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The traditional insurance pricing model is based on the loss distribution model, which forms the basis for Marquee's contract insurance pricing. The loss distribution model is developed by analyzing historical data and using the central limit theorem and the law of large numbers to estimate the probability density function of a specific claim and the corresponding expected claim cost. Marquee's model also incorporates supply and demand factors and considers the purchasing behavior of each contract type and the cash flow of the entire pool to adjust the final price.
The loss distribution model is defined as follows:
Let X be the random variable of the loss distribution. If
are sampled from the population data and when n is sufficiently large, the empirical distribution function F(x) satisfies:
This F(x) can be called the empirical distribution function. For the average amount of claims beyond the deductible d and for a specific loss x, they are both functions of the deductible d. The loss distribution in non-life insurance usually has the characteristics of a thick-tailed distribution, and the average excess function reflects the tail situation of the random variable. In the actual calculation and implementation process, the above definition is usually converted to the following empirical average excess function:
Based on the loss distribution model and the contract-specific characteristics, Marquee's pricing model is developed by optimizing the relevant parameters to calculate the premium, as follows:
Since insurance is in effect options and hence, the insurance premium can be calculated using Monte Carlo method. In the risk-free environment, suppose the underlying asset price follows geometric Brownian motion:
In the discrete-time form, μ is the expected return, and σ is its standard deviation and
follows a standard normal distribution.
Following the Taylor expansion, we can find the price path of the insurance premium:
Where μ = r is the risk-free rate, σ is the yearly standard deviation of the underlying asset's return, and T - t is the time to maturity,
is the price of the underlying asset at period t. The key variable that controls
which can be obtained by sampling
and simulating the price path of the underlying asset. This allows us to calculate the insurance premium.