Cramer Lundberg

Definitions

Cramer-Lundberg models (CLs) are the foundational model resulting from ruin theory.
Ruin theory focuses on the 'surplus process' of an insurance company, which is the difference between its accumulated premiums and claims up to a certain time. Essentially, ruin theory tracks whether an insurance/leveraged firm is approaching insolvency.
The primary concern of ruin theory is the 'probability of ruin,' which is the chance that an insurer's reserve becomes negative. The Cramer-Lundberg model provides an approximation for the infinite time-horizon probability of ruin. This is crucial to dynamically measuring and managing risk.
CLs aid in deciding suitable premium rates that strike a balance between maintaining a low probability of ruin and offering competitive insurance products. This contribution to premium calculation and risk management is at the heart of ruin theory.
While the classic Cramér-Lundberg model deals with the probability of ruin over an infinite time horizon, variations of the model also study the probability of ruin over a finite time period. This allows for more specific risk assessment in real-world scenarios, further connecting the model to practical applications.

It should be relatively clear that CLs are directly applicable to Concrete’s use case. Concrete’s coverage policies are well-modeled by insurance-related models, as are crypto market dynamics.
The difficulty in applying CLs to Concrete’s use case is the modeling of crypto market crashes as Poisson distributed events. This will require some finagling/modification of the base model.

Key assumptions
- The process of claims arrivals follows a Poisson process.
- Claim sizes are independent and identically distributed random variables following an exponential distribution.
- The inter-arrival times of claims are exponentially distributed and independent of the claim sizes.
- Premiums are received continuously at a constant rate.
Ruin theory
- Ruin theory is concerned with the scenario in which Ud(n) < 0 and therefore, the firm is insolvent. We let the probability, over an infinite time horizon, that the firm becomes insolvent can be formally described by
- One of the fundamental results in ruin theory is the Lundberg inequality, which provides an upper bound on the probability of ruin. For a discrete-time model with a constant premium rate, the inequality is given as:
The base CL model can be described by the following formula: where X_0=x is the initial surplus or the surplus known at a giving or starting instant, X_t represents the dynamics of surplus of an insurance company up to time t, c is the premium income per unit time, assumed deterministic and fixed, and Qt the total claim amount process. {U_i} where i≥1 is a sequence of i.i.d. random variables with common cumulative distribution F, with F(0)=0. This base model is modified heavily for Concrete’s use case.