An introduction to compound distributions

Compound distributions have many natural applications. We motivate the notion of compound distributions with an insurance application. In an individual insurance setting, we wish to model the aggregate claims during a fixed policy period for an insurance policy. In this setting, more than one claim is possible. Auto insurance and property and casualty insurance are examples. In a group insurance setting, we wish to model the aggregate claims during a fixed policy period for a group of insureds that are independent. In other words, we discuss distributions that can either model the total claims for an individual insured or a group of independent risks over a fixed period such that the claim frequency is uncertain (no claim, one claim or multiple claims). Note that in a previous post (More insurance examples of mixed distributions), we discussed a specific type of compound distribution with the simplifying assumption of having at most one claim. We now discuss models for aggregate claims where the claim frequency includes the possibility of having multiple claims. We first define the notion of compound distributions. We then discuss some general properties. We present some examples to illustrate the calculations discussed in Some examples of compound distributions.

The random variable Y is said to have a compound distribution if Y is of the following form

\displaystyle Y=X_1+X_2+\cdots + X_N

where (1) the number of terms N is uncertain, (2) the random variables X_i are independent and identically distributed (with common distribution X) and (3) each X_i is independent of N.

The sum Y as defined above is sometimes called a random sum. If N=0 is realized, then we have Y=0. Even though this is implicit in the definition, we want to call this out for clarity.

In our insurance contexts, the variable N represents the number of claims generated by an individual policy or a group of indpendent insureds over a policy period. The variable X_i represents the i^{th} claim. Then Y represents the aggregate claims over the fixed policy period.

We discuss the following properties of compound distributions:

  1. Distribution function.
  2. Mean and higher moments.
  3. Variance.
  4. Moment generating function and cumulant generating function.
  5. Skewness.

The random sum Y is a mixture. Thus many properties such as distribution function, expected value and moment generating function of Y can be expressed as a weighted average of the corresponding items for the basic distributions.

1. Compound Distribution – Distribution Function
By the law of total probability, the distribution function of Y is given by the following:

\displaystyle F_Y(y)=\sum \limits_{n=0}^{\infty} G_n(y) \thinspace P[N=n]

where for n \ge 1, G_n(y) is the distribution function of the independent sum X_1+ \cdots + X_n and G_0(y) is the distribution function of the point mass at y=0.

We can also express F_Y in terms of convolutions:

\displaystyle F_Y(y)=\sum \limits_{n=0}^{\infty} F^{*n}(y) \thinspace P[N=n]

where F is the common distribution function for X_i and F^{*n} is the n-fold convolution of F.

If the common claim distribution X is discrete, then the aggregate claims Y is discrete. On the other hand, if X is continuous and if P[N=0]>0, then the aggregate claims Y will have a mixed distribution, as is often the case in insurance applications.

2. Compound Distribution – Mean and Higher Moments
The mean aggregate claims E[Y] is:

\displaystyle E[Y]=E[N] \thinspace E[X]

The expected value of the aggregate claims has a natural interpretation. It is the product of the expected number of claims and the expected individual claim amount. This makes intuitive sense. The following is the derivation:

\displaystyle E[Y]=E_N[E(Y \lvert N)]= E_N[E(X_1+ \cdots +X_N \lvert N)]

\displaystyle =E_N[N \thinspace E(X)]=E[N] \thinspace E[X]

The higher moments of the aggregate claims Y do not have a intuitively clear formula as the first moment. However, we can obtain the higher moments by using the first principle.

\displaystyle E[Y^n]=E_N[E(Y^n \lvert N)]= E_N[E(\lbrace{X_1+ \cdots +X_N}\rbrace^n \lvert N)]

\displaystyle = E[Z_1^n] \thinspace P[N=1]+E[Z_2^n] \thinspace P[N=2]+ \cdots

where \displaystyle Z_n=X_1+ \cdots +X_n.

3. Compound Distribution – Variance
The variance of the aggregate claims Var[Y] is:

\displaystyle Var[Y]=E[N] \thinspace Var[X]+Var[N] \thinspace E[X]^2

The variance of the aggregate claims also has a natural interpretation. It is the sum of two components such that the first component stems from the variability of the individual claim amount and the second component stems from the variability of the number of claims. The variance of the aggregate claims can be derived by using the total variance formula:

\displaystyle Var[Y]=E_N[Var(Y \lvert N)]+Var_N[E(Y \lvert N)]

\displaystyle =E_N[Var(X_1+ \cdots +X_N \lvert N)]+Var[E(X_1+ \cdots +X_N \lvert N)]

\displaystyle =E_N[N \thinspace Var(X)]+Var[N \thinspace E(X)]

\displaystyle =E[N] \thinspace Var[X]+Var[N] \thinspace E[X]^2

4. Compound Distribution – Moment Generating Function and Cumulant Generating Function

The moment generating function M_Y(t) is: \displaystyle M_Y(t)=M_N[ln \thinspace M_X(t)] where the function ln is the natural log function. The following is the derivation.

\displaystyle M_Y(t)=E[e^{tY}]=E_N[E(e^{t(X_1+ \cdots +X_N)} \lvert N)]

\displaystyle =E_N[E(e^{tX_1} \cdots e^{tX_N} \lvert N)]

\displaystyle =E_N[E(e^{tX_1}) \cdots E(e^{tX_N}) \lvert N]=E_N[M_X(t)^N]

\displaystyle =E_N[e^{N \thinspace ln M_X(t)}]=M_N[ln \thinspace M_X(t)]

Cumulant Generating Function
For any random variable Z, the cumulant generating function of Z is defined as: \Psi_Z(t)=ln M_Z(t). It can be shown that the cumulant generating function characterizes the second and third moments. We will use this fact to derive the skewness of the aggregate claims Y.

\displaystyle \Psi_Z^{(k)}(0)=\left\{\begin{matrix}E[Z]&\thinspace k=1\\{Var[Z]=E[(Z-\mu_Z)^2]}&k=2\\{E[(Z-\mu_Z)^3]}&k=3\end{matrix}\right.

Based on the definition of cumulant generating function, for the aggregate claims Y, M_Y(t)=M_N[\Psi_X(t)]. Thus we have:

\displaystyle \Psi_Y(t)=ln M_Y(t)=ln \thinspace M_N[\Psi_X(t)]=\Psi_N[\Psi_X(t)]

5. Compound Distribution – Skewness
The skewness for any random variable Z is defined as:

\displaystyle \gamma_Z=E\biggl[\biggl(\frac{Z-\mu_Z}{\sigma_Z}\biggr)^3\biggr]=\sigma_Z^{-3} \thinspace E\biggl[\biggl(Z-\mu_Z\biggr)^3\biggr].

Since \Psi_Z^{(3)}(0)=E[(Z-\mu_Z)^3], we have \gamma_Z=\sigma_Z^{-3} \thinspace \Psi_Z^{(3)}(0) and \Psi_Z^{(3)}(0)= \sigma_Z^3 \thinspace \gamma_Z.

From the section 4, \Psi_Y(t)=\Psi_N[\Psi_X(t)]. Taking the third derivative of \Psi_Y(t) and evaluate at t=0, we have:

\displaystyle \Psi_Y^{(3)}(0)=\gamma_N \thinspace \sigma_N^3 \thinspace \mu_X^3+3 \thinspace \sigma_N^2 \thinspace \mu_X \thinspace \sigma_X^2+\mu_N \thinspace \gamma_X \thinspace \sigma_X^3

Thus, the following is the skewness of the aggregate claims Y:

\displaystyle \gamma_Y=\frac{\gamma_N \thinspace \sigma_N^3 \thinspace \mu_X^3+3 \thinspace \sigma_N^2 \thinspace \mu_X \thinspace \sigma_X^2+\mu_N \thinspace \gamma_X \thinspace \sigma_X^3}{(\mu_N \thinspace \sigma_X^2+\sigma_N^2 \thinspace \mu_X^2)^{\frac{3}{2}}}

Refer to Some examples of compound distributions for illustrations of the calculations discussed in this post.


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