# Aggregate Models

An insurance company will sell insurance policies that enable the purchaser to make a claim when a certain well-defined event occurs within a specified time period. An actuary will be called upon to examine the distributional (stochastic) properties of portfolios of such policies. We consider two basic approaches: the individual risk model and the collective risk model. The individual risk model is less granular than the collective risk model since it doesn't try to model claim frequency and claim size (severity) separately. We will denote by $S$ the random variable representing the aggregate claims (total loss to the insurer) associated with a portfolio of policies.

## The Individual Risk Model

In the individual risk model, we let $X_i$ represent the claim (loss) size associated with the $i\textrm{th}$ policy in a portfolio of $n$ policies:

[$]S = \sum_{i=1}^n X_i \, .[$]

It is assumed that $X_1,\ldots,X_n$ are mutually independent but not necessarily identically distributed.

## The Collective Risk Model

In the collective risk model, we let $N$ denote the random variable representing the number of individual claims and let $Y_i$ represent the claim (loss) size associated with the $i\textrm{th}$ individual claim:

[$]S = \sum_{i=1}^N Y_i \,.[$]

The following additional constraints are also assumed:

• $Y_1,\ldots,Y_n$ are mutually independent and identically distributed.
• $N$ is independent of each $Y_i$.

### Relation to Compound Distributions

A compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with the parameters of that distribution being assumed to be themselves random variables. The compound distribution is the result of marginalizing over the intermediate random variables that represent the parameters of the initial distribution. An important type of compound distribution occurs when the parameter being marginalized over represents the number of random variables in a summation of random variables as is the case with the collective risk model.

### Mean and Variance

Mean and variance of the compound distribution derive in a simple way from the law of total expectation and the law of total variance. The mean equals

[$] \mu_S = \operatorname{E}[S] = \operatorname{E}[N]\operatorname{E}[Y_i] = \mu_N \mu_Y [$]

and the variance equals

[] \begin{align*} \sigma^2_S = \operatorname{Var}[S] &= \operatorname{E}[N]\operatorname{Var}[Y_i] + \operatorname{E}[Y_i]^2 \operatorname{Var}[N] \\ &=\mu_N \sigma^2_Y + \mu^2_Y \sigma_N^2. \end{align*} []

### Moment Generating Function

The moment generation function of $S$ can be expressed in terms of the moment generating functions of $Y$ and $N$:

[$] M_{S}(t) = M_N\left(\ln\left(M_Y(t)\right)\right). [$]

## Compound Poisson Distribution

If the claim frequency $N$ has a poisson distribution with mean $\theta$, then $S$ is said to have a compound poisson distribution.

### Properties

We have (see Mean and Variance)

[$] \mu_S = \mu_N \mu_Y,\,\sigma^2_S = \mu_N \sigma^2_Y + \mu^2_Y \sigma_N^2. [$]

Since $\operatorname{E}[N] = \operatorname{Var}[N]$, these formulae can be reduced to:

[$]\mu_S = \theta \mu_Y,\,\sigma^2_S = \theta \operatorname{E}\left[Y^2\right].[$]

The probability generating function of $S$ has a simple representation in terms of the probability generation function of $Y$:

[$]P_S(t) = \textrm{e}^{\lambda(P_Y(t) - 1)}.[$]

## Compound Negative Binomial Distribution

If the claim frequency $N$ has a negative binomial distribution with parameters $r$ and $\beta$, then $S$ is said to have a compound negative binomial distribution.

### Properties

Since $\mu_N = r\beta$ and $\sigma^2_{N} = \mu_N (1 + \beta)$, then

[$]\mu_S = r\beta \mu_Y,\,\sigma^2_S = r\beta \left(\beta \operatorname{E}[Y]^2 + \operatorname{E}[Y^2] \right).[$]

The probability generating function of $S$ equals

[$]P_S(t) = [1 - \beta(P_Y(t) - 1)]^{-r}.\,[$]

## Compound Binomial Distribution

If the claim frequency $N$ has a binomial distribution with size parameter $m$ and success parameter $q$, then $S$ is said to have a compound binomial distribution.

### Properties

Since $\mu_N = pn$ and $\sigma^2_{N} = \mu_N (1-p)$, then

[$]\mu_S = np \mu_Y,\,\sigma^2_S = np \left(\operatorname{E}[Y^2] - p \operatorname{E}[Y]^2 \right).[$]

The probability generating function of $S$ can be computed:

[$]P_S(t) = [1 + p(P_Y(t) - 1)]^{n}.\,[$]

## Normal Approximation

It would be useful to consider a simple method that approximates the distribution function for $S$, which is usually difficult to compute. One basic approach is to use a normal approximation for the distribution of $S$. To understand why such an approximation is used, we first consider the classical central limit theorem.

### Classical Central Limit Theorem

Suppose $X_1,X_2,\ldots$ is a sequence of independent and identically distributed random variables with $\operatorname{E}[X_i] = \mu$ and $\operatorname{Var}[X_i] = \sigma_2 \lt \infty$. If $S_n = \sum_{i=1}^nX_i$ and $Z$ is a standard normal, then

[$] $$\label{clt} \operatorname{P}\left( \frac{S_n - n\mu}{\sigma \sqrt{n}} \leq c \right) \rightarrow \operatorname{P}\left( Z \leq c \right)\quad \text{as} \,\,n\rightarrow \infty$$ [$]

In other words, if we take the random sum $S_n$, center it by subtracting by its expected value $n\mu$, and then divide by its standard deviation $\sigma \sqrt{n}$, we obtain a random variable which is approximately normally distributed.

### The Method

The conditions required for the classical central limit theorem to hold aren't necessarily met in both the individual risk model and the collective risk model. The normal approximation method for $S$ tries to mimic the approach in the classical central limit theorem by centering $S$ and dividing by its standard deviation:

Normal Approximation
1. Compute $\mu_S$ and $\sigma_S^2$.
2. Set $Z = \frac{S - \mu_S}{\sigma_S}$.
3. Use the standard normal distribution table to get approximate percentiles for $Z$.

## References

• Wikipedia contributors. "Central limit theorem". Wikipedia. Wikipedia. Retrieved 8 June 2019.