### Mathematical models for insurance payments – part 0 – introduction

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Insurance losses depend on two random variables. The first one is the number of losses that will occur in a specified period in connection to an insured (or a group of insureds). This is commonly referred to as the loss frequency or claim frequency. Its probability distribution is called the loss (claim) frequency distribution. The second random variable is the amount of the loss (if a loss has occurred). The amount of loss is usually referred to as the severity and its probability distribution is called the severity distribution. Then putting these two distributions together will lead to the total loss distribution.

In the several posts that follow, the focus is on the severity distribution, i.e. we will focus on the size of the loss or size of the claim. Once the methodology for the severity is discussed in a fair amount of details, we will add claim frequency.

Given that a loss has occurred and that the amount is $X$, the insurance company will make a payment to the insured to cover the loss $X$. The insurance payment is a random quantity since $X$ is random. Due to the presence of policy provisions such as deductible and limit, the insurance payment is likely less than $X$. The focus is on determining the distributional quantities of the probability model of the insurance payment to the insured. First and foremost, we would like to calculate the mean and variance of the distribution of payment as well as probability density function and cumulative distribution function and other distributional quantities.

The subsequent post will show, from a mathematical standpoint, the random variable of the insurance payment is a truncated and/or censored variable of the loss $X$ due to policy provisions such as deductible and policy limit.

In practice, the mean and variance of the insurance payments are often estimated from historical observations rather than using a hypothesized distribution of losses. However, the parametric distribution approach is a good starting point of the discussion of estimation of insurance losses.

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$\copyright$ 2017 – Dan Ma