

• Discourages Excessive Coverage: The steeper curve for high-coverage requests shall act as a natural disincentive for protocols to seek unreasonably large insurance amounts.

Conclusion:

While a logarithmic curve might seem appealing due to its potential for lower initial pool requirements, it will compromise the AVS's ability to handle significant claims in the long run. The exponential curve, with its focus on capital adequacy, is a more suitable choice for an insurance AVS where robust financial backing is crucial for fulfilling its risk mitigation role.

2. Protocol Premium

In the scenario described earlier, we can calculate the premiums for Protocol A and Protocol B using the following formula derived from the exponential bonding curve concept :

$Premium (P) = C * (K*R + F * R)$

Variables:

• $P$: Premium amount (LST value) the protocol pays $(0 < P <= C)$
• $C$: Coverage amount requested by the protocol (LST value)
• $R$: Risk score assigned to the protocol by the AVS risk assessment engine $(0 <= R <= 1)$
• $K$: Overall risk factor of the AVS pool (set by AVS governance) $(0 <= K <= 1)$
• $F$: Factor representing the influence of risk score on premium (set by AVS governance)

Example Calculations:

1. Protocol A (Low Risk, R = 0.3):

• Coverage amount (C) = 100,000 LST

• Overall risk factor (K) = 0.2

• Risk score factor (F) = 0.5

P = 100,000 * (0.2 + 0.5 * 0.3) P ≈ 35,000 LST

This means Protocol A would pay a premium of approximately 35,000 LST for 100,000 LST in coverage.

1. Protocol B (High Risk, R = 0.8):

• Coverage amount (C) = 900,000 LST (adding to the total coverage)

• Overall risk factor (K) = 0.2

• Risk score factor (F) = 0.5

P = 900,000 * (0.2 + 0.5 * 0.8) P ≈ 600,000 LST

Here, Protocol B pays a significantly higher premium of approximately 600,000 LST for 900,000 LST in coverage due to its higher risk score (R = 0.8).

Key Points:

• This premium formula directly calculates the premium amount (LST value) a protocol needs to pay based on its requested coverage and risk profile.

• The formula ensures a base premium (KR) is paid by all protocols, reflecting the inherent risk of the AVS pool.

• The risk score factor (F * R) adds a variable premium based on the individual protocol's risk, making high-risk protocols contribute more to the pool.

Benefits:

• Risk-Based Pricing: Protocols with higher perceived risk contribute proportionally more to the pool, ensuring sufficient funds for potential claims.

• Fairness: Low-risk protocols benefit from lower premiums, while high-risk protocols pay a fair share commensurate with the potential payout burden they represent.